Which of the following indicates RMS velocity of a gas?

a) $\sqrt{\frac{2\mathrm{RT}}{\mathrm{M}}}$

b) $\sqrt{\frac{3\mathrm{P}}{\mathrm{d}}}$

c) $\sqrt{\frac{8\mathrm{RT}}{\mathrm{\pi M}}}$

d) $1.58\times \sqrt{\frac{\mathrm{T}}{\mathrm{M}}}\times {10}^4 \mathrm{cm}/\sec$

Formula Used: V_rms=$\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$ 

Detailed Solution:

We know that 

V_rms=$\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$ 

The equation $\sqrt{\frac{3\mathrm{P}}{\mathrm{d}}}$can be obtained from ideal gas equation i.e. 

$$\begin{gathered} \mathrm{PV}=\mathrm{nRT} \\ \frac{\mathrm{P}}{\mathrm{d}}=\frac{\mathrm{RT}}{\mathrm{M}} \end{gathered}$$

V_rms=$$\begin{gathered} \sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}=\sqrt{\frac{3\times 8.314\times {10}^7\mathrm{ergs}\times \mathrm{T}}{\mathrm{N}}} \\ =1.58\times \sqrt{\frac{\mathrm{T}}{\mathrm{M}}}\times {10}^4\mathrm{cm}/\sec \end{gathered}$$

Hence, b), and d) are correct.

Therefore, option(B) is the answer.