Question

Consider a system of \( N \) particles obeying classical statistics, each of which can have an energy 0 or \( E \). The system is in thermal contact with a reservoir maintained at a temperature \( T \). Let \( k \) denote the Boltzmann constant. Which one of the following statements regarding the total energy \( U \) and the heat capacity \( C \) of the system is correct?

• A. \( U = \frac{NE}{1 + e^{E/kT}} \) and \( C = k \frac{NE}{kT} \)
• B. \( U = kT \left( 1 + e^{E/kT} \right) \) and \( C = k \frac{NE}{kT} \)
• C. \( U = \frac{NE}{1 + e^{E/kT}} \) and \( C = k \frac{NE}{kT^2} \)
• D. \( U = \frac{NE}{1 + e^{E/kT}} \) and \( C = k \frac{NE}{kT^2} \)

Hint:

For systems with discrete energy levels, the total energy and heat capacity can be derived using the Boltzmann distribution and its temperature dependence.

Answer 1

The Correct Option is C

Step 1: Understanding the total energy.
For a system obeying classical statistics with energy levels \( 0 \) or \( E \), the total energy \( U \) is given by the expected energy of the system, which involves the Boltzmann factor \( e^{E/kT} \). The formula for the total energy is \( U = \frac{NE}{1 + e^{E/kT}} \).
Step 2: Heat capacity.
The heat capacity is given by \( C = \frac{dU}{dT} \). Differentiating the expression for \( U \) with respect to \( T \) gives the heat capacity as \( C = k \frac{NE}{kT^2} \).
Step 3: Conclusion.
Thus, the correct answer is option (C) because both the expressions for \( U \) and \( C \) match those derived.