To determine the entropy [tex]\( S \)[/tex] of a system where [tex]\( W = 4 \)[/tex] using the statistical definition of entropy, we use the formula:
[tex]\[ S = k_B \ln(W) \][/tex]
where:
- [tex]\( S \)[/tex] is the entropy,
- [tex]\( k_B \)[/tex] is the Boltzmann constant (approximately [tex]\( 1.38 \times 10^{-23} \)[/tex] joules/kelvin),
- [tex]\( W \)[/tex] is the number of possible microstates.
Let's proceed with the calculation:
1. Identify the given values:
- [tex]\( W = 4 \)[/tex]
- [tex]\( k_B = 1.38 \times 10^{-23} \)[/tex] joules/kelvin
2. Determine the natural logarithm of [tex]\( W \)[/tex]:
[tex]\[ \ln(4) \][/tex]
3. Multiply [tex]\( k_B \)[/tex] by [tex]\( \ln(4) \)[/tex]:
[tex]\[ S = 1.38 \times 10^{-23} \times \ln(4) \][/tex]
Performing this calculation, we find the entropy [tex]\( S \)[/tex]. The result of this calculation is approximately:
[tex]\[ S \approx 1.91 \times 10^{-23} \text{ joules/kelvin} \][/tex]
Therefore, the correct answer is:
C. [tex]\( 1.91 \times 10^{-23} \)[/tex] joules/kelvin
