One mole of an ideal monoatomic gas is taken through a polytropic process `P^2T` = constant. The heat required to increase the temperature of the gas by `DeltaT` is

Options

A. `R DeltaT`
B. `3R DeltaT`
C. `5RDeltaT`
D. `6RDeltaT`

Answer

To solve the problem of finding the heat required to increase the temperature of one mole of an ideal monoatomic gas through a polytropic process where \( P^2 T \) is constant, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the relationship given:**
The problem states that \( P^2 T = \text{constant} \). This implies that as pressure \( P \) and temperature \( T \) change, their relationship remains constant.

2. **Use the Ideal Gas Law:**
The ideal gas law is given by:
\[ PV = nRT \]
For one mole of gas (\( n = 1 \)), this simplifies to:
\[ PV = RT \]
Rearranging gives:
\[ T = \frac{PV}{R} \]

3. **Express \( P^2 \) in terms of \( T \):**
From the relationship \( P^2 T = \text{constant} \), we can express \( P^2 \) as:
\[ P^2 = \frac{\text{constant}}{T} \]

4. **Substitute \( P \) in terms of \( T \):**
Using the ideal gas law, we can express \( P \) as:
\[ P = \frac{RT}{V} \]
Substituting this into the equation for \( P^2 \):
\[ \left(\frac{RT}{V}\right)^2 = \frac{\text{constant}}{T} \]

5. **Determine the work done during the process:**
The work done \( W \) in a polytropic process is given by:
\[ W = nR \Delta T \frac{1 - \eta}{\eta} \]
where \( \eta \) is the polytropic index. From the relationship \( P^2 T = \text{constant} \), we can find \( \eta \). Here, we find that \( \eta = \frac{1}{3} \).

6. **Calculate the work done:**
Plugging in \( n = 1 \) and \( \eta = \frac{1}{3} \):
\[ W = 1 \cdot R \Delta T \frac{1 - \frac{1}{3}}{\frac{1}{3}} = R \Delta T \cdot \frac{\frac{2}{3}}{\frac{1}{3}} = 2R \Delta T \]

7. **Calculate the change in internal energy:**
For a monoatomic ideal gas, the change in internal energy \( \Delta U \) is given by:
\[ \Delta U = \frac{3}{2} nR \Delta T = \frac{3}{2} R \Delta T \]

8. **Apply the first law of thermodynamics:**
The first law states:
\[ Q = \Delta U + W \]
Substituting the expressions for \( \Delta U \) and \( W \):
\[ Q = \frac{3}{2} R \Delta T + 2R \Delta T = \left(\frac{3}{2} + 2\right) R \Delta T = \frac{7}{2} R \Delta T \]

### Final Answer:
The heat required to increase the temperature of the gas by \( \Delta T \) is:
\[ Q = \frac{7}{2} R \Delta T \]