Question

Imagine a gas of N spherical particles that are free to distribute throughout either a large volume or a small one. Why does the gas spread out into a large volume? The tendency to spread out is pressure and can be explained in terms of multiplicity or increased probability. To illustrate this, let's create a simpler model that we can visualize, solve exactly, and will illustrate the essence of the problem without the mathematical complexity. For each of the following, calculate the number of distinguishable arrangements of the vacancies and occupancies of the Nparticles in the M sites.

1. Consider N=3 spherical particles spread out inM=5 equal size boxes that are arranged in a line. The boxes are just big enough to hold at most one particle.

2. Now consider N=3 spherical particles spread out in M=4 boxes arranged in a line.

3. Finally consider N=3 particles spread in onlyM=3 boxes.

4. Based on these results why does a gas spread out into a larger volume?

Answer 1

Case1: when N=3 particles and M =5 boxes are present and each box is big enough to accomodate at most 1 particles

then the no of ways to arrange  these 3 particles in 5 boxes is 5C3 = 10

Case 2 : when N= 3 and M = 4

then the no of ways to arrange these 3 particles in 4 boxes is 4C3 = 4

Case 3 : when N =3 and M = 3

then the no of ways to arrange these 3 particles in 3 boxes is 3C3 = 1

since the no of ways of arranging the particles is higher in case 1 , the probability of

spreading out in case 1 will also be higher which implies gas always spread out into

a larger volume.