Relating the entropy of an ideal gas with partial derivatives

AI Thread Summary
The discussion focuses on relating the entropy of an ideal gas to partial derivatives and the extensible properties of the system. Participants explore the application of Euler's homogeneous theorem to derive relationships involving energy, entropy, volume, and particle number. Key equations are presented, including the connection between entropy and energy changes, leading towards the Gibbs-Duhem equation. The importance of rewriting terms using differential relationships is emphasized for clarity in understanding these concepts. Overall, the conversation aims to clarify the mathematical framework linking thermodynamic properties.
Mayan Fung
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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.
 
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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
 
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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##
 
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