A carnot engine has efficiency `1//5` . Efficiency becomes `1//3` when temperature of sink is decreased by 50 K What is the temperature of sink ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Efficiency of a Carnot Engine
The efficiency (η) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_2}{T_1} \]
where \(T_1\) is the temperature of the heat source (hot reservoir) and \(T_2\) is the temperature of the sink (cold reservoir).

### Step 2: Set Up the Initial Condition
From the problem, we know that the efficiency is initially \( \frac{1}{5} \). Therefore, we can write:
\[ \frac{1}{5} = 1 - \frac{T_2}{T_1} \]
Rearranging this gives:
\[ \frac{T_2}{T_1} = 1 - \frac{1}{5} = \frac{4}{5} \]
Thus, we can express \(T_1\) in terms of \(T_2\):
\[ T_1 = \frac{5}{4} T_2 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Condition After Decreasing the Sink Temperature
When the temperature of the sink is decreased by 50 K, the new efficiency becomes \( \frac{1}{3} \). The new temperature of the sink is \(T_2 - 50\). Therefore, we can write:
\[ \frac{1}{3} = 1 - \frac{T_2 - 50}{T_1} \]
Rearranging this gives:
\[ \frac{T_2 - 50}{T_1} = 1 - \frac{1}{3} = \frac{2}{3} \]
Thus, we can express \(T_1\) in terms of \(T_2\) again:
\[ T_1 = \frac{3}{2}(T_2 - 50) \quad \text{(Equation 2)} \]

### Step 4: Equate the Two Expressions for \(T_1\)
Now we have two expressions for \(T_1\):
1. \(T_1 = \frac{5}{4} T_2\)
2. \(T_1 = \frac{3}{2}(T_2 - 50)\)

Setting these equal to each other:
\[ \frac{5}{4} T_2 = \frac{3}{2}(T_2 - 50) \]

### Step 5: Solve for \(T_2\)
Cross-multiplying to eliminate the fractions:
\[ 5 \cdot 2 T_2 = 3 \cdot 4 (T_2 - 50) \]
This simplifies to:
\[ 10 T_2 = 12 T_2 - 600 \]
Rearranging gives:
\[ 10 T_2 - 12 T_2 = -600 \]
\[ -2 T_2 = -600 \]
\[ T_2 = 300 \, \text{K} \]

### Step 6: Find \(T_1\)
Now substitute \(T_2 = 300\) K back into Equation 1 to find \(T_1\):
\[ T_1 = \frac{5}{4} \times 300 = 375 \, \text{K} \]

### Final Answer
The temperature of the sink \(T_2\) is:
\[ \boxed{300 \, \text{K}} \]