Question

Give the laws of Boyle, Charles, Gay-Lussac and Dalton by writing the postulates of kinetic theory of gases.

Answer 1

The main assumptions of kinetic theory of gases are:
1. A gas is made up of a large number of sub-microscopic particles called atoms.
2. Gases consist of particles in constant random motion. They continue to move in a straight line until they collide with each other on the walls of their container.
3. Gas pressure is due to the moleiMIeS colliding with the walls of the container. All of these collisions are perfectly elastic. This means that there is no change in energy of either the particles or the wall upon collision. So, no energy is lost or gained from collisions.
4. No molecular forces are at work. The potential energy of molecules is zero. So, whole of the energy. in an ideal gas is kinetic energy only. There is no attraction or repulsion between the particles.
5. The time taken for the collision is negligible as compared with the time between collisions. 6. The kinetic energy of a gas is a measure of its kelvin temperature. Each gas molecule has different speed but the temperature and kinetic energy of gas refer to the average of these speeds.
7. The average kinetic energy of an atom of gas is directly proportional to the temperature. An increase in temperature increases the speed in which the gas molecules move.
8. All gases at a given temperature have same average kinetic energy.
9. fighter molecules of a gas move faster than the heavier molecules.
According to the kinetic theory, the average kinetic energy of molecules of a gas is directly proportional to the absolute temperature of the gas. So, from equation(14.21),
$\frac{1}{2}m\stackrel{-}{v^2}=\alpha T$ ..........(14.24)
where $\alpha$ is a proportionality constant.
From equation (14.17), we get
$PV=\frac{2}{3}{N}_A\alpha T$
If temperature (T) is constant, then
$PV=constant$
i.e., the product of pressure and volume of gas at constant temperature is constant. This is Boyle's law.
From equation (14.25), if pressure (P) of a gas is constant, then
$V\propto T$ .........(14.26)
i.e., for a gas of a given mass, volume of gas is directly proportional to the temperature at constant pressure. This is Charles law.
From equation (14.17),
$P=\frac{1}{3}\frac{mN}{V}\stackrel{-}{v^2}$ .........(14.27)
Since, from equation (14.20),
$\stackrel{-}{v^2}=\frac{3k_{B}T}{m}$
By putting the value of $\stackrel{-}{v^2}$ in equation (14.27), we have
$P=\frac{1}{3}\frac{mN}{V}\times \frac{3k_{B}T}{m}$
$P=\frac{N{k}_{B}T}{V}$
$\Rightarrow P\propto T$ .........(14.28)
So, at constant volume for a gas of given mass, the pressure of a gas is directly proportional to the temperature of gas. This is Gay Lussac’s law.
According to Dalton’s law of partial pressure, in thermal equilibrium state, the total pressure of each gas is equal to the sum of the different pressures of each gas for a mixture of unreactive gases filled in a container, i.e., if $P_1,P_2,P_3,...$ respectively are the different pressure of gases. Then, the total pressure (P)
$P=P_1+P_2+P_3+.....$ (14,41)
Let the gases be filled in a container with volume V in which the numb er of molecules are $n_1,n_2,n_3,..$ respectively, $m_1,m_2,m_3,..$ be the mass of each molecule, and $\stackrel{-}{v_1^2},\stackrel{-}{v_2^2},\stackrel{-}{v_3^2},..$ respectively be the mean square speed.Now, according to the kinetic theory, the pressure on the first gas
$P_1=\frac{1}{3}\frac{m_1{n}_1}{V}\stackrel{-}{v_1^2}$
$P_{1}V=\frac{1}{3}{m}_1{n}_1\stackrel{-}{v_1^2}$ ........(14.42)
Similarly, for the second gas,
$P_{2}V=\frac{1}{3}{m}_2{n}_2\stackrel{-}{v_2^2}$ ........(14.43)
and for the third gas,
$P_{3}V=\frac{1}{3}{m}_3{n}_3\stackrel{-}{v_3^2}$ ........(14.44)
Similarly, we can write equation for other gases. Since the mixture of all-the gases are in thermal equilibrium state, i.e., the temperature is same. Then, average kinetic energy of each molecule of a gas will be equal, i.e.,
$\frac{1}{2}{m}_1\stackrel{-}{v_1^2}=\frac{1}{2}{m}_2\stackrel{-}{v_2^2}=\frac{1}{2}{m}_3\stackrel{-}{v_3^2}=......=\frac{1}{2}m\stackrel{-}{v^2}$ ....(14.45)
Then, the sum of pressure of all gases
$P_1+P_2+P_3+...........=\frac{1}{3}\frac{m_1{n}_1}{V}\stackrel{-}{v_1^2}+\frac{1}{3}\frac{m_2{n}_2}{V}\stackrel{-}{v_2^2}+\frac{1}{3}\frac{m_3{n}_3}{V}\stackrel{-}{v_3^2}+..$ .....(14.46)
From equations (14.45) and (14.46),
$P_1+P_2+P_3+....=\frac{1}{3}(n_1+n_2+n_3+..{.}_m\stackrel{-}{v^2}$ ....(14.47)
If the total pressure of container is P and the total number of molecules are $(n_1+n_2+n_3+...)$, then the total applied pressure,
$P=\frac{1}{3}(n_1+n_2+n_3+....)m\stackrel{-}{v^2}$.......(14.48)
i.e., $P=P_1+P_2+P_3+...$
This is Dalton's law of partial pressure.