Boltzmann’s formula: S=K\hW
, where W is the number of microstates. The number of microstates indicates a) The magnitude of total energy in a group of particles b) The range of energy in a system of particles c) Number of energy levels with same energy values d) How many different ways a system of particles can arrange for a same constant energy
The Correct Answer and Explanation is:
Correct Answer: d) How many different ways a system of particles can arrange for a same constant energy
Explanation:
Boltzmann’s formula is a fundamental equation in statistical mechanics and thermodynamics, written as:S=klnWS = k \ln WS=klnW
Where:
- SSS is the entropy of a system,
- kkk is the Boltzmann constant (1.38×10−23 J/K1.38 \times 10^{-23} \, \text{J/K}1.38×10−23J/K),
- WWW is the number of microstates corresponding to the macroscopic state.
A microstate refers to a specific microscopic configuration of a system — that is, the detailed arrangement of individual particles (like atoms or molecules), including their positions and momenta. In contrast, a macrostate is defined by macroscopic properties such as temperature, pressure, and volume. For a given macrostate, there can be many possible microstates.
The variable WWW in Boltzmann’s formula represents the total number of microstates compatible with the macroscopic conditions of the system. The higher the value of WWW, the greater the number of ways the system’s particles can be arranged without changing the overall energy or other macroscopic properties.
This means that entropy, SSS, is a measure of disorder or randomness in a system. When a system has more microstates — that is, more possible arrangements of its particles — it has higher entropy.
Thus, Boltzmann’s formula shows that entropy increases with the number of microstates. Importantly, these arrangements all correspond to the same total energy for the macrostate, not different energy levels. Therefore, the correct interpretation of WWW is the number of distinct microscopic arrangements that a system can assume while maintaining the same overall energy.
Hence, the best answer choice is:
d) How many different ways a system of particles can arrange for a same constant energy.
