Use the ideal gas law in molecular form

For a gas, $PV = NkT$ where:

• $P$ = pressure
• $V$ = volume
• $N$ = number of molecules
• $k$ = Boltzmann constant
• $T$ = temperature

Given conditions

• Same gas in both vessels
• Same volume: $V_1 = V_2$
• Same temperature: $T_1 = T_2$
• Number of molecules ratio: $N_1:N_2 = 1:4$

Compare pressures

From $P = \frac{NkT}{V}$ since $k$, $T$, and $V$ are same for both vessels, $P \propto N$ Therefore, $P_1:P_2 = N_1:N_2 = 1:4$

So statement B is true and statement C is false.

Compare r.m.s. speeds

The r.m.s. speed of gas molecules is $v_{\mathrm{rms}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

Since both vessels contain the same gas and are at the same temperature, $m$ and $T$ are same in both cases.

Hence, $v_{\mathrm{rms,1}} = v_{\mathrm{rms,2}}$

So statement A is true and statement D is false.

Identify correct combination

True statements: A and B only.

Therefore, the correct option is: $\boxed{\text{C}}$

Comparison with stored answer

Stored correct answer: C

My derived answer is also C, so they agree.