[QUE/SM-03007] 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 \(\epsilon\).

1. Find the number of micro states with total energy \(U\).
2. 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 \((T >> kT)\)?