Entropy of Mixing

a) A 2-L container is divided in half: One half contains oxygen at 1 atm, the other nitrogen at the same pressure, and both gases may be considered ideal. The system is in an adiabatic enclosure at a temperature T = 293 K. The gases are allowed to mix. Does the temperature of the system change in this process? If so, by how much? Does the entropy change? If so, by how much?

b) How would the result differ if both sides contained oxygen?

c) Now consider one half of the enclosure filled with diatomic molecules of oxygen isotope ¹⁶O and the other half with ¹⁸O Will the answer be different from parts (a) and (b)?

SOLUTION

a) The energy of the mixture of ideal gases is the sum of energies of the two gases (since we assume no interaction between them). Therefore the temperature will not change upon mixing. The pressure also remains unchanged. The entropy of the mixture is simply the sum of the entropies of each gas (as if there is no other gas) in the total volume. We may write the total entropy S , as

(1)

where N₁ and N₂ are the number of molecules of each gas in the mixture. V is the total volume of the mixture (V = V₁ + V₂). The entropy of the gases before they are allowed to mix is

(2)

Therefore, the change in entropy, ∆S, is given by

(3)

In our case V₁ = V₂ = V/2, and

So, (3) becomes

(4)

In conventional units we find

(5)

The entropy increased as it should because the process is clearly irreversible.

b) If the gases are the same, then the entropy after mixing is given by

(6)

and so ∆S. In the case of identical gases, reversing the process only requires the reinsertion of the partition, whereas in the case where two dissimilar gases are mixed, some additional work has to be done to separate them again.

c) The same arguments as in (a) apply for a mixture of two isotopes,¹⁶O and ¹⁸O. The Gibbs free energy can be written in the form

(7)

where μ₀₁ and μ₀₂ are the chemical potentials of pure isotopes. Therefore, the potential (7) has the same form as in the mixture of two different gases, and there is no correction to the result of (a). This is true as long as (7) can be written in this form, and it holds even after including quantum corrections to the order of h².