Question

Boltzmann's equation is:

• A. \( S = kT \ln P \)
• B. \( S = k \ln W \)
• C. \( S = \frac{PV}{T} \)
• D. \( S = \frac{Q}{T} \)

Hint:

Remember that Boltzmann's equation links the microscopic state (number of microstates \( W \)) to the macroscopic property of entropy (\( S \)) in statistical mechanics. It forms the foundation of entropy in terms of probability and statistical ensembles.

Answer 1

The Correct Option is B

Boltzmann's equation relates the entropy \( S \) to the number of accessible microstates of a system. The equation is expressed as: \[ S = k \ln W \] where:
- \( S \) is the entropy of the system,
- \( k \) is the Boltzmann constant, and
- \( W \) is the number of possible microstates (also called the "thermodynamic probability").
This equation is fundamental in statistical mechanics, linking the macroscopic property of entropy to the microscopic behavior of particles in a system. The other options listed are incorrect representations of entropy in thermodynamics:
- Option (1) \( S = kT \ln P \) is related to a form of entropy in thermodynamics but is not Boltzmann's equation.
- Option (3) \( S = \frac{PV}{T} \) is a form of the equation for the entropy of an ideal gas but does not represent Boltzmann's equation.
- Option (4) \( S = \frac{Q}{T} \) is the equation for the heat added to a system divided by the temperature, but not directly Boltzmann's equation.
Therefore, the correct answer is Option (2) \( S = k \ln W \).