# Maxwell-Boltzmann Distribution

1. Consider a collection of N noninteracting atoms with a single excited state at energy E. Assume the atoms obey the Maxwell-Boltzmann statistics, and take both the ground state and the excited state to be nondegenerate. a.) At temperature T, what is the ratio of the number of atoms in the excited state to the number in the ground state? b.) What is the average energy of an atom in this system? c.) What is the total energy of this system? d.) What is the heat capacity of this system?

2. f(E)=(A^-1)*[e^(-E/kT)]
p(E)=g(E)f(E)
p(E2)/P(E1)=[g(E2)/g(E1)]*{e^[-(E2-E1)/kT]}

3. The answer to part "a" is: e^(-E/kT) which I understand since both the ground state and the excited states are nondegenerate.
The answer to part "b" is: E/[1+e^(E/kT)]. This is the part I do not understand. Can you please explain why this is the answer and how to get this?

## Answers and Replies

nrqed
Science Advisor
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1. Consider a collection of N noninteracting atoms with a single excited state at energy E. Assume the atoms obey the Maxwell-Boltzmann statistics, and take both the ground state and the excited state to be nondegenerate. a.) At temperature T, what is the ratio of the number of atoms in the excited state to the number in the ground state? b.) What is the average energy of an atom in this system? c.) What is the total energy of this system? d.) What is the heat capacity of this system?

2. f(E)=(A^-1)*[e^(-E/kT)]
p(E)=g(E)f(E)
p(E2)/P(E1)=[g(E2)/g(E1)]*{e^[-(E2-E1)/kT]}

3. The answer to part "a" is: e^(-E/kT) which I understand since both the ground state and the excited states are nondegenerate.
The answer to part "b" is: E/[1+e^(E/kT)]. This is the part I do not understand. Can you please explain why this is the answer and how to get this?

the average energy is equal to

$$E_0 P(E_0) + E_1 P(E_1)$$

where the probability of obtaining each energy is simply

$$P(E_i) = \frac{e^{-E_i /kT}}{e^{-E_0/kT} + e^{-E_1/kT}}$$

Try this