Question

Positional Entropy:

Starting from our equation S = kB ln W, we can work out the entropy associated with the simple position of molecules within a space. Let us assume we have a collection of N molecules, each one of volume Vm located within a volume V such that N*Vm << V.

First, let us think about how many possible ways we can place one molecule within the volume V. There will be V/Vm possible locations. How many ways are there to arrange two molecules in this volume? V/Vm for the first and (V-Vm)/Vm for the second. Here is where our condition becomes important. Since the total volume of molecules is so small, we can approximate the number of possible positions for the second molecule as V/Vm. So for the two together the total number of possible position locations will be V/Vm*V/Vm or (V/Vm)2.

Q6: How many possible position combinations are there for N molecules?

Q7: What is the entropy for N molecules of size Vm located in a volume V? Take W to be the total possible positions.

Q8: What would the entropy be when N molecules are contained in volume V/2? What would the entropy be when N molecules are contained in volume V? This is, in a sense, what we allowed to happen in this section. Is the entropy higher in the second case?

Answer 1

6. Continuing like for two molecules, No. of possible position combinations for Ň molecules V N bill be thee Vm is the volume occupied by one molecule 7. Here S = kalnu N 55-ko ln S = Nkq ln (4)

8 N for volume = V/2, no. of possible positions for single me molecule 2vm V while for N molecules will be am - Entropy would be Szkolen () » S = NKB ln (27) So NkB (lov - ln 2 -ln um In Ist case S = NK Inv-laven) So entropy in second case is lower.