Title: Find an expression for S(T,x) from tension and specific heat
Tags: thermodynamics;statistical-mechanics;entropy;maxwell-relations
Question: I'm working on a problem from a Statistical Mechanics lecture series online, and on the homework, I hit a bump in this problem.
Here is the problem set, and I'm asking about #1.c.
Short version, we are given that the force to move the filament is $J=ax-bT+cTx$ and that the specific heat for a constant displacement is $C_x=A(x)T$, and in part be it is shown that $A$ is just a constant.
My first thought was to write the entropy as follows
$$S(T,x)=\Delta S+S_0=S_0+\int_{(0,0)}^{(T,x)}\frac{\delta Q_{rev}}{T}.$$
But I get stuck around trying to find a reliable expression for $\delta Q_{rev}$.
I also tried using Maxwell Relations to find $\frac{\partial S}{\partial x}$ and $\frac{\partial S}{\partial T}$ and integrate the two to find $S(T,x)$. That's about all I can think of for solution methods. Any guidance would be appreciated!

Here is the accepted answer: The route of finding $\frac{\partial S}{\partial x}$ and $\frac{\partial S}{\partial T}$ is the more practical route. $\frac{\partial S}{\partial x}$ can be found using a simple Maxwell relation for the Helmholtz free energy:
$$dA=dU-d(TS)=Jdx-SdT$$
$$\left. \frac{\partial A}{\partial x}\right|_T = J \, \, \mathrm{and} \, \, \left. \frac{\partial A}{\partial T}\right|_x = -S.$$
So then $-\frac{\partial J}{\partial T} = \frac{\partial S}{\partial x}=b-cx$. The other relation is obtained from the equation for specific heat:
$$C=\frac{\delta Q}{\partial T}=C_x=AT$$
$$TdS= \delta Q = ATdT$$
$$\frac{\partial S}{\partial T} = A.$$
Integrating both expressions gives $S(T,x) = bx -\frac{cx^2}{2} + f(T)=AT+g(x)=AT+bx-\frac{cx^2}{2}+C$. Lastly, you simply put in $(T,x)=(0,0)$ and get that $C=S_0$.
Hope that helps anyone struggling with similar problems!


{'source': 'StackExchangeDataset', 'subset': 'physics.stackexchange.com', 'metadata': {'AnswerCount': 1, 'CommentCount': 0, 'ContentLicense': 'CC BY-SA 3.0', 'CreationDate': '2015:00:07/05/15', 'FavoriteCount': None, 'PostTypeId': 1, 'Tags': '<thermodynamics><statistical-mechanics><entropy><maxwell-relations>', 'Title': 'Find an expression for S(T,x) from tension and specific heat', 'ViewCount': 65, 'id': 192726, 'user_AboutMe': "<p>Hello all! My name is Jordan, and I'm currently a community college student studying computer science (namely networking and security aspects), hopefully graduating with 4 associates and a certificate within the year. I'm a huge fan of all things mathematical and physical! I've been fascinated by the universe from a very young age, so I've pursued a lot of knowledge in the field, as well as the mathematics which govern it. One of my favorite topics is that of classical rotation and other rotational problems. I'm trying my hardest not to ramble here, so I'll wrap it up there. Have fun, and never stop wondering!!</p>\n", 'user_AccountId': 6298552.0, 'user_DisplayName': 'JoDraX', 'user_Reputation': 347.0, 'user_Views': 56.0, 'user_id': 81086.0}}