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(B) The root mean square speed of gas molecules is given by the formula: $\upsilon_{rms} = \sqrt{\frac{3P}{\rho}} = \sqrt{\frac{3PV}{M}}$,where $P$ is

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$2P_0$

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(B) The root mean square speed of gas molecules is given by the formula: $\upsilon_{rms} = \sqrt{\frac{3P}{ho}} = \sqrt{\frac{3PV}{M}}$,where $P$ is pressure,$V$ is volume,and $M$ is the total mass of the gas.Since volume $V$ is constant,we have $\upsilon_{rms} \propto \sqrt{\frac{P}{M}}$.Let the initial state be $(\upsilon_1, P_1, M_1) = (\upsilon, P_0, M)$ and the final state be $(\upsilon_2, P_2, M_2) = (2\upsilon, P_2, M/2)$.Using the proportionality: $\frac{\upsilon_1}{\upsilon_2} = \sqrt{\frac{P_1}{P_2} 	imes \frac{M_2}{M_1}}$.Substituting the values: $\frac{\upsilon}{2\upsilon} = \sqrt{\frac{P_0}{P_2} 	imes \frac{M/2}{M}}$.$\frac{1}{2} = \sqrt{\frac{P_0}{P_2} 	imes \frac{1}{2}}$.Squaring both sides: $\frac{1}{4} = \frac{P_0}{P_2} 	imes \frac{1}{2}$.$\frac{1}{4} = \frac{P_0}{2P_2} \Rightarrow 2P_2 = 4P_0 \Rightarrow P_2 = 2P_0$.

$A$ gas is contained in a vessel at pressure $P_0$. If the mass of all molecules is halved and their speed is doubled,what will be the final pressure?

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