2 moles of the same gas are enclosed in two containers A and B at `27^(@)C` and at pressure 2 and 3 atms respectively. The rms velocity .

Options

A. will be same in both the container 'A' and 'B'
B. will be more in container 'B'
C. will be more in container 'A'
D. can't be said as Volume of the container is not mentioned.

Answer

To solve the problem of finding the root mean square (rms) velocity of the gas in two containers A and B, we will follow these steps:

### Step 1: Understand the Given Information
We have:
- 2 moles of the same gas in both containers.
- Temperature (T) = 27°C = 300 K (convert to Kelvin by adding 273).
- Pressure in container A (P_A) = 2 atm.
- Pressure in container B (P_B) = 3 atm.

### Step 2: Use the Formula for RMS Velocity
The formula for the root mean square velocity (v_rms) of a gas is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- R = universal gas constant = 0.0821 L·atm/(K·mol)
- T = absolute temperature in Kelvin
- M = molar mass of the gas.

### Step 3: Relate RMS Velocity to Pressure
We can also express the rms velocity in terms of pressure and volume using the ideal gas law, \( PV = nRT \):
\[ v_{rms} = \sqrt{\frac{3PV}{nM}} \]
Since n (number of moles) is constant (2 moles) and M (molar mass) is constant for the same gas, we can simplify our calculations.

### Step 4: Analyze the Relationship
From the ideal gas law, we know that:
\[ PV = nRT \]
This implies that for both containers A and B, the product of pressure and volume is constant. Thus, we can express the rms velocity as:
\[ v_{rms} \propto \sqrt{P} \]
This means that the rms velocity depends on the square root of the pressure when the number of moles and the molar mass are constant.

### Step 5: Compare the RMS Velocities
- For container A: \( v_{rms,A} \propto \sqrt{P_A} = \sqrt{2} \)
- For container B: \( v_{rms,B} \propto \sqrt{P_B} = \sqrt{3} \)

### Step 6: Conclusion
Since \( \sqrt{3} > \sqrt{2} \), we conclude that:
- The rms velocity in container B is greater than that in container A.

### Final Answer
The rms velocity will be more in container B.