Question

A particle can occupy two possible states E1 and E2, where E2>E1. If the particle is in equilibrium with a thermal reservoir at temperature T, write an expression for the probability of finding the particle in state 2.

Answer 1

Solution:

The probability of a particle in a particular microstate is directly proportional to exp (- E_i / kT ) where k is the Boltzmann's constant and T is the temperature of that state , E_i is the energy of that state

Since probability can be normalized we have where p_iis the probability of finding the particle in each state with energy E_i

Therefore,

where Z is the partition function of the system

which is defined as     where n is the total number of microstates in the system

Here,   

Therefore, the probability of finding the particle in state 2 with energy E₂ is,

where