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In this chapter you will learn about quantities that change, such as the height of a tree. As the tree grows, the height changes. A quantity that changes is called a variable quantity or just a variable.

It is often the case that when one quantity changes, another quantity also changes. For example, as you make more and more calls on a phone, the total cost increases. In this case, we say there is a relationship between the amount of money you have to pay and the number of calls you make. You will learn how to describe a relationship between two quantities in different ways.

## Constant and variable quantities

### Looking for connections between quantities

Consider the following seven situations. There are two quantities in each situation. For each quantity, state whether it is constant (always the same number) or whether it changes. Also state, in each case, whether one quantity has an influence on the other. If it has, try to say how the one quantity will influence the other quantity.

2. The number of calls you make and the airtime left on your cellphone
3. The length of your arm and your ability to finish Mathematics tests quickly
4. The number of identical houses to be built and the number of bricks required
5. The number of learners at a school and the length of the school day
6. The number of learners at a school and the number of classrooms needed
7. The number of matches in each arrangement here, and the number of triangles in the arrangement

A quantity that changes is called a variable quantity or just a variable.

If one variable quantity is influenced by another, we say there is a relationship between the two variables. It is sometimes possible to find out what value of the one quantity, in other words what number, is linked to a specific value of the other quantity.

1. Look at the match arrangements in question 7. If you know that there are 3 triangles in an arrangement, can you say with certainty how many matches there are in that specific arrangement?
2. How many matches are there in the arrangement with 10 triangles?
3. Is there another possible answer for question (b)?
1. Complete the flow diagram by filling in all the missing numbers. Do you see any connections between the situation in question 7 and this flow diagram? If so, describe the connections.

### Completing some flow diagrams

A relationship between two quantities can be shown with a flow diagram, such as those below. Unfortunately, only some of the numbers can be shown on a flow diagram.

1. Calculate the output numbers for the flow diagram below. Some input numbers are missing. Choose and insert your own input numbers.

Each input number in a flow diagram has a corresponding output number. The first (top) input number corresponds to the first output number. The second input number corresponds to the second output number and so on.

We call +5 the operator.

1. What type of numbers are the given input numbers?
2. In the above flow diagram, the output number 8 corresponds to the input number 3. Complete the following sentences:

In the relationship shown in the above flow diagram, the output number ______ corresponds to the input number -1.

The input number ______ corresponds to the output number 7.

If more places are added to the flow diagram, the input number ______ will correspond to the output number 31.

1. Complete this flow diagram.

2. Compare this flow diagram to the flow diagram in question 1. What link do you find between the two?
2. Complete the flow diagrams below. You have to find out what the operator for (b) is, and fill it in yourself.
1. What number can you add in (a), instead of subtracting 5, that will produce the same output numbers?
2. What number can you subtract in (b), instead of adding a number, that will produce the same output numbers?
3. Complete the flow diagram:

A completed flow diagram shows two kinds of information:

• It shows what calculations are done to produce the output numbers.
• It shows which output number is connected to which input number.

The flow diagram that you have completed in question 4 shows the following information:

• Each input number is multiplied by 6, then 40 is added to produce the output numbers.
• The input and output numbers are connected as shown in the table below.
 Input numbers -1 -2 -3 -4 -5 Output numbers 34 28 22 16 10
1. Describe in words how the output numbers below can be calculated.

2. Use the table below to show which output numbers are connected to which input numbers in the above flow diagram.
1. The following information is available about the floor space and cost of new houses in a new development. The cost of an empty stand is R180 000.
 Floor space in square metres 90 120 150 180 210 Cost of house and stand 540 000 660 000 780 000 900 000 1 020 000
1. Represent the above information on the flow diagram below.

2. Show what the houses only will cost, if you get the stand for free.

3. Try to figure out what the cost of a house and stand will be, if there are exactly one hundred 1 m by 1 m sections of floor space in the house.

## Different ways to describe relationships

A relationship between red dots and blue dots

Here is an example of a relationship between two quantities:

In each arrangement there are some red dots and some blue dots.

1. How many blue dots are there if there is one red dot?
2. How many blue dots are there if there are two red dots?
3. How many blue dots are there if there are three red dots?
4. How many blue dots are there if there are four red dots?
5. How many blue dots are there if there are five red dots?
6. How many blue dots are there if there are six red dots?
7. How many blue dots are there if there are seven red dots?
8. How many blue dots are there if there are ten red dots?
9. How many blue dots are there if there are twenty red dots?
10. How many blue dots are there if there are one hundred red dots?
11. Which of the descriptions correctly describe the relationship between the number of blue dots and the number of red dots in the above arrangements? Test each description thoroughly for all the above arrangements. List them on the dotted line below. Write only the letters, for example (d).

Are there different possibilitiesfor the number of blue dots if there are 3 red dots in the arrangement?

Are there different possibilities for the number of blue dots if there are 2 red dots in the arrangement?

Are there different possibilities for the number of blue dots if there are 20 red dots in the arrangement?

1. the number of red dots the number of blue dots
2. to calculate the number of blue dots you multiply the number of red dots by 2, add 1 and multiply the answer by 2
3. number of blue dots = 2 $$\times$$ the number of red dots + 4
4.  Number of red dots 1 2 3 4 5 6 Number of blue dots 6 10 14 18 22 26
5. number of blue dots = 4 $$\times$$ the number of red dots + 2
6. number of blue dots $$= 2 \times (2 \times$$ the number of red dots + 1) (Remember that the calculations inside the brackets are done first.)

The descriptions in (c), (g) and (h) above are called word formulae.

### Translating between different languages of description

A relationship between two quantities can be described in different ways, including the following:

• a table of values of the two quantities
• a flow diagram
• a word formula
• a symbol formula (or symbolic formular)

You will learn about symbolic formulae in section 5.3.

1. The relationship between two quantities is described as follows:

The second quantity is always 3 times the first quantity plus 8.

The first quantity varies between 1 and 5, and it is always a whole number.

1. Describe this relationship with the flow diagram.

2. Describe the relationship between the two quantities with this table.
3. Describe the relationship between the two quantities with a word formula.
2. The relationship between two quantities is described as follows:

The input numbers are the first five odd numbers.

value of the one quantity the corresponding value of the other quantity

1. Describe this relationship with a table.
2. Describe the relationship with a word formula.

## Algebraic symbols for variables and relationships

Describing procedures in different ways

1. In each case do four things:
• Complete the table.
• Describe the relationship with a word formula.
• Describe the input numbers in words.
• Describe the output numbers in words.
1. input number output number
 Input number 5 10 15 20 25 30 Output number

the output number =

2. input number output number
 Input number 5 10 15 20 25 30 Output number

3. input number output number
 Input number 5 10 15 20 25 30 Output number

Formulae with symbols

Instead of writing "input number" and "output number" in formulae, one may just write a single letter symbol as an abbreviation.

Mathematicians have long ago adopted the convention of using the letter symbol $$x$$ as an abbreviation for "input number", and the letter symbol $$y$$ as an abbreviation for "output number".

Other letter symbols than $$x$$ and $$y$$ are also used to indicate variable quantities.

The word formula you wrote for question 1(a) can be written more shortly as

$$y = 10 \times x + 15$$

Mathematicians have also long ago agreed that one may leave the \times-sign out when writing symbolic formulae.

So, instead of $$y = 10 \times x + 15$$ we may write $$y = 10x + 15$$.

It is not at all wrong to use the multiplication sign in symbolic formulae.

1. Rewrite your word formulae in questions 1(b) and 1(c) as symbolic formulae.
2. Write a word formula for each of the following relationships:
1. $$y = 7x + 10$$
2. $$y = 7(x + 10)$$
3. $$y = 7(2x + 10)$$

### Writing symbolic formulae

Describe each of the following relationships with a symbolic formula:

1. To calculate the output number, the input number is multiplied by 4 and 7 is subtracted from the answer.
2. To calculate the output number, 7 is subtracted from the input number and the answer is multiplied by 5.
3. To calculate the output number, 7 is subtracted from the input number, the answer is multiplied by 5 and 3 is added to this answer.


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# Analyze raw tick data

I'd like to work with raw tick data and naturally this data is unevenly spaced (for example, a couple of quotes are at the same second etc.)

For example

10:12:35 - 14.44
10:12:35 - 14.45
10:12:35 - 14.47
10:12:36 - 14.46
10:12:36 - 14.49
10:12:37 - 14.50

My question is regarding how to set this data to "fixed" intervals for valid math calculation. Though I read here couple of suggestions on what should be done, I'm not sure its clear to me:

1. Do I have to manipulate the raw data to have "clean" timestamps as the x-axis to work with common technical indicators?

2. Or can I refer to the index as the x-axis (and just ignore the timestamp)? Can I look at it as a "stream" of ticks to anaylze?

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If that is how your data looks, it looks more like trades than quotes. For it to be a quote you need a bid and an ask. – Louis Marascio May 5 '13 at 0:30

If you just want to run some simplistic technical analysis on quotes, then select the last quote for each unique timestamp. That will ensure that you don't have duplicate timestamps. If you must have it evenly spaced (i.e. no gaps from one second to another), then you can reuse the previous quote to fill-in the missing value.

-
1. so in anyway, you don't recommend just to use the "stream" of ticks (and ignore the timestamp)? to refer the X-Axis as the tick index (tick 1, tick 2, tick 3) and the Y-Axis as its value? – user1025852 May 4 '13 at 17:53
2. I wonder how all the platforms address this issue (metatrader etc.) - all those platform that have technical indicator out-of-the-box – user1025852 May 4 '13 at 17:54
BTW - it's not "no gaps from one second to another" it's the oppisite - I have multiple quotes per second.... – user1025852 May 4 '13 at 17:56

To help you understand why you need to follow recipes (like chrisaycock's) just have a look at your tick data. You will find ticks clustered at some points in time while they seem scarce at others.

If you proceed with your recipe 2, you will lose those clusters of activity and stretch them out. In periods of low activity you will condense the market.

Most indicators you mentioned will expect to work over a window of time, simply because the results are so much more meaningful. In theory, you could apply the same algorithms to stretched or condensed indexed ticks, but the results will also be valid for those periods.

That means for a cluster of activity that makes the price jump from X to X+10 within 1000 ticks but only one second of time your indicator might tell you to sell on the 400th tick and buy on the 800th. But to implement this you would have to execute a round-trip within less than a second.

Now also this indicator's results are less comparable. Because in a period of low market activity the indicator might give you the same result (sell on 400, buy on 800), but now it's stretched over, say, hours, and it's much more likely that you can realise this trade.

-

It really depends on what you're trying to do, a solution as simple as agging the data to x-sec OHLC bars may suffice and from the sounds of things that's what you need. Now if you need to work with order book dynamics then tick data is fairly crucial, then what I'd do for analysis is reconstruct the orderbook from the ticks then just take snapshots of the book (again depends on your requirements).

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I agree with ast4. And if you're analyzing at the tick level then it seem like you would want to analyze bid size, ask size and trade price in addition to bid price and ask price... – Al De Los Santos May 30 '13 at 19:37

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# Really need help with combinatorial proof!

I have spent the last few hours trying to crack this combinatorial proof and I was really hoping someone would be able to help out.

$$\sum_{i=0}^r \left(\!\!{m\choose i}\!\!\right)= \binom{m+r}{r}$$

Thank you! I have tried thinking of a counting question to pose but none of them have really seemed to fit what is given above.

Note: On the left is a multiset of m and i.

• Why do you add the parenthesis? I've never seen that done before. – S.C.B. Feb 16 '17 at 6:57
• To represent a multisite of m and i – xonian Feb 16 '17 at 6:58
• And what is a multisite? Google returns no meaningful results. – S.C.B. Feb 16 '17 at 6:59
• I'm sorry, multiset - my autocorrect messed that one up – xonian Feb 16 '17 at 6:59
• Oh, that's what it is. I would think a better $\LaTeX$ for that would be \left(\!\!{n\choose k}\!\!\right). – S.C.B. Feb 16 '17 at 7:01

## 2 Answers

Use the formula that $\left(\!{n\choose k}\!\right)=\binom{n+k-1}{k}$ to get

$$\binom{m-1}{0}+\binom{m}{1}+\binom{m+1}{2}+\cdots+\binom{m+r-1}{r}.$$

Now note that $\binom{m-1}0=\binom m0$, so $\binom{m-1}0+\binom{m}1=\binom m0+\binom m1=\binom{m+1}1$. Keep using this identity and the whole sum will collapse.

 A combinatorial proof:

The RHS is the number of ways to choose a multiset of size $r$ from $n+1$ objects. There is a one-to-one correspondence between such multisets and possible multisets of size at most $r$ taken from the first $n$ objects -- just throw away any copies of object $n+1$ we may have chosen. This is what the LHS counts.

• that's actually how I went about it originally, but I need to prove it with combinatorially, or with some sort of counting question, rather than algebraic expansion or manipulation, which is where I am stuck. Thank you though! – xonian Feb 16 '17 at 7:08
• Ah, I see. Have added a combinatorial reason. – Especially Lime Feb 16 '17 at 7:21
• Awesome, thank you so much. – xonian Feb 16 '17 at 7:33

\begin{align} \sum_{i=0}^r \left(\!\!{m\choose i}\!\!\right) &=\sum_{i=0}^r\binom{m+i-1}{m-1}\\ &=\sum_{i=0}^r\binom{m-1+i}{m-1}\\ &=\binom{m-1+r+1}{m-1+1}\\ &= \binom{m+r}{m}\\ &=\binom {m+r}r \end{align}


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# Electronic Spectroscopy: Application

Electronic Absorption and Fluorescence spectroscopy are both analytical methods that center around the idea that when one perturbs a known or unknown solution with a spectrum of energetic photons, those photons that have the correct energy to interact with the molecules in solution will do so, and those molecules under observation will always interact with photons of energies characteristic to that molecule. These methods have tremendously helped scientists elucidate and characterize the physical properties of a variety of molecules by giving a means to probe the fundamental electronic structure of those molecules.

## Introduction

The theory behind Electronic Absorption and Fluorescence was described in a previous text. If you are unfamiliar with electronic spectroscopy, browsing the theory might help paint a better picture of what will be discussed in this module.But even if you understand what a Jablonski diagram represents or the mathematical description behind the transition dipole moment operator, eventually you want to know how to apply the information you've learned to your world.

What this means to you is that you now have the ability that allow you to find the reason why carrots are orange or why plants are green. You could take leaves and carrots, chop them into fine pieces, extract and separate the molecules inside of them, and use Electronic Absorption or Fluorescence spectroscopy to figure out that the orange color in carrots is due to a family of molecules known as the carotenoids and that the green color in the leaves is due to a variety of different chlorophyll molecules. So excited by this knowledge, you could take the carotenoids that you now know are responsible for the color orange and dye cloth to make yourself a carrot costume!

But what if you wanted a different color? What if you wanted a lighter shade of orange? What if you wanted a very exact and very specific color? Before you can go explore the world (or lab) to find (or design) a molecule that has those properties you need to be able to understand the instruments that allow you to do the work so you can be certain you get the results you desire.

The remainder of this wikitext describes the design and components of a photospectrometer to measure the the Ultra Violet through Visable (UV-Vis) region of the electromagnetic spectrum. Basic instrumental design and theory behind a spectrofluorometer will also be discussed along with the errors and limits known to plague these instruments. By the end of this text, you should be fully able to design a a photospectrometer in order to characterize the electronic behavior of molecules in order to prove to yourself that carotenoids are responsible for the color orange (building of a carrot costume is optional)

## Basic Design

As stated in the introduction, the purpose of a spectrophotometer is to characterize how molecules react with photons of varying wavelengths. This easiest way to do this is to design an instrument that can control what type of light is in contact with the sample and the amount of that light that makes it through. Shown below is a simple spectrophotometer that does just what was described. A light source produces light that can be separated and controlled by the monochrometer which then allows only the desired light to impinge upon the sample. The light that travels through the sample is detected and sent to a recorder. Keep this overall picture in mind when thinking about the components and what each of their parts are in whole experiment.

### Sources

As shown above, the first piece of equipment needed is a source of photons (light). Since we are interested in the UV-Vis region of the electromagnetic spectrum, we need to assure that this source is able to produce photons with the correct wavelengths (180-780 nm) that are characteristic to the spectrums of interest so that the experiment can be completed. Typical sources come in two forms, either in the form of a lamp, or in the form of a laser. Lamps can provide a wide spectrum of light that can be used for almost any purpose at a very reasonable cost, while lasers provide very intense monochromatic light and tend can be used for fluorescence and other specialized experiments due to their high costs.

### Tungsten Filament Lamps

A common lamp source that might be familiar is the tungsten filament lamp or sometimes referred to as the incandescent light bulb. This type of lamp produces a spectrum of light perfect for the visible spectrum (320 nm - 3.5 µm)1 by applying a voltage through a thin filament of tungsten until it heats up enough to produce black body radiation. The Tungsten-Halogen lamp is a significant improvement over the normal tungsten filament lamp due to its longer lifetime and the wider range of light it produces. This is achieved by incasing the filament with quartz instead of silica so the filament can be forced to higher temperatures (thus increasing the range of light produced) and by trapping a halogen gas (normally Iodine) that can react with sublimated tungsten atoms to form WI2 that can then redeposit the tungsten by coming into contact with the filament (thus increasing the lifespan).

### Hydrogen/Deuterium Lamps

This type of lamp is used to obtain a spectrum of light in the Ultra Violet region (190-400 nm). The spectrum is obtained by trapping Hydrogen at low pressures inside of a glass tube and creating an electrical arc inside of the cell. This electrical arc creates high energy hydrogen gas atoms that dissociate into two hydrogen atoms along with producing photons. The wavelength of the photons created from the dissociation is dependent on the resulting kinetic energies of the two hydrogen atoms after the dissociation. Since the kinetic energies of the dissociated atoms can vary from zero to the original energy of the excited atom, the photon produced can also vary which leads to a spectrum of photons. Deuterium is often used in lieu of hydrogen because it can produce more a more intense spectrum and has a longer lifetime. Both types of this lamp are housed within a quartz shell since silica absorbs in the Ultraviolet range.

### Lasers

Tunable dye and fixed wavelength lasers that produce photons within the UV-Vis spectrum are commonly used for fluorescence experiments where the intensity of the resulting fluorescence is directly proportional to the intensity of the source. Lasers are inherently focused which provides the benefit of being able to work with very small or dilute samples. The caveats of using a laser are that the spectrum produced is physically limited and that care must also be taken to avoid photodegredation of your samples by attenuating the power of the laser. A description of the construction and theory behind lasers won't be discussed here, but can be found through a variety of sources.1,2,3,4,5,6

## Filters and Monochrometers

Unless a laser is used, a device is needed to restrict and isolate the wavelengths that our sources provide in order to control the experiment. If a isolated wavelength is needed or isn't needed, interference or absorption filters can be used. If a spectrum of wavelengths needs to be passed through the sample, monochrometers provide the ability to separate and control the light passed through the sample while preserving the available spectrum.

### Filters

Intereference filters are designed to provide constructive or destructive interference of light by taking advantage of the refraction of light through different materials. As light passes from one medium to the other the direction and wavelength of light can be changed based on the index of refraction of both mediums involved and the angle of the incident and exiting light (For more look at Snell's Law). Due to this behavior, constructive and destructive interference can be controlled by varying the thickness (d) of a transparent dielectric material between two semi-reflective sheets and the angle the light is shined upon the surface. As light hits the first semi-reflective sheet, a portion is reflected, while the rest travels through the dielectric to be bent and reflected by the second semi-reflective sheet. If the conditions are correct, the reflected light and the initial incident light will be in phase and constructive interference occurs for only a particular wavelength.

Another common filter is the Absorption Filter. Absorption filters work on the premise of being able to filter light by absorbing all other wavelengths that aren't of interest. They are normally constructed from a colored glass that absorbs over a wide range. Although normally cheaper than interference filters, absorbance filters tend to be less precise at filtering for a selected wavelength and also have the added penalty of absorbing some of the selected light thus lowering the intensity.

### Monochrometers

Monochrometers, as previously mentioned, are used to control and separate light so that a sample can be subjected to a span of wavelengths. Light entering a monochrometer is filtered by a thin slit. The filtered light is then focused by a mirror onto a dispersing element that separates the light into its different wavelengths. The separated light is then focused again and angled toward an exit slit which then filters against all wavelengths except the desired one. The wavelength selected can then easily be changed by rotating the dispersing element to support the transmission of the new desired wavelength. Though monochrometers nearly all have the same practical design, the difference normally is determined by the type of dispersing element.

One type of dispersing element is a prism which disperses light by refraction through two angled surfaces. In order to separate the light into different wavelengths, the prism needs to be made of material that has a change in the index of refraction with respect to wavelength so that each wavelength is bent at a different angle. The larger the difference in the index of refraction the better the separation is between wavelengths. For the UV-Vis range, a typical prism is cut from left handed quartz at a thirty degree angle and attached to another piece of quarts cut the same way except from right handed quartz to make a Conru Prism. The pitfall of prism based dispersing elements is the fact that the index of refraction for most materials varies nonlinearly when compared to wavelength which results in a smaller degree of separation at longer wavelengths.

More commonly used due to less expensive fabrication costs and its ability to separate light in a linear fashion are grating dispersing elements. Gratings are designed to diffract light which is a separation based on the angle of the incident light to the grating normal and the spacing between groves. When beams of light impinge upon the grating some beams travel farther than others and this causes an effect akin to a interference filter which allows for constructive and destructive interference and provides the means to reflect a specific wavelength based on the angle of incidence. The most common type of grating is the echellette grating in which the grooves are angled in order to provide maximum reflection of the incident light into a single order of reflected light.

## Sample Cell

Care must be taken when considering a sample container to avoid unwanted absorption in the range of interest. While glass might be perfect for the visible range, it absorbs in the UV range. Quartz can be used for both but most likely would cost considerably more. Disposable Plastic cuvets created from polystyrene or polymethyl methacrylate are in common use today as they are cheap to purchase and eliminate the need for cleaning cuvettes in order to analyze multiple samples. The shape of the sample holder is also very important as unwanted scattering of light should be minimized and pathlength through the cell should remain constant . Square (1 cm) cuvettes with frosted sides have been designed to minimize scattering and provide a surface to hold the cuvette, thereby reducing smudges or smears of the optical window all the while keeping a fixed path length. Cuvettes for fluorescent experiments cannot allow the frosted sides due to the 90° angle of most instruments and have to be carefully cleaned as bodily oils have the ability to fluoresce. Placement of the sample within either a absorption or fluorescence device is imperative and is normally fixed by a sample holder to insure reproducible results.

## Detectors

After the light has passed through the sample, we want to be able to detect and measure the resulting light. These types of detectors come in the form of transducers that are able to take energy from light and convert it into an electrical signal that can be recorded, and if necessary, amplified.Photomultiplier tubes are a common example of a transducer that is used in a variety of devices. The idea is that when a photon hits the top of the tube electrons are released, which are pulled toward the other end of the tube by an electric field. The way the electrons are multiplied is due to the fact that along the length of the tube there are several dynodes that have a slightly less negative potential than surface before it which causes the electrons traveling down the tube to hit each surface which then in turn produces more electrons until at the very end there is a large amount of electrons (~1,000,000) representing the one photon that started the cascade.

Another type of transducer is a Charge injection device (CID). This device works by having a p-type and n-type semiconductor next to each other in which the n-type material is separated from the anodes by a silica layer that acts as a capacitor. As a photon interacts with the n-type layer, an electron migrates to the p-type semiconductor and a positive charge is generated that migrates toward the silica capacitor. This process continues to happen until the potential is measured by means of comparing the more negative of the anodes to ground (Vi) . The charge is then transferred to the other anode and measured (Vf). The difference between Vf and Vi is directly proportional to the amount of photons that collided with the n-type layer. The positive charges are then repelled when the anodes switch momentarily to having a positive charge and then the cycle can be repeated. Thousands of these little detectors can be aligned and used to describe the light that interacts with it and can rival the performance of the photomultiplier tube.

### Signal Processor

In the end of the experiment all of the data collected needs to be stored or written down. In the past printers have been used to record the transmittance as the experiment progresses. These days, computers are used to record, store, and even manipulate data along with controlling the devices within the spectrometer. This ability allowed by having a computer that can quickly integrate, compare, or annotate spectra dramatically decreases the learning curve needed to operate the instrument as well as reduces the labor associated with simple or even complex experiments. Instead of having to record the transmittance, convert it to absorbance and then subtract the background noise, the computer can accomplish all that for you in the matter of a few seconds .

### Instrumental Designs

Now that we've covered the instrument and its components in some detail, there are some different modifications that can be made in order to tailor the device to the experiment it'll be performing. The device introduced in the beginning of this text is the basic single beam spectrometer. This device can measure the transmittance or absorbance of a particular analyte at a given wavelength provided by the source and monochrometer. Background subtraction in these machines needs to be done separately before the analyte is inserted and can be stored and subtracted by the signal processor attached. Although this is the simplest design, the cost of these types of instruments can vary greatly depending on the components that the machine is comprised of.

For fluorescence spectroscopy the equivalent to a single beam spectrometer has a few slight modifications in that the detector is perpendicular to the source, and that there is an additional monochrometer that can be used to vary the wavelength detected. In this fashion, the source light is unable to interfere with the fluorescence light being measured, and the fluorescent light produced can be separated and described as the resulting wavelengths of fluorescence as a function of incident light.

A double beam spectrometer takes a more analytical approach to the design. Because of fluctuations in the source intensity and inconsistencies in the transducer, the source light is split into two beams, one which travels through the sample, and another that is sent through a blank or standard solution. Both beams are then read by separate transducers and the difference between the two is recorded as the corrected transmittance. This allows for quick screening of analytes and negates the need for two separate scans to complete a background subtraction. The same idea can be applied to a fluorometer as well to obtain the same benefit

The latest instrument designs use a multichannel detector, such as an array of CID's, that allow for the spectrum of an analyte to be gathered in seconds due to the fact that the light transmitted through the sample can be split and a spectrum of wavelengths can be monitored simultaniously instead of individually. Also fiber optic cables can be used to transfer the light from the source to the sample or from the sample to the dispersion grating and negate the need to consider the slit when thinking about resolution.

### Instrumental Noise

As with any instrument, the measurements we make are limited by the tools we use. In spectrophotometers, the only measurement they are designed to monitor is the transmittance of light through the sample. Thus any noise associated with the transmittance will correlate with errors in our analyses. The analysis of the noise in spectrophotometers has been done7 and is outlined by Skoog and others in "Principles of Analytical Chemistry"3. In the outline they summarize that the types of errors commonly experienced fall into three cases and are listed below.

In the first case, the standard deviation of the transmittance is a constant.

$S_T=k_1$

Errors associated with this situation are due to detectors with limited readout resolution and dark currant and amplifier noise. Limited readout resolution keeps the deviation constant because the measurements more precise than the readout can not be expressed or displayed by the machine. Dark Currant and Amplifier noise are only an issue when the lamp intensity and transducer sensitivity are low and random fluctuations in the currant become the dominant source of error.

In the second case, the deviation in transmittance varies with the equation.

$S_T=K_2 \sqrt{T^2+T}$

Errors associated with this equation are those related to shot noise or the transfer of electrons across any sort of junction or barrier like those found in the photomultiplier tube in the detector. Since the currant is dependant on this transfers, which happen randomly, the currant becomes a random distribution that centers around and average. This error becomes considerable when dealing with either really low or really high transmittances.

The last situation deals with errors that relate proportionally to the transmittance. Errors related to this type of noise have to do with source flickering along with cell positioning. Both of these errors can be easily corrected either by attaching a constant voltage source on the lamp or by placing a fixed cuvette holde

$S_T=k_3T$

Other big sources of error can be introduced from the entrance and exit slits not having the appropriate separation or if the experiment being conducted is outside or near the limit of the instrumentation. In older machines, a scan time that was faster than the recording printer could be a source of error as the printer would not be able to record the data as fast it was being given.

## References

1. G. W Ewing. "Instrumental Methods of Chemical Analysis", 5th Edition, McGraw-Hill 1985
2. H. H. Willard, Et Al. "Instrumental Methods of Analysis", 7th Edition, Wadsworth Inc. 1988
3. D. A. Skoog, Et Al. "Principles of Instrumental Analysis" 6th Edition, Thomson Brooks/Cole. 2007
4. R. A. Serway, J. W. Jewett, "Physics: For Scientists and Engineers with modern physics", Thomson Brooks/Cole, 2004
5. G. W. Ewing, J. Cazes, "Ewing's Analytical Instrumentation Handbook", 3rd Edition, Marcel Dekker, 2005
6. E. D. Stokes, Et Al. Optics Communications, Vol. 5, Num. 4, 267-270, 1972
7. L.D. Rothman, S. R. Crouch, J. D. Ingle. Analytical Chemistry, Vol 47, Num. 8, 1226-1233, 1975