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. `2RT_0`
B. `3/2RT_0`
C. `RT_0`
D. `1/2RT_0`

Explanation

To solve the problem step by step, we will follow the principles of thermodynamics and the behavior of an ideal gas.

### Step 1: Understand the Process
We are given that one mole of an ideal monoatomic gas expands slowly according to the law \( \frac{P}{V} = \text{constant} \). This indicates that the process is a polytropic process where \( PV^x = \text{constant} \).

### Step 2: Identify the Value of \( x \)
From the equation \( \frac{P}{V} = \text{constant} \), we can rewrite it as \( PV^{-1} = \text{constant} \). This means that \( x = -1 \) in the polytropic process equation \( PV^x = \text{constant} \).

### Step 3: Determine the Heat Capacity \( C \)
For a polytropic process, the heat capacity \( C \) can be expressed as:
\[ C = C_v + \frac{R}{1 - x} \]
For a monoatomic gas, the molar heat capacity at constant volume \( C_v \) is \( \frac{3R}{2} \). Substituting \( x = -1 \):
\[ C = \frac{3R}{2} + \frac{R}{1 - (-1)} = \frac{3R}{2} + \frac{R}{2} = 2R \]

### Step 4: Calculate the Change in Temperature \( \Delta T \)
The initial temperature is \( T_0 \) and the final temperature is \( 2T_0 \). Therefore, the change in temperature \( \Delta T \) is:
\[ \Delta T = 2T_0 - T_0 = T_0 \]

### Step 5: Calculate the Heat Supplied \( Q \)
Using the formula for heat supplied in a polytropic process:
\[ Q = nC\Delta T \]
Here, \( n = 1 \) mole, \( C = 2R \), and \( \Delta T = T_0 \):
\[ Q = 1 \cdot 2R \cdot T_0 = 2RT_0 \]

### Final Answer
The heat supplied to the gas is:
\[ Q = 2RT_0 \]