A copper block of mass 'm' at temperature `'T_(1)'` is kept in the open atmosphere at temperature `'T_(2)'` where `T_(2) gt T_(1)`. The variation of entropy of the copper block with time is best illustrated by

Options

A.
B.
C.
D.

Answer

To solve the problem of how the entropy of a copper block changes over time when it is placed in an environment with a higher temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- We have a copper block with mass 'm' at an initial temperature \( T_1 \).
- The block is placed in an open atmosphere at a higher temperature \( T_2 \) (where \( T_2 > T_1 \)).
- The initial entropy of the copper block at time \( t = 0 \) is denoted as \( S_0 \).

**Hint**: Remember that entropy is a measure of disorder or randomness in a system, and it is affected by temperature changes.

2. **Heat Transfer**:
- Since \( T_2 > T_1 \), heat will flow from the surroundings (at temperature \( T_2 \)) into the copper block (at temperature \( T_1 \)).
- This heat transfer will cause the temperature of the copper block to increase over time.

**Hint**: Consider the direction of heat flow and how it affects the temperature of the block.

3. **Change in Entropy**:
- As the copper block absorbs heat, its entropy will increase. The relationship between heat transfer and entropy change is given by:
\[ \Delta S = \frac{Q}{T} \]
- Here, \( Q \) is the heat absorbed by the copper block, and \( T \) is the temperature at which the heat transfer occurs. As the temperature of the block increases, the entropy change will also increase.

**Hint**: Think about how entropy changes with temperature and how it is related to heat transfer.

4. **Equilibrium Condition**:
- The process continues until the temperature of the copper block equals the temperature of the surroundings (\( T_1 = T_2 \)). At this point, the system reaches thermal equilibrium.
- At equilibrium, there is no further heat transfer, and thus the entropy of the copper block will remain constant.

**Hint**: Identify the point at which the system reaches equilibrium and how that affects the entropy.

5. **Graphical Representation**:
- The graph of entropy (y-axis) versus time (x-axis) will start at a finite value \( S_0 \) and will increase continuously as the block absorbs heat until it reaches equilibrium.
- After reaching equilibrium, the entropy will remain constant, indicating no further change.

**Hint**: Visualize how the graph would look based on the behavior of the system over time.

### Conclusion:
The correct illustration of the variation of entropy of the copper block with time would show an initial increase in entropy that continues until the block reaches thermal equilibrium with the surrounding atmosphere, after which the entropy remains constant.

### Final Answer:
The correct option is **C**, which illustrates the increase in entropy over time until equilibrium is reached.