Question

Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is 1 : 4, then

A. The r.m.s. velocity of gas molecules in two vessels will be the same.

B. The ratio of pressure in these vessels will be 1 : 4.

C. The ratio of pressure will be 1 : 1.

D. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of 1 : 4.

Choose the correct answer from the options given below.

• A. A and C only
• B. B and D only
• C. A and B only
• D. C and D only

Answer 1

The Correct Option is C

The problem states that the same gas is filled in two vessels of the same volume at the same temperature and the number of gas molecules in the two vessels is in the ratio 1:4. We need to determine the correct statements regarding the r.m.s. velocity and the pressure in the vessels from the given options.

1. Understanding the relationship between pressure and the number of molecules:
   • According to the ideal gas law, the pressure P of a gas is given by:

   PV = nRT

   • Since the volume V, gas constant R, and temperature T are constant, pressure P is proportional to the number of moles n.

   P \propto n

   • For vessels with the number of molecules in the ratio 1:4, the pressure ratio will also be 1:4.
2. Understanding the relationship of r.m.s. velocity with temperature:
   • The root mean square (r.m.s.) velocity of gas molecules is given by the formula:

   v_{rms} = \sqrt{ \frac{3kT}{m} }

   • where k is the Boltzmann constant, T is the temperature, and m is the molecular mass of the gas.
   • Since both vessels have the same gas and are at the same temperature, the r.m.s. velocity v_{rms} will be the same for both vessels despite the different number of molecules.

Conclusion:

• Option A: "The r.m.s. velocity of gas molecules in two vessels will be the same." This is correct as explained above.
• Option B: "The ratio of pressure in these vessels will be 1 : 4." This is correct because pressure depends directly on the number of molecules.
• Option C: "The ratio of pressure will be 1 : 1." This is incorrect as explained.
• Option D: "The r.m.s. velocity of gas molecules in two vessels will be in the ratio of 1 : 4." This is incorrect since the r.m.s. velocity depends only on the temperature, not the number of molecules.

The correct answer is therefore: A and B only.