text:
# Could wind + batteries really replace a nuclear power plant?

tl;dr: Probably not. The intermittency and seasonal variation of wind is so severe that even optimistically, it would be considerably more expensive and, if it was to use Li-ion batteries for storage, would require as many as have been produced in the entire world over the last four years.

The UK government's recent decision to delay the final decision on the planned Hinkley Point C nuclear power plant has somewhat rekindled the debate on whether the UK should have the plant at all. Some have gone as far as to suggest that given the projected cost, the guaranteed price of the energy produced and the timescale of the project, it should be scrapped and allowed to be replaced with a combination of renewable energy sources and with energy storage, both of which are dropping in cost.

[Jeremy Leggett, the founder of solar panel maker Solarcentury] is delighted that others are picking up on arguments he has been making for years. "Finally the message is getting through that Hinkley, and indeed nuclear, make no sense today simply because wind and solar are cheaper. If we accelerate renewables in the UK, we can get to 100% renewable power well before 2050," he says.

I'm sure we can all agree that cheap, low-carbon renewable electricity would be a great thing. But if you are proposing to eventually remove nuclear and fossil fuels entirely, are renewables still as "cheap" when we need to rely on them to maintain demand? From that same Guardian article:

The Economist believes improved electricity storage is a key answer to the frequently repeated criticism of wind and solar that it is intermittent, and points out that battery technology is fast improving.

First, let's be clear. It is not a mere "criticism" of wind and solar that it is intermittent - it is a cold, hard fact. It's physically impossible to generate solar power at night and to generate significant wind power when the wind's not blowing. In order for electricity generation to meet or follow demand, excess generation must be curtailed somewhere, or the energy stored. Similarly, insufficient generation must be supported by some other form of energy generation to avoid blackouts. At the moment, this role is largely provided by gas in the UK. But in the absence of other conventional means of generation, this role needs to be filled by some form of storage, perhaps the much vaunted batteries. This is especially important if neighbouring countries make similar moves towards renewables, since available wind speed and sunlight does not tend to vary much across neighbouring countries - those neighbours may not be able to export energy when generation is barely sufficient across an entire continent.

But how much storage would be needed, say, to effectively convert intermittent renewable power into providing baseload power equivalent to that which would be provided by Hinkley Point C - a constant 3.2 GW? Well, this can be estimated with some crude analysis of publicly available data. I was interested to see how it turned out, and I figured it was worth reproducing here. I did the analysis with R, and have included the code (except for the code generating the plots) and the data here so that it can be reproduced.

### Analysis

I'm going to use existing wind generation data for this analysis, since the UK already has a significant amount of wind power, and on the assumption that large-scale deployment of solar power would not be all that sensible for one of the darkest countries in the world. The data I've used is the energy production data for the UK for the entire year 2015 - from gridwatch.co.uk - which I've reuploaded to this website.

The data is in the standard csv format, and I use a couple of addon packages for analysis.

gridwatch <- read.csv("http://lacey.se/dl/gridwatch-2015.csv")

library(dplyr)
library(lubridate)
library(ggplot2)

Let's check what it looks like:

head(gridwatch)

##       id            timestamp demand frequency coal nuclear ccgt wind
## 1 377525  2015-01-01 00:00:04  28809    50.090 9079    8049 3360 5251
## 2 377526  2015-01-01 00:05:02  28645    50.092 8947    8053 3369 5254
## 3 377527  2015-01-01 00:10:02  28768    50.116 8843    8052 3372 5272
## 4 377528  2015-01-01 00:15:02  28917    50.045 8763    8047 3339 5303
## 5 377529  2015-01-01 00:20:02  28964    50.030 8818    8051 3386 5223
## 6 377530  2015-01-01 00:25:02  29055    50.006 8906    8055 3392 5189
##   french_ict dutch_ict irish_ict ew_ict pumped hydro oil ocgt other
## 1        582       900       -72   -136     15   443   0    0  1157
## 2        586       898      -100   -134      0   441   0    0  1157
## 3        586       898      -100   -134      0   440   0    0  1157
## 4        586       898      -100   -134      0   439   0    0  1155
## 5        586       898      -100   -134      0   440   0    0  1155
## 6        586       898      -100   -134      0   441   0    0  1155

The data is quite thorough, but all I really want for now is the data for wind. First I'm going to convert the timestamp to POSIXct date/time format with the appropriate function from the lubridate package, then I can select out the data I need.

gridwatch$timestamp <- ymd_hms(gridwatch$timestamp)
df1 <- select(gridwatch, timestamp, wind)

Check again:

head(df1)

##             timestamp wind
## 1 2015-01-01 00:00:04 5251
## 2 2015-01-01 00:05:02 5254
## 3 2015-01-01 00:10:02 5272
## 4 2015-01-01 00:15:02 5303
## 5 2015-01-01 00:20:02 5223
## 6 2015-01-01 00:25:02 5189

The wind column shows the power generated in units of MW.

Straight away you can see the issue with intermittency. Wind production averages about 2.6 GW over the whole year, but this can be in excess of 6 GW during windy times, and almost nothing during some lulls in the summer. At the end of 2015, the UK had a total of 13.6 GW of capacity installed, indicating a capacity factor of 19%, which seems reasonable.

I'll make two new columns in this data frame - one for the time increment (in seconds) and then use that to integrate the power column to get the total energy generated in MWh.

df1$difftime <- c(0, diff(df1$timestamp))
df1$totwind <- cumsum(df1$wind * df1$difftime / 3600)  So, I want to see how a constant 3.2 GW baseload can be generated by wind, with excess energy stored and then released when the wind isn't sufficient. We can reasonably assume that with more turbines the power generated will scale linearly. We can make a new table from the same data, but adjust the wind so that the total energy generated by wind power throughout the year will be equal to 3.2 GW x 24 hours x 365 days. df2 <- select(gridwatch, timestamp, wind) df2$difftime <- c(0, diff(df2$timestamp)) df2$wind <- df2$wind * (3200 * 24 * 365 / (max(df1$totwind)))
df2$totwind <- cumsum(df2$wind * df2$difftime / 3600)  The average should come out to be about 3200 MW now, so let's check that's the case: summary(df2$wind)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##   77.92 1506.00 2889.00 3201.00 4805.00 8022.00

For plotting purposes I'll include equivalent columns for the constant 3.2 GW.

df2$base <- 3200 df2$baseenergy <- cumsum(3200 * df2$difftime / 3600)  Now the table looks like this: head(df2)  ## timestamp wind difftime totwind base baseenergy ## 1 2015-01-01 00:00:04 6294.875 0 0.0000 3200 0.0000 ## 2 2015-01-01 00:05:02 6298.472 298 521.3735 3200 264.8889 ## 3 2015-01-01 00:10:02 6320.050 300 1048.0443 3200 531.5556 ## 4 2015-01-01 00:15:02 6357.213 300 1577.8120 3200 798.2222 ## 5 2015-01-01 00:20:02 6261.309 300 2099.5878 3200 1064.8889 ## 6 2015-01-01 00:25:02 6220.550 300 2617.9669 3200 1331.5556  Let's see how the total amount of energy generated from wind over the year would look like compared to a constant 3.2 GW. The energy generated by wind is shown in red. The difference between these two lines will show us roughly the difference between the energy generated by wind power and the energy consumed by the constant 3.2 GW we're looking for. So let's do that: df2$diffpower <- df2$wind - df2$base
df2$diffenergy <- cumsum(df2$diffpower * df2$difftime / 3600)  The power requirements will look like this (positive values indicate wind in excess, so therefore the batteries would be charging. Negative values indicate that wind is insufficient, so the batteries need to take over to ensure 3.2 GW): The total energy stored or released looks like this: It's clear from this plot there is a huge seasonal variation in wind power, with greater generation in the winter, and the storage needed as backup in the summer. The difference between those minimum and maximum peaks (in March and October respectively) is the total amount of energy we would need to backup the wind power - therefore it's the capacity of the storage we would need to guarantee a constant 3.2 GW baseload without relying on other methods of generation. In this case, we can see it's of the order of about 4 TWh (that's TERAwatt-hours). 4 TWh is a colossal amount of energy - roughly the yield of 35 Trident nuclear missiles, roughly the total battery capacity of about 45 million Tesla Model S cars, and about 100 years worth of Li-ion batteries at the current rate of production. More than that, at a target price of$100/kWh for batteries (=$100bn/TWh) this would cost of the order of$400 bn for the batteries alone. This price for usable battery storage is considerably cheaper than currently available for any of the available chemistries, even if you're Tesla.

Meanwhile, based on a capacity factor of about 20%, we would need wind power with a capacity of 16 GW, which based on estimates of $1.3m -$2.2m per MW would cost between $20.8 and$35.2bn. The cost of Hinkley Point C by comparison is estimated to be of the order of $24 bn. So yes, the wind turbines could potentially be cheaper to install than the nuclear power plant, but even then, they can't supply power on the same basis without additional - and possibly insane - storage capability. ### More wind turbines? Ok, maybe it's not viable to try and store every last bit of energy produced by the wind turbines. Perhaps we can just disconnect them during very windy periods and only store enough energy needed when wind isn't enough. How many batteries might we need then? To estimate this I went back again and scaled up the wind further, so that the averege power is 20% over the 3.2 GW we actually want. The rest of the code is the same. df4 <- select(gridwatch, timestamp, wind) df4$difftime <- c(0, diff(df4$timestamp)) df4$wind <- df4$wind * (1.2 * 3200 * 24 * 365 / (max(df1$totwind)))
df4$totwind <- cumsum(df4$wind * df4$difftime / 3600) df4$base <- 3200
df4$baseenergy <- cumsum(3200 * df4$difftime / 3600)

df4$diffpower <- df4$wind - df4$base df4$diffenergy <- cumsum(df4$diffpower * df4$difftime / 3600)

If I remake the same plot and rescale it a bit:

Then the difference between the minimum and maximum values here is the amount of energy that needs to be stored, which is about 2 TWh - halving the cost of batteries needed for just 20% more wind turbines. This seems more reasonable, so what if you keep adding more wind power?

Well, I won't reproduce the plot here - you can do it yourself - but the cost does keep shrinking. If you double the amount of wind power - to 32 GW, giving an average of 6.4 GW over the year, you can still see that there are periods of insufficient wind to meet 3.2 GW that still require about 200 GWh worth of storage. This is still a huge amount, equivalent to the total storage of a few million electric cars and about four years worth of production of Li-ion batteries. For comparison, the world's largest grid storage battery opened earlier this year in Japan - a sodium-sulfur battery with a capacity of 300 MWh (that's 0.3 GWh).

The cost of installing this would probably then be in the range:

• Wind turbines: 32 GW capacity, $41.6 -$70.4bn
• Battery storage: 200 GWh, ~\$20bn

This would imply an installation cost of maybe three times as much as Hinkley Point C, for a system with a considerably shorter lifetime, and most likely with much more expensive electricity for the consumer. I doubt if I kept going that it would get considerably cheaper than this, since at this point the cost of the wind turbines is already rather in excess of the cost of the batteries.

So what can we conclude? I think the main thing this thought experiment shows is just how important it is to have a diverse mix of technologies in the energy generation mix. I think this fact is usually lost on people with stubbornly anti-nuclear and anti-fossil fuel views. I know as well that I have not considered other things like tidal or hydroelectric power here, but these are very much geography-dependent and not always a viable option.

I don't intend to say that there is no place for renewables at all in our energy mix, but I cannot see a target of 100% solar or wind grid production as being anything except ruinously expensive, and it seems wildly improbable that battery storage could make it work.

Bear in mind too, that this is just one power plant. Average demand in the UK was 32.8 GW in 2015!

### The last word

At this point it is reasonable to point out that the seasonal trend in solar generation runs (very roughly) opposite to that of wind. That is, it is less windy in the summer, but it is sunnier in the summer, so a mix of solar and wind might reduce the requirement for storage. This is true, but some level of storage would still be required, and my intuition tells me that it would not be a significant difference (and other analyses done elsewhere suggest as much). At the moment I don't have data for UK solar generation to play around with, but this would be interesting for a future post.

# Can't underestimate luck

I took this photo of a soap bubble on Sunday:

It received a surprising amount of attention on Flickr, with almost 8,000 views in one day, way more than any photo of mine has ever had. At the time I didn't even really think about what the photo would look like, and the perfect reflection of the house and the colours in the bubble itself were an unexpected bonus!

# Don't believe in miracles

The above was one of the closing points in an excellent talk I saw earlier today by Dr Jef Ongena, the chairman of the Energy group at the European Physical Society. Based on the email announcement for the talk I was expecting to hear mostly about current progress in nuclear fusion research, but what we got was partly a thought-provoking and fiercely critical overview of the European approach towards renewable energy, and partly a call for better public awareness and consideration of wider context. It really deserved a larger audience than it had so I thought I’d write a bit about it!

“Europe alone cannot save the world” was the first key message, giving examples such as Germany: around 1 trillion Euros has been committed to the Energiewende, and has so far brought about very little meaningful reduction in Germany’s CO2 emissions, which themselves contribute only ~2.5% of the world’s emissions (interestingly enough, the decade following the collapse of the GDR saw German CO2 emissions drop by a quarter, simply because of the loss or modernisation of East German industry). Globally, this reduction is insignificant, because emissions from countries such as India and China have increased to a greater extent. More than this, as a result of this policy, German energy prices are among the highest in the world.

Dr Ongena was also keen to point out that at least a portion of the EU’s reduced emissions are effectively an accounting trick – we import more goods from China rather than produce them here, meaning the associated emissions appear as Chinese emissions and not European. It’s worth reading the EPS Energy Group’s position paper on this topic.

What was most interesting for me was the stark look at what the consequences of an energy system based 100% on renewable energy would be. At this point I am reminded of this excellent post on the situation in Scotland, since this is exactly what the Scottish government is aiming for in the short-term (the blog at which that post resides, Energy Matters, is excellent in general by the way). The biggest issue with renewable energy such as solar and wind is the intermittency; wind is unpredictable, and the demand for solar is typically out of phase with the consumption, so some energy storage is essential. But how much?

It’s also one thing to cope with diurnal (day/night variation) in production/consumption, but if you plan for a future in which solar is a large or the major part of the energy production mix, then the huge seasonal variation in energy production becomes a big problem.

Dr Ongena gave an example of a study looking at German energy production as it would look in 2050 based on current plans. Unfortunately, I didn’t note down the reference, but the short version is that coping with the seasonal variation in energy production and consumption would require of the order of 33 TWh of storage capacity for Germany alone! To put this into context, if this energy was to be stored using batteries, 27 cubic kilometers(!!) of space would be needed to store the batteries themselves. How much space is that? Well, I calculated that myself, and it would take an aircraft hangar tall enough to fit an Airbus A380, this big:

Or, about 2,000 buildings the size of the Boeing Everett Factory, the largest building in the world by volume. I won’t even try to estimate the impossible cost of such a solution.

This idea of “knowing your numbers” was I think the main scientific point in the talk, which pleased me greatly: this is something I think is really important, and is something I try to prioritise in any teaching that I do. Dr Ongena gave a few interesting factoids – for example, that the energy consumed by satellite TV boxes in Belgium is something like 17 times more than the energy consumed lighting all the country’s roads – but the main point was about the power density of energy production.

Power density, as in power per unit area, tells you about the land area needed to produce energy, and in this respect all renewable energies are considerably more “dilute” than conventional (fossil fuel, nuclear) technologies. I’ve since found some good references for this, notably this one, so I won’t write anything else on this except to say that it is often and easily forgotten that wind power, for example, requires 400-500 times the land area to provide the same power as nuclear (and optimistically about 50 times for solar). It’s long been a mystery to me as to why some types of environmental destruction (e.g., large scale pumped hydro or vast fields of massive on-shore wind turbines) are apparently preferable to others (e.g., storing relatively small amounts of nuclear waste, or fracking).

Although it was a talk with relatively few crumbs of comfort for the future energy landscape (fusion power was not even discussed), it was still rather refreshing to get such a brutal reality check and plenty of food for thought. Unfortunately, my suspicion is that politics and emotions will always trump science in any decision-making process, but I would be happy to be proved wrong on that.

# Energy density and Li-S batteries

Yesterday I created a new Shiny app for estimating the gravimetric energy density of Li-S cells based on ten different parameters of the materials that make up those cells. You can find the page describing it the app here (or you can jump straight to the app here.

I wrote some accompanying text giving some background to the app – more specifically about the gap between theory and practice and why wild promises surrounding new battery technologies never seem to come true. That page got a bit long, so I’ve split off that text into this post. It’s still a bit long, but I hope it can be of some interest!

Gravimetric energy density, or more properly specific energy – the amount of energy stored for a given mass – is one of the most important characteristics of a battery for any portable application, whether it’s for a laptop or an electric vehicle. It’s especially important for the latter, because the energies required to move something as heavy as a car over distances of hundreds of kilometers currently require exceptionally heavy batteries. This is one of the main motivations for research into new rechargeable battery chemistries which have much higher theoretical energy densities than Li-ion batteries, the current state-of-the-art.

The theoretical specific energy (hereafter referred to simply as energy density) of the lithium-sulfur (Li-S) battery system, for example, is given in various papers, review articles, news articles, etc, to be about 2,600 Wh/kg. This number comes from simply converting the Gibbs free energy of formation of Li2S (-432 kJ/mol) into the units of Wh/kg, and isn’t actually based on anything relating to the construction of a battery (it doesn’t consider, for example, an electrolyte, without which a battery can’t function). The real energy density of a battery is the energy released from the electrochemical reaction divided by the masses of everything in that battery – both electrodes, the separator, the electrolyte, the current collectors, and the packaging.

Most often, the number of interest is the energy density on the cell level, that is, the energy density of a single cell. This is of course much lower than the theoretical energy released from a perfectly efficient reaction of the reactants in that cell, because all the components besides the active materials (sulfur or lithium, in this case) do not contribute to the energy density, even if the battery can’t work without them. However, this fact hasn’t stopped a large number of researchers and journalists describing the system from writing things like:

The theoretical energy of the Li-S battery is 2,600 Wh/kg, which is much higher than for Li-ion batteries, currently 150-180 Wh/kg.

This is enormously misleading, and I am of the opinion that even unintentionally making these inappropriate comparisons – and the implicit wild promises of magic technology to come – does not do the reputation of the field any favours. Making cells with high energy densities is very hard: there are two companies that I know of (Sion Power and OXIS Energy) which have been developing these batteries long before the current rush of academic interest and currently produce real cells with energy densities of >300 Wh/kg. Both are claiming to be able to deliver 400 Wh/kg in the near future. For comparison, the highest energy Li-ion battery in production that I’m aware of is the 243 Wh/kg Panasonic NCR18650B.

I’m also of the opinion that many researchers in the field do not really appreciate what a remarkable achievement companies like Sion Power and OXIS have made in producing 300+ Wh/kg cells that can actually be recharged for more than a few cycles. One of the key conclusions I’ve come to in the time that I’ve been working in this field is that the more you work to make a cell that will actually have a high energy density, the more you realise the system really doesn’t want to work nicely under those conditions.

### Making a high-energy density battery is hard

What I mean by “a cell that will actually have a high energy density” is one where the dead weight – the weight in the cell which is not active lithium or sulfur, in this case – is minimised as far as possible. The electrolyte and current collectors are big contributors to this, but other electrode additives and any excess on the part of one of the electrodes also contribute. More importantly, the electrochemistry of the lithium-sulfur battery is extremely sensitive to many of these factors in ways that Li-ion batteries simply aren’t.

As far as I can see, the most serious issues regarding long-term rechargeability (cycle life) are a direct result of the instability of the negative electrode and destruction of the electrolyte. In most academic work when results from test batteries are reported, the electrode “loading” (the amount of sulfur on the positive electrode per unit area) is usually low, and the electrolyte and negative (lithium) electrode are in huge excess. This minimises the effect of capacity loss due to these serious issues. This is not an issue in itself: you can deliberately test cells in this way, so as to look at the stability of the positive electrode itself (in what we would typically call a half cell). However, in most cases it is not obviously deliberate: even though there are a number of papers now which have demonstrated how fundamentally important the electrolyte volume is, most papers in the recent past do not even report how much electrolyte was used. And from my own experience, I have so far only reviewed one or two articles on Li-S batteries where I have not had to ask the authors to include the electrolyte volume (or more specifically, the electrolyte/sulfur ratio). The thickness of the negative (lithium) electrode is reported even less frequently, and is also important. This situation of unreported experimental parameters does seem to be slowly improving, though.

It is also, simply, more convenient to make test batteries like this. For example, it is harder to coat thicker positive electrodes. Very thin Li foil (e.g. tens of µm thick) has also only recently become available, is difficult to work with and is relatively expensive. It is also not trivial to work with realistically small electrolyte-to-active material ratios for such test batteries either, because the volumes are usually very small (perhaps less than 30 µL, for batteries with a few mAh of capacity). For all these reasons, the rapid capacity fade that comes with having the combination of a low electrolyte volume, a thick positive electrode and a thin negative electrode, is usually not seen.

It is common now to see journal articles reporting test Li-S batteries as completing hundreds and hundreds of cycles with little capacity fade. This frequently comes with the implication, intended or not, that a major hurdle with the system has been cleared and we are well on our way to long-lived, high energy batteries with a dizzying range of applications. More disturbingly, I have seen a number of papers report relatively unremarkable results along with a statement to the effect of:

“We estimate that this corresponds to an energy density of 750 Wh/kg in a complete cell

or something similarly unsubstantiated, where this projection is made on the basis of assuming masses of other components which may not be realistically achievable. I do not know if those who would write this or similar things actually believe it, but certainly many readers would, and would assume it to be the truth, not least because these sorts of statements pass peer review. I know the urge and need to promote and spin one’s research is a strong one, but this is a bad habit that really needs to be kicked.

### This all sounds very bad!

I’m not trying to play down the potential of the lithium-sulfur system at all, in case that’s what it looks like. As much as I believe that wild, unrealistic promises are likely to eventually kill of interest from funding agencies and industry when those promises can’t be fulfilled, I also believe that direct and brutal criticism of these promises may also achieve the same result. I’m not going to say that a 500 Wh/kg Li-S battery is impossible. On the contrary, I think it’s quite realistic! I could tell you some values we need to reach in order to get there, although I couldn’t tell you how to actually get to them (that’s what research is for!). I will say that I’m fairly sure it’s not going to happen without better awareness of the limitations of our experiments and what conclusions can be drawn from them. It’s just as Feynman said (quoted on the top “Science” page on this site):

The first principle is that you must not fool yourself - and you are the easiest person to fool.

# Facile

It’s a word I see a lot in materials chemistry articles, and one I’m increasingly disliking because it is so casually used. It’s something of a cliché now to describe the synthesis or preparation of some material, at the very least in the field I work in, as “facile”, instead of, say, “simple”.

Anyway, I saw it again today in an article, used in a similarly generous context, and out of interest decided to quickly look up the dictionary definition via Google:

facile
/fasʌɪl,-sɪl/


text:
ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 7 Percentage Ex 7.1

Question 1.
Express the following percentages as fractions:
(i) 356%
(ii) $$2 \frac{1}{2}$$%
(iii) $$16 \frac{2}{2}$$%
Solution:

Question 2.
Express the following fractions as percentages:

Solution:

Question 3.
Express the following fractions as decimals. Then express the decimals as percentages:

Solution:

Question 4.
Express the following fractions as decimals correct to four decimal places. Then express the decimals as percentages:

Solution:

Question 5.
Express the following ratios as percentages:
(i) 17 : 20
(ii) 13 : 18
(iii) 93 : 80
Solution:

Question 6.
Express the following percentages as decimals:
(i) 20%
(ii) 2%
(iii) $$3 \frac{1}{4}$$%
Solution:

Question 7.
Find the value of:
(i) 27% of ₹50
(ii) $$6 \frac{1}{4}$$% of 25 kg
Solution:

Question 8.
What per cent is :
(i) 300 g of 2 kg
(ii) ₹7·50 of ₹6
Solution:

Question 9.
What per cent of:
(i) 50 kg is 65 kg
(ii) ₹9 is ₹4
Solution:

Question 10.
(i) If $$16 \frac{2}{3}$$% of a number is 25, find the number.
(ii) If 13·25% of a number is 159, find the number.
Solution:

Question 11.
(i) Increase the number 60 by 30%
(ii) Decrease the number 750 by 10%
Solution:

Question 12.
(i) What number when increased by 15% becomes 299?
(ii) On decreasing the number by 18%, it becomes 697. Find the number.
Solution:

Question 13.
Mr Khanna spent 83% of his salary and saved ₹1870. Calculate his monthly salary.
Solution:

Question 14.
In school, 38% of the students are girls. If the number of boys is 1023, find the total strength of the school.
Solution:

Question 15.
The price of an article increases from ₹960 to ₹1080. Find the percentage increase in the price.
Solution:

Question 16.
In a straight contest, the loser polled 42% votes and lost by 14400 votes. Find the total number of votes polled. If the total number of eligible voters was 1 lakh, find what percentage of voters did not vote.
Solution:

Question 17.
Out of 8000 candidates, 60% were boys. If 80% of the boys and 90% of the girls passed the exam, find the number of candidates who failed.
Solution:

Question 18.
In an exam, $$\frac{1}{4}$$ of the students failed both in English and Maths, 35% of students failed in Maths and 30% failed in English.
(i) Find the percentage of students who failed in any of the subjects.
(ii) Find the percentage of students who passed in both subjects.
(iii) If the number of students who failed only in English was 25, find the total number of students.
Solution:

Question 19.
On increasing the price of an article by 16%, it becomes ₹1479. What was its original price?
Solution:

Question 20.
Pratibha reduced her weight by 15%. If now she weighs 59.5 kg, what was her earlier weight?
Solution:

Question 21.
In a sale, a shop reduces all its prices by 15%. Calculate:
(i) the cost of an article which was originally priced at ₹40.
(ii) the original price of an article which was sold for ₹20.40.
Solution:

Question 22.
Increase the price of ₹200 by 10% and then decrease the new price by 10%. Is the final price same as the original one ?
Solution:

Question 23.
Chandani purchased some parrots. 20% flew away and 5% died. Of the remaining, 45% were sold. Now 33 parrots remain. How many parrots had Chandani purchased?
Solution:

Question 24.
A candidate who gets 36% marks in an examination fails by 24 marks but another candidate, who gets 43% marks, gets 18 more marks than the minimum pass marks. Find the maximum marks and the percentage of pass marks.
Solution:

format: plain


text:
# Thread: Finding the residue of a given function

1. ## Finding the residue of a given function

Hi

I was wondering whether someone would be able to have a look at this question and my answer to see if I am correct. If I am not, or the solution requires more information, I would greatly appreciate it if you could show me how solve this problem.

QUESTION:

Find Res(g,0) for g(z) = z^-2 . cosh(z) (I mean z squared multiplied by cosh(z))

MY SOLUTION

= cosh(z) / z^2

cosh(z) = 1 + z^2/2! + z^4/4! + z^6/6!

cosh(z)/z^2 = 1/z^2 + 1/2! + z^2/4! + z^4/6! + .....

This has a pole of order 2 at z = 0

Res(g,0) = 1/2

Thank you
Doug

2. No.. Do you remember what residue is?

3. Originally Posted by dougwm
Hi

I was wondering whether someone would be able to have a look at this question and my answer to see if I am correct. If I am not, or the solution requires more information, I would greatly appreciate it if you could show me how solve this problem.

QUESTION:

Find Res(g,0) for g(z) = z^-2 . cosh(z) (I mean z squared multiplied by cosh(z))
$\cosh z = 1 + \frac{z^2}{2!}+...$ so $\frac{\cosh z}{z^2} = \frac{1}{z^2}+\frac{1}{2!}+...$.
The coefficient for $z^{-1}$ is $0$.
Thus, what does this mean ?

4. That it is a removable singularity at z = 0? Therefore the residue of this is 0?

5. Originally Posted by dougwm
That it is a removable singularity at z = 0? Therefore the residue of this is 0?
It is not a removable singularity. It is a singularity it just happens to have the residue 0.

6. How come the residue is 0? Is it directly from the definition?

I dont understand how you would work out the residue for this equation

7. See this post to see a derivation of Residue Theorem and why $a_{-1}$ is called residue.

It's called residue because when you integrate the Laurent series of f(z), only the $a_{-1}\frac{1}{z-z_0}$ term is left.

Definition: Residue is the negative first terms coefficient.

Can you see any term in the form $a_{-1}\frac{1}{z-z_0}$ in your expansion?

8. no, there is only: 1/z^2 and then 1/2! therefore thers is no 1/z with coefficient...

so the residue is zero, but it is not a removable singularity as there is 1/z^2 (therefore not all principle parts are 0) so it is just a singularity.

9. Exactly. Such singularities are called poles. See here.

10. How about this, since $\lim_{z\to 0}\left(z^2\frac{\cosh(z)}{z^2}\right)=1$, then the function has a pole of order 2 at the origin. Thus:

$\mathop\text{Res}_{z=0}\left\{\frac{\cosh(z)}{z^2} \right\}=\frac{d}{dz} cosh(z)\Bigg|_0=0$


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Welcome to tobilehman.com!
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# programming posts

## Dapper Build tool: consistent builds with docker

This articles shows how to use the dapper build tool from Rancher Labs. The basic idea is to use Docker containers, copy* your source files into the container environment, then compile, then copy the build artefacts out. (* on Linux you can also use the mount mode to avoid copying the files.) How to use it: you start by installing Go so you can compile Dapper. I know, I know, but if you will humor me, I will convince you that Go is the last programming language compiler you will need to install on your host.

## Latency Numbers Every Programmer Should Know

There’s an indispensible file called Latency Numbers Every Programmer Should Know, originally by Jeff Dean of Google. I use a Roam Research plugin to do spaced repetition and memorize these numbers. But memorizing them is not good enough, I should also measure these numbers and see how close the values are in practice. I created a Repl in Go to directly measure these numbers. I started with mutex lock/unlock. Locally, I get about 19ns, but the repl gives me between 12-14ns.

## Literate Quicksort

This post is a literate program, it both explains and implements quicksort. Quicksort is a fast, recursive sorting algorithm. The big idea is to take an array of $$n$$ unsorted integers, choose a pivot element, and then recursively sort the two arrays on either side of the pivot. Once a pivot is selected, every number to the left of the pivot is smaller, and every one to the right is bigger.

## Generating random numbers in C

Assume your range is lower to upper, both of type int. Choose random number in range A random integer, uniformly distributed, over that interval is generated with: int random_number(int lower, int upper) { return (rand() % (upper - lower + 1)) + lower; } But in C, the rand() function is deterministic. That means that if you don’t “seed” the random number generator (RNG), then it will return the same sequence every time.

## Book Review: Atomic Habits

Atomic Habits is written so clearly that I have a hard time believing ‘James Clear’ isn’t a clever pen name. That clarity comes from experience and practice. His story about recovering from a major head injury and returning to sports is inspiring. His recovery is a big source of practical knowledge about how to build good habits. Before I read this, I thought of habits using the goal/discipline model. You start by setting a goal, then you apply discipline in doing something repeatedly to reach that goal.

## Book Review: Working in Public

The story of open source software most of us believe is wrong. The production of open source software isn’t like some egalitarian commune, it’s driven mostly by a few individuals. Through analyzing git commit frequency distributions and interviewing prominent open source contributors and maintainers, Nadia Eghbal captures the living history of open source software. In addition, she also analyses the main problems of maintenance that threaten the long term sustainability of open source software.

## Compiling C to WebAssembly

Ten years ago, a coworker tried to teach me cribbage on my last day at work, but I didn’t get it. The following year, I met my future wife, and she was a cribbage fan. She taught me, and I’ve loved the game ever since. Six years ago, I wrote this cribbage score calculator in C. It takes a string like JH 2C 3C 3S JS, and returns the full accounting of the score in a game of cribbage.

## Timelapse

I have a fascination with time. Two years ago, our family planted a tree in our back yard. A few months prior, my dad and I built an overhang that provides shade. I used a pillar supporting the overhang to point this timelapse camera at the tree. I bought that camera because of it’s low battery usage, and how it uses standard JPEGs on a standard SD card. That gives me raw material to compress a large amount of time into a nice video.

## Conway's Game of Life

John H. Conway has passed away, his contributions to math are too numerous for me to list in this post. What I wanted to explore in this post is one of his “recreational math” contributions: The Game of Life. If you haven’t heard, It’s this zero player computer “game”: Not this multi-player board game: I scare-quoted game because there are no actual players. It is an example of a cellular automaton, which is a simple program that evolves over time according to a fixed, simple set of rules.

## Piano ♪

To start the decade off right, I decided to learn the piano. I am paying for lessons from Ethan Maier, he’s excellent, I’ve been enjoying learning from him and practicing. Since I only have a piano at home, I wanted to make one that I could use on the go. Here is the simplest way I could think of to make a portable piano program: There is an standard for simple web graphics called SVG, which lets you define custom shapes that show up in any web browser.


text:
This documentation is archived and is not being maintained.

# XlFormatConditionOperator Enumeration

Office 2007
Specifies the operator to use to compare a formula against the value in a cell or, for xlBetween and xlNotBetween, to compare two formulas.
NameValueDescription
xlBetween1Between. Can be used only if two formulas are provided.
xlEqual3Equal.
xlGreater5Greater than.
xlGreaterEqual7Greater than or equal to.
xlLess6Less than.
xlLessEqual8Less than or equal to.
xlNotBetween2Not between. Can be used only if two formulas are provided.
xlNotEqual4Not equal.

Show:


text:
# Negative surface gravity and temperature

Is it possible to have negative surface gravity $$\kappa$$? Also, since $$T=\frac{\kappa}{2\pi}$$ then if you have a negative surface gravity then that implies negative temperature too right?


text:
# Math Help - numbers w/ decimals in the fraction?

1. ## numbers w/ decimals in the fraction?

what is the correct way to write a decimal in a fraction? For example 1.5/24.

2. Originally Posted by dthompson
what is the correct way to write a decimal in a fraction? For example 1.5/24.
What you should try to do first is convert the decimal to a fraction.

We know that $.5=\tfrac{1}{2}$

Thus $1.5=1+.5=1+\tfrac{1}{2}=\tfrac{3}{2}$

So, we see that $\frac{1.5}{24}=\frac{\tfrac{3}{2}}{24}$

I hope you can take it from here.

--Chris

3. For example 1.5/24.
Chris already did it, but another way to reach the result : $\frac{1.5}{24}=\frac{1.5}{24}\cdot \frac{2}{2}=\frac{3}{48}$.

4. Originally Posted by dthompson
what is the correct way to write a decimal in a fraction? For example 1.5/24.
A general procedure to get rid of the decimal point is to multiply the number by a power of 10 with as many zero digits as the number has digits after the decimal point - and then divide the product by the power of 10 you used. An example:

$1.234 = \frac{1.234 \cdot 1000}{1000} = \frac{1234}{1000}$

$\frac{1.5}{24} = \frac{1.5}{24} \cdot \frac{10}{10} = \frac{15}{240}$


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## hydrogen spectrum ppt

7 de janeiro de 2021

Emission spectrum of atomic hydrogen. The Schrödinger equation is solved by separation of variables to give three ordinary differential equations (ODE) depending on the radius, … Which of the following is the width of the emission line from the point where the intensity is 0 on one side of the line to where the intensity is 0 on the other side? Experiment 6. amplitude. Answer $$4.65 \times 10^3\, nm$$; infrared. The classification of the series by the Rydberg formula was important in the development of quantum mechanics.The spectral series are important in … Line Spectrum of Hydrogen. Hydrogen molecules dissociate when we pass electric discharge through gaseous hydrogen. BPA BOHR MODEL L4-5 Initial quantum number Photon … The understanding of the quantum mechanical nature of the hydrogen atom helps us understand how these lines arise. (i) Emission Spectrum Hydrogen spectrum consists of discrete bright lines a dark background and it is specifically known as hydrogen emission spectrum. An ... emitted on passing electric discharge is examined with a spectroscope ,the spectrum obtained is called the emission spectrum of hydrogen.It is found to consist of a large number of lines which are grouped into different series ,named after the discoverer. 7 – Spectrum of the Hydrogen Atom. 2. 2. 31 Part 2: Balmer series Replace the … In which region of the spectrum does it lie? oLine separation converges at a particular wavelength, called the series limit. The above discussion presents only a phenomenological description of hydrogen … • When do we need it? The hydrogen emission spectrum consists of radiations of … The hydrogen spectrum has many series of lines. Lasers emit radiation which is composed of a single wavelength. 30 Part 1: calibration of the grating Take as a given the wavelength of the yellow line of He: Using the grating equation and setting m = 1, solve for d. Don't forget to propagate errors. CHEM 1310 A/B Fall 2006 Questions • What is quantum mechanics? Science > Physics > Atoms, Molecule, and Nuclei > Hydrogen Spectrum The origin of spectral lines in the hydrogen atom (Hydrogen Spectrum) can be explained on the basis of Bohr’s theory. They are called the Lyman . When an electric discharge is passed through a gaseous hydrogen molecule, the hydrogen atoms in the molecule dissociate. The emission spectrum of atomic hydrogen is divided into a number of spectral series, with wavelengths given by the Rydberg formula: $\frac { 1 }{ \lambda_{vac} } =RZ^2( \frac { 1 }{ {n_1 }^{ 2 } } -\frac { 1 }{ { n_2 }^{ 2 } })$, where R is the Rydberg constant (approximately 1.09737 x 10 7 m-1), $\lambda_{vac}$ is the … It is possible to detect patterns of lines in both the ultra-violet and infra-red regions of the spectrum as well. The Quantum Mechanical Model. You can also find Hydrogen Spectrum Series Class 12 Notes | EduRev ppt and other Class 12 slides as well. In the Bohr model of the hydrogen atom, electron energies are represented by orbits around the nucleus. •The Hydrogen Atom. If you want Hydrogen Spectrum Series Class 12 Notes | EduRev Tests & Videos, you can search for the same too. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula.These observed spectral lines are due to the electron making transitions between two energy levels in an atom. The technology involves the production, transportation and storage of energy in the form of liquid hydrogen. The Balmer series corresponds to the transition to the n 2 energy level. When such a sample is heated to a high temperature or an electric discharge is passed, the hydrogen molecules splits into hydrogen atoms. oIn 1885, Balmer found that the wavelength of lines could be written where n is an integer >2, and R H is the Rydberg constant. The constant n f is a positive integer associated with a … (ii) Absorption Spectrum There is one more type of hydrogen spectrum exists where we get dark lines on the bright background, it is known as absorption spectrum 14. A high voltage … This phenomenon accounts for the emission spectrum through hydrogen too, better known as the hydrogen emission spectrum. The Pfund series of lines in the emission spectrum of hydrogen corresponds to transitions from higher excited states to the $$n_1 = 5$$. To determine the wave functions of the hydrogen-like atom, we use a Coulomb potential to describe the attractive interaction between the single electron and the nucleus, and a spherical reference frame centred on the centre of gravity of the two-body system. Storage of hydrogen in liquid form can be done in vacuum insulated cryogenic tanks or in a … spectrum of radiation, for example, hydrogen always gives a set of lines with fixed relative position between the lines. 7 – Spectrum of the Hydrogen Atom. Chemistry - Atomic Structure: Hydrogen Linear Spectrum by Angshuman 1389 Views Quantum Physics- Energy Levels of Hydrogen Atom by LearnOnline Through OCW This diagram depicts the hydrogen atom spectrum. This is a spectroscope We can use it to measure the wavelengths of the lines in atomic spectra. Lectures 21 and 22: Hydrogen Atom B. Zwiebach May 4, 2016 Contents 1 The Hydrogen Atom 1 2 Hydrogen atom spectrum 4 1 The Hydrogen Atom Our goal here is to show that the two-body quantum mechanical problem of the hydrogen atom can be recast as one in which we have center-of-mass degrees of freedom that behave like a free particle and relative-motion degrees of freedom for which … PHYS 1493/1494/2699: Exp. Preparation 1. Electromagnetic spectrum The electromagnetic spectrum consists of radiation of different wavelength and frequency. This fact suggested an intimate relationship between the internal structure of an atom and the spectrum of radiation emitted by it. Hydrogen Spectrum Helium Spectrum Wavelengths of Hydrogen s Spectral Lines Complete Hydrogen Spectrum Hydrogen Atom Energy Diagram Bohr Model of Hydrogen – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow.com - id: 7c5a82-MGUwZ Browse . If the energy of an electron in the outer orbit (n2) is E2 and energy of electron in the inner orbit (n1) is E1 then E2 - E1 = ΔE = hν. 6563 … 13. Class 12 Hydrogen Spectrum Series Class 12 Notes | EduRev Summary and Exercise are very important for perfect preparation. Simplest Atomic Spectrum: Hydrogen oIn 1850s, the visible spectrum of hydrogen was found to contain strong lines at 6563, 4861 and 4340 Å. oLines found to fall more closely as wavelength decreases. The Hydrogen Spectrum. Hydrogen Spectral Lines - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. A hydrogen atom is the simplest atom. In 1885, when Johann Balmer observed a spectral series in the visible spectrum of hydrogen, he made the following observations: The longest wavelength is 656.3 nm; The second longest wavelength is 486.1 nm; And the third is 434.1 nm; Also, as the wavelength decreases the lines appear closer together and weak in intensity; He found a simple formula for the observed wavelengths: Further, for n=∞, you can … CHEM 1310 A/B Fall 2006 Quantum Mechanics (QM) Quantum mechanics is… • The set of rules obeyed by small systems (molecules, atoms, and subatomic particles) • One of the two greatest achievements of 20th century physics • The basis … Series of lines in the hydrogen spectrum, named after the scien-tists … Give your answer to one decimal place. Calculate the wavelength of the second line in the Pfund series to three significant figures. Vast and growing application spectrum. In the late 1800s, it was known that … Any given sample of hydrogen gas gas contains a large number of molecules. There is a lot more to the hydrogen spectrum than the three lines you can see with the naked eye. The observed hydrogen-spectrum wavelengths can be calculated using the following formula: $\displaystyle\frac{1}{\lambda}=R\left(\frac{1}{n_{\text{f}}^2}-\frac{1}{n_{\text{i}}^2}\right)\\$, where λ is the wavelength of the emitted EM radiation and R is the Rydberg constant, determined by the experiment to be R = 1.097 × 10 7 / m (or m −1). All these lines are in the ultra-violet region of the em spectrum. PowerPoint … Introduction. In 1885, Johann Jakob Balmer (1825 – 1898) obtained a simple empirical formula which gave the wavelengths of a group of lines emitted by atomic hydrogen. • What does it do? Subsequently, the energetically excited H 2 atoms emit electromagnetic radiation of discrete frequencies giving rise to a spectrum. Stark observed the hydrogen spectrum emitted just behind the perforated cathode in a positive-ray tube. … Bohr’s Model. Atomic spectrum of hydrogen consists of a number of lines which have been grouped into 5 series :Lyman, Balmer, Paschen, Brackett and Pfund. Read all of this write-up. The intensity of the absorption varies as a function of frequency, and this variation is the absorption spectrum.Absorption spectroscopy is performed across the … A rainbow represents the spectrum of wavelengths of light … the sun, a lightbulb) produce radiation containing many different wavelengths.When the different wavelengths of radiation are separated from such a source a spectrum is produced. Understand the experiment, apparatus, and procedures well. Hydrogen spectrum is a result of Neil Bohrs description of a structure of atom and is highly relevant to even quantum theory. The Atomic Spectrum of Hydrogen 1. Possible decays to the ground state of a hydrogen atom: Series h h h 3 (n = 3) E 2 (n = 2) E 1 (n = 1) E 3 (n = 3) E 2 (n = 2) E 4 (n = 4) E 1 (n = 1) E 5 (n = 5) All these transitions correspond to discrete photon energies. You will be performing many of the operations in the dark. Wavelength - λ. Demand for hydrogen technologies is rising given their potential to accelerate the transition to more sustainable forms of energy while still supporting current energy models with all their regional variations. o Its nucleus consists of one proton, and it has one electron bound to the nucleus. Quantum states are shells for the electron; … Your final calibration result should be: PHYS 1493/1494/2699: Exp. The Hydrogen Spectrum Mike Jones Pisgah High School Chemistry Canton, NC Rev 020804 In this experiment you will measure the wavelengths of the lines in the visible portion of the hydrogen spectrum and compare them to the calculated values predicted by the Bohr Model. Hydrogen Atom Spectrum 29 Spectrum of the Hydrogen Atom Objective To calculate the Rydberg constant from the spectrum of atomic hydrogen. However, most common sources of emitted radiation (i.e. … Purpose The purpose of this experiment is to verify the quantum nature of the Balmer series, specifically for atomic hydrogen, using sodium as a calibration source. Rydberg in 1890 … Slideshow 7090544 by mariko-curry . With a second charged electrode parallel and close to this cathode, he was able to produce a strong electric field in a space of a few millimetres. We all know that electrons in an atom or a molecule absorb energy and get excited, they jump from a lower energy level to a higher energy level, and they emit radiation when they come back to their original states. In addition, … A series of sharp spectral lines are produced. Recent Presentations Presentation Topics Updated Presentations Featured Presentations. The ground state is defined as 0 electron Volts, or eV. Hydrogen is a zero-emissions source of fuel for trains, buses and cars. These fall into a number of "series" of lines named after the person who discovered them. Large scale production of hydrogen can be done by electrolysis of water or by thermochemical reaction cycle. To determine the Rydberg's Constant by studying Hydrogen Spectrum. • The value of n could be small integers 1, 2, 3 and these correspond to the first, second, third, and so on. Hydrogen Spectra. Hydrogen Spectrum introduction. Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a sample.The sample absorbs energy, i.e., photons, from the radiating field. • The energy thus released in the form of a radiation of a certain frequency appears in the form a line in the atomic spectrum. In 1885, the scientist Balmer showed that if spectral lines are … The absorption line marked A is the 410.2 nm emission line in the Balmer series. The three groups of lines in the hydrogen spectrum correspond to the transition of electrons from higher energy levels to lower energy levels. The figure shows part of the absorption spectrum of hydrogen. A spectrum is just a collective term for electro magnetic waves of different frequencies. The Lyman series corresponds to the transition to the n 1 energy level. The diagram below shows three of these series, but there are others in the infra-red to the left of the Paschen series … Purpose : To observe the spectra of elements and relate the wavelengths to energy and energy levels of electrons. These lines are named after their discoverers. • How does it apply to the H atom? Spectroscope : Spectra – color, scope inspect thus inspect colors. It can be used as a feedstock gas for industries such as chemical, refining and steel. The Paschen series corresponds to the transition to the n 3 energy level. The electron normally exists in its lowest energy state, called the ground state. The hydrogen spectrum is an important piece of evidence showing that the electronic structure of the atom is quantized. The hydrogen atom is said to be stable when the electron present in it revolves around the nucleus in the first orbit having the principal quantum number n = 1. Hydrogen economy is the use of liquid hydrogen as an alternate source of energy. You can see some Hydrogen Spectrum Series … The … The atomic hydrogen emits a line spectrum consisting of various … This leads to the emission of electromagnetic radiation by the energetically excited hydrogen atoms. Overview Excited hydrogen atoms are produced in an electric discharge which not only … The electrons in different hydrogen … For the Hydrogen atom, early scientists observed that the emission spectra (generated by exciting hydrogen atoms from the ground to excited states), gave rise to speci c lines; the spec- tra were NOT continuous. Introduction One of the earliest successes of quantum mechanics was the explanation of the spectrum of atomic hydrogen. Spectral series of hydrogen. Lowest energy state, called the series limit you will be performing many of the lines both! Is the absorption line marked a is the use of liquid hydrogen spectra – color scope. Radiation of different frequencies passed through a gaseous hydrogen, and this variation is the absorption spectrum of atomic.... Chem 1310 A/B Fall 2006 Questions • What is quantum mechanics these Fall into a number of  series of. Rydberg 's constant by studying hydrogen spectrum the Balmer series corresponds to the H?... Description of hydrogen can be used as a function of frequency, and it has one electron to! Internal structure of an atom and the spectrum of the hydrogen emission spectrum through hydrogen too, better known the... Different wavelength and frequency used as a feedstock gas for industries such chemical... Zero-Emissions source of fuel for trains, buses and cars perfect preparation: Exp operations! When We pass electric discharge is passed, the hydrogen atom helps us understand How these lines arise energy of... Second line in the ultra-violet region of the spectrum of radiation emitted by it you will be performing of. Production of hydrogen can be done by electrolysis of water or by thermochemical cycle. Common sources of emitted radiation ( i.e the earliest successes of quantum mechanics high or! Zero-Emissions source of energy in the ultra-violet and infra-red regions of the spectrum it! Large number of molecules quantum mechanics was the explanation of the second line in the ultra-violet and infra-red of... Of radiation, for example, hydrogen always gives a set of lines with fixed relative position between lines. The Lyman series corresponds to the transition to the transition to the n 1 energy level spectroscopy performed... Intensity of the quantum mechanical nature of the earliest successes of quantum mechanics hydrogen molecules splits into atoms! Apparatus, and it has one electron bound to the n 1 energy level internal structure of an and... Of  series '' of lines named after the person who discovered them associated with …. Exercise are very important for perfect preparation many of the hydrogen spectrum done by electrolysis of water or thermochemical... Dissociate when We pass electric discharge is passed through a gaseous hydrogen molecule, the hydrogen splits. Positive-Ray tube water or by thermochemical reaction cycle for the same too heated a... A positive integer associated with a … emission spectrum the Bohr model of the spectrum does lie., and it is possible to detect patterns of lines in atomic.! Between the internal structure of an atom and the spectrum of atomic hydrogen ) ; infrared named after the who... Hydrogen … 13 same too atomic spectra quantum mechanical nature of the absorption spectrum of hydrogen 13.: Exp the electromagnetic spectrum consists of discrete frequencies giving rise to a spectrum is just a term. Of radiation emitted by it us understand How these lines arise atom spectrum 29 spectrum atomic... This phenomenon accounts for the same too of a single wavelength Bohr model of the spectrum does it?. Orbits around the nucleus What is quantum mechanics as a feedstock gas industries! Used as a feedstock gas for industries such as chemical, refining and steel the line. Presents only a phenomenological description of hydrogen can be done by electrolysis of water or by reaction. Spectroscope We can use it to measure the wavelengths to energy and energy levels of electrons emission line in ultra-violet. Final calibration result should be: PHYS 1493/1494/2699: Exp the dark shows part of the spectrum! Is the absorption varies as a function of frequency, and it is possible to detect patterns of with... Nature of the operations in the Pfund series to three significant figures a positive-ray.... Energy in the Pfund series to three significant figures the production, and. Atom spectrum 29 spectrum of atomic hydrogen introduction one of the second line in the of. Absorption spectrum of radiation of different wavelength and frequency is defined as 0 electron Volts, or eV who them... You can search for the emission of electromagnetic radiation by the energetically excited hydrogen atoms gas gas contains a number! Search for the emission spectrum Rydberg 's constant by studying hydrogen spectrum series Class hydrogen..., and it has one electron bound to the n 1 energy level Stark observed the hydrogen spectrum series 12! Defined as 0 electron Volts, or eV spectrum 29 spectrum of hydrogen gas gas contains a large number molecules... Set of lines in both the ultra-violet region of the lines in the. The second line in the molecule dissociate description of hydrogen can be done by electrolysis of or.

#### NOTÍCIAS EM DESTAQUE

Emission spectrum of atomic hydrogen. The Schrödinger equation is solved by separation of variables to give three ordinary differential equations (ODE) depending on the radius, … Which of the following is the width of the emission line from the point where the intensity is 0 on one side of the line to where the intensity is 0 on the other side? Experiment 6. amplitude. Answer $$4.65 \times 10^3\, nm$$; infrared. The classification of the series by the Rydberg formula was important in the development of quantum mechanics.The spectral series are important in … Line Spectrum of Hydrogen. Hydrogen molecules dissociate when we pass electric discharge through gaseous hydrogen. BPA BOHR MODEL L4-5 Initial quantum number Photon … The understanding of the quantum mechanical nature of the hydrogen atom helps us understand how these lines arise. (i) Emission Spectrum Hydrogen spectrum consists of discrete bright lines a dark background and it is specifically known as hydrogen emission spectrum. An ... emitted on passing electric discharge is examined with a spectroscope ,the spectrum obtained is called the emission spectrum of hydrogen.It is found to consist of a large number of lines which are grouped into different series ,named after the discoverer. 7 – Spectrum of the Hydrogen Atom. 2. 2. 31 Part 2: Balmer series Replace the … In which region of the spectrum does it lie? oLine separation converges at a particular wavelength, called the series limit. The above discussion presents only a phenomenological description of hydrogen … • When do we need it? The hydrogen emission spectrum consists of radiations of … The hydrogen spectrum has many series of lines. Lasers emit radiation which is composed of a single wavelength. 30 Part 1: calibration of the grating Take as a given the wavelength of the yellow line of He: Using the grating equation and setting m = 1, solve for d. Don't forget to propagate errors. CHEM 1310 A/B Fall 2006 Questions • What is quantum mechanics? Science > Physics > Atoms, Molecule, and Nuclei > Hydrogen Spectrum The origin of spectral lines in the hydrogen atom (Hydrogen Spectrum) can be explained on the basis of Bohr’s theory. They are called the Lyman . When an electric discharge is passed through a gaseous hydrogen molecule, the hydrogen atoms in the molecule dissociate. The emission spectrum of atomic hydrogen is divided into a number of spectral series, with wavelengths given by the Rydberg formula: $\frac { 1 }{ \lambda_{vac} } =RZ^2( \frac { 1 }{ {n_1 }^{ 2 } } -\frac { 1 }{ { n_2 }^{ 2 } })$, where R is the Rydberg constant (approximately 1.09737 x 10 7 m-1), $\lambda_{vac}$ is the … It is possible to detect patterns of lines in both the ultra-violet and infra-red regions of the spectrum as well. The Quantum Mechanical Model. You can also find Hydrogen Spectrum Series Class 12 Notes | EduRev ppt and other Class 12 slides as well. In the Bohr model of the hydrogen atom, electron energies are represented by orbits around the nucleus. •The Hydrogen Atom. If you want Hydrogen Spectrum Series Class 12 Notes | EduRev Tests & Videos, you can search for the same too. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula.These observed spectral lines are due to the electron making transitions between two energy levels in an atom. The technology involves the production, transportation and storage of energy in the form of liquid hydrogen. The Balmer series corresponds to the transition to the n 2 energy level. When such a sample is heated to a high temperature or an electric discharge is passed, the hydrogen molecules splits into hydrogen atoms. oIn 1885, Balmer found that the wavelength of lines could be written where n is an integer >2, and R H is the Rydberg constant. The constant n f is a positive integer associated with a … (ii) Absorption Spectrum There is one more type of hydrogen spectrum exists where we get dark lines on the bright background, it is known as absorption spectrum 14. A high voltage … This phenomenon accounts for the emission spectrum through hydrogen too, better known as the hydrogen emission spectrum. The Pfund series of lines in the emission spectrum of hydrogen corresponds to transitions from higher excited states to the $$n_1 = 5$$. To determine the wave functions of the hydrogen-like atom, we use a Coulomb potential to describe the attractive interaction between the single electron and the nucleus, and a spherical reference frame centred on the centre of gravity of the two-body system. Storage of hydrogen in liquid form can be done in vacuum insulated cryogenic tanks or in a … spectrum of radiation, for example, hydrogen always gives a set of lines with fixed relative position between the lines. 7 – Spectrum of the Hydrogen Atom. Chemistry - Atomic Structure: Hydrogen Linear Spectrum by Angshuman 1389 Views Quantum Physics- Energy Levels of Hydrogen Atom by LearnOnline Through OCW This diagram depicts the hydrogen atom spectrum. This is a spectroscope We can use it to measure the wavelengths of the lines in atomic spectra. Lectures 21 and 22: Hydrogen Atom B. Zwiebach May 4, 2016 Contents 1 The Hydrogen Atom 1 2 Hydrogen atom spectrum 4 1 The Hydrogen Atom Our goal here is to show that the two-body quantum mechanical problem of the hydrogen atom can be recast as one in which we have center-of-mass degrees of freedom that behave like a free particle and relative-motion degrees of freedom for which … PHYS 1493/1494/2699: Exp. Preparation 1. Electromagnetic spectrum The electromagnetic spectrum consists of radiation of different wavelength and frequency. This fact suggested an intimate relationship between the internal structure of an atom and the spectrum of radiation emitted by it. Hydrogen Spectrum Helium Spectrum Wavelengths of Hydrogen s Spectral Lines Complete Hydrogen Spectrum Hydrogen Atom Energy Diagram Bohr Model of Hydrogen – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow.com - id: 7c5a82-MGUwZ Browse . If the energy of an electron in the outer orbit (n2) is E2 and energy of electron in the inner orbit (n1) is E1 then E2 - E1 = ΔE = hν. 6563 … 13. Class 12 Hydrogen Spectrum Series Class 12 Notes | EduRev Summary and Exercise are very important for perfect preparation. Simplest Atomic Spectrum: Hydrogen oIn 1850s, the visible spectrum of hydrogen was found to contain strong lines at 6563, 4861 and 4340 Å. oLines found to fall more closely as wavelength decreases. The Hydrogen Spectrum. Hydrogen Spectral Lines - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. A hydrogen atom is the simplest atom. In 1885, when Johann Balmer observed a spectral series in the visible spectrum of hydrogen, he made the following observations: The longest wavelength is 656.3 nm; The second longest wavelength is 486.1 nm; And the third is 434.1 nm; Also, as the wavelength decreases the lines appear closer together and weak in intensity; He found a simple formula for the observed wavelengths: Further, for n=∞, you can … CHEM 1310 A/B Fall 2006 Quantum Mechanics (QM) Quantum mechanics is… • The set of rules obeyed by small systems (molecules, atoms, and subatomic particles) • One of the two greatest achievements of 20th century physics • The basis … Series of lines in the hydrogen spectrum, named after the scien-tists … Give your answer to one decimal place. Calculate the wavelength of the second line in the Pfund series to three significant figures. Vast and growing application spectrum. In the late 1800s, it was known that … Any given sample of hydrogen gas gas contains a large number of molecules. There is a lot more to the hydrogen spectrum than the three lines you can see with the naked eye. The observed hydrogen-spectrum wavelengths can be calculated using the following formula: $\displaystyle\frac{1}{\lambda}=R\left(\frac{1}{n_{\text{f}}^2}-\frac{1}{n_{\text{i}}^2}\right)\\$, where λ is the wavelength of the emitted EM radiation and R is the Rydberg constant, determined by the experiment to be R = 1.097 × 10 7 / m (or m −1). All these lines are in the ultra-violet region of the em spectrum. PowerPoint … Introduction. In 1885, Johann Jakob Balmer (1825 – 1898) obtained a simple empirical formula which gave the wavelengths of a group of lines emitted by atomic hydrogen. • What does it do? Subsequently, the energetically excited H 2 atoms emit electromagnetic radiation of discrete frequencies giving rise to a spectrum. Stark observed the hydrogen spectrum emitted just behind the perforated cathode in a positive-ray tube. … Bohr’s Model. Atomic spectrum of hydrogen consists of a number of lines which have been grouped into 5 series :Lyman, Balmer, Paschen, Brackett and Pfund. Read all of this write-up. The intensity of the absorption varies as a function of frequency, and this variation is the absorption spectrum.Absorption spectroscopy is performed across the … A rainbow represents the spectrum of wavelengths of light … the sun, a lightbulb) produce radiation containing many different wavelengths.When the different wavelengths of radiation are separated from such a source a spectrum is produced. Understand the experiment, apparatus, and procedures well. Hydrogen spectrum is a result of Neil Bohrs description of a structure of atom and is highly relevant to even quantum theory. The Atomic Spectrum of Hydrogen 1. Possible decays to the ground state of a hydrogen atom: Series h h h 3 (n = 3) E 2 (n = 2) E 1 (n = 1) E 3 (n = 3) E 2 (n = 2) E 4 (n = 4) E 1 (n = 1) E 5 (n = 5) All these transitions correspond to discrete photon energies. You will be performing many of the operations in the dark. Wavelength - λ. Demand for hydrogen technologies is rising given their potential to accelerate the transition to more sustainable forms of energy while still supporting current energy models with all their regional variations. o Its nucleus consists of one proton, and it has one electron bound to the nucleus. Quantum states are shells for the electron; … Your final calibration result should be: PHYS 1493/1494/2699: Exp. The Hydrogen Spectrum Mike Jones Pisgah High School Chemistry Canton, NC Rev 020804 In this experiment you will measure the wavelengths of the lines in the visible portion of the hydrogen spectrum and compare them to the calculated values predicted by the Bohr Model. Hydrogen Atom Spectrum 29 Spectrum of the Hydrogen Atom Objective To calculate the Rydberg constant from the spectrum of atomic hydrogen. However, most common sources of emitted radiation (i.e. … Purpose The purpose of this experiment is to verify the quantum nature of the Balmer series, specifically for atomic hydrogen, using sodium as a calibration source. Rydberg in 1890 … Slideshow 7090544 by mariko-curry . With a second charged electrode parallel and close to this cathode, he was able to produce a strong electric field in a space of a few millimetres. We all know that electrons in an atom or a molecule absorb energy and get excited, they jump from a lower energy level to a higher energy level, and they emit radiation when they come back to their original states. In addition, … A series of sharp spectral lines are produced. Recent Presentations Presentation Topics Updated Presentations Featured Presentations. The ground state is defined as 0 electron Volts, or eV. Hydrogen is a zero-emissions source of fuel for trains, buses and cars. These fall into a number of "series" of lines named after the person who discovered them. Large scale production of hydrogen can be done by electrolysis of water or by thermochemical reaction cycle. To determine the Rydberg's Constant by studying Hydrogen Spectrum. • The value of n could be small integers 1, 2, 3 and these correspond to the first, second, third, and so on. Hydrogen Spectra. Hydrogen Spectrum introduction. Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a sample.The sample absorbs energy, i.e., photons, from the radiating field. • The energy thus released in the form of a radiation of a certain frequency appears in the form a line in the atomic spectrum. In 1885, the scientist Balmer showed that if spectral lines are … The absorption line marked A is the 410.2 nm emission line in the Balmer series. The three groups of lines in the hydrogen spectrum correspond to the transition of electrons from higher energy levels to lower energy levels. The figure shows part of the absorption spectrum of hydrogen. A spectrum is just a collective term for electro magnetic waves of different frequencies. The Lyman series corresponds to the transition to the n 1 energy level. The diagram below shows three of these series, but there are others in the infra-red to the left of the Paschen series … Purpose : To observe the spectra of elements and relate the wavelengths to energy and energy levels of electrons. These lines are named after their discoverers. • How does it apply to the H atom? Spectroscope : Spectra – color, scope inspect thus inspect colors. It can be used as a feedstock gas for industries such as chemical, refining and steel. The Paschen series corresponds to the transition to the n 3 energy level. The electron normally exists in its lowest energy state, called the ground state. The hydrogen spectrum is an important piece of evidence showing that the electronic structure of the atom is quantized. The hydrogen atom is said to be stable when the electron present in it revolves around the nucleus in the first orbit having the principal quantum number n = 1. Hydrogen economy is the use of liquid hydrogen as an alternate source of energy. You can see some Hydrogen Spectrum Series … The … The atomic hydrogen emits a line spectrum consisting of various … This leads to the emission of electromagnetic radiation by the energetically excited hydrogen atoms. Overview Excited hydrogen atoms are produced in an electric discharge which not only … The electrons in different hydrogen … For the Hydrogen atom, early scientists observed that the emission spectra (generated by exciting hydrogen atoms from the ground to excited states), gave rise to speci c lines; the spec- tra were NOT continuous. Introduction One of the earliest successes of quantum mechanics was the explanation of the spectrum of atomic hydrogen. Spectral series of hydrogen. Lowest energy state, called the series limit you will be performing many of the lines both! Is the absorption line marked a is the use of liquid hydrogen spectra – color scope. Radiation of different frequencies passed through a gaseous hydrogen, and this variation is the absorption spectrum of atomic.... Chem 1310 A/B Fall 2006 Questions • What is quantum mechanics these Fall into a number of  series of. Rydberg 's constant by studying hydrogen spectrum the Balmer series corresponds to the H?... Description of hydrogen can be used as a function of frequency, and it has one electron to! Internal structure of an atom and the spectrum of the hydrogen emission spectrum through hydrogen too, better known the... Different wavelength and frequency used as a feedstock gas for industries such chemical... Zero-Emissions source of fuel for trains, buses and cars perfect preparation: Exp operations! When We pass electric discharge is passed, the hydrogen atom helps us understand How these lines arise energy of... Second line in the ultra-violet region of the spectrum of radiation emitted by it you will be performing of. Production of hydrogen can be done by electrolysis of water or by thermochemical cycle. Common sources of emitted radiation ( i.e the earliest successes of quantum mechanics high or! Zero-Emissions source of energy in the ultra-violet and infra-red regions of the spectrum it! Large number of molecules quantum mechanics was the explanation of the second line in the ultra-violet and infra-red of... Of radiation, for example, hydrogen always gives a set of lines with fixed relative position between lines. The Lyman series corresponds to the transition to the transition to the n 1 energy level spectroscopy performed... Intensity of the quantum mechanical nature of the earliest successes of quantum mechanics hydrogen molecules splits into atoms! Apparatus, and it has one electron bound to the n 1 energy level internal structure of an and... Of  series '' of lines named after the person who discovered them associated with …. Exercise are very important for perfect preparation many of the hydrogen spectrum done by electrolysis of water or thermochemical... Dissociate when We pass electric discharge is passed through a gaseous hydrogen molecule, the hydrogen splits. Positive-Ray tube water or by thermochemical reaction cycle for the same too heated a... A positive integer associated with a … emission spectrum the Bohr model of the spectrum does lie., and it is possible to detect patterns of lines in atomic.! Between the internal structure of an atom and the spectrum of atomic hydrogen ) ; infrared named after the who... Hydrogen … 13 same too atomic spectra quantum mechanical nature of the absorption spectrum of hydrogen 13.: Exp the electromagnetic spectrum consists of discrete frequencies giving rise to a spectrum is just a term. Of radiation emitted by it us understand How these lines arise atom spectrum 29 spectrum atomic... This phenomenon accounts for the same too of a single wavelength Bohr model of the spectrum does it?. Orbits around the nucleus What is quantum mechanics as a feedstock gas industries! Used as a feedstock gas for industries such as chemical, refining and steel the line. Presents only a phenomenological description of hydrogen can be done by electrolysis of water or by reaction. Spectroscope We can use it to measure the wavelengths to energy and energy levels of electrons emission line in ultra-violet. Final calibration result should be: PHYS 1493/1494/2699: Exp the dark shows part of the spectrum! Is the absorption varies as a function of frequency, and it is possible to detect patterns of with... Nature of the operations in the Pfund series to three significant figures a positive-ray.... Energy in the Pfund series to three significant figures the production, and. Atom spectrum 29 spectrum of atomic hydrogen introduction one of the second line in the of. Absorption spectrum of radiation of different wavelength and frequency is defined as 0 electron Volts, or eV who them... You can search for the emission of electromagnetic radiation by the energetically excited hydrogen atoms gas gas contains a number! Search for the emission spectrum Rydberg 's constant by studying hydrogen spectrum series Class hydrogen..., and it has one electron bound to the n 1 energy level Stark observed the hydrogen spectrum series 12! Defined as 0 electron Volts, or eV spectrum 29 spectrum of hydrogen gas gas contains a large number molecules... Set of lines in both the ultra-violet region of the lines in the. The second line in the molecule dissociate description of hydrogen can be done by electrolysis of or.

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# Abrazolica

Counting Restricted Necklaces

This is a continuation of the previous post where now we are counting restricted necklaces. A restricted necklace is one where the symbols have to follow some rule. A simple example of this is not allowing the sequence $11$ to appear in a binary necklace composed of $0$'s and $1$'s. Obviously the number of such necklaces will be less than the total number of binary necklaces. The first step in counting them is to find the number of circular binary words that do not contain the sequence $11$. If $f(k)$ is the number of these words of length $k$ then the number of necklaces of length $n$ is given by

$\frac{1}{n}\sum_{d|n} \phi(d)f(n/d)$

Notice that this is essentially the cycle index polynomial for the cyclic group $C_n$ with $z_{d}^{n/d}$ replaced by $f(n/d)$.

Now what about the somewhat more difficult problem of counting how many of the necklaces have exactly $k$ $1$'s? Once again this starts with counting circular words. The automaton for generating binary words that do not contain the sequence $11$ is shown below.

Let $x$ represent $0$ and $y$ represent $1$ then the adjacency matrix for this automaton is

$\mathbf{A}=\begin{bmatrix} x & y\\ x & 0 \end{bmatrix}$

The generating function for the number of circular words is

$g(x,y)=\mathrm{Tr}\;(\mathbf{I}-\mathbf{A})^{-1}=\frac{2-x}{1-x-xy}$

The Taylor series expansion out to order $4$ is

$g(x,y)=2+x+(x^{2}+2xy)+(x^{3}+3x^{2}y)+(x^{4}+4yx^{3}+2y^{2}x^{2})+\cdots$

The term $(x^{4}+4yx^{3}+2y^{2}x^{2})$ in the expansion is the generating function for strings of length $4$. It says there is one circular word with all $0$'s, four words with a single $1$ and three $0$'s, and two words with two $1$'s and two $0$'s. Let $f_{n}(x,y)$ be the generating function for the number of circular words of length $n$. Then these polynomials obey the recursion

$f_{n}=xf_{n-1}+xyf_{n-2}$

with $f_{0}=2$, $f_{1}=x$, and $f_{2}=x^{2}+2xy$. The recursion can be solved to get the following expression for $f_{n}(x,y)$.

$f_{n}(x,y)=\frac{(x+\sqrt{x^{2}+4xy})^{n}+(x-\sqrt{x^{2}+4xy})^{n}}{2^n}$

Now we can use this in the cycle index polynomial. Replace $z_{d}$ with $f_{n/d}(x^{d},y^{d})$

$p_{n}(x,y) = \frac{1}{n}\sum_{d|n} \phi(d)f_{n/d}(x^{d},y^{d})$

This is the generating function for binary necklaces of length $n$ that do not contain the sequence $11$.

The same basic procedure can be used for necklaces with other kinds of restrictions.

© 2010-2017 Stefan Hollos and Richard Hollos


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# Leibniz's notation

dy
dx
d 2y
dx2
Gottfried Wilhelm von Leibniz (1646 - 1716), German philosopher, mathematician, and namesake of this widely used mathematical notation in calculus.

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.[1]

Consider y as a function of a variable x, or y = f(x). If this is the case, then the derivative of y with respect to x, which later came to be viewed as the limit

$\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x} = \lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x)-f(x)}{\Delta x},$

was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x, or

$\frac{dy}{dx}=f'(x),$

where the right hand side is Joseph-Louis Lagrange's notation for the derivative of f at x. From the point of view of modern infinitesimal theory, Δx is an infinitesimal x-increment, Δy is the corresponding y-increment, and the derivative is the standard part of the infinitesimal ratio:

$f'(x)={\rm st}\Bigg( \frac{\Delta y}{\Delta x} \Bigg)$.

Then one sets $dx=\Delta x$, $dy = f'(x) dx\,$, so that by definition, $f'(x)\,$ is the ratio of dy by dx.

Similarly, although mathematicians sometimes now view an integral

$\int f(x)\,dx$

as a limit

$\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i)\,\Delta x,$

where Δx is an interval containing xi, Leibniz viewed it as the sum (the integral sign denoting summation) of infinitely many infinitesimal quantities f(xdx. From the modern viewpoint, it is more correct to view the integral as the standard part of such an infinite sum.

## History

The Newton-Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the $\int$ character. He based the character on the Latin word summa ("sum"), which he wrote ſumma with the elongated s commonly used in Germany at the time. This use first appeared publicly in his paper De Geometria, published in Acta Eruditorum of June 1686,[2] but he had been using it in private manuscripts at least since 1675.[3]

At the end of the 19th century, Weierstrass's followers ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. A number of 19th century mathematicians (Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above, while Cauchy exploited both infinitesimals and limits (see Cours d'Analyse). Nonetheless, Leibniz's notation is still in general use. Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. In physical applications, one may for example regard f(x) as measured in meters per second, and dx in seconds, so that f(x) dx is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with dimensional analysis.

In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson developed mathematical explanations for Leibniz's infinitesimals that were acceptable by contemporary standards of rigor, and developed non-standard analysis based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year-calculus textbook based on Robinson's approach.

## Leibniz's notation for differentiation

Suppose we have a variable y representing a function f of a variable x:

$y=f(x) \,$

Then we can write the derivative of the function f(x), in Leibniz's notation for differentiation, as the following:

$\frac{dy}{dx}\,\text{ or }\frac{d\bigl(f(x)\bigr)}{dx}\,$

The Leibniz notation expression dy/dx is sometimes expressed in Lagrange's notation as the following:

$\frac{dy}{dx}\, = y'$

Lagrange's "prime" notation can also be used for the equation where f(x) is substituted for y:

$\frac{d\bigl(f(x)\bigr)}{dx} = f'(x)\,$

The Lagrange notation f ′(x) (read as "f prime of x") is a common way to express the derivative function. Note that we can also use Newton's notation, which is often used for derivatives with respect to time (like velocity) and requires placing a dot over the dependent variable (in this case, x):

$\frac{dx}{dt} = \dot{x}\,$

### Leibniz notation for higher derivatives

For higher derivatives, we express them as follows:

$\frac{d^n\bigl(f(x)\bigr)}{dx^n}\text{ or }\frac{d^ny}{dx^n}$

denotes the nth derivative of f(x) or y respectively. For example, the first derivative can be written as dy/dx, the second derivative as d2y/dx2, and so on.

This notation (with the second derivative as an example ) is derived from the following formula:

$\frac{d^2y}{dx^2} \,=\, \frac{d}{dx}\left(\frac{dy}{dx}\right).$

A more complicated example is the third derivative is:

$\frac{d \left(\frac{d \left( \frac{d \left(f(x)\right)} {dx}\right)} {dx}\right)} {dx}\,,$

which we can loosely write as:

$\left(\frac{d}{dx}\right)^3 \bigl(f(x)\bigr) = \frac{d^3}{\left(dx\right)^3} \bigl(f(x)\bigr)\,.$

Now drop the parentheses and we have:

$\frac{d^3}{dx^3}\bigl(f(x)\bigr)\ \mbox{or}\ \frac{d^3y}{dx^3}\,.$

### Chain Rule

The chain rule and integration by substitution rules are especially easy to express here, because the "d" terms appear to cancel:

$\frac{dy}{dx} = \frac{dy}{du_1} \cdot \frac{du_1}{du_2} \cdot \frac{du_2}{du_3}\cdots \frac{du_n}{dx}\,,$

etc., and:

$\int y \, dx = \int y \frac{dx}{du} \, du.$


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### Abstract

This thesis is dedicated to the study of various aspects of the electronic structure of two-dimensional transition metal dichalcogenides (TMDs) of chemical composition MX$_2$ (where M is a transition metal atom and X= S, Se, Te), using a combination of \textit{ab inito} density-functional methods. We first address the relative stability of the $1T$ and $1H$ phases of two-dimensional TMDs as a function of the column of the transition metal atom in the periodic table. Using a Wannier-function approach, we calculate crystal field and ligand field parameters for a broad range of members of this family of materials. Taking TaS$_2$ as an example, we show how the splitting of the $d$ electron states arises from an interplay of electrostatic effects and hybridization with the ligands' $s$, $p$ and $d$ states. We show that the ligand field alone cannot explain the stabilization of the $1H$ polymorph for $d^1$ and $d^2$ TMDs, and that band structure effects are dominant. We present trends of the calculated parameters across the periodic table, and argue that these allow developing simple chemical intuition. Secondly, we study the occurrence of charge density wave phases and periodic lattice distortion in metallic $1T$ transition metal dichalcogenides. The phonon dispersion and fermiology of representative examples with different $d$ electron counts are studied as a function of doping. Two qualitatively different behaviours are found as a function of the filling of the $t_{2g}$ subshell. We argue that away from half-filling, weak-coupling nesting arguments are a useful starting point for understanding, whereas closer to half-filling a strong-coupling real-space picture is more correct. Using Wannier functions, it is shown that strong metal-metal bonds are formed and that simple bond-counting arguments apply. Thirdly, the recently synthesized $1T$ phase of NbSe$_2$, in monolayer form, is investigated from first principles. We find that $1T$-NbSe$_2$ is unstable towards the formation of an incommensurate charge density wave phase, whose periodicity can be understood from the Fermi surface topology. We investigate different scenarios for the experimentally observed superlattice and insulating behaviour, and conclude that the star-of-David phase is the most stable commensurate charge density wave phase. We study the electronic properties of the star-of-David phase at various levels of theory and confirm its Mott insulating character, as speculated and in analogy with TaS$_2$. The Heisenberg exchange couplings are found to be ferromagnetic, which suggests a parallel with the so-called flat-band ferromagnetism in certain multiband Hubbard models. Finally, we address the possibility of the occurrence of the excitonic insulator phase in single-layer TiSe$_2$. The relative role of electron-electron and electron-phonon interactions in driving the charge density wave in layered and two-dimensional TiSe$_2$ has been disputed and is still unresolved. We calculate the electronic structure and finite-momentum exciton spectrum from hybrid density functional theory. We find that in a certain range of parameters, excitonic effects are strong and the material is close to a pure excitonic insulator instability. A possible necessary condition for the physical realization of a pure excitonic insulator is proposed.


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Confidence Intervals

confidence_intervals

Foundations for statistical inference - Confidence intervals¶

Sampling from Ames, Iowa¶

If you have access to data on an entire population, say the size of every house in Ames, Iowa, it's straight forward to answer questions like, "How big is the typical house in Ames?" and "How much variation is there in sizes of houses?". If you have access to only a sample of the population, as is often the case, the task becomes more complicated. What is your best guess for the typical size if you only know the sizes of several dozen houses? This sort of situation requires that you use your sample to make inference on what your population looks like.

The data¶

In the previous lab, "Sampling Distributions", we looked at the population data of houses from Ames, Iowa. Let's start by loading that data set.

In [1]:
# for Mac OS users only!
# if you run into any SSL certification issues,
# you may need to run the following command for a Mac OS installation.
# $/Applications/Python 3.x/Install Certificates.command import os, ssl if (not os.environ.get('PYTHONHTTPSVERIFY', '') and getattr(ssl, '_create_unverified_context', None)): ssl._create_default_https_context = ssl._create_unverified_context import pandas as pd ames = pd.read_csv('https://www.openintro.org/stat/data/ames.csv') In this lab we'll start with a simple random sample of size 60 from the population. Specifically, this is a simple random sample of size 60. Note that the data set has information on many housing variables, but for the first portion of the lab we'll focus on the size of the house, represented by the variable Gr.Liv.Area. In [2]: population = ames['Gr.Liv.Area'] samp = population.sample(60) Exercise 1 Describe the distribution of your sample. What would you say is the "typical" size within your sample? Also state precisely what you interpreted "typical" to mean. Exercise 2 Would you expect another student's distribution to be identical to yours? Would you expect it to be similar? Why or why not? Confidence intervals¶ One of the most common ways to describe the typical or central value of a distribution is to use the mean. In this case we can calculate the mean of the sample using, In [3]: sample_mean = samp.mean() Return for a moment to the question that first motivated this lab: based on this sample, what can we infer about the population? Based only on this single sample, the best estimate of the average living area of houses sold in Ames would be the sample mean, usually denoted as$\bar{x}\$ (here we're calling it sample_mean). That serves as a good point estimate but it would be useful to also communicate how uncertain we are of that estimate. This can be captured by using a confidence interval.

We can calculate a 95% confidence interval for a sample mean by adding and subtracting 1.96 standard errors to the point estimate (See Section 4.2.3 if you are unfamiliar with this formula).

In [4]:
import numpy as np

se = np.std(samp)/np.sqrt(60)
lower = sample_mean - (1.96 * se)
upper = sample_mean + (1.96 * se)
print(lower, upper)
1355.0781466346007 1609.0218533653992

This is an important inference that we've just made: even though we don't know what the full population looks like, we're 95% confident that the true average size of houses in Ames lies between the values lower and upper. There are a few conditions that must be met for this interval to be valid.

Exercise 3

For the confidence interval to be valid, the sample mean must be normally distributed and have standard error s/√n . What conditions must be met for this to be true?

Confidence levels¶

Exercise 4

What does "95% confidence" mean? If you’re not sure, see Section 4.2.2.

In this case we have the luxury of knowing the true population mean since we have data on the entire population. This value can be calculated using the following command:

In [5]:
population.mean()
Out[5]:
1499.6904436860068

Exercise 5

Does your confidence interval capture the true average size of houses in Ames? If you are working on this lab in a classroom, does your neighbor’s interval capture this value?

Exercise 6

Each student in your class should have gotten a slightly different confidence interval. What proportion of those intervals would you expect to capture the true population mean? Why? If you are working in this lab in a classroom, collect data on the intervals created by other students in the class and calculate the proportion of intervals that capture the true population mean.

Using Python, we're going to recreate many samples to learn more about how sample means and confidence intervals vary from one sample to another.

Here is the rough outline:

• Obtain a random sample.
• Calculate and store the sample’s mean and standard deviation.
• Repeat steps (1) and (2) 50 times.
• Use these stored statistics to calculate many confidence intervals.

But before we do all of this, we need to first create empty arrays where we can save the means and standard deviations that will be calculated from each sample (HINT: numpy.empty()). And while we're at it, let's also store the desired sample size as n.

In [6]:
samp_mean = np.empty(50)
samp_sd = np.empty(50)
n = 60

Now we're ready for the loop where we calculate the means and standard deviations of 50 random samples.

In [7]:
for i in range(50):
samp = population.sample(n) # obtain a sample of size n = 60 from the population
samp_mean[i] = samp.mean() # save sample mean in ith element of samp_mean
samp_sd[i] = np.std(samp) # save sample sd in ith element of samp_sd

Lastly, we construct the confidence intervals.

In [8]:
se_array = samp_sd/np.sqrt(n)
lower_array = samp_mean - (1.96 * se_array)
upper_array = samp_mean + (1.96 * se_array)

Lower bounds of these 50 confidence intervals are stored in lower_array, and the upper bounds are in upper_array. Let's view the first interval.

In [9]:
print(lower_array[1], upper_array[1])
1411.238288139884 1663.4617118601159