One mole of an ideal gas (mono-atomic) at temperature `T_0` expands slowly according to law `P^2 = c T` (c is constant). If final temperature is `2T_0` heat supplied to gas is:

Options

A. `2 RT_0`
B. `3/2 RT_0`
C. `RT_0`
D. `(RT_0)/2`

Explanation

To solve the problem, we need to find the heat supplied to one mole of an ideal monoatomic gas that expands according to the law \( P^2 = cT \) from an initial temperature \( T_0 \) to a final temperature \( 2T_0 \).

### Step-by-Step Solution:

1. **Understanding the Given Law**:
The law given is \( P^2 = cT \). This implies a relationship between pressure \( P \) and temperature \( T \) during the expansion.

2. **Using the Ideal Gas Law**:
From the ideal gas equation, we know:
\[ PV = nRT \]
For one mole of gas (\( n = 1 \)), this simplifies to:
\[ PV = RT \]
We can express \( T \) in terms of \( P \) and \( V \):
\[ T = \frac{PV}{R} \]

3. **Substituting into the Given Law**:
Substitute \( T \) into the given law:
\[ P^2 = c \left(\frac{PV}{R}\right) \]
Rearranging gives:
\[ P^2 = \frac{cPV}{R} \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ P = \frac{cV}{R} \]

4. **Identifying the Polytropic Process**:
The equation \( PV^{-1} = \text{constant} \) suggests that this is a polytropic process with \( n = -1 \).

5. **Finding the Molar Specific Heat**:
The formula for the molar specific heat \( C \) in a polytropic process is given by:
\[ C = C_V + \frac{R}{1 - n} \]
For a monoatomic ideal gas, the molar specific heat at constant volume \( C_V \) is:
\[ C_V = \frac{3}{2}R \]
Substituting \( n = -1 \):
\[ C = \frac{3}{2}R + \frac{R}{1 - (-1)} = \frac{3}{2}R + \frac{R}{2} = 2R \]

6. **Calculating the Change in Temperature**:
The change in temperature \( \Delta T \) is:
\[ \Delta T = T_f - T_i = 2T_0 - T_0 = T_0 \]

7. **Calculating the Heat Supplied**:
The heat supplied \( Q \) can be calculated using:
\[ Q = nC\Delta T \]
Substituting \( n = 1 \), \( C = 2R \), and \( \Delta T = T_0 \):
\[ Q = 1 \cdot (2R) \cdot T_0 = 2RT_0 \]

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