7.0 Gas Speed

The motion of molecules in a gas is characterized by anyone of the following three types:

1. Root Mean Square Speed (${C_{rms}}$): It is defined as the square root mean of squares of the speed of different molecules i.e. $${C_{rms}} = \sqrt {\frac{{C_1^2 + C_2^2 + ... + C_N^2}}{N}} = \sqrt {\overline {{C^2}} } = \sqrt {mean\ square\ speed} $$$$PV = \frac{1}{3}mNC_{rms}^2 $$

Thus, $$ {C_{rms}} = \sqrt {\frac{{3PV}}{{mN}}} = \sqrt {\frac{{3PV}}{M}} = \sqrt {\frac{{3P}}{\rho }}\ \ \ \ \left[ {\because \frac{M}{V} = \rho } \right]$$

Also, $$ {C_{rms}} = \sqrt {\frac{{3N RT}}{M}} = \sqrt {\frac{{3M}}{{{M_o}}}\frac{{RT}}{M}} = \sqrt {\frac{{3RT}}{{{M_o}}}} $$ where, $M_0$= Molecular mass

Also,$$ {C_{rms}} = \sqrt {\frac{{3N RT}}{{mN}}} = \sqrt {\frac{{3N}}{{{N_A}}}\frac{{RT}}{{mN}}} = \sqrt {\frac{{3R}}{{{N_A}}}\frac{T}{m}} = \sqrt {\frac{{3kT}}{m}}$$

Thus, we have $${C_{rms}} = \sqrt {\frac{{3PV}}{M}} = \sqrt {\frac{{3P}}{\rho }} = \sqrt {\frac{{3RT}}{{{M_o}}}} = \sqrt {\frac{{3kT}}{m}}$$

2. Most Probable Speed (${C_{mp}}$): The speed possessed by maximum number of gas molecules in a sample at a given temperature is called Most Probable Speed.

In Maxwell's Speed Distribution curve, the speed at which $\frac{{dN}}{{dC}}$ is maximum , is the most probable speed at a given temperature.

It's value has been calculated as $${C_{mp}} = \sqrt {\frac{{2kT}}{m}}$$

Thus, $$\begin{equation} \begin{aligned} {C_{mp}} = \sqrt {\frac{{2kT}}{m}} = \sqrt {\frac{{2RT}}{{{M_o}}}} = \sqrt {\frac{{2PV}}{M}} = \sqrt {\frac{{2P}}{\rho }} = \sqrt {\frac{2}{3}} {C_{rms}} = 0.816{C_{rms}} \\ \\\end{aligned} \end{equation} $$

3. Average Speed (${C_{av}}$): The average value of gas speeds of all gas molecules is called average speed i.e.

$$ {C_{av}} = \frac{{{C_1} + {C_2} + ... + {C_N}}}{N} = \sqrt {\frac{{8kT}}{{\pi m}}} $$

Thus, $$ {C_{av}} = \sqrt {\frac{{8kT}}{{\pi m}}} = \sqrt {\frac{{8RT}}{{\pi {M_o}}}} = \sqrt {\frac{{8PV}}{{\pi M}}} = \sqrt {\frac{{8P}}{{\pi \rho }}} = \sqrt {\frac{8}{{3\pi }}} {C_{rms}} = 0.92{C_{rms}}$$