The average speed of gas molecules is v at pressure P, If by keeping temperature constant the pressure of gas is doubled, then average speed will become

Options

A. `sqrt(2) V`
B. `v`
C. `2v`
D. `(v)/(sqrt2)`

Answer

To solve the problem, we need to analyze how the average speed of gas molecules changes when the pressure is doubled while keeping the temperature constant.

### Step-by-Step Solution:

1. **Understanding Average Speed of Gas Molecules**:
The average speed of gas molecules can be expressed using the formula:
\[ v_{avg} = \sqrt{\frac{8RT}{\pi m}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( m \) is the mass of a gas molecule.

2. **Given Conditions**:
We are given that the average speed of gas molecules is \( v \) at pressure \( P \) and temperature \( T \). We need to analyze the situation when the pressure is doubled (i.e., \( P' = 2P \)) while keeping the temperature constant.

3. **Using Ideal Gas Law**:
According to the ideal gas law:
\[ PV = nRT \]
If we keep the temperature constant and double the pressure, the volume must decrease to maintain the equation. This means:
\[ P'V' = nRT \quad \text{(where \( V' \) is the new volume)} \]
Since \( P' = 2P \), we have:
\[ 2PV' = nRT \]
Thus, the new volume \( V' \) will be:
\[ V' = \frac{nRT}{2P} \]
This indicates that the volume has decreased.

4. **Effect on Density**:
Since density (\( \rho \)) is defined as mass per unit volume, if the volume decreases while the mass remains constant, the density must increase. Therefore, we can conclude:
\[ \rho' = \frac{m}{V'} > \rho \]
(where \( \rho' \) is the new density).

5. **Revisiting Average Speed**:
The average speed formula can also be expressed in terms of pressure and density:
\[ v_{avg} = \sqrt{\frac{8R}{\pi} \cdot \frac{P}{\rho}} \]
Since we are keeping temperature constant, the term \( \frac{8R}{\pi} \) remains constant. When pressure is doubled, we have:
\[ v_{avg}' = \sqrt{\frac{8R}{\pi} \cdot \frac{2P}{\rho'}} \]
However, as we have established, increasing pressure leads to an increase in density. The increase in pressure and density will offset each other in the average speed equation.

6. **Conclusion**:
Since the average speed depends only on temperature and not on pressure or density, we can conclude that:
\[ v_{avg}' = v_{avg} \]
Thus, the average speed remains unchanged when the pressure is doubled at constant temperature.

### Final Answer:
The average speed will remain \( v \) even if the pressure is doubled.