Entropy, free energy and chemical potential of mixtures

  1. Hi,

    1. The problem statement, all variables and given/known data
    Consider a mixture of different gases with [tex]N_i[/tex] molecules each (i=1...k denotes the species).
    For ideal gases the following relation yields:

    [tex]S(T,V,N_1,...N_k)=\sum_{i=1}^k S_i(T,V,N_i)[/tex]

    a)Give explicit expressions for the entropy, the internal energy, Helmholtz free energy and Gibbs free energy

    b) What is the change in entropy, if the k components are initially separated by moveable and diathermal seperators which get removed.

    c) Calculate for this mixture the chemical potential [tex]\mu_i[/tex] for each component and show that the following relation holds:
    [tex]\mu_i(p,T,N_1,...,N_k)=\mu_{i,0}(p,T) + kT\ln(c_i).[/tex]
    Where [tex]c_i := N_i/N[/tex] (with [tex]N = \sum_i N_i[/tex]) is the concentration of the i-th component and [tex]\mu_{i,0}(p,T)[/tex] the chemical potential of the i-th component in unmixed state.


    2. Relevant equations



    3. The attempt at a solution
    I have no idea at all, how to solve this exercise. Here is my attempt:

    a)

    Entropy:
    I know from here that the entropy of an ideal gas is given by

    [tex]S_i(T,V,N_i) = S_0 + N_i k \ln\left(\left(\frac{T}{T_0}\right)^{3/2}\frac{V}{V_0}\right)[/tex]

    So the entropy of the mixture (which we look for) may be given by the sum over this expression. Correct?

    Internal energy:
    I know that the internal energy is an extensive property, so
    [tex]U = \sum_i U_i[/tex] with [tex]U_i = \frac{3}{2}\cdot N_i \cdot k \cdot T[/tex]
    But I think, i should derive the internal energy of the mixture from the given equation [tex]S(T,V,N_1,...N_k)=\sum_{i=1}^k S_i(T,V,N_i)[/tex].

    Helmholtz free energy:
    Helmholtz free energy is given by
    [tex]A = U - T\cdot S[/tex]
    But how should I give an explicit expression for the mixture.

    Gibbs free energy:
    It is given by:
    [text]G = H - T\cdot S[/tex]
    Same problem as for the Helmholtz free energy, I don't know how to give an explicit expression for the mixture.


    b)

    I think, due to the diathermal seperators, I can assume, that each component has the same temperature. But I don't know how to take into account that the seperators are moveable.
    To be honest, I have no clue at all, how to solve this part...


    c)

    According to the definition in our lecture, the chemical potential is given by:

    [tex]\mu_i = \left(\frac{\partial U}{\partial N_m}\right)_{(S,V,N_1,...,N_k)}[/tex]
    where U is the internal energy and N_m the number of particles of species m.

    So i probably should derivate
    [tex]U_i = \frac{3}{2}\cdot N_i \cdot k \cdot T[/tex]
    with respect to N_i, to get [tex]\mu_i[/tex]
    However, I see to chance how to show with this derivation, that the following relation holds:
    [tex]\mu_i(p,T,N_1,...,N_k)=\mu_{i,0}(p,T) + kT\ln(c_i).[/tex]

    (Sorry for my english, it's not my native language)


    Best,
    derivator
     
    Last edited: Mar 28, 2010
  2. jcsd
  3. *push*
     
  4. ok folks, lets look at b), please:

    Entropy will change, because the available volume for one species will change. So I can take my formula for entropy

    [tex]
    S_i(T,V,N_i) = S_0 + N_i k \ln\left(\left(\frac{T}{T_0}\right)^{3/2}\frac{V}{V_0}\right)
    [/tex]

    and calculate

    [tex]
    \Delta S = \sum_i{S_i(T,V,N_i)} -\left(S_0 + N_i k \ln\left(\left(\frac{T}{T_0}\right)^{3/2}\frac{V_i}{V_0}\right)\right)
    [/tex]

    and simplify it.

    But I didn't took notice of those movable separators. Do I have to take notice of them?

    --derivator
     
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?