Helmholtz entropy of ideal gas mixture is additive?

In summary: I believe I included the answer to both in my OP. Indeed, it seems like the problem I'm having is massaging a sum (on ##j##) over the first equation in my OP to be of the second equation's form, hence why I'm not sure if I'm making a mistake at the outset.Maybe something like this:$$F_j((T,V,N_j)=N_ju_j(T_0,v_0)+N_jC_{v,j}(T-T_0)-$$$$T\left[N_js_j(T_0,v_0)+N_jC_{v,j}\ln{(T/
  • #1
EE18
112
13
In his classic textbook, Callen remarks that
(1) The Helmholtz potential of a mixture of simple ideal gases is the sum of the Helmholtz potentials of each individual gas:$$ F(T,V,N_1, ... ,N_m)=F(T,V,N_1)+ ··· +F(T,V,N_m). $$
(2) An analogous additivity does not hold for any other potential expressed in terms of its natural variables.
I have labelled the claims (1) and (2). I am not sure about either. For the first, I have tried to proceed as follows (all equations are from Callen's second edition and all 0 subscripts are with respect to some reference state of an ideal gas):
We begin by noting that for a single component of the mixture (in the volume ##V## of the overall gas per the given formula in this problem) we have from (3.34) that
$$F_i(T,V,N_i) = U(T,V,N_i) - TS(T,V,N_i) $$
$$= c_iN_iRT -T\left(N_is_{i0} + c_iN_iR \ln (T/T_0) + N_iR \ln(V/V_0) - N_iR(c_i+1) \ln (N_i/N_0) \right)$$
Now for the mixture we have
$$F = U - TS $$
$$\stackrel{(1)}{\equiv} (\sum N_j c_j)RT - T\left( \sum N_js_{j0} + (\sum N_j c_j)R \ln (T/T_0) + (\sum N_j) R \ln(V/V_0) -R \sum N_j \ln(N_i/N)\right).$$
where (1) is from (3.39) and (3.40) (i.e. from the definition of ##F## that for a mixture of ideal gases (i.e. in the same vessel), we form ##U-TS## for ##U## and ##S## of the entire system).
I can't see how to go further in terms of identifying one with the other

But even supposing I can show that, what does claim (2) mean? Is Callen saying that there is no other thermodynamic potential (partial Legendre transform of the energy ##U##) which is such that this additivity holds in terms of natural variables? Obviously ##U = \sum N_jc_j RT## which is additive, but I guess this isn't a counter example since ##T## isn't a natural variable of ##U##?
 
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  • #2
I think you mean Helmholtz free energy, not Helmholtz entropy.

I think it might help to consider the following questions:

What is Gibbs theorem for the partial molar properties of the components in an ideal gas mixture?

What is the equation for the partial pressure of a gas component in an ideal gas mixture in terms of Nj, R, T, and V.

What is the equation for the partial molar entropy of a gas component in an ideal gas mixture as a function of temperature and partial pressure?

What is the equation for the partial molar Helmholtz free energy of a gas component in an ideal gas mixture in terms of temperature and partial pressure?
 
  • #3
Chestermiller said:
I think you mean Helmholtz free energy, not Helmholtz entropy.
Oy, yes I did mean Helmholtz (free) energy -- I've been thinking too much about the entropy because of another question I had clearly! If you do get the chance, that question is here. I've been very grateful for your insights on thermodynamics as I work through Callen, both now and in the past!

For these:
What is Gibbs theorem for the partial molar properties of the components in an ideal gas mixture?

What is the equation for the partial pressure of a gas component in an ideal gas mixture in terms of Nj, R, T, and V.

"The contribution to a property of a mixture of ideal gases is the sum of the properties that each component gas would have if it alone were to occupy the volume V at temperature T"
and
##P_i = N_iRT/V.##

I don't think either of these is relevant for this particular question though?

For these:
What is the equation for the partial molar entropy of a gas component in an ideal gas mixture as a function of temperature and partial pressure?

What is the equation for the partial molar Helmholtz free energy of a gas component in an ideal gas mixture in terms of temperature and partial pressure?
I believe I included the answer to both in my OP. Indeed, it seems like the problem I'm having is massaging a sum (on ##j##) over the first equation in my OP to be of the second equation's form, hence why I'm not sure if I'm making a mistake at the outset.
 
  • #4
Maybe something like this: $$F_j((T,V,N_j)=N_ju_j(T_0,v_0)+N_jC_{v,j}(T-T_0)-$$$$T\left[N_js_j(T_0,v_0)+N_jC_{v,j}\ln{(T/T_0)}+N_jR\ln{\left(\frac{V}{N_jv_0}\right)}\right]$$where ##T_0## and ##v_0## are the temperature and molar volume in a reference state, and ##s_j## and ##u_j## are the molar entropy and molar internal energy of species j in the reference state. Then substitute $$\ln\left(\frac{V}{N_jv_0}\right)=\ln\left(\frac{V}{x_jNv_0}\right)=\ln\left(\frac{V}{Nv_0}\right)-\ln{x_j}$$
 
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Related to Helmholtz entropy of ideal gas mixture is additive?

What is Helmholtz entropy in the context of an ideal gas mixture?

The Helmholtz entropy is a thermodynamic property that quantifies the disorder or randomness of a system. For an ideal gas mixture, it is derived from the Helmholtz free energy and relates to the number of microstates accessible to the gas particles at a given temperature and volume.

Why is the Helmholtz entropy of an ideal gas mixture considered additive?

The Helmholtz entropy is considered additive for an ideal gas mixture because the total entropy of the system is the sum of the entropies of the individual gas components. This additivity arises from the fact that the gases in the mixture do not interact with each other and each gas behaves independently, contributing its own entropy to the total.

How do you calculate the Helmholtz entropy for an ideal gas mixture?

The Helmholtz entropy for an ideal gas mixture can be calculated by summing the entropies of the individual gases. For each gas component, the entropy can be derived from its Helmholtz free energy, which depends on the number of particles, temperature, and volume. The total entropy S is given by S = Σ S_i, where S_i is the entropy of the i-th gas component.

What assumptions are made when considering the Helmholtz entropy of an ideal gas mixture?

The primary assumptions are that the gases in the mixture are ideal and do not interact with each other, meaning they obey the ideal gas law. Additionally, it is assumed that the mixture is in thermal equilibrium, and the properties of each gas component can be treated independently.

Can the additivity of Helmholtz entropy be applied to non-ideal gas mixtures?

No, the additivity of Helmholtz entropy generally cannot be directly applied to non-ideal gas mixtures. In non-ideal mixtures, interactions between gas particles must be considered, which complicates the calculation of entropy. The presence of intermolecular forces means that the entropy of the mixture is not simply the sum of the entropies of the individual components.

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