Question

Consider the distribution of 1000 ideal gas molecules among three bulbs (A, B, and C) of equal volume. For each of the following states, determine the number of ways (W) that the state can be achieved, and use Boltzmann’s equation to calculate the entropy of the state.


a) 1000 molecules in bulb A

b) 1000 molecules randomly distributed among bulbs A, B, and C

Answer 1

Solution.

The number of ways that N molecules can occupy B bulbs is W= B^N

S=k$\sdot$ lnW, k=1.38$\sdot$ 10^-23J/K.

a) 1000 molecules in bulb A

W=B^N=1¹⁰⁰⁰=1;

S=( 1.38⋅ 10^-23J/K )$\sdot$ ln(1)=0;

b)1000 molecules randomly distributed among bulbs A, B, and C

W=3¹⁰⁰⁰;

S=( ( 1.38⋅ 10^-23J/K )⋅ ln(3¹⁰⁰⁰) =( 1.38⋅ 10^-23J/K )⋅1000$\sdot$ ln(3)=1.5$\sdot$ 10^-20J/K.

Answer:

a) W=1; S=0;

b) W=3¹⁰⁰⁰; S=1.5⋅ 10^-20J/K.