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Homework Help: Helmholtz free energy

  1. Mar 23, 2012 #1
    Is defined as F = U - TS and it is the energy needed to create a system minus the heat it can get for free.

    But why is the heat that it can get for free given by:
    Q = TS

    I know the thermodynamic relation:

    S = Q/T

    So in that way I understand it. But how do you know, that the system will be in thermal equilibrium with surroundings, when it has received a total energy of Q = TS as heat? I mean, the point where heat stops interesting is when temperaturs are equal, i.e. when dS/dU is the same for both systems. How do you know that holds when a total of heat of Q=TS has entered?
  2. jcsd
  3. Mar 23, 2012 #2
    Hint :

    This formula holds for reversible processes only.

    Remember, entropy is defind as [dQ (reversible)/dT ].
  4. Mar 27, 2012 #3
    hmm you gotta help me more on this: Why do you know that the temperature of the system is equal to the surroundings after having received Q = TS
  5. Mar 27, 2012 #4
    What happens during a reversible process?

    How are the temperatures(or some other quantities like pressure) of system and surrounding related during each step in such a process?

    (Hint: It has something to do with equilibrium.)
    Last edited: Mar 27, 2012
  6. Mar 28, 2012 #5
    They are in equilibrium at all times. It's like a pile of sand on a piston of a cylinder with a gas, removing a tiny grain each time.
    So should the two systems be in thermal equilibrium at all times? Well making a system out of nothing, how will the temperature then be the same as the environment?
  7. Mar 28, 2012 #6

    Equilibrium is of various types, like mechanical , thermal, chemical etc.

    When we speak of it in general we mean all the above are satisfied meaning the two bodies are in thermal equilibrium as well.

    in fact the Helmholtz and Gibbs equations are obtained for the conditios that The system and surrounding are in thermal equilibrium at all stages.

    A more applicable and conveying form of the equation is dA=dU-TdS,

    with dA<=0 for spontaneous change.
    (we will later see why)

    This equality holds only for reversible processes and the Clausius inequality sets in for determining it for any process.

    The interpretation you make on why the temperatures become equal after Q=TdS is exchanged is unheard of , by me and (unless it was stated in your text book where the author maybe conveying something else) I think it's wrong.

    As i said The system and surrounding are in thermal equilibrium at all stages, and the process stops when WE WISH to end the infinitely small difference between driving and opposing force


    Take the interpretation of Helmholtz energy as this.


    Here, dS is entropy change of the system and -dU/T is the entropy change of surroundings (since system has a constant volume, use first law).

    For a spontaneous change this total entropy change of universe(system + surrounding) should be positive and the total should tend to maximum.

    So Using Clausius inequality we need to have dA <0 for a spontaneous change.

    Of all models available this IMHO is the best possible interpretation.

    Also As mentioned this is obtained for the system and surrounding in thermal equilibrium.

    If thats not the case then,
    Since dA is a state function, we can create an equivalent reversible process to account for dS of system which will be same for both processes as S is a state function.
    Last edited: Mar 28, 2012
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