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 $$1:4$$.

Same gas in two vessels of same volume and same temperature, with number of molecules in ratio 1:4. We check statements A, B, C, D.

Statement A: RMS velocity is the same in both vessels.

The RMS velocity of gas molecules is:

$$v_{\text{rms}} = \sqrt{\frac{3k_BT}{m}}$$

Since the gas is the same (same molecular mass $$m$$) and the temperature $$T$$ is the same, the RMS velocity is identical in both vessels regardless of the number of molecules.

Statement A is correct.

Statement B: Ratio of pressures is 1:4.

From the ideal gas law: $$PV = Nk_BT$$

Since $$V$$ and $$T$$ are the same:

$$P \propto N$$

$$\frac{P_1}{P_2} = \frac{N_1}{N_2} = \frac{1}{4}$$

Statement B is correct.

Statement C: Ratio of pressure is 1:1.

This contradicts Statement B. Since pressure is proportional to the number of molecules, the ratio is 1:4, not 1:1.

Statement C is incorrect.

Statement D: RMS velocity ratio is 1:4.

As shown in Statement A, RMS velocity is the same for both (ratio 1:1, not 1:4).

Statement D is incorrect.

Statements A and B are correct.

The correct answer is Option C: A and B only.