Le Chatelier and Le Chatelier Braun

[Telemachus]

Hi there. I'm trying to solve this problem, which says:

A system is in equilibrium with its envornment at a common temperature and a common pressure. The entropy of the system is increased slightly (by a fluctation in which heat flows into the system, or by the purposeful injection of heat into the system). Explain the implication of both the Le Chatelier and the Le Chatelier-Braun principles to the ensuing processes, proving your assertions in detail.

I don't know how to start with this.

I know form the book that if I have a spontaneous fluctuation dX_1^f in my composite system then this fluctuation will be accompanied by a change in intensive parameter P_1 of the subsystem, and then I have this equation:

dP_1^f=\frac{\partial P_1}{\partial X_1}dX_1

And that the fluctuation dX_1^f also alters intensive parameter P_2

dP_2^f=\frac{\partial P_2}{\partial X_1}dX_1

This is how the book explains it, then it makes a reasoning based on the responses. The thing is I don't know how to relate this with the specific problem I'm dealing with, I don't know how to use the fluctuation on the entropy, and I don't know which kind of responses that will produce.

Would you help me?

Thanks in advance. Bye there!

 

[Telemachus]

Anyone please?

May I have in consideration that S(U,V,N)?

 

[Telemachus]

Ok. Some time have passed, anyway, I'm facing this problem again.

This is my approach now. At first, about the notation: (r) corresponds to response, (f) to fluctuation, and (res) is for reservoir.
I've considered that the fluctuation on the entropy has a primary effect on temperature

dT^f=\left ( \frac{\partial T}{\partial s} \right )_v ds^f=\frac{1}{c_v}dQ_1^f

And then that there is also consequently an effect over the pressure:

dP^f=\frac{\partial P}{\partial s}ds^f=\frac{\alpha}{c_v k_t}dQ^f

Then for the response:
d(U+U^{res})=(T-T^{res})ds^r+(P-P^{res})dv^r \leq 0
dT^fds^r+dP^fdv^r\leq 0

Because the two members at the right are independent this leads to:
dT^fds^r=\frac{\partial T}{\partial s}ds^fds^r \leq 0
Because of the convexity criterion this can be written as:

\frac{\partial T}{\partial s}ds^f \frac{\partial T}{\partial s} ds^r=dT^fdT^r \leq 0

And
dP^fdv^r=\left ( \frac{\partial P}{\partial s} \right )_v ds^f dv^r \leq 0

Now, I've used the Maxwell relations on this, and here is the problem I think.
\left ( \frac{\partial P}{\partial s}\right )_v=-\left ( \frac{\partial T}{\partial v}\right )_s
The minus sign there is what bothers me. Because the response should attenuate the fluctuation, and written like that I get to:
\left ( \frac{\partial T}{\partial s} \right )_v ds^f \left(-\left ( \frac{\partial T}{\partial v}\right )_s dv^r \right ) \leq 0
So, it doesn't seems to be opposing to the change.

Perhaps I'm missing something here, or perhaps there is something wrong about the whole thing. Any suggestion or help will be thanked.

PD: By the way, I think this should go on advanced Physics. Please, move it if you can.