Question

Let's say we have 4 small interacting systems. I've already calculated the number of microstates for each independent system which are as follows; Ohm_1 = 6 Times 10^5, Ohm_2 = 3 Times 10^6, Ohm_3 = 5 Times 10^5, and Ohm_4 = 8 Times 10^6. a) When the 4 systems are interacting what is the total number of states (microstates, Ohm_T) accessible? b) What are S_1, S_2, S_3, and S_4 (entropy of each system independently) in terms of Boltzmann's Constant, k? c) Compute the entropy of the combined system (S_T) two ways (again in units of Boltzmann's constant): 1) By using the answer to part a). 2) By using the answer to part b).

Answer 1

a] Total number of microstates accessible = $\Omega_{total} = \Omega_{1} \Omega_{2} \Omega_{3} \Omega_{4}$

substitute the individual values and find their product.

b] Entropy of the individual system will be:

$\\ S_1 = kln\Omega_{1} \\S_2 = kln\Omega_{2} \\ S_3 = kln\Omega_{3} \\S_4 = kln \Omega_{4}$

substitute the values from above and k = 1.38 x 10^-23 J/K to get S.

c]

1] Use $S_{total} = k ln\Omega _{total}$ to get the total entropy.

2] Use to get total entropy.