One mole of an ideal monoatomic gas at temperature `T_0` expands slowly according to the law `p/V` = constant. If the final temperature is `2T_0`, heat supplied to the gas is

Options

A. a.`2RT_(0)`
B. b.`RT_(0)`
C. c.`(3)/(2)RT_(0)`
D. d.`(1)/(2)RT_(0)`

Explanation

To solve the problem, we need to determine the heat supplied to one mole of an ideal monoatomic gas that expands slowly according to the law \( \frac{p}{V} = \text{constant} \), with the initial temperature \( T_0 \) and final temperature \( 2T_0 \).

### Step-by-Step Solution:

1. **Identify the Process**:
The given process is described by the equation \( \frac{p}{V} = \text{constant} \). This can be rewritten as:
\[ pV^{-1} = \text{constant} \]
This indicates a polytropic process where \( n = -1 \).

**Hint**: Recognize that the relationship \( pV^n = \text{constant} \) helps identify the type of thermodynamic process.

2. **Determine Specific Heat**:
The specific heat \( C \) for a polytropic process is given by:
\[ C = C_v + \frac{R}{1 - n} \]
For a monoatomic gas, the specific heat at constant volume \( C_v = \frac{3R}{2} \). Since \( n = -1 \):
\[ C = \frac{3R}{2} + \frac{R}{1 - (-1)} = \frac{3R}{2} + \frac{R}{2} = 2R \]

**Hint**: Use the specific heat formula for polytropic processes to find the effective specific heat.

3. **Calculate the Change in Temperature**:
The change in temperature \( \Delta T \) is:
\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 2T_0 - T_0 = T_0 \]

**Hint**: Always calculate the change in temperature by subtracting the initial temperature from the final temperature.

4. **Calculate the Heat Supplied**:
The heat supplied \( \Delta Q \) can be calculated using the formula:
\[ \Delta Q = mC \Delta T \]
Here, \( m = 1 \) mole, \( C = 2R \), and \( \Delta T = T_0 \):
\[ \Delta Q = 1 \cdot (2R) \cdot T_0 = 2RT_0 \]

**Hint**: Remember the formula for heat transfer in terms of mass, specific heat, and temperature change.

5. **Final Result**:
Therefore, the heat supplied to the gas is:
\[ \Delta Q = 2RT_0 \]

### Conclusion:
The heat supplied to the gas during the expansion is \( 2RT_0 \).