Consider black body radiation contained in a cavity whose walls are at temperature T. The radiation is in equilibrium with the walls of the cavity. If the temperature of the walls is increased to 2T and the radiation is allowed to come to equilibrium at the new temperature, the entropy of the radiation increases by a factor of:a)8b)16c)4d)2Correct answer is option 'A'. Can you explain this answer?

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Consider black body radiation contained in a cavity whose walls are at temperature T. The radiation is in equilibrium with the walls of the cavity. If the temperature of the walls is increased to 2T and the radiation is allowed to come to equilibrium at the new temperature, the entropy of the radiation increases by a factor of:

a)
8
b)
16
c)
4
d)
2

Correct answer is option 'A'. Can you explain this answer?

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Consider black body radiation contained in a cavity whose walls are at...

Explanation:
When the radiation is in equilibrium with the walls of the cavity, it is in thermal equilibrium and the entropy is maximized. Let's consider the initial equilibrium state of the radiation at temperature T.

Step 1: Initial State
- The radiation is in thermal equilibrium with the walls of the cavity at temperature T.
- The entropy of the radiation in this state is S1.

Step 2: Increasing the Temperature of the Walls
- The temperature of the walls is increased to 2T.
- The radiation is allowed to come to equilibrium with the new temperature.
- The entropy of the radiation in this new equilibrium state is S2.

Step 3: Comparing Entropies
- We want to compare the entropies S1 and S2.
- Entropy is an extensive property, which means it is proportional to the amount of the system.
- When the temperature is doubled, the radiation can occupy twice as many energy levels, resulting in a doubling of the number of microstates available to the radiation.
- Since the entropy is proportional to the logarithm of the number of microstates, doubling the number of microstates will result in an increase in entropy by a factor of 2.

Step 4: Repeating the Process
- If we repeat the process and double the temperature again, the number of microstates will again double, leading to another factor of 2 increase in entropy.
- This process can be repeated until the temperature reaches 2T.

Step 5: Calculating the Total Increase in Entropy
- Since the process of doubling the temperature and increasing entropy can be repeated 3 times (T to 2T, 2T to 4T, and 4T to 8T), the total increase in entropy is given by (2^3) = 8.
- Therefore, the entropy of the radiation increases by a factor of 8 when the temperature of the walls is increased to 2T.

Hence, the correct answer is option 'A' - 8.

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Consider black body radiation contained in a cavity whose walls are at...

For black body radiation energy is given as,

Where, σ is the Stefan's constant.
The entropy of the radiation is given as,

Now, the temperature of the walls is increased to 2T, from equation (1) entropy is,

From equation (1) and (2),

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