A monoatomic ideal gas is used in a carnot engine as the working substance. If during the adiabatic expansion the volume of the gas increases from `(V)/(8)` to V, the efficiency of the engine is

Options

A. `0.50`
B. `0.25`
C. `0.75`
D. `0.40`

Answer

To find the efficiency of the Carnot engine using a monoatomic ideal gas during an adiabatic expansion, we can follow these steps:

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

### Step 2: Relate Temperature and Volume in an Adiabatic Process
For an adiabatic process involving an ideal gas, we can use the relation:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
where \( \gamma \) (gamma) is the heat capacity ratio. For a monoatomic ideal gas, \( \gamma = \frac{5}{3} \).

### Step 3: Identify Initial and Final Volumes
Given:
- Initial volume \( V_1 = \frac{V}{8} \)
- Final volume \( V_2 = V \)

### Step 4: Substitute Volumes into the Temperature Relation
From the adiabatic relation, we can express the ratio of temperatures:
\[ \frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{\gamma - 1} \]
Substituting the values:
\[ \frac{T_2}{T_1} = \left(\frac{\frac{V}{8}}{V}\right)^{\frac{5}{3} - 1} = \left(\frac{1}{8}\right)^{\frac{2}{3}} = \frac{1}{4} \]

### Step 5: Substitute Temperature Ratio into Efficiency Formula
Now we can substitute \( \frac{T_2}{T_1} \) into the efficiency formula:
\[ η = 1 - \frac{T_2}{T_1} = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 6: Convert Efficiency to Percentage
To express the efficiency as a percentage:
\[ η = \frac{3}{4} \times 100\% = 75\% \]

### Final Answer
The efficiency of the Carnot engine is \( 75\% \).

---