A carnot engine has the same efficiency between (i) 100 K and 500 K and (ii) T and 900 K. Find T.

Options

A. 180 K
B. 90 K
C. 270 K
D. 360 K

Answer

To solve the problem, we need to find the temperature \( T \) such that the efficiency of a Carnot engine operating between two sets of temperatures is the same. The two sets of temperatures are:

1. Between \( 100 \, K \) (cold reservoir) and \( 500 \, K \) (hot reservoir).
2. Between \( T \) (cold reservoir) and \( 900 \, K \) (hot reservoir).

### Step-by-Step Solution:

1. **Write the formula for efficiency of a Carnot engine:**
The efficiency \( \eta \) of a Carnot engine is given by:
\[ \eta = 1 - \frac{T_c}{T_h} \]
where \( T_c \) is the temperature of the cold reservoir and \( T_h \) is the temperature of the hot reservoir.

2. **Calculate the efficiency for the first set of temperatures (100 K and 500 K):**
Here, \( T_c = 100 \, K \) and \( T_h = 500 \, K \).
\[ \eta_1 = 1 - \frac{100}{500} = 1 - 0.2 = 0.8 \]

3. **Set up the efficiency for the second set of temperatures (T and 900 K):**
Here, \( T_c = T \) and \( T_h = 900 \, K \).
\[ \eta_2 = 1 - \frac{T}{900} \]

4. **Set the efficiencies equal to each other:**
Since the efficiencies are the same, we can set them equal:
\[ 0.8 = 1 - \frac{T}{900} \]

5. **Solve for \( T \):**
Rearranging the equation gives:
\[ \frac{T}{900} = 1 - 0.8 \]
\[ \frac{T}{900} = 0.2 \]
Multiplying both sides by 900:
\[ T = 0.2 \times 900 = 180 \, K \]

### Final Answer:
The value of \( T \) is \( 180 \, K \).

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