Le Chatelier and Le Chatelier Braun

1. Jun 26, 2011

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 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?

2. Jun 29, 2011

Telemachus

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

3. Feb 25, 2012

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.

Last edited: Feb 25, 2012