In an ideal gas at temperature T , the average force that a molecule applies on the walls of a closed container depends on `T "as" T^(q)` . A good estimate for q is :-

Answer

To solve the problem, we need to determine how the average force that a molecule applies on the walls of a closed container depends on the temperature \( T \) in an ideal gas. We are looking for a relationship of the form \( F \propto T^q \) and need to estimate the value of \( q \).

### Step-by-Step Solution:

1. **Understanding Pressure in an Ideal Gas**:
The average pressure \( P \) exerted by an ideal gas on the walls of a container is given by the formula:
\[ P = \frac{1}{3} \frac{m n}{V} v_{\text{rms}}^2 \]
where:
- \( m \) is the mass of a gas molecule,
- \( n \) is the number of molecules,
- \( V \) is the volume of the container,
- \( v_{\text{rms}} \) is the root mean square speed of the gas molecules.

2. **Relating Average Force to Pressure**:
The average force \( F \) that the gas molecules apply on the walls can be expressed as:
\[ F = P \cdot A \]
where \( A \) is the area of the wall. Substituting the expression for pressure:
\[ F = \left(\frac{1}{3} \frac{m n}{V} v_{\text{rms}}^2\right) \cdot A \]

3. **Substituting Volume**:
The volume \( V \) can be expressed as \( V = A \cdot L \) (where \( L \) is the length of the container). Thus, we can rewrite the force as:
\[ F = \frac{1}{3} \frac{m n}{A \cdot L} v_{\text{rms}}^2 \cdot A \]
Simplifying this gives:
\[ F = \frac{1}{3} \frac{m n}{L} v_{\text{rms}}^2 \]

4. **Relating \( v_{\text{rms}} \) to Temperature**:
The root mean square speed \( v_{\text{rms}} \) is related to the temperature \( T \) by the equation:
\[ v_{\text{rms}}^2 \propto T \]
Therefore, we can say:
\[ v_{\text{rms}}^2 = \frac{3k_B T}{m} \]
where \( k_B \) is the Boltzmann constant.

5. **Substituting Back**:
Substituting \( v_{\text{rms}}^2 \) back into the expression for force:
\[ F \propto \frac{m n}{L} \cdot T \]
Since \( m \), \( n \), and \( L \) are constants, we can conclude that:
\[ F \propto T \]

6. **Finding \( q \)**:
From the relationship \( F \propto T^q \), we find that:
\[ q = 1 \]

### Final Answer:
The value of \( q \) is \( 1 \).