Explain the following terms:
$(i) u_{mp}$
$(ii) u_{av}$
$(iii) \overline{u}^2$
$(iv) u_{rms}$

$(i)$ Most probable speed $(u_{mp})$: It is the speed possessed by the maximum number of molecules of a gas at a given temperature. $u_{mp} = \sqrt{\frac{2RT}{M}} = 0.816 \times u_{rms}$.
$(ii)$ Average speed $(u_{av})$: It is the arithmetic mean of the speeds of all the molecules of a gas. $u_{av} = \frac{u_1 + u_2 + u_3 + \dots + u_n}{n} = \sqrt{\frac{8RT}{\pi M}}$,where $u_1, u_2, \dots$ are individual speeds and $n$ is the total number of molecules.
$(iii)$ Mean square speed $(\overline{u}^2)$: It is the arithmetic mean of the squares of the speeds of all the molecules of a gas. $\overline{u}^2 = \frac{u_1^2 + u_2^2 + \dots + u_n^2}{n} = \frac{3RT}{M}$.
$(iv)$ Root mean square speed $(u_{rms})$: It is the square root of the mean of the squares of the speeds of all the molecules of a gas. $u_{rms} = \sqrt{\overline{u}^2} = \sqrt{\frac{u_1^2 + u_2^2 + u_3^2 + \dots}{n}} = \sqrt{\frac{3RT}{M}}$.