Root mean square velocity of a molecule is \(v\) at pressure \(P\). If pressure is increased two times, then the rms velocity becomes (A) \(3 \mathrm{~V}\) (B) \(2 \mathrm{v}\) (C) \(0.5 \mathrm{~V}\) (D) \(\mathrm{v}\)

The root mean square speed (rms speed) is given by the following formula: \(v_{rms} = \sqrt{\frac{3kT}{m}}\) where: - \(v_{rms}\) is the root mean square speed - \(k\) is the Boltzmann constant - \(T\) is the temperature in Kelvin - \(m\) is the mass of one molecule of the gas

We will use the ideal gas law to relate pressure to the rms speed. The ideal gas law is given by: \(PV = nRT\) where: - \(P\) is the pressure - \(V\) is the volume - \(n\) is the number of moles - \(R\) is the gas constant - \(T\) is the temperature in Kelvin Since the number of molecules \(N\) is related to the number of moles \(n\) by the formula \(N = nN_A\), where \(N_A\) is Avogadro's number, we can write the ideal gas law as: \(PV = \frac{NkT}{N_A}\) Now, we can write the pressure formula in terms of the rms speed: \(P = \frac{Nkm}{N_A} \cdot \frac{v_{rms}^2}{3}\) where we have replaced the temperature \(T\) by the rms speed using the formula \(v_{rms}^2 = 3kT/m\)

Let's call the new pressure and rms speed \(P'\) and \(v'_{rms}\), respectively. The new pressure \(P'\) is two times the initial pressure \(P\): \(P' = 2P\) Now, with the new pressure and rms speed, the pressure formula becomes: \(P' = \frac{Nkm}{N_A}\cdot \frac{(v'_{rms})^2}{3}\) Since we already know that \(P' = 2P\), we can equate the two pressure formulas and solve for the new rms speed: \(\frac{Nkm}{N_A}\cdot \frac{(v'_{rms})^2}{3} = 2 \cdot \frac{Nkm}{N_A} \cdot \frac{v_{rms}^2}{3}\) By simplifying the equation, we get: \((v'_{rms})^2 = 2\cdot(v_{rms})^2\) Finally, we take the square root of both sides: \(v'_{rms} = \sqrt{2}\cdot v_{rms}\) Since \(\sqrt{2}\) is not equal to any of the given options, we can conclude that none of the given options (A, B, C, or D) are correct.