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 the Average Speed Formula**:
The average speed \( v \) of gas molecules is given by the formula:
\[ v = \sqrt{\frac{8RT}{\pi M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

2. **Identifying Constants**:
In this scenario, we are told that the temperature \( T \) and the molar mass \( M \) remain constant. Therefore, \( R \), \( T \), and \( M \) do not change when the pressure changes.

3. **Effect of Pressure on Density**:
According to the ideal gas law, pressure \( P \) is related to density \( \rho \) by the equation:
\[ P = \rho RT \]
If the pressure is doubled (i.e., \( P' = 2P \)), the density will also double (i.e., \( \rho' = 2\rho \)) when temperature remains constant.

4. **Revising the Average Speed Formula**:
We can express the average speed in terms of pressure and density:
\[ v = \sqrt{\frac{8P}{\pi \rho}} \]
If the pressure is doubled and the density is also doubled, we can write:
\[ v' = \sqrt{\frac{8(2P)}{\pi(2\rho)}} = \sqrt{\frac{8P}{\pi \rho}} = v \]

5. **Conclusion**:
Since the average speed \( v' \) after doubling the pressure remains equal to the original average speed \( v \), we conclude that:
\[ v' = v \]

Thus, the average speed of gas molecules remains unchanged when the pressure is doubled at constant temperature.

### Final Answer:
The average speed will remain \( v \).

---