Root mean square velocity of a gas is x `ms^(-1)` at a pressure p atm and temperature TK. If pressure is made 2p under isothermal condition, root mean square speed becomes

Options

A. 2x
B. 4x
C. x/2
D. x

Answer

To solve the problem of finding the root mean square (RMS) velocity of a gas when the pressure is doubled under isothermal conditions, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for RMS Velocity**:
The root mean square velocity (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature in Kelvin, and \( M \) is the molar mass of the gas.

2. **Identify Initial Conditions**:
Initially, the RMS velocity is given as \( v_{rms} = x \, \text{m/s} \) at pressure \( P \) and temperature \( T_K \).

3. **Recognize Isothermal Conditions**:
Since the process is isothermal, the temperature \( T \) remains constant. Therefore, any change in pressure will not affect the temperature.

4. **Apply the Ideal Gas Law**:
According to the ideal gas law:
\[ PV = nRT \]
For one mole of gas, this simplifies to:
\[ PV = RT \]
This implies that at constant temperature, the product of pressure and volume remains constant.

5. **Consider the Change in Pressure**:
When the pressure is increased from \( P \) to \( 2P \), we need to find the new volume \( V' \) under the condition that \( PV = P'V' \) remains constant. Therefore:
\[ P \cdot V = 2P \cdot V' \]
From this, we can deduce that:
\[ V' = \frac{V}{2} \]

6. **Relate RMS Velocity to Pressure**:
The RMS velocity can also be expressed in terms of pressure and volume:
\[ v_{rms} \propto \sqrt{\frac{P}{M}} \]
Since \( M \) (molar mass) remains constant, we can say:
\[ v_{rms} \propto \sqrt{P} \]

7. **Calculate New RMS Velocity**:
When the pressure is doubled (from \( P \) to \( 2P \)):
\[ v_{rms}' = \sqrt{\frac{3R(2T)}{M}} = \sqrt{2} \cdot v_{rms} \]
However, since temperature is constant, we can directly relate the new RMS velocity to the old one:
\[ v_{rms}' = \sqrt{2} \cdot v_{rms} \]
Since \( v_{rms} = x \):
\[ v_{rms}' = \sqrt{2} \cdot x \]

8. **Final Result**:
Therefore, the new root mean square velocity when the pressure is doubled under isothermal conditions is:
\[ v_{rms}' = x \]