In a carnot engine, for `eta=1`, which of the following is true? (Symbols have their usual meaning)

Options

A. `T_(1)=T_(2)`
B. `T_(1)=0`
C. `T_(2)=0`
D. `eta` is independent of `T_(1) and T_(2)`.

Answer

To solve the problem regarding the Carnot engine and its efficiency (η), we start by recalling the formula for the efficiency of a Carnot engine:

\[ \eta = 1 - \frac{T_2}{T_1} \]

where:
- \( T_1 \) is the absolute temperature of the hot reservoir,
- \( T_2 \) is the absolute temperature of the cold reservoir.

Given that \( \eta = 1 \), we can substitute this value into the efficiency equation:

\[ 1 = 1 - \frac{T_2}{T_1} \]

To isolate \( \frac{T_2}{T_1} \), we rearrange the equation:

\[ 1 - 1 = -\frac{T_2}{T_1} \]

This simplifies to:

\[ 0 = -\frac{T_2}{T_1} \]

From this, we can conclude that:

\[ \frac{T_2}{T_1} = 0 \]

This implies that \( T_2 = 0 \) (the cold reservoir temperature must be absolute zero).

Now, let's analyze the options provided in the question:

1. \( T_1 = T_2 \)
2. \( T_1 = 0 \)
3. \( T_2 = 0 \)
4. \( \eta \) is independent of \( T_1 \) and \( T_2 \)

From our conclusion, we see that the only true statement is:

**\( T_2 = 0 \)**

Thus, the answer to the question is:

**\( T_2 = 0 \)**