An isolated box is partitioned into two compartments, each containing an ideal gas of different molecular species. Assume that each compartment has the same temperature, but different no. of moles, different pressure, and different volume. The partition is suddenly removed and the system is allowed to reach equilibrium. a) What are the final temperature and pressure? b)What is the change in entropy?  Attempt:  a) The temperature still remains the same. I assumed that energy of the system equals the energy of individual compartment. On equating, I got the same temperature. For pressure, I used  $PV=nRT$ where temperature is still the same. b) Since, initial and final temperature are same, I assumed that there is no change in entropy.  I would like to know if it correct or not? Thank you for any help :)
Entropy increases when two substances mix with each other.  For example, the entropy of mixing of two different gases are given by $$\Delta S= 2Nk\ln\frac{V_f}{V_i}\;.$$  But, the entropy doesn't increase when the two gases mixing are same.  This is pointed by Daniel V Schroeder:     It's important to note that this result applies only if the two gases are different like helium and argon. If you start with the same gas on both sides, the entropy doesn't increase at all when you remove the partition.   Now, why is this so? Both the gases, though the same, increase their individual entropies when they expand, don't they?  So, why did Schroeder say there is no entropy of mixing? What actually happens such that there is no change in entropy when the gases are the same?
How can I prevent inline math formulas from overflowing into the margin? When I compile my document in LaTeX, some formulas in the document in math mode, between $ $, go into the right margin area beyond all the text.  Is there a way to fix this aesthetic problem? Any solutions to Veritasium's hydrodynamic levitation? So this Youtube video has been around for a while now  . Also this  may be related.  I just wanted to know if there is any standard hydrodynamic terms to explain it. Derek clearly states it is not about Bernoulli principle, as in this case the ball rotation is important.  Has any recent paper/videos discussed what happens if you do the same experiment with hydrophobic spheres? Can it be predicted with laminar flow theories or does it need turbulence?  Too many questions have been posed, but not so many responses on the internet yet.  Edit: I said Derek stated "it wasn't Coandă effect", yet what he says is that it is not "just Bernoulli's principle". Coandă effect is the typical hairdryer ping-pong ball levitation which is explained by Bernoulli's principle.   Edit2: rephrasing.  Additional comment: There seems to be two explanations "water adhere to the sphere and is ejected conserving horizontal momentum" and "Magnus effect" (ME). The later is not so obvious to me as the system is in equilibrium and there are two fluids, water and air, (does ME means air pushes the ball horizontally?).   Edit3: DESCRIPTION To avoid rot links and people that do not want to see the video, I'll try to explain the phenomena. You have a sphere and a water jet. The water jet is vertical (opposite to gravity), and touches the sphere almost tangentially. The ball not only rotates due to the water jet but also stays at constant height from the ground. The angular velocity can be defined in the direction of the cross product between the position of the contact point between the jet and the sphere (origin in the center of the sphere) and the direction of the jet. Some water drops fly away at the top of the sphere away from the jet.  Edit4: New question, would this phenomenon occur if there was no air? Strange behaviour of binomial coefficient's delimiters This question led to a new package:      While writing down my Combinatorics exercises I found that the binomial coefficient's delimiters weren't displayed as indicated inside the amsmath documentation.  By subsequent trials and errors I was able to reduce the volume of my document to this simpler and fully compilable MWE:  \documentclass[11pt]{article} % \usepackage{amsmath} % \makeatletter % \usepackage{lmodern} \usepackage{lipsum} % \makeatother % \begin{document} % \lipsum[1] \begin{equation} 2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2} \end{equation} \lipsum[1] % \end{document}   Which gives (using TeXworks 0.44 on MikTeX 2.9.4521 updated 09/10/13) this un-expected result:        While the amsmath doc shows that with the loaded packages inside the MWE the result would actually be like this:        From here we can see that the delimiters on the first example are a little bit stretched vertically as they surpass the arguments inside the binom. coeff.  The more strange event is that when I comment the line \usepackage{lmodern} the result on delimiter's shape in my example looks exactly as the amsmath doc:  \documentclass[11pt]{article} % \usepackage{amsmath} % \makeatletter % %\usepackage{lmodern} \usepackage{lipsum} % \makeatother % \begin{document} % \lipsum[1] \begin{equation} 2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2} \end{equation} \lipsum[1] % \end{document}   Output:        From here I deduce that the problem lies inside lmodern, but I'd like to keep using this package and fix this wierd "issue" with the math mode. Is there some clue that I've missed or this is in fact an update-process bug infecting amsmath's delimiter control? What do the supercharges in extended supersymmetry do? What do the supercharges in  do?  In ${\cal N}=1$ supersymmetry there are a certain number of fermions and and equal number of bosons. You can transform all fermions to the bosons (and vice versa) in a 1 to 1 fashion using a single supercharge, $Q$.  So what happens when you have, for example, ${\cal N}=2$ supersymmetry with 8 supercharges? Since $Q$ is a generator of supersymmetry transformations, is it a linear combination of these supercharges that act on the particles? In which case could one particle be acted on by two separate linear combinations of $Q$? Or is it strictly one linear combination of $Q$ per fermion/boson?  Also, what does ${\cal N}$ mean physically? What difference does ${\cal N}=2$ have to ${\cal N}=1$ other than more supercharges? Or is that the only difference between the two theories? "Reality" of EM waves vs. wavefunction of individual photons - why not treat the wave function as equally "Real"? In thinking how to ask this question (somewhat) succinctly, I keep coming back to a Microwave Oven.  A Microwave Oven has a grid of holes over the window specifically designed to be smaller in diameter than the wavelength of the microwaves it produces, yet larger than the wavelengths of the visible light spectrum - this is so you can watch your food being heated without getting an eye full of microwaves.  The "realness" of electromagnetic waves seems indisputable - both from the microwave example above, and also because if I want to broadcast a radio wave with a certain wavelength, then I need to make sure I have an antenna of corresponding length to produce the wave I'm looking for.  Furthermore, we discuss and treat these waves as real, measurable "objects" that exist and can be manipulated.     Now, if I want to describe the behavior of my Microwave Oven in the framework of QM (let's pretend my oven is going to only produce 1 photon of energy corresponding to a wavelength of the microwave spectrum for simplicity) then I'll describe the behavior of that photon as a wavefunction that evolves over time and gives a probability distribution within my microwave that similarly does not allow the photon to pass through the safety grid and exit the oven cavity giving me a retinal burn.  The difference is, the wavefunction is never treated as something "real" in this description.  When the safety grid is described as working to protect you because it has holes smaller than the wavelength of the classical waves it's blocking, this is a useful description that seems to describe "real" objects/be-ables.  While it is possible to describe why an individual photon has low probability of passing through the same grid but extended physical properties such as wavelength (in space) are treated as non-real because we're dealing with a point particle and with behavior described by something we also treat as non-real (the wavefunction); it seems unclear to me why we insist this wave function which predicts behavior of physical measurements so well is somehow "non-real."    Put another way, if we have no problem treating EM waves as "Real," then why do we insist on treating the wavefunction that describes the same behavior as "unreal?"  I understand there is recent research (Eric Cavalcanti and his group for one) trying to argue this point, but as every respectable physics professor, I've ever encountered, has treated the wavefunction as an indisputably non-real mathematical tool, I needed to ask this community for an answer. I'm trying to answer the following question from the book :  Let $X_1,X_2,\dots$ be a sequence of sub-gaussian random variables, which are not necessarily independent. Show that   $E\bigg[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \bigg] \le CK$,  where $K = \max_i \|X_i\|_{\psi_2}$. Deduce that for ever $N \ge 2$ we have   $E\bigg[ \max_{i \le N} |X_i| \bigg] \le CK \sqrt{\log N}$.  I've tried to figure out what is the distribution of the maximum of Gaussians, but I'm reaching only inequalities that that don't help me answer the question.   I've also seen a similar question .  Does anyone have a clue or something to start with in order to answer this question?  Thanks! Transpose Ranges repeated by row I would like to convert several range of columns into rows, and keeping the association to the original col/row source  Sample source sheet  --------------------------------------------------- | Data 1 | 1/1/2018 | 1/2/2018 | 1/3/2018 | etc...365 days | --------------------------------------------------- | name 1  |    25    |   30     |    5     | etc...|  --------------------------------------------------- | name 2  |    2     |          |    4     | etc...| --------------------------------------------------- | etc ... 100 rows   Expected Results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- | name 1  |  1/2/2018   |   30     |    --------------------------------------------------- | name 1  |  1/3/2018   |   5      |    --------------------------------------------------- | name 1  |  etc.. 365 days --------------------------------------------------- | name 2  |  1/1/2018   |   2      |    --------------------------------------------------- | name 2  |  1/2/2018   |          |    --------------------------------------------------- | name 2  |  1/3/2018   |   4      |    --------------------------------------------------- | name 2  |  etc.. 365 days --------------------------------------------------- | etc ... for each change of name pull all the col dates/values   Current results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- |         |  1/2/2018   |   30     |    --------------------------------------------------- |         |  1/3/2018   |   5      |    --------------------------------------------------- |         |  etc.. 365 days                      I have to copy down to bring in "name 1" for each date  --------------------------------------------------- | name 2  |  1/1/2018   |   2      |             now update the formula to bring in the second item for the sequence of dates --------------------------------------------------- |         |  1/2/2018   |          |    --------------------------------------------------- |         |  1/3/2018   |   4      |    --------------------------------------------------- |         |  etc.. 365 days --------------------------------------------------- | etc ...     =Sheet1!A:A I can bring in the name from the source list, but I have to copy it down for the number of dates.  How can I have it match the number of dates?  and be dynamic to update as dates/values are added Edit: trying to use split and rept to count the number of times to repeat, then transpose.. =transpose(rept(split(arrayformula(concatenate(Sheet1!A:A&amp;"1")),"1",True, True),COUNTA(Sheet1!I1:ZZ1)))  =TRANSPOSE(Sheet1!$C$1:$Z$1) I can use transpose to get the dates/values, but it's fixed range.  How can I make it dynamic to import dates as they are added? Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$ Which 1970s UK fantasy kids TV show has an older brother die? List sorting by custom field I dont know how to sort a list of SObject Case.  I have a List&lt;Case&gt; newCases which i iterate over in for cycle, where i save Cases to my new List&lt;Case&gt; casesList.  for (Case newCase : newCases) {             if(newCase.Case_Content__c == '111' &amp;&amp; newCase.Status == '24'){                 String cuid = newCase.CUID__c;                 cuidSet.add(cuid);                 casesList.add(newCase);                 casesList.sort(); //TODO sorting by CUID__c             }         }   What i need to do is, to sort this new list by field CUID__c. So the Cases with same CUID are together.  Thank you for any help :) Does DTIME(O(n)) = REGULAR? I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere. Should I add a 'Home' Button to the navigation? I know that it is common behavior to have the logo of a web page be a link to the home page, but should I also have a 'home' link in the navigation bar for it as well? Can I boil milk unattended (and still keep the pan clean)? What is the plural form of "Software"? Is it correct to use softwares, as in the following sentence?  There are various video conversion softwares in the market. Polarization vectors of massive and massless particles This is a really trivial question, but I've seen different conventions and some edits regarding which is correct, so I'm curious:  Should question titles be title-cased or sentence-cased? Zero visitors? What happened to site analytics about a week ago? This is the views section of the site analytics visible to 25k users on one of my sites:    That's, uh, not correct.  Our site has had its usual activity level during this time.  I checked a couple other sites, and the pattern is consistent.  This looks like a problem in the analytics data that's coming in; did Google change their API?  Update: Same site a week later; note that in addition to the brief spike of accuracy, page views have dropped to (near-)zero for the whole period, unlike the earlier report.  (I know this question has been closed as a dupe, but I thought having the two screen shots together might help somebody.) Where Android apps store data? Could you list all possible directories where Android apps may store data, providing description what kind of data are stored in each directory? Why does pressing a transform hotkey not display the axes on the selected object? I open Blender 2.80 (official release, not a beta or RC) and get the new scene with the usual 3 objects present (camera, light and cube). I select the cube, press 't' for the tools menu and click on the Move tool. The usual 3 axis glyph appears at the cube's origin as expected. But if I use the hotkey 'g' to activate the move transform instead of selecting the tool from the menu with the mouse this glyph does not appear. The exact same problem exists if I use the a hotkey for rotate or scale as well. Why? Is there some clickable checkbox setting hidden somewhere that could have disabled this?