Derivator
Mar28-10, 11:33 AM
Hi,
1. The problem statement, all variables and given/known data
Consider a mixture of different gases with N_i molecules each (i=1...k denotes the species).
For ideal gases the following relation yields:
S(T,V,N_1,...N_k)=\sum_{i=1}^k S_i(T,V,N_i)
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 \mu_i for each component and show that the following relation holds:
\mu_i(p,T,N_1,...,N_k)=\mu_{i,0}(p,T) + kT\ln(c_i).
Where c_i := N_i/N (with N = \sum_i N_i) is the concentration of the i-th component and \mu_{i,0}(p,T) 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 (http://books.google.com/books?id=12DKsFtFTgYC&lpg=PP1&dq=greiner%20thermodynamics&pg=PA42#v=onepage&q=&f=false) that the entropy of an ideal gas is given by
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)
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
U = \sum_i U_i with U_i = \frac{3}{2}\cdot N_i \cdot k \cdot T
But I think, i should derive the internal energy of the mixture from the given equation S(T,V,N_1,...N_k)=\sum_{i=1}^k S_i(T,V,N_i).
Helmholtz free energy:
Helmholtz free energy is given by
A = U - T\cdot S
But how should I give an explicit expression for the mixture.
Gibbs free energy:
It is given by:
G = H - T\cdot S
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:
\mu_i = \left(\frac{\partial U}{\partial N_m}\right)_{(S,V,N_1,...,N_k)}
where U is the internal energy and N_m the number of particles of species m.
So i probably should derivate
U_i = \frac{3}{2}\cdot N_i \cdot k \cdot T
with respect to N_i, to get \mu_i
However, I see to chance how to show with this derivation, that the following relation holds:
\mu_i(p,T,N_1,...,N_k)=\mu_{i,0}(p,T) + kT\ln(c_i).
(Sorry for my english, it's not my native language)
Best,
derivator
1. The problem statement, all variables and given/known data
Consider a mixture of different gases with N_i molecules each (i=1...k denotes the species).
For ideal gases the following relation yields:
S(T,V,N_1,...N_k)=\sum_{i=1}^k S_i(T,V,N_i)
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 \mu_i for each component and show that the following relation holds:
\mu_i(p,T,N_1,...,N_k)=\mu_{i,0}(p,T) + kT\ln(c_i).
Where c_i := N_i/N (with N = \sum_i N_i) is the concentration of the i-th component and \mu_{i,0}(p,T) 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 (http://books.google.com/books?id=12DKsFtFTgYC&lpg=PP1&dq=greiner%20thermodynamics&pg=PA42#v=onepage&q=&f=false) that the entropy of an ideal gas is given by
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)
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
U = \sum_i U_i with U_i = \frac{3}{2}\cdot N_i \cdot k \cdot T
But I think, i should derive the internal energy of the mixture from the given equation S(T,V,N_1,...N_k)=\sum_{i=1}^k S_i(T,V,N_i).
Helmholtz free energy:
Helmholtz free energy is given by
A = U - T\cdot S
But how should I give an explicit expression for the mixture.
Gibbs free energy:
It is given by:
G = H - T\cdot S
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:
\mu_i = \left(\frac{\partial U}{\partial N_m}\right)_{(S,V,N_1,...,N_k)}
where U is the internal energy and N_m the number of particles of species m.
So i probably should derivate
U_i = \frac{3}{2}\cdot N_i \cdot k \cdot T
with respect to N_i, to get \mu_i
However, I see to chance how to show with this derivation, that the following relation holds:
\mu_i(p,T,N_1,...,N_k)=\mu_{i,0}(p,T) + kT\ln(c_i).
(Sorry for my english, it's not my native language)
Best,
derivator