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 solve the problem of finding the temperature of the sink (T2) for a Carnot engine, we can use the relationship between the heat absorbed (Q1), the heat rejected (Q2), and the temperatures of the hot and cold reservoirs (T1 and T2). The formula we will use is:

\[ \frac{Q_2}{Q_1} = \frac{T_2}{T_1} \]

### Step-by-Step Solution:

1. **Identify the given values:**
- Heat absorbed by the engine (Q1) = 300 cal
- Heat rejected by the engine (Q2) = 150 cal
- Temperature of the hot reservoir (T1) = 500 K

2. **Set up the equation using the Carnot relation:**
\[ \frac{Q_2}{Q_1} = \frac{T_2}{T_1} \]
Substituting the known values:
\[ \frac{150 \text{ cal}}{300 \text{ cal}} = \frac{T_2}{500 \text{ K}} \]

3. **Simplify the left side of the equation:**
\[ \frac{150}{300} = \frac{1}{2} \]
So, we can rewrite the equation as:
\[ \frac{1}{2} = \frac{T_2}{500} \]

4. **Cross-multiply to solve for T2:**
\[ T_2 = 500 \times \frac{1}{2} \]

5. **Calculate T2:**
\[ T_2 = 250 \text{ K} \]

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

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