Two vessels of the same volume contain the same gas at same temperature. If the pressure in the vessels is in the ratio of 1 : 2, then

Options

A. the ratio of the average kinetic energy is 1 : 2
B. the ratio of the root mean square velocity is 1 : 1
C. the ratio of the average velocity is 1 : 2
D. the ratio of number of molecules is 1 : 2

Explanation

To solve the problem, we need to analyze the relationship between pressure, average kinetic energy, root mean square velocity, average velocity, and the number of molecules in two vessels containing the same gas at the same temperature but with different pressures.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have two vessels of the same volume (V).
- The same gas is present in both vessels at the same temperature (T).
- The pressures in the vessels are in the ratio of 1:2 (P1:P2 = 1:2).

2. **Using the Kinetic Theory of Gases**:
- According to the kinetic theory of gases, the pressure (P) of an ideal gas is related to the average kinetic energy (KE) of its molecules.
- The average kinetic energy (KE) of a gas molecule is given by:
\[ KE = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature.
- Since both vessels have the same gas at the same temperature, the average kinetic energy is the same for both vessels.

3. **Ratio of Average Kinetic Energy**:
- Since the average kinetic energy does not depend on pressure, the ratio of average kinetic energy (KE1 : KE2) is:
\[ KE1 : KE2 = 1 : 1 \]

4. **Root Mean Square Velocity (RMS Velocity)**:
- The root mean square velocity (C) is related to pressure by the formula:
\[ P = \frac{1}{3} \frac{M N C^2}{V} \]
where \( M \) is the mass of the molecule, \( N \) is the number of molecules, and \( V \) is the volume.
- From the pressure ratio \( P1 : P2 = 1 : 2 \), we can write:
\[ \frac{C1^2}{C2^2} = \frac{P1}{P2} = \frac{1}{2} \]
- Taking the square root gives:
\[ \frac{C1}{C2} = \frac{1}{\sqrt{2}} \]

5. **Average Velocity**:
- The average velocity (V_avg) is related to the root mean square velocity by:
\[ V_{avg} = \frac{C}{\sqrt{3}} \]
- Therefore, the ratio of average velocities will also be the same as the ratio of root mean square velocities:
\[ \frac{V_{avg1}}{V_{avg2}} = \frac{1}{\sqrt{2}} \]

6. **Number of Molecules**:
- The number of molecules in each vessel can be related to pressure and volume:
\[ P \propto N \]
- Thus, the ratio of the number of molecules is the same as the ratio of pressures:
\[ \frac{N1}{N2} = \frac{P1}{P2} = \frac{1}{2} \]

### Final Results:
- Ratio of Average Kinetic Energy: **1:1**
- Ratio of Root Mean Square Velocity: **1:√2**
- Ratio of Average Velocity: **1:√2**
- Ratio of Number of Molecules: **1:2**

### Summary:
- The only correct answer is that the ratio of the number of molecules is **1:2**.