8. In this problem we will investigate the entropy of two different gases in the

Question

8. In this problem we will investigate the entropy of two different gases in the same box using the lattice model discussed in class. Consider the system consisting of k oxygen molecules and k2 nitrogen molecules in a box. The box contains n possible locations where a gas molecule can reside, and at most one molecule can be located in each location. All oxygen molecules are indistinguishable, and all nitrogen molecules are indistinguishable, but oxygen and nitrogen molecules can be distinguished from each other What is the entropy of this system? Prove that your answer is correct

Explanation / Answer

First, we need to compute the multiplicity of the system, which is the number of possible microstates.
Label the Oxygen molecules as 1, the Nitrogen molecules as 2 and the empty sites as 0. A microstate can be represented like this 111...122....200...0 where we have k 1's, k 2's and n - k - k 0's. The total number of possible microstates is given by = n!/(k!k!(n-k-k)!). We arrive at the later expression using the fact that the number of permutations of a string with similar class of objects is the total number of permutations divided by the number of permutations of each class of object. Note that is the multiplicity of the system.
The Entropy of a system is given by S = k ln , where k is the Boltzmann constant and ln is the natural logarithm (logarithm with base e).
Thus S = k ln n!/(k!k!(n-k-k)!). [It is possible to express the answer in other forms using properties of logarithms ln ab = ln a + ln b and ln a/b = ln a - ln b.]

Related Questions

Navigate

• Browse (All)
• Browse 8
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.