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) `3/2RT_0`
C. (c) `RT_0`
D. (d) `1/2RT_0`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Process
We have one mole of an ideal monoatomic gas expanding according to the law \( \frac{p}{V} = \text{constant} \). This indicates a polytropic process where \( pV^n = \text{constant} \) with \( n = -1 \).

**Hint:** Identify the type of thermodynamic process and the relationship between pressure and volume.

### Step 2: Determine the Value of \( x \)
From the law \( \frac{p}{V} = \text{constant} \), we can express it in the form \( pV^x = \text{constant} \). Here, we have \( x = -1 \).

**Hint:** Relate the given expression to the general form of a polytropic process.

### Step 3: Calculate the Molar Specific Heat Capacity
For a polytropic process, the molar specific heat capacity \( C \) is given by:

\[ C = C_v + \frac{R}{1 - x} \]

For a monoatomic ideal gas, \( C_v = \frac{3}{2}R \). Substituting \( x = -1 \):

\[ C = \frac{3}{2}R + \frac{R}{1 - (-1)} = \frac{3}{2}R + \frac{R}{2} = \frac{3}{2}R + \frac{1}{2}R = 2R \]

**Hint:** Use the specific heat capacity formula for polytropic processes and substitute the known values.

### Step 4: Calculate the Change in Temperature
The initial temperature \( T_i = T_0 \) and the final temperature \( T_f = 2T_0 \). The change in temperature \( \Delta T \) is:

\[ \Delta T = T_f - T_i = 2T_0 - T_0 = T_0 \]

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

### Step 5: Calculate the Heat Supplied to the Gas
The heat supplied \( Q \) to the gas can be calculated using the formula:

\[ Q = nC\Delta T \]

Substituting the values we have:

\[ Q = 1 \cdot (2R) \cdot (T_0) = 2RT_0 \]

**Hint:** Use the heat transfer formula for an ideal gas and substitute the number of moles, specific heat, and temperature change.

### Final Answer
The heat supplied to the gas is \( Q = 2RT_0 \).