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The chemical potential of an ideal monoatomic gas should be: $\mu = \tau ln \frac {n}{n_Q}$ http://web.mit.edu/ndhillon/www/Teaching/Physics/bookse5.html

I get this result if I derive it using the definition of Gibbs Free Energy: G=U+PV-TS

However, if I use the differential equation of enthalpy, H: dH=TdS+VdP+$\mu$dN

I get the result: $\mu = \tau ln (\frac {n}{n_Q}) - \frac 5 2 \tau $

This is the proof, which is where I'm trying to figure where the error is:

dH=$\frac 5 2 Nk_BdT $ = TdS+VdP+$\mu$dN

At constant T and P, -TdS =$\mu$dN

$\mu$ = -T$\frac{dS}{dN}$

For a monoatomic gas (Sackur Tetrode), S = $Nk_Bln(\frac{V}{N\lambda^3}) + \frac{5}{2}Nk_B$

Since $\frac{P}{k_BT} = \frac{N}{V}$

$\mu$=-T$\frac{dS}{dN}$ = -$k_BTln(\frac{k_BT}{P\lambda^3}) - \frac{5}{2}k_BT$ at constant P and T.

Rearrange, $\mu = \tau ln (\frac {n}{n_Q}) - \frac 5 2 \tau $

Just trying to figure out what I am doing wrong with this proof. Shouldn't the answers be the same if I start from G vs. H?

Ted Yu
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  • Please note that Physics.StackExchange is not a homework help site. Please read this Meta post on asking homework-like questions and this Meta post for "check my work" problems. – Kyle Kanos Jan 22 '16 at 11:56
  • I am a Professor. – Ted Yu Jan 23 '16 at 00:41
  • This is for a class I am teaching. – Ted Yu Jan 23 '16 at 00:43
  • I'm trying to understand why my questions get flagged for "homework questions" including this one which I resolved. – Ted Yu Jan 25 '16 at 07:32
  • http://physics.stackexchange.com/questions/199832/simpler-derivation-of-sackur-tetrode-equation – Ted Yu Jan 25 '16 at 07:32
  • I'm leaning that people who flagged me must hold such a high ideal for physics that the simple questions I ask cannot possibly be from someone who asks a serious question. – Ted Yu Jan 25 '16 at 07:32
  • I think the better question to ask yourself before you flag it as a homework question is, "Can I answer this question?" If the answer is no, you should not be flagging it as a homework question. Even though it is simple, there is subtlety in the question that is important to me. Feel free to flag it as a homework question if you think it is too easy. But if you can't answer it, DON'T FLAG it! – Ted Yu Jan 25 '16 at 07:34
  • If you want to check the legitimacy of my claim that I am a Professor, send an e-mail to this address: ted.yu@csulb.edu, which is also in this webpage: https://web.csulb.edu/colleges/coe/che/views/faculty/yu.shtml I already hold a PhD, it is not in my interest to come on some board to get homework done. – Ted Yu Jan 25 '16 at 07:41
  • I linked two relevant posts in my comment that specifically state that problematic like this are considered homework whether it actually is homework or not. We deal with specific physics concepts at this site; answering how to solve some problem isn't a concept question. It doesn't matter whether I am able to solve the problem or not, or whether I think it too easy; this is simply the type of question that we don't want asked here. – Kyle Kanos Jan 25 '16 at 11:03
  • @TedYu: I'd be inclined to also say that this question is very "homework-like," not to be pejorative but to indeed state that it seems to be mostly a "check my work" question. In this case, I can tell you that your math is probably flawed because the thermal wavelength is defined as $\lambda = \sqrt{3Nh^2\over 4\pi mU},$ and therefore ${\left({\partial\lambda\over\partial N}\right)}_{P,T}$ is probably not $0$. – CR Drost Jan 25 '16 at 20:55
  • Isn't there an N term in U? U = $\frac{3}{2}Nk_BT$. This would make the N terms cancel out. This would make the equation for λ the same as found here: https://en.wikipedia.org/wiki/Thermal_de_Broglie_wavelength which is independent of N. – Ted Yu Jan 27 '16 at 07:57
  • I just don't see how my question is that different from the following: http://physics.stackexchange.com/questions/128501/derivation-of-fr-field-equations-problem-with-integration-by-parts http://physics.stackexchange.com/questions/76210/deriving-the-hamiltonian-density-for-a-free-scalar-field http://physics.stackexchange.com/questions/204008/confusion-i-have-regarding-einsteins-1905-derivation-of-lt – Ted Yu Jan 27 '16 at 08:04
  • @Kyle Kanos: If you can recommend better sites where I can pose these types of questions, I would appreciate the recommendation. Honestly, as an engineering professor, there is not really a slew of colleagues to go to for these questions, so this is my only outlet for now. – Ted Yu Jan 27 '16 at 08:18
  • @CR Drost: And to clarify, the reason I'm asking is because if I use the simpler derivation of the Sackur-Tetrode Equation without the 5/2 term, I actually get the correct result for the chemical potential if I use the derivation from H. http://physics.stackexchange.com/questions/199832/simpler-derivation-of-sackur-tetrode-equation – Ted Yu Jan 27 '16 at 08:26
  • @TedYu: If you do want that kind of help, you can take a look at this thread for a list of free online homework help resources. – Kyle Kanos Jan 27 '16 at 11:14

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