Chem C130/MCB C100A. Fall 2010. Page 1
Problem Set 7 UC Berkeley
Hand in the five problems marked with a * on Thursday, October 21.
ANSWER KEY
1. What is the difference between an energy distribution and a microstate? Explain using
simple diagrams.
2.* Consider a system of three identical and independent molecules, with energy levels
that have values 0,1,2,3, …(arbitrary units of energy).
Consider two states of the system, A and B, shown below. States A and B have equal
energy (6 units). State A has one molecule in level 4 and two in level 1. State B has 1
molecule each in levels 1, 2 and 3. The multiplicity, W, of each state is given by
N! / (n1! x n2! x n3! …x nj! x….), where N is the total number of molecules and nj is the
number of molecules in state j.
(i) Calculate the relative probability of observing the molecules in configuration
A over configuration B.
(ii) What is the change in entropy in going from state A to state B (assume kb=1).
(iii) Assume that the 3 molecules in the system have access to only the 5 energy
levels shown above. When the total energy of all 3 molecules is increased to
, Chem C130/MCB C100A. Fall 2010. Page 2
Problem Set 7 UC Berkeley
the maximum possible value, explain whether the entropy increases or
decreases with respect to state B.
(iv) Next, assume that the system has an infinite number of possible energy levels
of increasing energy. In this case, when the total energy of the 3 molecules is
increased does the entropy always increase, or does it sometimes increase and
sometimes decrease? Explain your answer.
3. A more accurate version of Stirling’s approximation is given below:
ln N!= N ln N − N + ln( 2πN)
Estimate, by trial and error, how big N has to be for the error introduced by neglecting the
last term to be less than 1% of the value of ln N!
Problem Set 7 UC Berkeley
Hand in the five problems marked with a * on Thursday, October 21.
ANSWER KEY
1. What is the difference between an energy distribution and a microstate? Explain using
simple diagrams.
2.* Consider a system of three identical and independent molecules, with energy levels
that have values 0,1,2,3, …(arbitrary units of energy).
Consider two states of the system, A and B, shown below. States A and B have equal
energy (6 units). State A has one molecule in level 4 and two in level 1. State B has 1
molecule each in levels 1, 2 and 3. The multiplicity, W, of each state is given by
N! / (n1! x n2! x n3! …x nj! x….), where N is the total number of molecules and nj is the
number of molecules in state j.
(i) Calculate the relative probability of observing the molecules in configuration
A over configuration B.
(ii) What is the change in entropy in going from state A to state B (assume kb=1).
(iii) Assume that the 3 molecules in the system have access to only the 5 energy
levels shown above. When the total energy of all 3 molecules is increased to
, Chem C130/MCB C100A. Fall 2010. Page 2
Problem Set 7 UC Berkeley
the maximum possible value, explain whether the entropy increases or
decreases with respect to state B.
(iv) Next, assume that the system has an infinite number of possible energy levels
of increasing energy. In this case, when the total energy of the 3 molecules is
increased does the entropy always increase, or does it sometimes increase and
sometimes decrease? Explain your answer.
3. A more accurate version of Stirling’s approximation is given below:
ln N!= N ln N − N + ln( 2πN)
Estimate, by trial and error, how big N has to be for the error introduced by neglecting the
last term to be less than 1% of the value of ln N!
