Temperature of two moles of a monoatomic gas is increased by `300 K` in the process `p prop V`. Find
(a) molar heat capacity of the gas in the given process
(b) heat required in the given process.

Explanation

To solve the problem step by step, we will break it down into two parts: (a) finding the molar heat capacity of the gas in the given process, and (b) calculating the heat required in the process.

### Step-by-Step Solution

**(a) Finding the Molar Heat Capacity:**

1. **Understanding the Process:**
The question states that the process follows the relation \( P \propto V \). This implies that \( \frac{P}{V} = \text{constant} \) or \( PV^{-1} = \text{constant} \).

2. **Identifying the Value of \( x \):**
From the relation \( PV^{-x} = \text{constant} \), we can identify that \( x = -1 \).

3. **Using the Formula for Molar Heat Capacity:**
The molar heat capacity \( C \) can be expressed as:
\[ C = C_V + \frac{R}{1 - x} \]
Substituting \( x = -1 \):
\[ C = C_V + \frac{R}{1 - (-1)} = C_V + \frac{R}{2} \]

4. **Finding \( C_V \) for a Monoatomic Gas:**
For a monoatomic ideal gas, the molar heat capacity at constant volume \( C_V \) is given by:
\[ C_V = \frac{3R}{2} \]

5. **Calculating the Molar Heat Capacity:**
Now substituting \( C_V \) into the equation for \( C \):
\[ C = \frac{3R}{2} + \frac{R}{2} = \frac{4R}{2} = 2R \]

**(b) Calculating the Heat Required:**

6. **Using the Heat Formula:**
The heat \( Q \) required can be calculated using the formula:
\[ Q = nC\Delta T \]
where \( n \) is the number of moles, \( C \) is the molar heat capacity, and \( \Delta T \) is the change in temperature.

7. **Substituting the Values:**
We have:
- \( n = 2 \) moles
- \( C = 2R \)
- \( \Delta T = 300 \, K \)

Therefore:
\[ Q = 2 \times (2R) \times 300 = 1200R \]

### Final Answers:
- (a) The molar heat capacity of the gas in the given process is \( 2R \).
- (b) The heat required in the given process is \( 1200R \).