A carnot engine takes 300 cal. of heat at 500 k and rejects 150 cal of heat to the sink. The temperature (in k) of sink is __________.

Explanation

To find the temperature of the sink (T2) in a Carnot engine, we can use the relationship between the heat absorbed (Q1), the heat rejected (Q2), and the temperatures of the heat source (T1) and the sink (T2).

### Step-by-Step Solution:

1. **Identify Given Values:**
- Heat absorbed (Q1) = 300 cal
- Heat rejected (Q2) = 150 cal
- Temperature of the heat source (T1) = 500 K

2. **Use the Efficiency Formula:**
The efficiency (η) of a Carnot engine can be expressed as:
\[ \eta = 1 - \frac{Q2}{Q1} \]
This can also be expressed in terms of temperatures:
\[ \eta = \frac{T1 - T2}{T1} \]

3. **Set Up the Equation:**
From the two expressions for efficiency, we can equate them:
\[ 1 - \frac{Q2}{Q1} = \frac{T1 - T2}{T1} \]

4. **Substitute the Known Values:**
Substitute Q1, Q2, and T1 into the equation:
\[ 1 - \frac{150}{300} = \frac{500 - T2}{500} \]

5. **Simplify the Left Side:**
Calculate the left side:
\[ 1 - 0.5 = 0.5 \]
So, we have:
\[ 0.5 = \frac{500 - T2}{500} \]

6. **Cross Multiply to Solve for T2:**
Cross multiplying gives:
\[ 0.5 \times 500 = 500 - T2 \]
\[ 250 = 500 - T2 \]

7. **Rearrange to Find T2:**
Rearranging the equation gives:
\[ T2 = 500 - 250 \]
\[ T2 = 250 \text{ K} \]

### Final Answer:
The temperature of the sink (T2) is **250 K**.