Question

Thermodynamics and Statistical Mechanics problem:

A system comprises three distinguishable marbles (red, white, and blue) each of mass m which may be found on any step of a staircase whose steps are a distance h apart, the energy is not specified but the temperature is.

(a) What kind of ensemble is this?

(b) What temperature T of a heat bath will maximize the probability of finding all three marbles on the first step above ground level?

(c) What is this maximum probability?

Answer 1

a) The number of the particles, volume and temperature of the system is specified. Therefore it is a canonical ensemble.

b)

The energy of the n^th step is

The single-particle partition function is

where

T is the temperature and k is the Boltzmann constant

The probability of finding one particle in the first step (n = 1) is

This probability will be maximum when

or

or

or

or

or

or

or

is the temperature that will maximize the probability of finding all three marbles on the first step above ground level.

c) At this temperature the maximum probability of finding one particle on the first step is

The probability of finding all 3 marbles on the first step is