Question

Calculate the rms speed of the gas $C{O_2}$ at ${27^ \circ }$ C?

Answer 1

Root mean square speed is defined as the square root of the mean of the squares of the speeds of all the molecules present in the given sample of the gas. We will understand the concept widely in the solution.

Complete step by step answer:
According to kinetic theory of gases we know that gas molecules are in a continuous random motion with high velocities and there is no loss of kinetic energy during collision as the collision is perfectly plastic. Therefore kinetic gas equation for one mole of a gas is given by
$PV = \dfrac{1}{3}m{N_A}{c^2}$ Where c is the molecular speed.
Root mean square speed is the hypothetical speed possessed by all the gas molecules when total kinetic energy is equally distributed amongst them. Total kinetic energy of a sample containing n number of molecules is
$= \dfrac{1}{2}mc_1^2 + \dfrac{1}{2}mc_2^2 + .... + \dfrac{1}{2}mc_n^2$ (Where mass of every molecule is m)
If velocity possessed by all the molecules is same and that is c then the total kinetic energy is equal to
= $n \times \dfrac{1}{2}m{c^2}$.
Equating both the expressions of total kinetic energies we will get
$n \times \dfrac{1}{2}m{c^2}$ $= \dfrac{1}{2}mc_1^2 + \dfrac{1}{2}mc_2^2 + .... + \dfrac{1}{2}mc_n^2$
Now take (1/2)m common from both sides and it will get cut on both the sides
{c^2} = \dfrac{{c_1^2 + c_2^2 + .... + c_n^2}}{n} \\\ c = \sqrt {\dfrac{{c_1^2 + c_2^2 + .... + c_n^2}}{n}} \\\
The value of c can be determined from the kinetic gas equation $PV = \dfrac{1}{3}m{N_A}{c^2}$. We get
$c = \sqrt {\dfrac{{3PV}}{M}} = \sqrt {\dfrac{{3RT}}{M}}$ (remember ideal gas equation PV=nRT)
Now in the question we are given to calculate root mean square speed for $C{O_2}$ at ${27^ \circ }$C or 300K and molar mass(M)=12+32=44g, R( Rydbergs constant) = 8.314kJ/mol. Therefore let us put the values in the above equation,
$c = \sqrt {\dfrac{{3 \times 8.314 \times 300}}{{44}}} = 13.04m/s$
This is the rms speed for $C{O_2}$.

Note: There are three types of molecular speeds :Root mean square speed, Most probable speed and average speed. Average velocity is the arithmetic mean of the velocities of different molecules at a given temperature.