Kinetic Theory of Gases

Introduction

The Kinetic Theory of Gases is a fundamental model that explains the macroscopic properties of gases in terms of the motion and interactions of their molecules. This theory posits that gas particles are in constant, random motion, and their collisions with each other and the walls of their container are perfectly elastic. According to the kinetic theory, the temperature of a gas is directly related to the average kinetic energy of its molecules, which is proportional to the temperature in Kelvin. Gas molecules are in constant, random motion. The volume of individual gas molecules is negligible compared to the volume of the container. Molecules interact with each other only through elastic collisions, meaning no net loss of kinetic energy during collisions. The average kinetic energy of the gas molecules is proportional to the absolute temperature of the gas. These postulates help explain various properties of gases, such as pressure, temperature, and volume, and form the basis for deriving the ideal-gas law. The kinetic theory also provides insights into phenomena like diffusion, effusion, and the behavior of gases at different temperatures and pressures. Understanding the kinetic theory is essential for both theoretical studies and practical applications in physics, chemistry, and engineering. It offers a microscopic explanation for the macroscopic behaviors of gases, bridging the gap between molecular dynamics and observable gas properties.

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This Story also Contains

1. Introduction
2. For the better understanding of the topic Kinetic Theory of Gases ,watch the video:
3. Some Solved Examples
4. Summary

Kinetic Energy and Molecular Speeds
As you have studied in the previous section the molecules of a gas are always in motion and are colliding with each other and with the walls of the container. Due to these collisions, the speeds and the kinetic energies of the individual molecules keep on changing. However, at a given temperature, the average kinetic energy Of the gas molecules remains constant.

If at a given temperature, n₁, molecules have speed u₁, n₂, molecules have speed u₂, n₃ molecules have speed u₃, and so on. Then, the total kinetic energy (EK) of the gas at this temperature is given by:
$E_K=\frac{1}{2} m\left(n_1 v_1^2+n_2 v_2^2+n_3 v_3^2+\ldots \ldots \ldots\right)$
where m is the mass of the molecule. The corresponding average kinetic energy $\overline{E_k}$ of the gas will be:

$
\overline{\mathrm{E}_{\mathrm{K}}}=\frac{1}{2} \frac{\mathrm{m}\left(\mathrm{n}_1 \mathrm{v}_1^2+\mathrm{n}_2 \mathrm{v}_2^2+\mathrm{n}_3 \mathrm{v}_3^2+\ldots \ldots \ldots\right)}{\left(\mathrm{n}_1+\mathrm{n}_2+\mathrm{n}_3+\ldots \ldots \ldots\right)}
$

If the term $\frac{\left(n_1 v_1^2+n_2 v_2^2+n_3 v_3^2+\ldots \ldots \ldots\right)}{\left(n_1+n_2+n_3+\ldots \ldots\right)}=\bar{v}^2$ then the average kinetic energy is given by :
$
\overline{\mathrm{E}_\kappa}=\frac{1}{2} \mathrm{mv}^2
$
where $v$ is given by
$
v=\sqrt{\frac{\left(n_1 v_1^2+n_2 v_2^2+n_3 v_3^2+\ldots \ldots \ldots\right)}{\left(n_1+n_2+n_3+\ldots \ldots \ldots\right)}}
$

This 'v' is known as root-mean-square speed u_rms

Maxwell-Boltzmann Distribution of speeds

According to it

• Molecules have different speeds due to frequent molecular collisions with the walls and among themselves.
• Rare molecules have either very high or very low speed.
• Maximum number of molecules of the gas have maximum velocity which is called most probable velocity and after Vmp velocity decreases.
• Zero velocity is not possible.
• All these velocities increase with the increase in temperature but fraction of molecules having these velocities decreases.

Average Speed, u_av
It is the average of different velocities possessed by the molecules

$\begin{aligned} & u_{a v}=\frac{u_1+u_2+u_3}{n} \\ & u_{a v}=\frac{n_1 u_1+n_2 u_2+n_3 u_3}{n_1+n_2+n_3}\end{aligned}$

Here n₁, n₂, n₃ are the number of molecules having u₁, u₂, u₃ velocities respectively.
Relation between U_av, temperature and molar mass is given as

$\mathrm{u}_{\mathrm{av}}=\sqrt{8 \mathrm{RT} / \pi \mathrm{M}}=\sqrt{8 \mathrm{PV} / \pi \mathrm{M}}$

Most Probable Speed, u_mp
The most probable speed(u_mp) of a gas at a given temperature is the speed possessed by the maximum number of molecules at that temperature. Unlike average speed and root mean square speed, the most probable speed cannot be expressed in terms of the individual molecular speeds.

The most probable speed(u_mp) is related to absolute temperature (T) by the expression:

$u_{m p}=\sqrt{\frac{2 R T}{M}}=\sqrt{\frac{2 P V}{M}}$

Root Mean Square Speed u_rms
It is the square root of the mean of the square of the velocities of different molecules.$\begin{aligned} & u_{r m s}=\frac{\sqrt{u_1^2+u_2^2+\ldots}}{n} \\ & =\frac{\sqrt{n_1 u_1^2+n_2 u_2^2+n_3 u_3^2}}{n_1+n_2+n_3} \\ & u_{r m s}=\sqrt{3 R T / M} \\ & u_{r m s}=\sqrt{3 P V / M}=\sqrt{3 P / d}\end{aligned}$

Relation between u_av, u_mp and u_rms
The three types of molecular speeds, namely, most probable speed(v_mp), average speed (v_av) and root mean square speed(v_rms) of a gas at a given temperature are related to each other as follows:
$\begin{aligned} & v_{m p}: v_{a v}: v_{m s}=\sqrt{\frac{2 R T}{M}}: \sqrt{\frac{8 R T}{\pi M}}: \sqrt{\frac{3 R T}{M}} \\ & v_{m p}: v_{a v}: v_{r m s}=1.414: 1.596: 1.732 \\ & v_{m p}: v_{a v}: v_{r m s}=1: 1.128: 1.224\end{aligned}$

For a particular gas, at a particular temperature:
v_mp < v_av < v_rms

It follows from the above relationship that:
Average speed(v_av) =0.921 x Root mean square speed(v_rms)
Most probable speed(v_mp) = 0.817 x Root mean square speed(v_rms)

All the gas laws that we have discussed like Boyle's law, Charles' Law, and Avogadro's Law are merely based on experimental evidence. There was no theoretical background to justify them. So, the scientists were curious to know why the gases behave in a peculiar manner under a certain set of conditions. From Charles' law we got to know that the gases expand on heating. But there was no theory to give the reason for such fact. So, there was a need for some theory that could tell about the happenings at the molecular level and so could answer all the questions arising regarding the behaviour of gases.

Later a theory was given called kinetic molecular theory of gases to provide a sound theoretical basis for various gas laws. The kinetic theory of gases is based on the following assumptions or postulates:

• The actual volume of gas molecules is negligible in comparison to the total volume of the gas: Postulate says that all the gases are made up of extremely small particles called molecules dispersed throughout the container. These particles are so small that they are regarded as point masses. As they are point masses, so the actual volume occupied by the gas molecules is negligible in comparison to the total volume of the gas.
  
  
  
• Support for the assumption. This assumption explains the great compressibility of gases because there is a lot of empty space between the gas molecule.
• No force of attraction between the gas molecules: As the distance between the gas molecules is very large, it is assumed that there is no force of attraction between the gas molecules at ordinary temperature and pressure.
  Support for the assumption: Due to the lack of force of attraction between the gas molecules, the gases easily expand and occupy all the space available to them when heating.
• Particles of gas are in constant random motion: Particles of gas are in a state of constant random motion.
  Support for the assumption: This assumption is supported by the fact that gases do not have a fixed shape because of their random motion.
• Particles of gas collide with each other and with the walls of the container: Particles of gas move in a straight line with high velocities in all possible directions. During their this motion, they collide with each other and with the walls of the container in which gas is enclosed and even change direction upon collisions.
  
  
  
• Support for the assumption. Gas exerts some pressure. The pressure of the gas exerted is just because of the collisions of particles with the walls of the container.
• Collisions are perfectly elastic: When the gas molecules collide with each other they pass on their energies. There is a transfer of energy from one colliding molecule to the other but the total energy of molecules before and after the collision remains the same therefore, the collisions are called perfectly elastic. So, there is no net loss of energy.
  Support for the assumption: As there is no loss of kinetic energy, therefore the motion of molecules do not cease so, the gases never settle down.
• Different particles of the gas, have different speeds: Different particles of gas possess different kinetic energies, therefore they have different speeds at a particular time.
  Support for the assumption: This postulate is reasonable as when the molecules collide, they change their speed. Even though the initial speeds are the same, after collisions, there is the transfer of energy from one molecule to the other. So, as the energy changes after the collisions, so do the speeds. However, the distribution of speeds remains constant at a particular temperature.
• The average kinetic energy of the gas molecules is directly proportional to the absolute temperature: As discussed in the above assumption the speed of a molecule changes with time, i.e. the speed of a molecule is variable. Therefore we talk about the average kinetic energy of the molecules, The Kinetic molecular theory of gases establishes a link between molecular motion and temperature. As the temperature increases, the kinetic energy also increases

For the better understanding of the topic Kinetic Theory of Gases ,watch the video:

Some Solved Examples

Example 1: The ratio among most probable velocity, mean velocity, and root mean square velocity is given by

1)$1: 2: 3$

2)$1: \sqrt{2}: \sqrt{3}$

3)$\sqrt{2}: \sqrt{3}: \sqrt{\frac{8}{\pi}}$

4) $\sqrt{2}: \sqrt{\frac{8}{\pi}}: \sqrt{3}$

Solution

Most probable speed of gas molecules -

$V_{m p}=\sqrt{2 R T / M}$

M- Molecular Mass, R- Gas Constant, T- Temperature
most probable velocity: mean velocity : $V_{r m s}$

$=\sqrt{\frac{2 R T}{M}}: \sqrt{\frac{8 R T}{\pi M}}: \sqrt{\frac{3 R T}{M}}=\sqrt{2}: \sqrt{\frac{8}{\pi}}: \sqrt{3}$

Hence, the answer is the option (4).

Example 2: Calculate the $u_{r m s}$ (in m/sec) of $\mathrm{O}_2$ if its density at 1 atm pressure and $0^{\circ} \mathrm{O}$ is 1.429g/l.

Answer upto one decimal places

1) 462.2

2)46.1

3)56.3

4)44.2

Solution

It is the square root of the mean of the square of the velocities of different molecules.$\begin{aligned} & u_{r m s}=\frac{\sqrt{u_1^2+u_2^2+\ldots \ldots}}{n} \\ & =\frac{\sqrt{n_1 u_1^2+n_2 u_2^2+n_3 u_3^2}}{n_1+n_2+n_3} \\ & u_{r m s}=\sqrt{3 R T / M} \\ & u_{r m s}=\sqrt{3 P V / M}=\sqrt{3 P / d}\end{aligned}$

We know that rms velocity is given as:

$u_{r m s}=\frac{3 P}{d}$

Now, $P=1 \mathrm{~atm}=101.3 \times 10^3 \mathrm{~Pa}$

And d = 1.42g/litre = 1.42kg/

$u_{r m s}=\sqrt{\frac{3 \times 101.32 \times 10^3}{1.429}}=462.21 \mathrm{~m} / \mathrm{sec}$

Hence, the answer is the option (1).

Identify the correct labels A. B and C in the following graph from the options given below:

Example 3: Root mean square speed $\left(V_{r m s}\right)$ ; most probable speed $\left(V_{m p}\right)$, Average speed $\left(V_{a v}\right)$

1) A - $\left(V_{m p}\right)$, B- $\left(V_{a v}\right)$, C- $\left(V_{r m s}\right)$

2)A - $\left(V_{m p}\right)$, B- $\left(V_{r m s}\right)$ , C- $\left(V_{r m s}\right)$

3)A - $\left(V_{a v}\right)$, B- $\left(V_{r m s}\right)$ ,C- $\left(V_{a v}\right)$

4)A - $\left(V_{r m s}\right)$ ,B- $\left(V_{m p}\right)$, C- $\left(V_{a v}\right)$

Solution
The three types of molecular speeds, namely, most probable speed(v_mp), average speed (v_av), and root mean square speed(v_rms) of a gas at a given temperature are related to each other as follows:
$\begin{aligned} & v_{m p}: v_{a v}: v_{m s}=\sqrt{\frac{2 R T}{M}}: \sqrt{\frac{8 R T}{\pi M}}: \sqrt{\frac{3 R T}{M}} \\ & v_{m p}: v_{a v}: v_{r m s}=1.414: 1.596: 1.732 \\ & v_{m p}: v_{a v}: v_{r m s}=1: 1.128: 1.224\end{aligned}$

For a particular gas, at a particular temperature:
v_mp < v_av < v_rms

It follows from the above relationship that:
Average speed(v_av) =0.921 x Root mean square speed(v_rms)
Most probable speed(v_mp) = 0.817 x Root mean square speed(v_rms)

C_RMS > C_Average > C_MPS.

A - V_MPS, B - V_Average , C - V_RMS

Therefore, Option(1) is correct.

Summary

The kinetic theory of gases explains the macroscopic properties with respect to the motion and interaction of individual molecules in the gas. According to this theory, these gas molecules are subject to continuous, random motion, where the impacts with each other and the walls of the container are perfectly elastic. The theory is founded on several key postulates: gas molecules occupy negligible volume compared with the container; they interact with each other only through elastic collisions; their average kinetic energy is proportional to the absolute temperature of the gas. Those principles thus explain the pressure exerted by gases on the walls of a container as a manifestation of molecular collisions. The temperature of a gas is proportional to the average kinetic energy of the molecules, which thereby provides a molecular interpretation of the ideal gas law, PV=nRT. More than that, kinetic theory explains the gas behavior during diffusion and effusion processes where the light molecules are comparatively faster than the heavy ones.