The efficiency of a Carnot's engine at a particular source and sink temperature is `(1)/(2)` .When the sink temperature is reduced by `100^(@)C`, the engine efficiency, becomes `(2)/(3)`. Find the source temperature.

Options

A. 600K
B. 300K
C. 500K
D. 550K

Answer

To solve the problem, we need to find the source temperature \( T_1 \) of a Carnot engine given its efficiencies at two different sink temperatures. Let's break it down step by step.

### Step 1: Set up the equations for efficiency
The efficiency \( \eta \) of a Carnot engine is given by the formula:

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

Where:
- \( T_1 \) is the source temperature.
- \( T_2 \) is the sink temperature.

From the problem, we know that the efficiency at the first condition is \( \frac{1}{2} \). Thus, we can write:

\[ \frac{1}{2} = \frac{T_1 - T_2}{T_1} \]

### Step 2: Rearranging the equation
Rearranging the equation gives us:

\[ 1 - \frac{T_2}{T_1} = \frac{1}{2} \]

This simplifies to:

\[ \frac{T_2}{T_1} = \frac{1}{2} \]

Let this be Equation (1).

### Step 3: Modify the sink temperature
The problem states that when the sink temperature \( T_2 \) is reduced by \( 100^\circ C \), the new sink temperature \( T_2' \) becomes:

\[ T_2' = T_2 - 100 \]

### Step 4: Set up the new efficiency equation
The new efficiency \( \eta' \) is given as \( \frac{2}{3} \). Thus, we can write:

\[ \frac{2}{3} = \frac{T_1 - T_2'}{T_1} \]

Substituting \( T_2' \):

\[ \frac{2}{3} = \frac{T_1 - (T_2 - 100)}{T_1} \]

### Step 5: Rearranging the new equation
This can be rewritten as:

\[ \frac{2}{3} = \frac{T_1 - T_2 + 100}{T_1} \]

This simplifies to:

\[ 1 - \frac{T_2 - 100}{T_1} = \frac{2}{3} \]

### Step 6: Further simplification
Rearranging gives:

\[ \frac{T_2 - 100}{T_1} = \frac{1}{3} \]

### Step 7: Substitute from Equation (1)
From Equation (1), we know \( \frac{T_2}{T_1} = \frac{1}{2} \). Therefore, we can express \( T_2 \) in terms of \( T_1 \):

\[ T_2 = \frac{1}{2} T_1 \]

Substituting this into our new equation gives:

\[ \frac{\frac{1}{2} T_1 - 100}{T_1} = \frac{1}{3} \]

### Step 8: Cross-multiply and solve for \( T_1 \)
Cross-multiplying leads to:

\[ \frac{1}{2} T_1 - 100 = \frac{1}{3} T_1 \]

Rearranging gives:

\[ \frac{1}{2} T_1 - \frac{1}{3} T_1 = 100 \]

Finding a common denominator (which is 6) gives:

\[ \frac{3}{6} T_1 - \frac{2}{6} T_1 = 100 \]

This simplifies to:

\[ \frac{1}{6} T_1 = 100 \]

### Step 9: Solve for \( T_1 \)
Multiplying both sides by 6 gives:

\[ T_1 = 600 \text{ K} \]

### Conclusion
The source temperature \( T_1 \) is \( 600 \text{ K} \).