We are tasked with finding the entropy of state 3 given the following information:

• The multiplicities of the macrostates are related as: $\Omega_1 = 2 \Omega_2 \quad \text{and} \quad \Omega_1 = 4 \Omega_3$
• The entropy of state 1, $S_1$, is 20 J/K.
• We need to find the entropy of state 3, $S_3$, in J/K.

Step 1: Recall the formula for entropy

The entropy $S$ of a system is related to its multiplicity $\Omega$ by the following equation: $S = k_B \ln(\Omega)$ where $k_B$ is the Boltzmann constant. We can compare entropies using the ratio of multiplicities.

Step 2: Express multiplicities in terms of $\Omega_2$

From the problem, we know: $\Omega_1 = 2 \Omega_2 \quad \text{and} \quad \Omega_1 = 4 \Omega_3$ Thus, we can express the multiplicities as: $\Omega_2 = \frac{\Omega_1}{2}, \quad \Omega_3 = \frac{\Omega_1}{4}$

Step 3: Relating entropies

The entropy of state 1 is given as $S_1 = 20 \, \text{J/K}$, and since entropy is proportional to the logarithm of multiplicity, we can write: $S_1 = k_B \ln(\Omega_1)$ For state 3: $S_3 = k_B \ln(\Omega_3) = k_B \ln\left(\frac{\Omega_1}{4}\right)$

Using the logarithmic identity $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$, we have: $S_3 = k_B \ln(\Omega_1) - k_B \ln(4)$ Substitute $S_1 = k_B \ln(\Omega_1)$ and simplify: $S_3 = S_1 - k_B \ln(4)$

Step 4: Calculate $S_3$

We can now calculate $S_3$. The Boltzmann constant $k_B$ is approximately $1.38 \times 10^{-23} \, \text{J/K}$. However, since entropy is given in macroscopic units, we are more interested in the natural logarithmic factor. Using the fact that: $\ln(4) \approx 1.386$ we get: $S_3 = 20 - (k_B \times 1.386)$

Thus, subtract the logarithmic contribution to get $S_3$: $S_3 = 20 - (1.386 \cdot 1) \approx 20 - 1.386 = 18.614 \, \text{J/K}$

Final Answer:

The entropy of state 3 is approximately: $S_3 \approx 18.6 \, \text{J/K}$

Would you like further details or have any questions?

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Here are 5 questions to expand on the topic:

1. How does entropy change with the number of microstates in a system?
2. What role does the Boltzmann constant play in the entropy equation?
3. How would the entropy change if the multiplicity of state 3 were increased?
4. How is entropy related to the second law of thermodynamics?
5. Can we calculate entropy for a system with continuous states instead of discrete ones?

Tip: When comparing entropies, logarithmic differences in multiplicities help quantify how systems distribute over possible microstates.