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Correlation analyses on TCGA data
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@mariokeller1988-13652
Last seen 4 months ago
Germany

Good morning,

from our experiments in cell lines we identified genes and exons differentially regulated upon knockdown of a splicing factor using DESeq2 and MAJIQ. Now we want to transfer these findings to TCGA patients with hepatocellular carcinoma (TCGA-LIHC) using the recount3 package. The aim is to (1st task) correlate the expression of the splicing factor with the expression of the differentially regulated genes and (2nd task) to correlate the expression of the splicing factor with the PSI values of the differentially regulated exons using all 371 TCGA-LIHC patients. The calculation of the PSI values works fine and is not relevant in this post. My main problem is choosing the correct normalization strategy for the gene expression estimation.

In a first attempt for the 1st task TPM values for all genes in each of the 371 LIHC patients was calculated.

  library(recount)
library(recount3)

human_projects <- available_projects()
project_info <- subset(human_projects, project == "LIHC" & file_source == "tcga")
rse_recount3 <- create_rse(project_info)
rse_recount3 <- rse_recount3[,rse_recount3$tcga.gdc_cases.samples.sample_type == "Primary Tumor"] #Create TPM values assay(rse_recount3, "counts") <- transform_counts(rse_recount3) TPM.m <- getTPM(rse_recount3, length_var = "score")  As a first check I made pairwise-correlations between my splicing factor and all other genes (log-transformed TPM values), followed by the generation of density distribution of all ~20k pearson correlation coefficients obtained from the pairwise-correlations. What I observed was a completely skewed distribution shifted to the right (median ~0.4 to 0.5), which indicates that my splicing factor shows a positive correlation with most of all other genes. These results looked quite suspicious. Therefore, I picked 10.000 random gene-pairs and correlated them, followed by the generation of the density distribution. And once again, although to a lesser extent, the distribution was also shifted to the right (median ~0.2 to 0.25). It seems that some confounder is affecting the correlation analysis, which can not be captured by TPM normalization. In a second attempt, I wanted to try another normalization strategy that takes into account differences in transcriptome composition, namely the combination of TMM and cpm from the edgeR package.  y <- DGEList(counts= assays(rse_recount3)$counts, remove.zeros=T)
y <- calcNormFactors(y)
CPM.m <- cpm(y, log=TRUE) %>% t

Like for the first attempt I made the correlations and generated the density distributions. Now both distributions (my splicing factor against all other genes and the 10k random pairs) are centered at around 0 and follow a normal distribution.

I also checked the correlation between my splicing factor and one gene of interest and observed a tremendous difference between the TPM and CPM normalization. For TPM I observed a quite strong positive correlation of 0.42, while for CPM I observed a weak negative correlation of -0.23.

So, here are my questions:

1) Which normalization strategy should I use? As far as I understand it TPM is good for intra-sample comparisons among genes but should not be used for inter-sample comparisons. In addition, the density distributions for the TPM approach were strongly skewed and shifted to the right, wich give me the feeling that I should go for the CPM normalization.

2) The TMM+CPM approach does not account for gene lengths. However, I think this should not be a problem for my correlation analyses. But what I could do is using the rpkm() method provided by edgeR, which converts my logCPM values to RPKM. Would this make sense?

Best, Mario

RNA-seq Normalization edgeR recount3 • 408 views
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Entering edit mode

Hi,

I don't think that this is a recount3 question per se. Though I see that you included other tags. Anyway, I'll keep an eye open in case there's anything related to recount3 that we can help with.

Best, Leo


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# Monday, April 30, 2018¶

When I wrote yesterday that “I fixed it!”, it was actually fixed “in my mind” but not yet in reality. I needed two several versions of Atelier before it was visible to the world.

Motivated by this action, I continued also today to work for getting the test suites to pass on Travis. (There’s is much to do outside as well, and it suited me to work in chunks: do some fix, let Travis run the tests while I was outside, then come back and see the results and do some more fixes.

I removed the ‘Sphinx<1.7.1’ requirement for Atelier, and it still caused that failure during inv bd in The Lino Book which made me add the requirement:

SphinxWarning: .../lino/lino/core/site.py:docstring of lino.core.site:1:Literal block expected; none found.

It took me some time to find the explanation: I had docstrings of type:

"""
Blabla bla.
Usage example::

site_config_defaults = dict(default_build_method='appypdf')

"""

And autosummary includes only the first paragraph of every docstring into its summary table. It’s indeed a syntax error when you announce a literal block which isn’t there. And Sphinx has obviously become better in finding this error.

I also fixed more failures in Python 3 environment and, believe it or not: the Lino Book test suite now passes on Travis in both Pythons! A historic day!

And then I attacked #2296, yet another mysterious failure in Lino Welfare. Happens only on Travis, not on my machine. It had to do with SITE.site_config <lino.core.site.Site.site_config, our cached copy of SiteConfig.

format: plain


text:
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Absolutely! In fact, in my opinion, the most important "math skill" that should be taught in conjunction with, and using, word problems is checking whether the answers make sense. This is an ...
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Don't these word problems seem designed to be confusing?

I think there's a countervailing issue that the book you're describing is trying to deal with. I teach college students, so I don't know what the particular approach it's taking is age appropriate ...
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What is the value of sentence type maths problems for primary school students?

Consider these 5 problems: What is $35 \div 10$ ? We are in the store and want to buy packages of party plates for a birthday party. The plates come in packages of 10. There will be 35 children ...
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Should word problems be reasonable?

The answers to a word problem should in my opinion make sense (within reasonable limits). The goal you mentioned that students should check their answers is one shared by many, as also witnessed by ...
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Where can I find realistic data for college-level elementary statistics problems?

DASL (pronounced "dazzle" and short for Data And Story Library) is an online collection of stories with matching data sets to be used for educational purposes. They are real data from real research. ...
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Unable to solve word problems, fluent in arithmetic

Does your colleague have any information about these students' reading comprehension? The reason I ask is that, in my opinion, difficulty with word problems is not always a matter of taking the ...
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Challenge questions for extremely bright kids

Here are some suggestions for problem sources in English. Some of them are appropriate for very bright students studying geometry or Algebra II, but might nonetheless prove too difficult for students ...
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Professional work in general, and scientific and mathematical work in particular, is done principally in writing. Requirements, specifications, orders, field reports, case studies, journals, grants, ...
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Interesting settings for exponential growth or decay

(edited) a. For all of these, I would think to expand the category into subcategories. So not just "bank account" but time value of money and NPV, bond details, etc. You learn something when doing ...
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Yes, there is evidence for the claim; for example, consult the following: Abedi, J., & Lord, C. (2001). The language factor in mathematics tests. Applied Measurement in Education, 14(3), 219-...
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I suggest you break it down into smaller parts. In fact, don't start on equations. Start on translation of terms: Double of a number, translation: $2x$ Sum of two numbers, translation: $x+y$ The ...
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Don't these word problems seem designed to be confusing?

I feel like I always post the same thing in these threads, but this again sounds like an issue of blocking vs interleaving. In this case, the textbook may have started interleaving different problems ...
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Physically (or even economically) unreasonable answers are confusing. The student needs to be shown that math is a tool that can solve practical problems. Answers that are unreasonable send the ...
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Have your students covered division and multiplication yet? To me, these problems can all be solved using the "meaning of division" and the "meaning of multiplication". "Meaning of multiplication": ...
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I would say that this book is trying to get students to think about what the calculations mean, rather than simply execute an algorithm, and I strongly believe that is something that should be done ...
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Where can I find realistic data for college-level elementary statistics problems?

(I came to mention DASL, but since it's already been mentioned, I'll give some other resources.) opendata.stackexchange.com often has mentions of useful sources of data, some of which are small (and ...
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Explaining the phrase "in terms of"

I do avoid using this phrase in all of my math classes. Not that I've ever thought of it as a particular goal, but I would want to reserve the word "term" to specify an addend. In your first example,...
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I think you have a very different understanding of what "mathematics" is than I do. Consider the following questions: What kinds of geometric properties are held by the figure you get when you join ...
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Recently, I got to teach a new high school class called "Essentials of College Math." It sounds like hard content, but it was basically just a transition course from Algebra II to Precal. Anyways, ...
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I only have anecdotal evidence but since you said it was welcome, I am adding my 2 cents from both the perspective of a teacher and an item writer, hired to write questions for standardized tests. ...
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I work with gifted elementary school students, but one of my favorite sites, nrich has challenging problems that you could use for older gifted students. Try looking at secondary problems for stages ...
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One of my colleagues introduced me to a strategy called "3 Reads" which he details in his blog Misteristhisright. This strategy has two major features: You remove the question from the problem. The ...

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This is not a good answer, but I'll mention anyway that Mathematica now connects to vast "curated" data sources. It would take some effort to master these sources, but they could provide endless ...
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There are three direct benefits, as far as I can tell. Word problems answer questions like "why do I need to know this". If you have a student who thinks that learning math is pointless then you can ...
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One basic model of atmospheric pressure has it decaying exponentially as a function of height above sea level. Two places to look for this are https://people.clas.ufl.edu/kees/files/...
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The website mathschallenge.net has pdfs of tiered problems with solutions. They are nicely posed problems, often with some humour, and are based on different stages of mathematical maturity. To begin ...
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Perhaps an example would clarify what you mean. Question 1: What number is obtained after reducing 40 by 5%? Question 2: Farmer John has lately been having problems selling his corn. He needs ...
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text:
# Question 1c985

The formula for the percentage by mass of a chemical species $x$ is given by
$\underline{\overline{| \stackrel{\text{ ")(" "%"mass of"color(white)(l)x = ("mass of"color(white)(l)x)/"total mass" xx 100%" }}{|}}}$
For example, if you wanted to find the mass percentage of $\text{NaCl}$ given that there are $24$ $\text{g NaCl}$ in a solution with mass $131$ $\text{g}$, we would have
%"mass NaCl" = (24cancel("g NaCl"))/(131cancel("g soln")) xx 100% = color(blue)(ul(18.3%#


text:
# I'm confused by this problem. Is the answer 1 or 0 OR 1/2? If the

D H
Staff Emeritus

This post contains replies to several recent posts.

Eilenberg-Mazur
Huh? Except for the connection made in very poorly written (and apparently biased) wikipedia article on the Eilenberg-Mazur swindle, there is no connection between the Grandi series and this swindle. The article equates fallacious proofs of 1=0 via the Grandi series to the Eilenberg-Mazur swindle. This is fallacious reasoning in and of itself. For one thing, the Eilenberg-Mazur swindle is valid.

For another, there's nothing special about the Grandi series here. Consider the alternating harmonic series, 1-1/2+1/3-1/4+... This is a convergent series: It converges to ln(2). Now let's rearrange the terms of the series. The algorithm for the rearrangement is a state machine that alternates between two states:
• In state 1 (the initial state), find the first remaining additive term (e.g., 1, 1/3, 1/5, ...). Pair this with the subtractive term that is half of the first term. Add the sum of the two as a new term in the rearranged series. Eliminate both terms in the original formulation from further consideration.
• In state 2, find the first remaining subtractive term. Add this term as a new term in the rearranged series and eliminate the found term in the original formation from further consideration.

The rearranged series is (1-1/2)-1/4+(1/3-1/6)-1/8+(1/5-1/10)-1/12+(1/7-1/14)-1/16+... = 1/2-1/4+1/6-1/8+1/10-1/12+1/14-1/16 = 1/2*(1-1/2+1/3-1/4+...). By rearranging the terms in the series we have halved the value of the series! In fact, given any number, it is possible to find a rearrangement of the alternating harmonic series that yields that number as a sum.

The problem here is that alternating harmonic series is a conditionally convergent series; it is not absolutely convergent. This magic of rearranging terms to come up with drastically different sums applies to any conditionally convergent series. What this means is that rearranging terms is invalid for such series. (It is a valid operation for absolutely convergent series.)

This may be elementary but why can't it be solved this way:

S = 1 - 1 + 1 - 1 + 1 - 1 + . . .

S = 0 + 1 - 1 + 1 - 1 + 1 - 1 + . . .

2S = 1

S = 1/2
The problem here is that you are manipulating terms of an alternating series that is not absolutely convergent. It is an illegal operation.

It doesn't work because there is not such S. In other words, it's nonsensical (in the sense of adding an infinite series with another).
There certainly is such an S. You just can't find it using elementary techniques. Cesaro summation, Abel summation, Borel summation, and zeta function regularization all yield the same answer, 1/2, to 1-1+1-1+...

It is perfectly fine if the series converge. It rests on the fact that if a_n --> a, b_n --> b, then a_n+b_n --> a+b.
As noted above, this is not a valid operation on conditionally convergent series.

disregardthat

There certainly is such an S. You just can't find it using elementary techniques. Cesaro summation, Abel summation, Borel summation, and zeta function regularization all yield the same answer, 1/2, to 1-1+1-1+....
Of course, I meant that $$\lim_{N \to \infty} \sum^N_{n=0} (-1)^n$$ does not exist. It should be obvious from the context.

As noted above, this is not a valid operation on conditionally convergent series.
It most certainly is valid. Note that I don't allow a rearrangement of the terms as you propose in your post. It's a matter of adding finite partial sums with well defined terms, something I earlier have stressed. Now I'm talking about converging series, not zeta function regularization or something else.

... For one thing, the Eilenberg-Mazur swindle is valid...
that's right, that is what I meant in my post #18:
scientists may debate for centuries ("liar" paradox [true/false]) when they are reluctant to accept the principle that "exception proves the rule"
...For another, there's nothing special about the Grandi series here...
Grandi has some analogy [convergent/divergent] with Russell's http://wikipedia.org./wiki/Present_King_of_France" [Broken] [true/false], if you consider it as the antecedent of Leibniz [a= -2]
$$\sum^{\infty}_{n=0} [a=-1]^{n}$$
It ought to be re-written, re-defined.
Is 1,0,0,0,... a series? is it convergent? is it divergent? what is Cesaro-sum?

the sense of post #18 was:
[so expresses logical implication $\Rightarrow$] : disregardthat has made a valid argument:
... they alternate between 0 and 1, and so will not converge at all...
values alternate $\Rightarrow$ series is not convergent. [$\Rightarrow$....]
This is a new argument, because swindle [is valid] had already proven that, but had the consequence ("tertium non datur") that series is convergent.

This is a better argument, as it applies to both "possibilities" and we must not wait for another swindle. If it is so, disregardthat has made a swindle-preventing discovery.

Last edited by a moderator:
pwsnafu

$$\sum^{\infty}_{n=0} [a=-1]^{n}$$
You need to define your notation. I've never seen that.

Is 1,0,0,0,... a series? is it convergent? is it divergent? what is Cesaro-sum?
It is convergent sequence, and not a series.
Cesaro is http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation" [Broken].

Last edited by a moderator:


text:
#jsDisabledContent { display:none; } My Account |  Register |  Help

# Log-normal distribution

Article Id: WHEBN0000102476
Reproduction Date:

 Title: Log-normal distribution Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Log-normal distribution

 Notation Probability density function Some log-normal density functions with identical location parameter \mu but differing scale parameters \sigma Cumulative distribution function Cumulative distribution function of the log-normal distribution (with \mu = 0 ) \ln\mathcal{N}(\mu,\,\sigma^2) \mu \in \mathbb R — location, \sigma > 0 — scale x \in (0, +\infty) \frac{1}{x\sigma\sqrt{2\pi}}\ e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}} \frac12 + \frac12\,\mathrm{erf}\Big[\frac{\ln x-\mu}{\sqrt{2}\sigma}\Big] e^{\mu+\sigma^2/2} e^{\mu}\, e^{\mu-\sigma^2} (e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2} (e^{\sigma^2}\!\!+2) \sqrt{e^{\sigma^2}\!\!-1} e^{4\sigma^2}\!\! + 2e^{3\sigma^2}\!\! + 3e^{2\sigma^2}\!\! - 6 \frac12 + \frac12 \ln(2\pi\sigma^2) + \mu defined only on the negative half-axis, see text representation \sum_{n=0}^{\infty}\frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2} is asymptotically divergent but sufficient for numerical purposes \begin{pmatrix}1/\sigma^2&0\\0&2/\sigma^2\end{pmatrix}

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = \ln(X) has a normal distribution. Likewise, if Y has a normal distribution, then X = \exp(Y) has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.[1] The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.[1]

A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain. The log-normal distribution is the maximum entropy probability distribution for a random variate X for which the mean and variance of \ln(X) are specified.[2]

## Contents

• Notation 1
• Characterization 2
• Probability density function 2.1
• Cumulative distribution function 2.2
• Characteristic function and moment generating function 2.3
• Properties 3
• Location and scale 3.1
• Geometric moments 3.1.1
• Arithmetic moments 3.1.2
• Mode and median 3.2
• Arithmetic coefficient of variation 3.3
• Partial expectation 3.4
• Conditional expectation 3.5
• Other 3.6
• Occurrence 4
• Maximum likelihood estimation of parameters 5
• Multivariate log-normal 6
• Related distributions 7
• Similar distributions 8
• Notes 10
• References 11

## Notation

Given a log-normally distributed random variable X and two parameters \mu and \sigma that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of X is normally distributed, and we can write X as

X=e^{\mu+\sigma Z}

with Z a standard normal variable.

This relationship is true regardless of the base of the logarithmic or exponential function. If \log_a(Y) is normally distributed, then so is \log_b(Y), for any two positive numbers a,b\neq 1. Likewise, if e^X is log-normally distributed, then so is a^{X}, where a is a positive number \neq 1.

On a logarithmic scale, \mu and \sigma can be called the location parameter and the scale parameter, respectively.

In contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted m, s.d., and v in this article. The two sets of parameters can be related as (see also Arithmetic moments below)[3]

\mu=\ln\left(\frac{m}{\sqrt{1+\frac{v}{m^2}}}\right), \sigma=\sqrt{\ln\left(1+\frac{v}{m^2}\right)} .

## Characterization

### Probability density function

A random positive variable x is log-normally distributed if the logarithm of x is normally distributed,

\mathcal{N}(\mbox{ln}x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac {(\mbox{ln}x - \mu)^{2}} {2\sigma^{2}}\right], \ \ x>0.

A change of variables must conserve differential probability. In particular,

\mathcal{N}(\mbox{ln}x)d\mbox{ln}x = \mathcal{N}(\mbox{ln}x)\frac{d\mbox{ln}x}{dx}dx = \mathcal{N}(\mbox{ln}x)\frac{dx}{x} = {\ln\mathcal{N}}(x) dx,

where

{\ln\mathcal{N}}(x;\mu,\sigma) = \frac{1}{ x\sigma \sqrt{2 \pi}}\exp\left[-\frac {(\mbox{ln}x - \mu)^{2}} {2\sigma^{2}}\right],\ \ x>0

is the log-normal probability density function.[1]

### Cumulative distribution function

\int_{0}^{x} {\ln\mathcal{N}}(\xi;\mu,\sigma) d\xi = \frac12 \left[ 1 + \operatorname{erf}\!\left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) \right] = \frac12 \operatorname{erfc}\!\left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) = \Phi\bigg(\frac{\ln x - \mu}{\sigma}\bigg),

where erfc is the complementary error function, and Φ is the cumulative distribution function of the standard normal distribution.

### Characteristic function and moment generating function

All moments of the log-normal distribution exist and it holds that: \operatorname{E}[X^n]=\mathrm{e}^{n\mu+\frac{n^2\sigma^2}{2}} (which can be derived by letting z=\frac{\ln(x) - (\mu+n\sigma^2)}{\sigma} within the integral). However, the expected value \operatorname{E}[e^{t X}] is not defined for any positive value of the argument t as the defining integral diverges. In consequence the moment generating function is not defined.[4] The last is related to the fact that the lognormal distribution is not uniquely determined by its moments.

Similarly, the characteristic function \operatorname{E}[e^{i t X}] is not defined in the half complex plane and therefore it is not analytic in the origin. In consequence, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.[5] In particular, its Taylor formal series \sum_{n=0}^\infty \frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2} diverges. However, a number of alternative divergent series representations have been obtained[5][6][7][8]

A closed-form formula for the characteristic function \varphi(t) with t in the domain of convergence is not known. A relatively simple approximating formula is available in closed form and given by[9]

\varphi(t)\approx\frac{\exp\bigg(-\dfrac{W^2(t\sigma^2e^\mu)+2W(t\sigma^2e^\mu)}{2\sigma^2}\bigg)}{\sqrt{1+W(t\sigma^2e^\mu)}}

where W is the Lambert W function. This approximation is derived via an asymptotic method but it stays sharp all over the domain of convergence of \varphi.

## Properties

### Location and scale

The location and scale parameters of a log-normal distribution, i.e. \mu and \sigma, are more readily treated using the geometric mean, \mathrm{GM}[X], and the geometric standard deviation, \mathrm{GSD}[X], rather than the arithmetic mean, \mathrm{E}[X], and the arithmetic standard deviation, \mathrm{SD}[X].

#### Geometric moments

The geometric mean of the log-normal distribution is \mathrm{GM}[X] = e^{\mu}, and the geometric standard deviation is \mathrm{GSD}[X] = e^{\sigma}.[10][11] By analogy with the arithmetic statistics, one can define a geometric variance, \mathrm{GVar}[X] = e^{\sigma^2}, and a geometric coefficient of variation,[10] \mathrm{GCV}[X] = e^{\sigma} - 1.

Because the log-transformed variable Y = \ln X is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median, \mathrm{Med}[X].[12]

Note that the geometric mean is less than the arithmetic mean. This is due to the AM–GM inequality, and corresponds to the logarithm being convex down. In fact,

\begin{align} \mathrm{E}[X] &= e^{\mu + \frac12 \sigma^2} &= e^{\mu} \cdot \sqrt{e^{\sigma^2}} &= \mathrm{GM}[X] \cdot \sqrt{\mathrm{GVar}[X]}. \end{align}

In finance the term e^{-\frac12\sigma^2} is sometimes interpreted as a convexity correction. From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.

#### Arithmetic moments

The arithmetic mean, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are given by

\begin{align} & \operatorname{E}[X] = e^{\mu + \tfrac{1}{2}\sigma^2}, \\ & \operatorname{Var}[X] = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2} = (e^{\sigma^2} - 1)(\operatorname{E}[X])^2, \\ & \operatorname{SD}[X] = \sqrt{\operatorname{Var}[X]} = e^{\mu + \tfrac{1}{2}\sigma^2}\sqrt{e^{\sigma^2} - 1} = \operatorname{E}[X] \sqrt{e^{\sigma^2} - 1}, \end{align}

respectively.

The location (\mu) and scale (\sigma) parameters can be obtained if the arithmetic mean and the arithmetic variance are known; it is simpler if \sigma is computed first:

\begin{align} \mu &= \ln(\operatorname{E}[X]) - \frac12 \ln\!\left(1 + \frac{\mathrm{Var}[X]}{(\operatorname{E}[X])^2}\right) = \ln(\operatorname{E}[X]) - \frac12 \sigma^2, \\ \sigma^2 &= \ln\!\left(1 + \frac{\operatorname{Var}[X]}{(\operatorname{E}[X])^2}\right). \end{align}

For any real or complex number s, the sth moment of a log-normally distributed variable X is given by[1]

\operatorname{E}[X^s] = e^{s\mu + \frac12s^2\sigma^2}.

A log-normal distribution is not uniquely determined by its moments \operatorname{E}[X^k] for k\geq1, that is, there exists some other distribution with the same moments for all k.[1] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.

### Mode and median

Comparison of mean, median and mode of two log-normal distributions with different skewness.

The mode is the point of global maximum of the probability density function. In particular, it solves the equation (\ln f)'=0:

\mathrm{Mode}[X] = e^{\mu - \sigma^2}.

The median is such a point where F_X=0.5:

\mathrm{Med}[X] = e^\mu\,.

### Arithmetic coefficient of variation

The arithmetic coefficient of variation \mathrm{CV}[X] is the ratio \frac{\mathrm{SD}[X]}{\mathrm{E}[X]} (on the natural scale). For a log-normal distribution it is equal to

\mathrm{CV}[X] = \sqrt{e^{\sigma^2} - 1}.

Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

### Partial expectation

The partial expectation of a random variable X with respect to a threshold k is defined as g(k) = \int_k^\infty \!x{\ln\mathcal{N}}(x)\, dx where {\ln\mathcal{N}}(x) is the probability density function of X. Alternatively, and using the definition of conditional expectation, it can be written as g(k)=\operatorname{E}[X|X>k] P(X>k). For a log-normal random variable the partial expectation is given by:

g(k) = \int_k^\infty \!x{\ln\mathcal{N}}(x)\, dx = e^{\mu+\tfrac{1}{2}\sigma^2}\, \Phi\!\left(\frac{\mu+\sigma^2-\ln k}{\sigma}\right).

Where Phi is the normal cumulative distribution function. The derivation of the formula is provided in the discussion of this WorldHeritage entry. The partial expectation formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.

### Conditional expectation

The conditional expectation of a lognormal random variable X with respect to a threshold k is its partial expectation divided by the cumulative probability of being in that range:

E[X|X
E[X|X\geqslant k]=e^{\mu +\sigma ^{2}/2}\cdot \frac{\Phi [\frac{(\mu +\sigma^{^{2}}-ln(k)}{\sigma} ]}{\frac{1}{2}-\frac{1}{2} erf[\frac{ln(k)-\mu }{\sqrt{2}\sigma }]}

### Other

A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).[13]

The harmonic H, geometric G and arithmetic A means of this distribution are related;[14] such relation is given by

H = \frac{G^2}{ A} .

Log-normal distributions are infinitely divisible,[15] but they are not stable distributions, which can be easily drawn from.[16]

## Occurrence

The log-normal distribution is important in the description of natural phenomena. The reason is that for many natural processes of growth, relative growth rate is independent of size. This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies. It can be shown that a growth process following Gibrat's law will result in entity sizes with a log-normal distribution.[17] Examples include:

• In biology and medicine,
• Measures of size of living tissue (length, skin area, weight);[18]
• For highly communicable epidemics, such as SARS in 2003, if publication intervention is involved, the number of hospitalized cases is shown to satistfy the lognormal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of entropy production.[19]
• The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth;
• Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations)[20]
Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see distribution fitting
Consequently, reference ranges for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.
• In colloidal chemistry and polymer chemistry
• In hydrology, the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.[21]
• In social sciences and demographics
• Technology
• In reliability analysis, the lognormal distribution is often used to model times to repair a maintainable system.[27]
• In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution." [28] Also, the random obstruction of radio signals due to large buildings and hills, called shadowing, is often modeled as a lognormal distribution.
• It has been proposed that coefficients of friction and wear may be treated as having a lognormal distribution [29]
• In spray process, such as droplet impact, the size of secondary produced droplet has a lognormal distribution, with the standard deviation ：\sigma=\frac{\sqrt{6}}{6} determined by the principle of maximum rate of entropy production[30] It is an open question whether this value of σ has some generality for other cases, though for spreading of communicable epidemics, σ is shown also to take this value.[19]
• Particle size distributions produced by comminution with random impacts, such as in ball milling
• The file size distribution of publicly available audio and video data files (MIME types) follows a log-normal distribution over five orders of magnitude.[31]
• The length of chess games tends to follow a log normal distribution.[32]

## Maximum likelihood estimation of parameters

For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. To avoid repetition, we observe that

L (x;\mu, \sigma) = \prod_{i=1}^n \left(\frac 1 x_i\right) \, \mathcal{N} (\ln x; \mu, \sigma)

where by L we denote the probability density function of the log-normal distribution and by \mathcal{N} that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

\begin{align} \ell_L (\mu,\sigma | x_1, x_2, \dots, x_n) & {} = - \sum _k \ln x_k + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n) \\ & {} = \operatorname {constant} + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n). \end{align}

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, \ell_L and \ell_N, reach their maximum with the same \mu and \sigma. Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the log-normal distribution it holds that

\widehat \mu = \frac {\sum_k \ln x_k} n, \widehat \sigma^2 = \frac {\sum_k \left( \ln x_k - \widehat \mu \right)^2} {n}.

## Multivariate log-normal

If \boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma) is a multivariate normal distribution then \boldsymbol Y=\exp(\boldsymbol X) has a multivariate log-normal distribution[33] with mean

\operatorname{E}[\boldsymbol Y]_i=e^{\mu_i+\frac{1}{2}\Sigma_{ii}} ,
\operatorname{Var}[\boldsymbol Y]_{ij}=e^{\mu_i+\mu_j + \frac{1}{2}(\Sigma_{ii}+\Sigma_{jj}) }( e^{\Sigma_{ij}} - 1) .

## Related distributions

• If X \sim \mathcal{N}(\mu, \sigma^2) is a normal distribution, then \exp(X) \sim \operatorname{\ln\mathcal{N}}(\mu, \sigma^2).
• If X \sim \operatorname{\ln\mathcal{N}}(\mu, \sigma^2) is distributed log-normally, then \ln(X) \sim \mathcal{N}(\mu, \sigma^2) is a normal random variable.
• If X_j \sim \operatorname{\ln\mathcal{N}}(\mu_j, \sigma_j^2) are n independent log-normally distributed variables, and Y = \textstyle\prod_{j=1}^n X_j, then Y is also distributed log-normally:
Y \sim \operatorname{\ln\mathcal{N}}\Big(\textstyle \sum_{j=1}^n\mu_j,\ \sum_{j=1}^n \sigma_j^2 \Big).
• Let X_j \sim \operatorname{\ln\mathcal{N}}(\mu_j,\sigma_j^2)\ be independent log-normally distributed variables with possibly varying \sigma and \mu parameters, and Y=\textstyle\sum_{j=1}^n X_j. The distribution of Y has no closed-form expression, but can be reasonably approximated by another log-normal distribution Z at the right tail.[34] Its probability density function at the neighborhood of 0 has been characterized[16] and it does not resemble any log-normal distribution. A commonly used approximation due to L.F. Fenton (but previously stated by R.I. Wilkinson and mathematical justified by Marlow[35]) is obtained by matching the mean and variance of another lognormal distribution:
\begin{align} \sigma^2_Z &= \ln\!\left[ \frac{\sum e^{2\mu_j+\sigma_j^2}(e^{\sigma_j^2}-1)}{(\sum e^{\mu_j+\sigma_j^2/2})^2} + 1\right], \\ \mu_Z &= \ln\!\left[ \sum e^{\mu_j+\sigma_j^2/2} \right] - \frac{\sigma^2_Z}{2}. \end{align}

In the case that all X_j have the same variance parameter \sigma_j=\sigma, these formulas simplify to

\begin{align} \sigma^2_Z &= \ln\!\left[ (e^{\sigma^2}-1)\frac{\sum e^{2\mu_j}}{(\sum e^{\mu_j})^2} + 1\right], \\ \mu_Z &= \ln\!\left[ \sum e^{\mu_j} \right] + \frac{\sigma^2}{2} - \frac{\sigma^2_Z}{2}. \end{align}
• If X \sim \operatorname{\ln\mathcal{N}}(\mu, \sigma^2), then X+c is said to have a shifted log-normal distribution with support x\in (c, +\infty). \operatorname{E}[X+c] = \operatorname{E}[X]+c, \operatorname{Var}[X+c] = \operatorname{Var}[X].
• If X \sim \operatorname{\ln\mathcal{N}}(\mu, \sigma^2), then a X \sim \operatorname{\ln\mathcal{N}}( \mu + \ln a,\ \sigma^2).
• If X \sim \operatorname{\ln\mathcal{N}}(\mu, \sigma^2), then \tfrac{1}{X} \sim \operatorname{\ln\mathcal{N}}(-\mu,\ \sigma^2).
• If X \sim \operatorname{\ln\mathcal{N}}(\mu, \sigma^2) then X^a \sim \operatorname{\ln\mathcal{N}}(a\mu,\ a^2 \sigma^2). for a \neq 0\,
• Lognormal distribution is a special case of semi-bounded Johnson distribution
• If X|Y \sim \mathrm{Rayleigh}(Y)\, with Y \sim \operatorname{\ln\mathcal{N}}(\mu, \sigma^2), then X \sim \mathrm{Suzuki}(\mu, \sigma)\, (Suzuki distribution)

## Similar distributions

A substitute for the log-normal whose integral can be expressed in terms of more elementary functions[36] can be obtained based on the logistic distribution to get an approximation for the CDF

F(x;\mu,\sigma) = \left[\left(\frac{e^\mu}{x}\right)^{\pi/(\sigma \sqrt{3})} +1\right]^{-1}.

This is a log-logistic distribution.

## Notes

1. ^ a b c d e Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), "14: Lognormal Distributions", Continuous univariate distributions. Vol. 1, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York:
2. ^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics (Elsevier) 150 (2): 219–230.
3. ^ "Lognormal mean and variance"
4. ^ Heyde, CC. (1963), "On a property of the lognormal distribution", Journal of the Royal Statistical Society, Series B (Methodological) 25 (2): 392–393,
5. ^ a b Holgate, P. (1989). "The lognormal characteristic function, vol. 18, pp. 4539–4548, 1989". Communications in Statistical – Theory and Methods 18 (12): 4539–4548.
6. ^ Barakat, R. (1976). "Sums of independent lognormally distributed random variables". Journal of the Optical Society of America 66 (3): 211–216.
7. ^ Barouch, E.; Kaufman, GM.; Glasser, ML. (1986). "On sums of lognormal random variables" (PDF). Studies in Applied Mathematics 75 (1): 37–55.
8. ^ Leipnik, Roy B. (January 1991). "On Lognormal Random Variables: I – The Characteristic Function". Journal of the Australian Mathematical Society Series B 32 (3): 327–347.
9. ^ S. Asmussen, J.L. Jensen, L. Rojas-Nandayapa. "On the Laplace transform of the Lognormal distribution", Thiele centre preprint, (2013).
10. ^ a b Kirkwood, Thomas BL (Dec 1979). "Geometric means and measures of dispersion". Biometrics 35 (4): 908–9.
11. ^ Limpert, E; Stahel, W; Abbt, M (2001). "Lognormal distributions across the sciences: keys and clues". BioScience 51 (5): 341–352.
12. ^ Daly, Leslie E.; Bourke, Geoffrey Joseph (2000). Interpretation and uses of medical statistics (5th ed.). Wiley-Blackwell. p. 89.
13. ^ Damgaard, Christian; Weiner, Jacob (2000). "Describing inequality in plant size or fecundity". Ecology 81 (4): 1139–1142.
14. ^ Rossman, Lewis A (July 1990). "Design stream ﬂows based on harmonic means". J Hydraulic Engineering 116 (7): 946–950.
15. ^ Olof Thorin (1977), ["On the Infinite Divisibility of the Log Normal Distribution"]. Scandinavian Actuarial Journal, 03/1977; 1977(3):121-148.
16. ^ a b Gao, X.; Xu, H; Ye, D. (2009), "Asymptotic Behaviors of Tail Density for Sum of Correlated Lognormal Variables". International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 630857. doi:10.1155/2009/630857
17. ^ Sutton, John (Mar 1997). "Gibrat's Legacy". Journal of Economic Literature 32 (1): 40–59.
18. ^ Huxley, Julian S. (1932). Problems of relative growth. London.
19. ^ a b Wang, WB; Wang, CF; Wu, ZN; Hu, RF (2013). "Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model" 56 (11). SCIENCE CHINA Physics, Mechanics & Astronomy. pp. 2143–2150.
20. ^ Makuch, Robert W.; D.H. Freeman; M.F. Johnson (1979). "Justification for the lognormal distribution as a model for blood pressure". Journal of Chronic Diseases 32 (3): 245–250.
21. ^ Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis (PDF). Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224.
22. ^ Clementi, Fabio; Gallegati, Mauro (2005) "Pareto's law of income distribution: Evidence for Germany, the United Kingdom, and the United States", EconWPA
23. ^ Wataru, Souma (2002-02-22). "Physics of Personal Income".
24. ^ Black, F.; Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637.
25. ^ Mandelbrot, Benoit (2004). The (mis-)Behaviour of Markets. Basic Books.
26. ^ Bunchen, P., Advanced Option Pricing, University of Sydney coursebook, 2007
27. ^ O'Connor, Patrick; Kleyner, Andre (2011). Practical Reliability Engineering. John Wiley & Sons. p. 35.
29. ^ Steele, C. (2008). "Use of the lognormal distribution for the coefficients of friction and wear". Reliability Engineering & System Safety 93 (10): 1574–2013.
30. ^ Wu, Zi-Niu (July 2003). "Prediction of the size distribution of secondary ejected droplets by crown splashing of droplets impinging on a solid wall". Probabilistic Engineering Mechanics 18 (3): 241–249.
31. ^ Gros, C; Kaczor, G.; Markovic, D (2012). "Neuropsychological constraints to human data production on a global scale". The European Physical Journal B 85 (28).
32. ^ http://chess.stackexchange.com/questions/2506/what-is-the-average-length-of-a-game-of-chess/4899#4899
33. ^ Tarmast, Ghasem (2001). Multivariate Log–Normal Distribution (PDF). ISI Proceedings: 53rd Session. Seoul.
34. ^ Asmussen, S.; Rojas-Nandayapa, L. (2008). "Asymptotics of Sums of Lognormal Random Variables with Gaussian Copula". Statistics and Probability Letters 78 (16): 2709–2714.
35. ^ Marlow, NA. (Nov 1967). "A normal limit theorem for power sums of independent normal random variables". Bell System Technical Journal 46 (9): 2081–2089.
36. ^ Swamee, P. K. (2002). "Near Lognormal Distribution". Journal of Hydrologic Engineering 7 (6): 441–444.

## References

• Crow, Edwin L.; Shimizu, Kunio (Editors) (1988), Lognormal Distributions, Theory and Applications, Statistics: Textbooks and Monographs 88, New York: Marcel Dekker, Inc., pp. xvi+387,
• Aitchison, J. and Brown, J.A.C. (1957) The Lognormal Distribution, Cambridge University Press.
• E. Limpert, W. Stahel and M. Abbt (2001) Log-normal Distributions across the Sciences: Keys and Clues, BioScience, 51 (5), 341–352.
• Eric W. Weisstein et al. Log Normal Distribution at MathWorld. Electronic document, retrieved October 26, 2006.
• Holgate, P. (1989). "The lognormal characteristic function". Communications in Statistics - Theory and Methods 18 (12): 4539–4548.


text:
# Usage of “such that” in logical statements

I have a simple question about the usage of "such that" in logical statements.

Let the statement S be: "For every $x \in A, f (x) > 5$."

In the negation of S:

There is an $x \in A$ such that $f (x) ≤ 5$.

The comma in S is converted to "such that" in the negation of S.

Can somebody clarify on the "such that" came in the negation of S.

• Usually you use "such that" in combination with $\exists$ (exists). Since to negate $\forall$ you use $\exists$, it's natural to use "such that" as well. – rubik Feb 20 '15 at 15:10

For every $x\in A$, $f(x)>5$ means $\forall\,x$ such that $x\in A$, $f(x)>5$, which in turn can be translated to $\forall\,x\quad x\in A\implies f(x)>5$.
• And likewise, the negation: There is an x such that ($x\in A$ AND $f(x) \leq 5)$. (This isn't directed at you, @Tim, nor am I insinuating and criticism. I'm just adding an observation (like you suggest) that such that appears often in the parsing of quantified statements.) – Namaste Feb 20 '15 at 15:31


text:
# Eclipse PTP Setup: FORTRAN

1. Dec 28, 2011

Alrighty then So in order to solve a problem in a previous thread, I simply reinstalled Eclipse, but this time i went with the Parallel Tools Platform since it is supposed to work with FOTRAN too.

I created a C++ project first and did the 'Hello World' thing to make sure it was working. C++ is working fine as far as I can tell.

So I switched to a FORTRAN project and I am having trouble Building. I have a screenshot below. Perhaps someone can help me out with what this might mean? I am used to using Digital Fortran (yeah, I know--old as dinosaurs) where I simply write code, build and run.

2. Dec 29, 2011

EDIT: I think I should include some more details:

I am using a Mac OS = Snow Leopard.

The exact steps I used to generate the error in the screenshot above:

1) Right-click in Package Explorer: New -> FORRAN Project ->Makefile Project ->Demo Hello World FORTRAN

2) Attempt to Build the source code that was generated by the Hello World Project creation.

*Note:

a) I don't know why I use the 'Makefile' projects, I just saw it in a tutorial. Can someone enlighten me?

b) When I select the Makefile project, there are options in the 'Toolchains' window. I think this might be related to my problem: When the box that says "Show project types and toolchains only if they are supported on the platform" is checked I only see the XLF Tool Chain. But if I uncheck it, I can see GCC Fortran. (See new screen shots below). Selecting the GCC Fortran does not help resolve the issue, but it raises my question of why does is it NOT available when the box IS checked?

With Box checked:

With Box Unchecked:

3. Dec 29, 2011

### I like Serena

In your first screenshot you can see a file named "Makefile".
Can you show its contents?

4. Dec 29, 2011

Hi ILS There actually are no contents in the makefile. Perhaps that is the problem. I don't know a lot (anything really) about makefiles. It seems like most "modern" languages don't need you to explicitly make a makefile. Am I correct in saying that? After some Googling, it seems like the types of things that a makefile does are the types of things that get done automatically when you build/link etc.

I have written plenty of FORTRAN code in my graduate work, but never once encountered a makefile. I think I should look into getting a different project type to work. For example, I think that I should be using the option "Executable (GNU Fortran on MacOS X)" would be a more "normal" option. Normal as in 'just type some source code and then run it'.

However I recall that project type gave me errors too. I'll post back momentarily with the new errors. Thanks!

5. Dec 29, 2011

### I like Serena

Hi! ;)

FYI, you always need a file specifying which files to compile and link.
Also where to find include files and libraries.
And in particular which compiler and linker to use.
This information is kept in either a project file or a makefile.

When you already have a makefile you can tell Eclipse to get this information from it by creating a Makefile project.

Otherwise you have to tell Eclipse this, although it can make an educated guess, after which Eclipse generates a makefile and uses that.

6. Dec 29, 2011

So if I follow these steps:

1) New -> Fortran Project -> Executable (Gnu Fortran on MacOS X)

2) In the Project: New -> source folder

3) In Source folder: New -> Named HelloWorld1.for -> Select from dropdown: fixed form template

Upon running a simple Hello World I get the following screen. Here is what is bothering me:

a) The source folder icon named HelloWorld1 in the package explorer on the left has a little superscripted "C" on it whereas the project folder icon has a superscripted "F". So I think Eclpipse is thinking this is C or something.

b) In the error window there is the C/C++ thing under 'type' which makes me think the same.

7. Dec 29, 2011

### I like Serena

This error shows that Eclipse does understand that you are compiling a fortran file.

So the question is: do you have gfortran on your system?
And if so, perhaps you need to configure in which directory it is.

In the Console tab-sheet you should be able to see which command is actually given.
Perhaps you can show what it says?

8. Dec 29, 2011

I do have gfortran installed. I confirmed this by typing: gfortran --version in a terminal window. The console tab contents are as follows.

Code (Text):

**** Build of configuration Debug for project HellowWorld1 ****

make all
Building file: ../HelloWorld1/HelloWorld1.for
Invoking: GNU Fortran Compiler
gfortran -funderscoring -O0 -g -Wall -c -fmessage-length=0 -o "HelloWorld1/HelloWorld1.o" "../HelloWorld1/HelloWorld1.for"
make: *** [HelloWorld1/HelloWorld1.o] Error 127

**** Build Finished ****

Thanks again

9. Dec 29, 2011

### I like Serena

So what happens if you type:

gfortran -funderscoring -O0 -g -Wall -c -fmessage-length=0 -o "HelloWorld1/HelloWorld1.o" "../HelloWorld1/HelloWorld1.for"

in a terminal window?

10. Dec 29, 2011

I get the following error:

Code (Text):
Last login: Thu Dec 29 14:51:40 on ttys000
Salad-Samurais-MacBook-Pro:~ saladsamurai$gfortran -funderscoring -O0 -g -Wall -c -fmessage-length=0 -o "HelloWorld1/HelloWorld1.o" "../HelloWorld1/HelloWorld1.for" gfortran: error: ../HelloWorld1/HelloWorld1.for: No such file or directory gfortran: fatal error: no input files compilation terminated. 11. Dec 29, 2011 ### I like Serena Can you change the current directory first to the "HelloWorld1" directory? That way the gfortran compiler should be able to find the source file "../HelloWorld1/HelloWorld1.for". Anyway, apparently you have a PATH environment problem, meaning that Eclipse cannot find gfortran in the default PATH, but from the terminal window you can. Did you already reboot your computer? 12. Dec 29, 2011 ### Saladsamurai Hi ILS. I changed to the correct directory (duh ) and I still get the same error. With regard to rebooting: Since when do you mean? If you mean since last night, yes. Also, I keep hearing about this PATH thingy, what is that? Is that an environmental variable? I think we are certainly honing in on the problem: The C++ code works fine, so I wonder why the gfortran is not. I guess we need a way of checking "where" the gfortran compiler and where Eclipse is expecting it to be. As of now, it seems that the problem is that the two are not equal. 13. Dec 29, 2011 ### I like Serena If you are in the correct directory, you should not get the same error... So apparently you are not in the correct directory. I meant rebooting since you installed fortran... Let's check this. Can you type the following commands in a terminal window? which gfortran echo$PATH

14. Dec 29, 2011

Oof! You're write. I only went to the parent 'project' folder. This is what I get:

Code (Text):
--MacBook-Pro:HellowWorld1 saladsamurai$cd HelloWorld1 saladsamurai-MacBook-Pro:HelloWorld1 saladsamurai$ gfortran -funderscoring -O0 -g -Wall -c -fmessage-length=0 -o "HelloWorld1/HelloWorld1.o" "../HelloWorld1/HelloWorld1.for"
/var/folders/9K/9KvTsFwDELOjojSdLl+IFU+++TI/-Tmp-//ccQVvBsP.s:unknown:FATAL:can't create output file: HelloWorld1/HelloWorld1.o
saladsamurais-MacBook-Pro:HelloWorld1 saladsamurai$When I type "which gfortran" I get: Code (Text): /usr/local/bin/gfortran ANd when I type "echo$PATH" I get:

Code (Text):
/usr/bin:/bin:/usr/sbin:/sbin:/usr/local/bin:/usr/texbin:/usr/X11/bin

15. Dec 29, 2011

### I like Serena

Can you close eclipse and then restart eclipse from the terminal window by typing:
eclipse

That way Eclipse will inherit the PATH environment variable from your terminal window effectively fixing the problem that gfortran cannot be found.

16. Dec 29, 2011

Hmmm. When I type eclipse in a terminal I get the error:

Code (Text):

17. Dec 29, 2011

Let me guess: eclipse is not on the PATH ? :tongue:

18. Dec 29, 2011

### I like Serena

Okay.
Try to find out where eclipse is and start it with the entire path.
On my linux system it is:
$/opt/eclipse/eclipse Perhaps you can find it with:$ locate eclipse

Right!

19. Dec 29, 2011

the locate command needs to generate a database as this is the 1st time I have used it. Before doing so, what exactly am I looking for? Eclipse is weird to me in that it's not installed in my applications folder like all other programs. It's just a 'folder' that I downloaded from the site. I run it by clicking on an icon inside of this folder.

The thing is, I have Eclipse PTP and Eclipse SDK. Both of them have their own folder on my desktop. I run each from within their respective folders. Is that helpul?

20. Dec 29, 2011

### I like Serena

Perhaps you can right-click on the icon and select "Properties" or something like that (I'm not a Mac user).
That should tell you what it's named and where it is...

Alternatively you can try:
$cd ~/Desktop$ ls
and see if you can find those folders...

Or perhaps you can drag and drop (or copy and paste) the icon in the Terminal window...

Of perhaps you can open some sort of file explorer and search for a file named eclipse...

Or...

Last edited: Dec 29, 2011


text:
Volume 19 (2015)

 Recent Volumes Volume 1, 1998 Volume 2, 1999 Volume 3, 2000 Volume 4, 2002 Volume 5, 2002 Volume 6, 2003 Volume 7, 2004 Volume 8, 2006 Volume 9, 2006 Volume 10, 2007 Volume 11, 2007 Volume 12, 2007 Volume 13, 2008 Volume 14, 2008 Volume 15, 2008 Volume 16, 2009 Volume 17, 2011 Volume 18, 2012 Volume 19, 2015
 The Series About this Series Ethics Statement Purchase Printed Copies Author Index ISSN (electronic): 1464-8997 ISSN (print): 1464-8989 MSP Books and Monographs Other MSP Publications
Decision problems for $3$–manifolds and their fundamental groups

Matthias Aschenbrenner, Stefan Friedl and Henry Wilton

Geometry & Topology Monographs 19 (2015) 201–236
 arXiv: 1405.6274
Abstract

We survey the status of some decision problems for $3$–manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (word problem, conjugacy problem, isomorphism problem), and also the homeomorphism problem for $3$–manifolds and the membership problem for $3$–manifold groups.

Keywords
3-manifolds, fundamental groups, decision problems
Primary: 57M05


text:
# Cs2 Vapor Pressure

The Macroscopic View. 0 °C is 96 torr. 0 g of cs2 vapor (molar mass = 76. 5k points) At 35°C, the vapour pressure of CS 2 is 512 mm Hg and that of acetone is 344 mm Hg. 1 Vapor-Liquid Equilibrium for Ideal Systems 5-1 Example 5. The equation was presented in 1888 by the French engineer Louis Charles Antoine (1825-1897). 5 degrees Celsius. 8 torr, respectively, at 296 K. pdf from CS 2 at Washington State University. View Recitation 6 - solution. How many grams of SO 2 can be produced from reaction of 10. 1-1: Bubble point temperature calculation 5-3 Example 5. (The vapor pressure of pure ethanol at 25. Vapor (110 ppm) a Vapor (23Oppm) * P<. 2 VAPOR-PRESSURE EQUATION CONSTANTS Table A. stronger ionic bonds. This is because, by definition, the vapor pressure of a substance at its normal boiling point is 760 mmHg. 3∘C Find Delta Hvap for carbon disulfide. Cs2 Vapor Pressure When the liquid in a closed container is heated, more molecules escape the liq. The handling of this chemical may incur notable safety precautions. 8 ´ 10 -3 J mol -1 correct incorrect. What is vapor pressure? Vapor pressure is the pressure exerted by the vapor molecules of a substance in a closed system. Emits highly toxic fumes of oxides of sulfur when heated to decomposition [Bretherick, 5th ed. Title: SKM_Css_Lib18021314071 Created Date: 2/14/2018 6:13:09 PM. 23 Boiling Pt, Melting Pt, Vapor Pressure Estimations (MPBPWIN v1. This substance is widely used for the manufacturing of perfumes and other fragrance products. Based on this information, CS2 likely has a melting point and a vapor pressure when compared with CCl4. In a fire or if heated, a pressure increase will occur and the container may burst, with the risk of a subsequent explosion. Self-test questions: Focus 04. What is the. 20ml of H2(g) at 22. The vapor pressure of CS2 as a function of temperature is given b\ Solution Base for enthalpies: liquid oil, liquid CS2, gaseous N2 at 297 K Assume T1 = 298. Physical state and appearance: colorless to faintly yellow liquid. What volume will it occupy at 0. The CS2-N2 mixture has a partial pressure of CS2 equal to 50 mm Hg at 24 C and is to be blown into the absorber at essentially standard atmospheric pressure at the rate of 50,000 ft3 / hr. The agreement between the. Correct answers: 2 question: There are three puddles of different sizes on a sidewalk: a small puddle of rainwater a medium-sized puddle of spilled hot water a large puddle of spilled cold water which puddle has the highest vapor pressure and why? has the highest vapor pressure because vapor pressure depends on choices for blank one: rain water choices for blank two: substances origin. The Macroscopic View. Study Resources. The highest vapor pressure is observed for which of the following liquid/temperature combinations? a. Calculate the entropy of unfolding of lysozyme at 25. 0 gm sulphur in 100 gm of CS2 has vapour pressure 848. What is the mass of the chlorine? 31. 2-213 and 287. 26 (20°C) Melting point: -108. 40m3/s (50,000 ft3/h). Using Raoult's law (since the liquid exhibits ideal behavior), this partial pressure must be equal to the molar fraction of CS2 in the liquid times the vapor pressure of pure CS2: P C S 2 = x C S 2 L P C S 2 o. An aqueous solution contains 28% phosphoric acid by mass. What is the dominant intermolecular force involved in the properties of the following species: (a) Potassium chloride, KCl Ionic bonding. The vapor pressure of a liquid is the equilibrium pressure of a vapor above its liquid (or solid); that is, the pressure of the vapor resulting from evaporation of a liquid (or solid) above a sample of the liquid (or solid) in a closed container. MSDS available from Mallinckrodt Baker. The CS2-N2 mixture has a partial pressure of CS2 equal to 50 mm Hg at 24 C and is to be blown into the absorber at essentially standard atmospheric pressure at the rate of 50,000 ft3 / hr. This is because, by definition, the vapor pressure of a substance at its normal boiling point is 760 mmHg. A) gases B) solids C) liquids D) salts E) solids and liquids Section D 25. , when the molecules are both vaporizing and condensing at the same rate at a particular pressure. What is vapor pressure? Vapor pressure is the pressure exerted by the vapor molecules of a substance in a closed system. what is the vapor pressure of a solution prepared by dissolving 12. Vapor pressure is highly dependent on the intra- and intermolecular interactions of the chemical species of inter-est. 0 gm sulphur in 100 gm of CS2 has vapour pressure 848. For more data or any further information please search the DDB or contact DDBST. The 60 fs laser pulses were centered at 620 nm. What is the vapor pressure of. Need more help! Using this graph of CS2 data, determine. The vapor pressure of liquid carbon disulfide, CS2, is 100. This means that _____. CS2, has a normal boiling point of 46(C and a heat of vaporization of 26. 2 g/mol) in 125. Using the graph, determine the following. Give the approximate strength of (a) a covalent bond (b) a hydrogen bond (c) a dipole-dipole interaction (a) 400 kJ/mol (b) 20 kJ/mol (c) 5 kJ/mol 5. What is the vapor pressure. CH 222 Practice Problem Set #5 This is a practice problem set and not the actual graded problem set that you will turn in for credit. The vapor pressure at the normal boiling point would be 760 mmHg or torr. 1 degrees Celsius and a normal boiling point of 46. It is volatile ie; evaporates at room temperature. The density of the vapor is more than two and one-half (2. Copy Sheet of paper on top of another sheet. The August equation describes a. 0 g of H 2 S and 10. Carbon disulfide has been used in a variety of industrial applications for well over 100 years. stronger ionic bonds. The compound is used frequently as a building block in organic chemistry as well as an industrial and chemical non-polar solvent. 137(e) none of these. 👍 Correct answer to the question The vapor pressures of pure carbon disulfide and carbon tetrachloride are 360 and 99. What is the vapor pressure of. CH 3CH 2CH 2NH 2 c. The equation was presented in 1888 by the French engineer Louis Charles Antoine (1825-1897). 4)] to determine the vapor pressure of a solution. Graph the following data for vapor pressure of ammonia using excel or another computer program and from the graph. At standard temperature and pressure, CO2 (s) changes directly to CO2(g). IUPAC Standard InChIKey: QGJOPFRUJISHPQ-UHFFFAOYSA-N. What is the vapor pressure of carbon disulfide at 35 degree C?. 0 C Carbon Disulfide, CS2, Has Pvap = 100mm At -5. What is the vapor pressure of CS2 at 0°C?a)447 torrb)4160 torrc)313 torrd)139 torre)5. Transport Phenomena (ChE 310) Nehal I. Answer to: Carbon disulfide, CS2, has a vapor pressure of 100 mmHg at -5. (a) In terms of sttengths of all the intermolecular forces in each compound, explain why the of boiling CS. 0 °C and 721 mm Hg. surface tension c. The vapor pressure of a liquid is the equilibrium pressure of a vapor above its liquid (or solid); that is, the pressure of the vapor resulting from evaporation of a liquid (or solid) above a sample of the liquid (or solid) in a closed container. CHEMICAL AND PHYSICAL PROPERTIES. Need more help! Using this graph of CS2 data, determine. 8 ´ 10 -3 J mol -1 correct incorrect. The sample consisted of a quartz cell containing CS2. The molecular - 3693098. *It depends on the air pressure. 5 degrees Celsius. Above this temperature it is water vapor (or steam), a gas. 60 (Mean or Weighted MP) VP(mm Hg,25 deg C): 4. Vapor pressure is highly dependent on the intra- and intermolecular interactions of the chemical species of inter-est. Vapor ignites in contact with aluminum powder or fluorine. 52 As 4O 6 (s monoclinic) ‐1309. Reported vapor pressures of common PAH range from 10-2 to 10-12 atmospheres at ambient conditions. 0 g of CS2 vapor (76. vapor pressure b. The important thing to remember is that at each temperature, water or any liquid has a specific equilibrium vapor pressure, plotted as in text Figure 11. Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. The density of the vapor is more than two and one-half (2. These equal rate produce stable vapor pressure that does not change as long as the temperature remains constant. 0 K Appro[imatel\ 12 ideal stages Gas outlet temperature = 298 K 5. What is the vapor pressure of a solution containing 50. Answer to: Carbon disulfide, CS2, has a vapor pressure of 100 mmHg at -5. Vapor pressure is defined as the pressure exerted by a gas in equilibrium with its liquid. CH 3CH 2CH 2NH 2 c. The agreement between the. 10 M in a can of lemon-lime soda? The Henry’s law constant for O 2 is 1. To find the vapor pressure at a given temperature, use the Clausius-Clapeyron equation: ln(P1/P2) = (ΔHvap/R)((1/T2) - (1/T1)). FREE Expert Solution. what is the vapor pressure of a solution prepared by dissolving 12. 14 g/mol, 1. Substance D 26. 2 VAPOR-PRESSURE EQUATION CONSTANTS Table A. At normal pressure, the. 2-213 and 287. The molecular - 3693098. The vapor content of the gas is to be reduced to 0. This means that _____. Sodium amide forms toxic and flammable H2S gas with CS2. Vapor pressure varies with the strength of the intermolecular forces in the liquid. The Antoine equation is derived from the Clausius-Clapeyron relation. 906 g/mol) occupies a volume of 50. Antoine Coefficients for almost 700 Components Antoine Coefficients from "To estimate vapor pressure easily. CH4 (g) + 2H2S (g) ↔ CS2 (g) + 4H2 (g) (a) Decrease the concentration of dihydrogen monosulfide (b) Increase the pressure on the system (c) Increase the temperature of the system (d) Increase the concentration of carbon disulfide (e) Decrease the concentration of methane. 0 °C is 96 torr. 180} {T+273. 2 gives the Antoine equation constants for logp ¼ A B T þC; pinmmHg; T in C ðA:1Þ We often see this equation as. 74 kJ/mol, and its normal boiling point is 46°C. A) vapor pressure B) surface tension C) viscosity D) density. 12 mg/m3 * CRC (1971), Handbook of Chemistry and Physics (Chemical. Substance C d. Predicted data is generated using the US Environmental Protection Agency’s EPISuite™. Calculate the Henry's law constant, in units of torr, for carbon disulfide. ← Prev Question Next Question → 0 votes. 121 atm(c) 0. So you're sitting here somewhere. 1 C and a normal boiling point of 46. #1: The volume of a gas is 120 L at 0. 1 degrees Celsius and a normal boiling point of 46. 005 mm Hg, and at discharge currents of 0. -2- INTRODUCTION Since the early 1970's, a number of factors have motivated efforts to improve. At 35°C, the vapour pressure of CS2 is 512 mm Hg and that of acetone is 344 mm Hg. 1-2: Bubble point pressure calculation 5-5 Example 5. 121 atm(c) 0. It comprises several orders of magnitude, therefore, a graphical representation usually represents only part of its characteristics unless a change of scale is used. agreeable odor. y = VX1 — X2 / (x-x1) + y1 (5) One must be careful not to interpolate across the vapor pressure. Answer to: Carbon disulfide, CS2, has a vapor pressure of 100 mmHg at -5. Stronger are the forces of attraction, lower is the vapor pressure as the molecules are more tightly bonded and less molecules goes to the gas phase. 52 g naphthalene (C10H8, 128. 198atm, while that of pure heptane (c7H16) is 0. answer choices. In microsomes, (CS2) metabolism was increased by phenobarbital pretreatment of the rats and decreased with pretreatment of the rats with cobaltous chloride. 0 g of heptane at 25°C, assuming hexane and heptane form an ideal solution. 3 × 10 −3 M/atm at 25 °C. 96 As 2O 5 (s) ‐924. stronger ionic bonds. 6 g mol -1 and a density of 13. 1 °C and its molar heat of vaporization is 30. The vapor pressures of CS2, and acetone at this temperature are, respectively, 512 mm Hg and 344 mm Hg. The absorption oil has av mol wt 180, viscosity 2 cP, and sp gr 0. Cs2 vapor pressure. Sulfur vapor is an equilibrium mixture of several molecular species, including S8, S6, and S2. Three typical factors that affect vapor push are surface ar area, intermolecular forces and also temperature. It was found that higher critical CS2 vapor pressures were needed to form crystal nuclei with increasing conjugation length and molecular weight of the P3AT derivatives; i. Sulfur vapor reacts with charcoal at temperatures of 750-900 °C to form carbon disulfide. 3 × 10 −3 M/atm at 25 °C. 829056 × 10 − 06 ( T + 273. What is the. 1 C and a normal boiling point of 46. Self-test questions: Focus 04. 9k and PQT12 15. ^ forethane, propane, n-butane,isobutane,n-pentane,ethylene, propyleneand 1-butene beused for the range 1500 to 10 mm Hg,and. 15 g/mol) at 70°C and 726 mm Hg?. Uses formula: log e  P m m H g = {\displaystyle \scriptstyle \log _ {e}P_ {mmHg}=} log e  ( 760 101. Our mission is to help you succeed in your Chemistry class. Correct answers: 2 question: There are three puddles of different sizes on a sidewalk: a small puddle of rainwater a medium-sized puddle of spilled hot water a large puddle of spilled cold water which puddle has the highest vapor pressure and why? has the highest vapor pressure because vapor pressure depends on choices for blank one: rain water choices for blank two: substances origin. Need more help! Using this graph of CS2 data, determine. Vapor pressure is highly dependent on the intra- and intermolecular interactions of the chemical species of inter-est. Vapor ignites in contact with aluminum powder or fluorine. 19 Ar (g) 0. 180} {T+273. For the vapor pressure/temperature diagram shown, approximate the normal boiling points for: a. 23 Boiling Pt, Melting Pt, Vapor Pressure Estimations (MPBPWIN v1. *It depends on the air pressure. i,(l) is higher than that of COS(l). Pramana 1992, 39 (4) , 367-380. The tables provided are at fine grids of temperature and pressure. So in this area, if I were to plot the real curve, I would. Reacts violently with azides, ethylamine ethylenediamine, ethylene imine. MSDS available from Mallinckrodt Baker. The agreement between the. Copy Sheet of paper on top of another sheet. C) temperature is equal to 273 K (standard temperature). IUPAC Standard InChI: InChI=1S/CS2/c2-1-3. Covering: Chapter Ten, Chapter Eleven and Chapter Guide Five Important Tables and/or Constants: R = 8. 120 seconds. 74 kJ/mol, and its normal boiling point is 46°C. 14 g/mol, 1. Which of the following statement(s) is/are correct?a)A mixture of 100 mL of CH3COCH3 and 100 mL of CS2 has a volume of 200 mLb)When CH3COCH3 and CS2 are mixed. The calculations are based on predictions of vapor/liquid equilibrium. 1˚C and 755 mm Hg at 197. If the vapor pressure of Ethylene glycol is 7. The signal was collected background-free at the phase matching geometry. Oct 07,2021 - At 35C, the vapour pressure of carbon disulphide (CS2)is 512 mm Hg and that of acetone (CH3COCH3)is 344 mm Hg. A solution of CS2 and CH3COCH3 in which mole fraction of CS2 is 0. The vapor pressure at the normal boiling point would be 760 mmHg or torr. For the vapor pressure/temperature diagram above, approximate the vapor pressures for: a. 62) times that of air, a factor to be considered in respect to distribution and concentration. Antoine's equation is a representation of the vapor-liquid equilibrium for a pure component. Carbon disulfide is formed by a direct reaction of carbon and sulfur. Solution for Carbon disulfide, CS2, has a normal boiling point of 46°C and a heat of vaporization of 26. At standard temperature and pressure, CO2 (s) changes directly to CO2(g). 23 mm Hg at 95. Problem: Using the graph,determine the following. As such, vapor pressures may vary by many orders of magnitude for organic compounds of comparable molecular size. 5k points) At 35°C, the vapour pressure of CS 2 is 512 mm Hg and that of acetone is 344 mm Hg. This table gives azeotropic data for a number of binary mixtures at normal atmospheric pressure ( Paz =101. larger molecules. a larger molecular mass. 1˚C and 755 mm Hg at 197. C)When a small amount of CS2 is added to an excess of acetone, BP of solution increases. The feed is saturated liquid with the flow rate of 100 kmol/h comprising 45 mol% methanol, 30% ethanol, and 25% butanol. Odor description: Commercial - a sweetish aromatic; industrial - rotten cabbage or radish. At temperatures above 279°C and pressures above 78 atm, CS2 can only occur as a _____. 261 g/mL)? General Chemistry. Sulfur vapor is an equilibrium mixture of several molecular species, including S8, S6, and S2. The impure carbon disulfide that is usually used in most industrial processes, however, is a yellowish liquid with an. 20ml of H2(g) at 22. A distillation column is separating methanol, ethanol, and butanol at a pressure of 2 atm. 1 6 CO carbon-monoxide 6. It has an "ether-like" odor, but commercial samples are typically contaminated with foul-smelling impurities. What pressure of carbon dioxide is needed to maintain a CO 2 concentration of 0. Report an issue. viscosity 35. Uses of CS2. The density of the vapor is more than two and one-half (2. What is the vapor pressure of. CS2 ' cos 0♦0 - Oxygen atom = 0 Boiling Point at 1 atm (K) 319 223 4. What is the dominant intermolecular force involved in the properties of the following species: (a) Potassium chloride, KCl Ionic bonding. 7 L at 752 mmHg and 20 C. 0 °C of a solution composed of 42. 137(e) none of these. Both experimentation and analysis are underway on the possible use of cesium vapor for a transformer laser to be optically pumped by a battery of Nd glass lasers. The vapour pressure of a solution having 2. The vapor/gas is heavier than air and will spread along the ground. Our mission is to help you succeed in your Chemistry class. log10 of Carbon disulfide vapor pressure. The agreement between the. Vapor pressure is constant when there is an equilibrium of water molecules moving between the liquid phase and the gaseous phase, in a closed container. 14 g/mol, 1. 1 6 CO carbon-monoxide 6. 0 g of CS2 vapor (76. Sulfur vapor reacts with charcoal at temperatures of 750-900 °C to form carbon disulfide. Solution for Carbon disulfide, CS2, has a normal boiling point of 46°C and a heat of vaporization of 26. 500 atm and 15. Vapor pressure varies with the strength of the intermolecular forces in the liquid. The vapor pressures of pentane, hexane and heptane are compared. A) vapor pressure is exactly 1 atmosphere. what is the vapor pressure of a solution prepared by dissolving 12. Table 1 PHYSICAL AND CHEMICAL PROPERTIES OF CS2 Molecular weight: 76. 250 atm and 10. Log Octanol-Water Partition Coef (SRC): Log Kow (KOWWIN v1. How many grams of SO 2 can be produced from reaction of 10. Based on this information, CS2 likely has a- melting point and a- vapor pressure when compared with. 1) The vapor pressure of any substance at its normal boiling point is A) 1 torr B) equal to atmospheric pressure C) equal to the vapor pressure of water D) 1 atm E) 1 Pa 1) 2) The property responsible for the ʺbeading upʺ of water is _____. Answer (1 of 2): The partial vapor pressure of a component in a mixture is equal to the vapor pressure of the pure component at that temperature multiplied by its mole fraction in the mixture. 005 mm Hg, and at discharge currents of 0. $\endgroup$ - Chet Miller. The calculations are based on predictions of vapor/liquid equilibrium. Vapor pressure is a concept that many students find difficult to grasp. Carbon Disulfide (CS2) - Carbon Disulfide is the chemical name of CS2. 43g CH3COCH3. 0 mL CS2 liquid (76. The absorption oil has av mol wt 180, viscosity 2 cP, and sp gr 0. At the interface, the partial pressure of the water vapor will be only slightly lower than the equilibrium vapor pressure, so evaporation can still occur. The vapor/gas is heavier than air and will spread along the ground. Sodium amide forms toxic and flammable H2S gas with CS2. What is the mass of the chlorine? 31. MSDS available from Mallinckrodt Baker. 9k and PQT12 15. Title: SKM_Css_Lib18021314071 Created Date: 2/14/2018 6:13:09 PM. The column has a total condenser and a partial re-boiler. Antoine Coefficients for almost 700 Components Antoine Coefficients from "To estimate vapor pressure easily. It comprises several orders of magnitude, therefore, a graphical representation usually represents only part of its characteristics unless a change of scale is used. 28 kJ K−1 mol−1 and can be assumed to be independent of temperature. Material Safety Data Sheet. 0 K Appro[imatel\ 12 ideal stages Gas outlet temperature = 298 K 5. Experimentally, the absorption spectrum of saturated cesium. Vapor (110 ppm) a Vapor (23Oppm) * P<. Vapor Pressure at 20. 15 Vapor Pressures of Benzene-Toluene Solutions Plots of the vapor pressures of benzene (C 6 H 6 ) and toluene (C 6 H 5 CH 3 ) versus the mole fractions at 20°C are straight lines. The vapor pressure is an exponential function of temperature, starting at the triple point and ending at the critical point. The Antoine equation is derived from the Clausius-Clapeyron relation. CS2 is miscible with alcohol, ether, and benzene. The vapour pressure of a solution having 2. Sulfur vapor is an equilibrium mixture of several molecular species, including S8, S6, and S2. Problem: At 25°C the vapor pressure of pure hexane (C 6H14) is 0. To find the vapor pressure at a given temperature, use the Clausius-Clapeyron equation: ln(P1/P2) = (ΔHvap/R)((1/T2) - (1/T1)). ppm to mg/m³ (at 25C and 760 mmHg) 1 ppm = 3. It is highly volatile, the vapor pressure being 294. 15)- {\frac {4563. The data represent a small sub list of all available data in the Dortmund Data Bank. Uses formula: log e  P m m H g = {\displaystyle \scriptstyle \log _ {e}P_ {mmHg}=} log e  ( 760 101. #1: The volume of a gas is 120 L at 0. What is the vapor pressure of carbon disulfide…. 0 K Appro[imatel\ 12 ideal stages Gas outlet temperature = 298 K 5. stronger ionic bonds. The molecule carbon disulfide (CS2) is nonpolar and has only London dispersion forces between the molecules. Need more help! Using this graph of CS2 data, determine. The vapor pressures of pentane, hexane and heptane are compared. vapor pressure b. Which of the following liquids would have the lowest vapor pressure, factoring in both the impact. The vapor pressure of any substance at its normal boiling point is. 005 mm Hg, and at discharge currents of 0. For the solvent-vapor treatment experiment, the fresh films were put into a container in which the vapor pressure of CS2 was slightly lower than the saturated vapor pressure of CS2 at the same temperature, since saturated CS2 vapor runs the high risk of dissolving the P3BT film completely, resulting in a damaged and thus discontinuous, film. mole fraction of solute in the solution. azeotropic point are the azeotropic temperature taz, pressure Paz, and liquid-phase composition, usually expressed as mole fractions. Study Resources. You have been warned! Problem #5: What is the vapor pressure of benzene at 25. The reflux ratio L D æöç÷ çç ÷÷. What pressure (in atm) is exerted by the gas? 32. 3 × 10 −3 M/atm at 25 °C. mass of S = 32) :- Fill nearest integer value) [Assume very dilute solution] Backspace Save & Next Save & Mark For Review Clear Response Mark For Review & Next BACK NEXT. 15 g/mol) at 70°c and 726 mm hg? l?. (a) In terms of sttengths of all the intermolecular forces in each compound, explain why the of boiling CS. Vapor Pressure at 20. Select the correct molecule with the lowest surface tension. Transport Phenomena (ChE 310) Nehal I. of molecules returning to liquid phase. Antoine's equation is a representation of the vapor-liquid equilibrium for a pure component. At the azeotropic point, the vapor-phase composition is the same as the liquid-phase composition. A nonlinear concent…. The vapor at equilibrium is removed, condensed and the condensate is heated back to 25OC. The data represent a small sub list of all available data in the Dortmund Data Bank. couple gauge which has been calibrated for CS2 against a McLeod gauge on a separate MCUU~ system. 15 ) − 4563. The CS2-N2 mixture has a partial pressure of CS2 equal to 50 mm Hg at 24 C and is to be blown into the absorber at essentially standard atmospheric pressure at the rate of 50,000 ft3 / hr. Vapor ignites in contact with aluminum powder or fluorine. 14 g/mol, 1. At the interface, the partial pressure of the water vapor will be only slightly lower than the equilibrium vapor pressure, so evaporation can still occur. Uses formula: log e  P m m H g = {\displaystyle \scriptstyle \log _ {e}P_ {mmHg}=} log e  ( 760 101. 1˚C and 755 mm Hg at 197. CS2 ' cos 0♦0 - Oxygen atom = 0 Boiling Point at 1 atm (K) 319 223 4. In microsomes, (CS2) metabolism was increased by phenobarbital pretreatment of the rats and decreased with pretreatment of the rats with cobaltous chloride. CS2 is miscible with alcohol, ether, and benzene. 829056 × 10 − 06 ( T + 273. 1) The vapor pressure of any substance at its normal boiling point is A) 1 torr B) equal to atmospheric pressure C) equal to the vapor pressure of water D) 1 atm E) 1 Pa 1) 2) The property responsible for the ʺbeading upʺ of water is _____. 41 As 2S 3 (s) ‐169. 00 g1 mL×1 mol18. Identify one physical property and one chemical property of CS2. At normal pressure, the. BSP retention with time after CS2 administration by various routes in mice. 6k with the highest. The vapour pressure of CS2 at 50°C is 854 torr and a solution of 2. 96 As 2O 5 (s) ‐924. Uses of CS2. A sample of CS, is placed in a closed, evacuated 562 mL container at a temperature of 268 K. Carbon disulfide has a vapor pressure of 363 torr at 25∘c and a normal boiling point of 46. 42): Boiling Pt (deg C): 482. Above this temperature it is water vapor (or steam), a gas. Why is cs2 nonpolar? Carbon de defiders (CS2) is a nonpolar. The Macroscopic View. The August equation describes a. What influence vapor pressure? Vapor pressure is the pressure caused by the evaporation of liquids. It is found that, at medium and large values of discharge current, an increase in nitrogen pressure produces a sharp decrease in the Cs2(+) production rate, while. Which of the following liquids would have the lowest vapor pressure, factoring in both the impact. 0 g to moles using the molar mass. 95g of CS2 and 2. of molecules leaving the liquid is equal to the no. 1007/BF02845821. The agreement between the. Antoine Coefficients for almost 700 Components Antoine Coefficients from "To estimate vapor pressure easily. CH 222 Practice Problem Set #5 This is a practice problem set and not the actual graded problem set that you will turn in for credit. Not indicating the above in this type of question is common. Answer: Convert the 2. The critical temperature and pressure of CS2 are 279 °C and 78 atm, respectively. Correct answers: 2 question: There are three puddles of different sizes on a sidewalk: a small puddle of rainwater a medium-sized puddle of spilled hot water a large puddle of spilled cold water which puddle has the highest vapor pressure and why? has the highest vapor pressure because vapor pressure depends on choices for blank one: rain water choices for blank two: substances origin. Films of poly(3-hexyl thiophene) (P3HT):[6,6]-phenyl C61-butyric acid methyl ester (PCBM) were controllably exposed to CS2 vapor in a column with a linear solvent vapor pressure gradient. This temperature in kelvin would be 319. to achieve a vapor pressure of 0. Vapor ignites in contact with aluminum powder or fluorine. 3 mm Hg, at a cesium vapor pressure of. Mercury has a molar mass of 200. Carbon tetrachloride (CCl4) is also nonpolar but is made of larger molecules that are more strongly affected by London dispersion forces. 47 As 4O 6 (s octahedral) ‐1313. Cs2 Vapor Pressure. Carbon disulfide is formed by a direct reaction of carbon and sulfur. Substance B c. 0°C, given that the difference in the constant-pressure heat capacities upon unfolding is 6. Self-test questions: Focus 04. 43g CH3COCH3. The signal was collected background-free at the phase matching geometry. CS1 uses a vapor-split control loop with four product temperature control loops, and CS2 incorporates the pressure-compensated scheme into the lower side-stream temperature control loop of CS1. Give the approximate strength of (a) a covalent bond (b) a hydrogen bond (c) a dipole-dipole interaction (a) 400 kJ/mol (b) 20 kJ/mol (c) 5 kJ/mol 5. what is the vapor pressure of a solution prepared by dissolving 12. R is the "gas constant" 0. The vapor at equilibrium is removed, condensed and the condensate is heated back to 25OC. What is the volume of 2. The boiling point and Kb of CS2 are 46. 0g of a solute X(molar mass 32gmol−1) in 100g of CS2 (vapour pressure 854 Torr) is 848. A) vapor pressure B) surface tension C) viscosity D) density. Note that glucose is a nonvolatile solute. Cs2 Vapor Pressure When the liquid in a closed container is heated, more molecules escape the liq. The Antoine equation is derived from the Clausius-Clapeyron relation. And you expect Raoult to work. BSP retention with time after CS2 administration by various routes in mice. Steam-stripping of ben]ene in a sieve-plate column. Flammable liquid and vapor. 6 torr at 25 degrees Celsius. The vapor pressure of a liquid is the point at which equilibrium pressure is reached, in a closed container, between molecules leaving the liquid and going into the gaseous phase and molecules leaving the gaseous phase and entering the liquid. Calculate the vapor pressure of a mixture of 17. Self-test questions: Focus 04. IUPAC Standard InChIKey: QGJOPFRUJISHPQ-UHFFFAOYSA-N. At 35°C, the vapour pressure of CS2 is 512 mm Hg and that of acetone is 344 mm Hg. The transport process involved is diffusion of the water vapor from the interface to the bulk air. At the interface, the partial pressure of the water vapor will be only slightly lower than the equilibrium vapor pressure, so evaporation can still occur. 0 C Carbon Disulfide, CS2, Has Pvap = 100mm At -5. Elemental iodine (I2) is a solid at room temperature. Determine the pressure of the gas (in mmHg) in the diagram below, given atmospheric pressure= 0. Substance A b. The vapor content of the gas is to be reduced to 0. 52 g naphthalene (C10H8, 128. A common form of the equation is. couple gauge which has been calibrated for CS2 against a McLeod gauge on a separate MCUU~ system. Vapor ignites in contact with aluminum powder or fluorine. Find step-by-step Chemistry solutions and your answer to the following textbook question: The vapor pressure of pure liquid CS2 is 0. Water boils at 100 degrees Celsius. the vapor pressure of carbon sulfide is 355. vapor pressure b. "Clutch really helped me by reinforcing the things I learned in class and making exam reviews a breeze. The absorption oil has a molecular weight of 180. Created Date: 12/13/2019 1:38:39 PM. Emits highly toxic fumes of oxides of sulfur when heated to decomposition [Bretherick, 5th ed. This is the vapor pressure of CS2 pure and CS2 goes from zero to one. 12 mg/m3 * CRC (1971), Handbook of Chemistry and Physics (Chemical. 5°C and the standard enthalpy of transition is 509 kJ mol−1. Cs2 Vapor Pressure When the liquid in a closed container is heated, more molecules escape the liq. Report an issue. 1 Vapor-Liquid Equilibrium for Ideal Systems 5-1 Example 5. At equilibrium, no. Covering: Chapter Ten, Chapter Eleven and Chapter Guide Five Important Tables and/or Constants: R = 8. to achieve a vapor pressure of 0. Uses of CS2. 13 g/mol) at 70. C)When a small amount of CS2 is added to an excess of acetone, BP of solution increases. For the vapor pressure/temperature diagram above, approximate the vapor pressures for: a. Carbon disulfide is formed by a direct reaction of carbon and sulfur. Flammable liquid and vapor. , the critical vapor pressures for P3DDT 13. i,(l) is higher than that of COS(l). Formula State H f 0 S0 G f 0 Am 2O 3 (s) ‐1757. In a fire or if heated, a pressure increase will occur and the container may burst, with the risk of a subsequent explosion. Uses formula: log e  P m m H g = {\displaystyle \scriptstyle \log _ {e}P_ {mmHg}=} log e  ( 760 101. 45 lb of CS2 are evaporated. What pressure (in atm) is exerted by the gas? 32. A solution of CS2 and CH3COCH3 in which mole fraction of CS2 is 0. 8 torr, respectively, at 296 K. So in this area, if I were to plot the real curve, I would. The highest vapor pressure is observed for which of the following liquid/temperature combinations? a. 6 atm, substance D must be heated to about 60˚C 25. Using the graph, determine the following. Substance C d. 3 K kg mol1, respectively. Carbon tetrachloride (CCl4) is also nonpolar but is made of larger molecules that are more strongly affected by London dispersion forces. 2 g of hexane and 25. 67 estimate) = 0. ppm to mg/m³ (at 25C and 760 mmHg) 1 ppm = 3. The August equation describes a. Vapor pressure is directly related to the kinetic energy of a molecule. These equal rate produce stable vapor pressure that does not change as long as the temperature remains constant. Substance A b. CAS Registry Number: 75-15-. 15 g/mol) at 70°c and 726 mm hg? l?. That enthalpy the vaporization is 26. A quantity of chlorine gas (70. For the vapor pressure/temperature diagram shown, approximate the normal boiling points for: a. 2 ppm (ACGIH, 1991) Molecular weight: 76. Transport Phenomena (ChE 310) Nehal I. agreeable odor. 2013 May 16;117(19):5996-6006. Which of the following liquids would have the lowest vapor pressure, factoring in both the impact. A common form of the equation is. The vapor pressure of a solution (𝑃1) is First we calculate the number of moles of glucose and water in the solution: 𝑛1water=460 mL×1. : Vapor-Liquid Equilibria in the Nitrogen + Carbon Dioxide + Propane System from 240 to 330 K at Pressures to 15 MPa. Table 1 PHYSICAL AND CHEMICAL PROPERTIES OF CS2 Molecular weight: 76. In both microsomes and hepatocytes, CS2 metabolism was inhibited by SKF-525A. #1: The volume of a gas is 120 L at 0. Chemical structure:. 0°C, given that the difference in the constant-pressure heat capacities upon unfolding is 6. 26 (20°C) Melting point: -108. CS2 ' cos 0♦0 - Oxygen atom = 0 Boiling Point at 1 atm (K) 319 223 4. This table gives azeotropic data for a number of binary mixtures at normal atmospheric pressure ( Paz =101. 198atm, while that of pure heptane (c7H16) is 0. 15 g/mol) at 70°C and 726 mm Hg?. Identify one physical property and one chemical property of CS2. Not indicating the above in this type of question is common. 0 2 4 6 8 10 12 14 16 18 Time After Treatment (hours) FIG. 1 Vapor-Liquid Equilibrium for Ideal Systems 5-1 Example 5. What pressure (in atm) is exerted by the gas? 32. So let's do Raoult. 1007/BF02845821. 00 atm, and that the partial pressure of O 2 is 0. If the vapor pressure of Ethylene glycol is 7. The feed is saturated liquid with the flow rate of 100 kmol/h comprising 45 mol% methanol, 30% ethanol, and 25% butanol. , when the molecules are both vaporizing and condensing at the same rate at a particular pressure. 6 mm Hg at 25 °C, and its log octanol/water partition coefficient (log K ow) is 1. y = VX1 — X2 / (x-x1) + y1 (5) One must be careful not to interpolate across the vapor pressure. B)BP of the azeotropic mixture is always less than the BP of pure CS2. what is the vapor pressure of carbon disulfide at its normal boiling point? The normal boiling point of carbon disulfide (CS2) is 46. The 60 fs laser pulses were centered at 620 nm. 5°C and the standard enthalpy of transition is 509 kJ mol−1. Carbon disulfide, also spelled as carbon disulphide, is a neurotoxic colorless volatile liquid with the formula CS 2. Need more help! Using this graph of CS2 data, determine. Carbon disulfide is formed by a direct reaction of carbon and sulfur. CS2, has a normal boiling point of 46(C and a heat of vaporization of 26. A straw oil used to absorb benzene from coke-oven gas is to be. For the solvent-vapor treatment experiment, the fresh films were put into a container in which the vapor pressure of CS2 was slightly lower than the saturated vapor pressure of CS2 at the same temperature, since saturated CS2 vapor runs the high risk of dissolving the P3BT film completely, resulting in a damaged and thus discontinuous, film. Carbon disulfide has been used in a variety of industrial applications for well over 100 years. A) 1 mL of this solution contains 28 g of phosphoric acid. 0 2 4 6 8 10 12 14 16 18 Time After Treatment (hours) FIG. Pramana 1992, 39 (4) , 367-380. Material Safety Data Sheet. It was found that higher critical CS2 vapor pressures were needed to form crystal nuclei with increasing conjugation length and molecular weight of the P3AT derivatives; i. 2013 May 16;117(19):5996-6006. (a) the approximate vapor pressure of CS2 at 30 °C, (b) the temperature at which the vapor pressure equals 300 torr,. Answer (1 of 2): The partial vapor pressure of a component in a mixture is equal to the vapor pressure of the pure component at that temperature multiplied by its mole fraction in the mixture. a larger molecular mass. The density of the vapor is more than two and one-half (2. The critical temperature and pressure of CS2 are 279 °C and 78 atm, respectively. mass of S = 32) :- Fill nearest integer value) [Assume very dilute solution] Backspace Save & Next Save & Mark For Review Clear Response Mark For Review & Next BACK NEXT. About Vapor Cs2 Pressure. Calculate the change in the molar Gibbs energy of liquid mercury when the pressure increases by 10. 6 torr at 25 degrees Celsius. The vapor/gas is heavier than air and will spread along the ground. 6 g mol -1 and a density of 13. 2 g/mol) in 125. 62 As 4 (g) 143. 48 250 9 CS2 carbon-disulfide 6. 1) The vapor pressure of any substance at its normal boiling point is A) 1 torr B) equal to atmospheric pressure C) equal to the vapor pressure of water D) 1 atm E) 1 Pa 1) 2) The property responsible for the ʺbeading upʺ of water is _____. The feed is saturated liquid with the flow rate of 100 kmol/h comprising 45 mol% methanol, 30% ethanol, and 25% butanol. A common form of the equation is. 62) times that of air, a factor to be considered in respect to distribution and concentration. 74 kJ/mol, and its normal boiling point is 46°C. What volume will be occupied by 2. The vapor content of the gas is to be reduced to 0. 26 (20°C) Melting point: -108. At the azeotropic point, the vapor-phase composition is the same as the liquid-phase composition. ← Prev Question Next Question → 0 votes. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The first operation of GEM gas gain elements in negative ion gas mixtures is reported. 42): Boiling Pt (deg C): 482. 96 As 2O 5 (s) ‐924. CS2 is miscible with alcohol, ether, and benzene. 74 kJ/mol, and its normal boiling point is 46°C. Using this graph of CS2 data, determine (a) the approximate vapor. 6 torr at 25 degrees Celsius. What is the molar enthalpy of. The August equation describes a. Search: Cs2 Vapor Pressure. Substance C d. 005 mm Hg, and at discharge currents of 0. Sulfur vapor is an equilibrium mixture of several molecular species, including S8, S6, and S2. Molecular weight: 76. mass of S = 32) :- Fill nearest integer value) [Assume very dilute solution] Backspace Save & Next Save & Mark For Review Clear Response Mark For Review & Next BACK NEXT. 12 mg/m3 * CRC (1971), Handbook of Chemistry and Physics (Chemical. 180} {T+273. 00 kg of CS2, the vapor pressure of CS2 decreases to 0. What is the vapor pressure of a solution containing 50. Carbon tetrachloride (CCl4) is also nonpolar but is made of larger molecules that are more strongly affected by London dispersion forces. Gains up to several thousand were obtained from single-stage GEMs in CS2 vapor at low pressure, and in mixtures of CS2 with Argon and Helium, some near 1 bar total pressure. The data represent a small sub list of all available data in the Dortmund Data Bank. A number of important unknown parameters of the Cs2 molecule have now been estimated, as a result of a detailed survey of existing data on the other alkali metal molecules. The vapor pressure that a molecule different at various temperatures. This is the vapor pressure of CS2 pure and CS2 goes from zero to one. 14 g/mol, 1. Structure and properties. 45 lb of CS2 are evaporated. The vapor pressure for carbon disulfide is 352. IUPAC Standard InChIKey: QGJOPFRUJISHPQ-UHFFFAOYSA-N. 63 (air = 1) Vapor pressure: 360 mm Hg (25^C) Solubility: 2. asked Jan 26, 2020 in Chemistry by Sarita01 (53. 5 °C? The normal boiling point of benzene is 80. 13 Specific gravity: 1. 1) The vapor pressure of any substance at its normal boiling point is A) 1 torr B) equal to atmospheric pressure C) equal to the vapor pressure of water D) 1 atm E) 1 Pa 1) 2) The property responsible for the ʺbeading upʺ of water is _____. 78 AmO 2 (s) ‐1005. Using the graph, determine the following. MSDS available from Mallinckrodt Baker. FREE Expert Solution. Carbon tetrachloride (CCl4) is also nonpolar but is made of larger molecules that are more strongly affected by London dispersion forces. 14 At 25OC, 1 mol of A having a vapor pressure of 100torr and 1 mol of B having a vapor pressure of 300 torr were mixed. 7%, respectively, and there were no nuclei of fibrils for pBTTT12 15. The vapour pressure of CS2 at 50°C is 854 torr and a solution of 2. Reacts violently with azides, ethylamine ethylenediamine, ethylene imine. 817221\log _ {e} (T+273. However, for more accurate cal culations the tables should be interpolated. Problem: Using the graph,determine the following.