A gaseous system undergoes a change in temperature and volume. What is the entropy change for a particle in this system if the final number of microstates is 0.503 times that of the initial number of microstates?
-9.48E-24K/J
The entropy change (ΔS) for a particle in a gaseous system undergoing a change in temperature and volume can be determined using Boltzmann's equation:
ΔS = k * ln(W_f / W_i)
Where:
- ΔS is the entropy change
- k is Boltzmann's constant (1.38 × 10^(-23) J/K)
- W_f is the final number of microstates
- W_i is the initial number of microstates
Given that the final number of microstates (W_f) is 0.503 times that of the initial number of microstates (W_i), we can rewrite it as:
ΔS = k * ln(0.503 * W_i / W_i)
Simplifying further:
ΔS = k * ln(0.503)
Now we just need to calculate the natural logarithm of 0.503. Using a scientific calculator or mathematical software, we find:
ln(0.503) ≈ -0.686
Substituting this value back into the equation:
ΔS ≈ 1.38 × 10^(-23) J/K * (-0.686) ≈ -9.51 × 10^(-24) J/K
Therefore, the entropy change for a particle in this system is approximately -9.51 × 10^(-24) J/K.
To calculate the entropy change for a particle in a gaseous system, we need to consider the ratio of the final number of microstates (Ω2) to the initial number of microstates (Ω1). The entropy change (ΔS) can be obtained using the formula:
ΔS = k * ln(Ω2/Ω1)
where k is the Boltzmann's constant.
Given that the final number of microstates is 0.503 times that of the initial number of microstates (Ω2 = 0.503 * Ω1), we can substitute this value into the formula:
ΔS = k * ln(0.503 * Ω1/Ω1)
ΔS = k * ln(0.503)
Thus, the entropy change for a particle in this gaseous system is ΔS = k * ln(0.503).