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Le Chatelier and Le Chatelier Braun

  1. Jun 26, 2011 #1
    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 [tex]dX_1^f[/tex] in my composite system then this fluctuation will be accompanied by a change in intensive parameter [tex]P_1[/tex] of the subsystem, and then I have this equation:

    [tex]dP_1^f=\frac{\partial P_1}{\partial X_1}dX_1[/tex]

    And that the fluctuation [tex]dX_1^f[/tex] also alters intensive parameter [tex]P_2[/tex]

    [tex]dP_2^f=\frac{\partial P_2}{\partial X_1}dX_1[/tex]

    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!
     
  2. jcsd
  3. Jun 29, 2011 #2
    Anyone please?

    May I have in consideration that [tex]S(U,V,N)[/tex]?
     
  4. Feb 25, 2012 #3
    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

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

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

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

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

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

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

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

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