Answer :
The statistical definition of entropy in thermodynamics is given by the formula:
[tex]\[ S = k_B \cdot \ln(W) \][/tex]
where:
- [tex]\( S \)[/tex] is the entropy,
- [tex]\( k_B \)[/tex] is the Boltzmann constant ([tex]\( 1.38 \times 10^{-23} \)[/tex] joules per kelvin),
- [tex]\( W \)[/tex] is the number of possible microstates of the system.
Given:
- [tex]\( W = 4 \)[/tex]
Step-by-step solution:
1. Identify the constants and values given:
- Boltzmann constant, [tex]\( k_B = 1.38 \times 10^{-23} \)[/tex] joules per kelvin.
- Number of microstates, [tex]\( W = 4 \)[/tex].
2. Calculate the natural logarithm of the number of microstates:
- [tex]\( \ln(4) \)[/tex].
3. Multiply the Boltzmann constant by the natural logarithm of the number of microstates:
- [tex]\( S = 1.38 \times 10^{-23} \cdot \ln(4) \)[/tex].
Performing this calculation, we get:
[tex]\[ S \approx 1.91 \times 10^{-23} \text{ joules per kelvin} \][/tex]
Therefore, the correct answer is:
C. [tex]\( 1.91 \times 10^{-23} \)[/tex] joules/kelvin
[tex]\[ S = k_B \cdot \ln(W) \][/tex]
where:
- [tex]\( S \)[/tex] is the entropy,
- [tex]\( k_B \)[/tex] is the Boltzmann constant ([tex]\( 1.38 \times 10^{-23} \)[/tex] joules per kelvin),
- [tex]\( W \)[/tex] is the number of possible microstates of the system.
Given:
- [tex]\( W = 4 \)[/tex]
Step-by-step solution:
1. Identify the constants and values given:
- Boltzmann constant, [tex]\( k_B = 1.38 \times 10^{-23} \)[/tex] joules per kelvin.
- Number of microstates, [tex]\( W = 4 \)[/tex].
2. Calculate the natural logarithm of the number of microstates:
- [tex]\( \ln(4) \)[/tex].
3. Multiply the Boltzmann constant by the natural logarithm of the number of microstates:
- [tex]\( S = 1.38 \times 10^{-23} \cdot \ln(4) \)[/tex].
Performing this calculation, we get:
[tex]\[ S \approx 1.91 \times 10^{-23} \text{ joules per kelvin} \][/tex]
Therefore, the correct answer is:
C. [tex]\( 1.91 \times 10^{-23} \)[/tex] joules/kelvin
