[QUE/SM-03001] SM-PROBLEM

Consider a system of $N$ atoms. Assuming that they can exist in two states only. The ground state having energy zero and an excited state having energy $\epsilon$.

• Find the number of micro states with total energy $U$.
• Write an expression for entropy and using Stirling approximation for the factorial $$ \ln (N!) \approx N \ln N - N$$ find the temperature of the system and hence show that $$ U = \frac{N\epsilon}{1+ e^{\epsilon/kT}}$$
• What is fraction of atoms are in the excited state at very large temperature $(T >> kT)$?