A monatomic gas undergoes a thermodynamic process such that `Tprop V^(-2/7)` (T = temperature and V = volume of the gas) As heat is supplied to the gas, choose INCORRECT option.

Options

A. ms velocity of gas molecules increases
B. rms velocity of gas molecules decreases
C. work done on the gas is positive
D. density of the gas increases

Answer

To solve the problem, we need to analyze the thermodynamic process of a monatomic gas under the given condition that temperature \( T \) is proportional to \( V^{-2/7} \). We will derive relationships and determine the incorrect option based on the implications of supplying heat to the gas.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
Given that \( T \propto V^{-2/7} \), we can express this as:
\[ T = k \cdot V^{-2/7} \]
where \( k \) is a constant.

2. **Rearranging the Equation**:
To eliminate \( T \), we can multiply both sides by \( V^{2/7} \):
\[ T \cdot V^{2/7} = k \]
This indicates that \( T \cdot V^{2/7} \) remains constant.

3. **Using the Ideal Gas Law**:
From the ideal gas law, we know:
\[ PV = nRT \]
Rearranging gives:
\[ T = \frac{PV}{nR} \]

4. **Substituting for \( T \)**:
We can substitute \( T \) from the ideal gas law into our earlier equation:
\[ \frac{PV}{nR} \cdot V^{2/7} = k \]
Simplifying gives:
\[ PV^{9/7} = nRk \]
This indicates that \( PV^{9/7} \) is also a constant.

5. **Analyzing Heat Transfer**:
When heat \( Q \) is added to the system, the first law of thermodynamics states:
\[ Q = \Delta U + W \]
where \( \Delta U \) is the change in internal energy and \( W \) is the work done by the gas.

6. **Change in Internal Energy**:
For a monatomic gas, the change in internal energy is given by:
\[ \Delta U = \frac{3}{2} nR \Delta T \]

7. **Work Done by the Gas**:
The work done by the gas can be expressed as:
\[ W = P \Delta V \]
Using the relationship from earlier, we can find how \( W \) changes with \( \Delta T \).

8. **Determining the Effect of Heat**:
As heat is supplied, the temperature \( T \) increases. However, since \( T \) is proportional to \( V^{-2/7} \), an increase in \( T \) implies a decrease in \( V \) (the volume must decrease to maintain the relationship).

9. **Density Consideration**:
Density \( \rho \) is defined as:
\[ \rho = \frac{m}{V} \]
If the volume decreases while mass remains constant, the density increases.

10. **Identifying the Incorrect Option**:
Based on the analysis:
- If the gas expands, the work done by the gas is positive.
- If the temperature decreases, the RMS velocity decreases.
- If the volume decreases, the density increases.

Therefore, the incorrect statement would be the one that suggests that the density of the gas decreases when heat is supplied.