Relating the entropy of an ideal gas with partial derivatives

Mayan Fung
Messages
131
Reaction score
14
Homework Statement
For an ideal gas, use ##dE=TdS-PdV+\mu dN## to prove
1. ##V(\frac{\partial P}{\partial T})_{\mu} = S##
2. ##V(\frac{\partial P}{\partial \mu})_T = N##
Relevant Equations
##dE=TdS-PdV+\mu dN##
It looks very easy at first glance. However, the variable S is a variable in the given expression. I have no clue to relate the partial derivatives to entropy and the number of particles.
 
Physics news on Phys.org
Using the extensible properties of the system as variables, we know that ##E (x \lambda) = \lambda E(X)## (Homogeneous function of degree a=1), so that we can say

##x * \nabla f = a f## (Euler's homogeneous theorem), where ##x = (x_{1},x_{2},...,x_{n})## is the vector with the variables.

So that
$$S( \partial E/ \partial S ) + V ( \partial E/ \partial V)+ N (\partial E/ \partial N )= a * E$$
$$ ST - PV + N \mu = E$$

The rest i think you can go on, eventually you will get the Gibbs Duhem equation
 
  • Like
Likes Mayan Fung and etotheipi
Just to add to what @LCSphysicist wrote, first try to re-write each term on the RHS according to ##x\mathrm{d}y = \mathrm{d}(xy) - y\mathrm{d}x##.
 
Thanks! This makes me recall the fact that ##G=\mu N## and ##G=U-TS+\mu N##
 
Mayan Fung said:
Thanks! This makes me recall the fact that ##G=\mu N## and ##G=U-TS+\mu N##
Careful, it's ##G := U -TS + pV = \mu N##
 
Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'
(Disclaimer: this is not a HW question. I am self-studying, and this felt like the type of question I've seen in this forum. If there is somewhere better for me to share this doubt, please let me know and I'll transfer it right away.) I am currently reviewing Chapter 7 of Introduction to QM by Griffiths. I have been stuck for an hour or so trying to understand the last paragraph of this proof (pls check the attached file). It claims that we can express Ψ_{γ}(0) as a linear combination of...
Back
Top