Solutions to Practice Problems on Statistical Thermodynamics

1. The entropy is given by the formula

   

   If we let the energy of the ground state ( ) be 0 (which we can do since we are free to set the zero of energy anywhere we want) then

   

   The partition function is

   

   so the entropy becomes

   

2. 1. Again, let the energy of the ground state be zero. (This is not necessary but considerably simplifies the calculations.) The probability of occupation of level i is

      

      For the ground state, we get P(0)=1/Z while for the excited state, because the level is nondegenerate. Let us say that the probability of occupation of level 1 is significant when the ratio becomes greater than 0.01. (This is arbitrary and we could have picked a different value. The result would have been quantitatively different, but not qualitatively different.) Thus we want to know at what temperature

      

      By taking a natural logarithm of both sides of this equation and rearranging, we get or

      

      This kind of calculation shows that an energy level only contributes significantly to the thermodynamic properties of a substance if its energy is comparable to kT.

   2. The partition function is

      

      The internal energy is therefore

      

      Thus the entropy is