An ideal carnot engine, whose efficiency is 40%, receives heat at 500 K. If its efficiency is 50%, then the intake temperature for the same exhaust temperature is:

Options

A. 800K
B. 900K
C. 600K
D. 700K

Explanation

To solve the problem, we will use the formula for the efficiency of a Carnot engine, which is given by:

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

where:
- \(\eta\) is the efficiency of the engine,
- \(T_1\) is the temperature of the heat source (intake temperature),
- \(T_2\) is the temperature of the heat sink (exhaust temperature).

### Step 1: Identify the known values
From the problem, we know:
- The efficiency of the first Carnot engine (\(\eta_1\)) = 40% = 0.4
- The intake temperature (\(T_1\)) = 500 K
- The efficiency of the second Carnot engine (\(\eta_2\)) = 50% = 0.5

### Step 2: Calculate the exhaust temperature for the first engine
Using the efficiency formula for the first engine:

\[ 0.4 = 1 - \frac{T_2}{500} \]

Rearranging this gives:

\[ \frac{T_2}{500} = 1 - 0.4 \]
\[ \frac{T_2}{500} = 0.6 \]
\[ T_2 = 0.6 \times 500 = 300 \text{ K} \]

### Step 3: Use the exhaust temperature to find the intake temperature for the second engine
Now we will use the exhaust temperature \(T_2 = 300 \text{ K}\) to find the intake temperature for the second engine using its efficiency:

\[ 0.5 = 1 - \frac{300}{T_1'} \]

Rearranging gives:

\[ \frac{300}{T_1'} = 1 - 0.5 \]
\[ \frac{300}{T_1'} = 0.5 \]
\[ T_1' = \frac{300}{0.5} = 600 \text{ K} \]

### Final Answer
The intake temperature for the second Carnot engine, when its efficiency is 50%, is **600 K**.

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