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# Kinetic Theory of Gases

Gas molecules move randomly in the given space. If they are taken in a container they undergo collision with neighboring molecules and the container walls. It builds the pressure, volume and temperature. To study the action among gas molecules we study the model known as kinetic theory of Gas. Lets see more about it.

 Related Calculators Calculate Kinetic Energy Calculate Kinetic Friction

## Definition

Kinetic Theory of gases states that the gas made up of extremely small particles called molecules which itself made up of further small particles called atoms are always in a constant, random motion. The molecules and atoms are always in motion and collide with each other and also with the walls of the container in which it is kept.

Few assumptions of Kinetic theory of gases are as follows:

The molecules of a particular gas are identical within themselves but are different from the molecules of another gas.

The collision among the molecules and the walls of the container is perfectly elastic with no energy loss.

The size of the molecules as compared to the molecular distance is negligible, around $10^{-9} m$.

Speed of the molecules varies from very low speed (zero) to vry high speed (infinity).

Number of collisions made per unit volume is always constant.

Transfer of kinetic energy among the molecules is in the form of heat.

## Factors Affecting Kinetic Theory

There are primarily four factors affecting the kinetic theory of gases:

a) Temperature

b) Pressure

c) Volume

d) Number of moles determining the amount of gas
When the amount of gas is increased n times keeping the volume and the temperature constant, the container being unable to expand the particles exerts n times the pressure on the walls of the container. So, as $n1 > n2,\ P1 > P2$ ($V$ and $T$ remain constant.)

When the volume is decreased $n$ times, the particles gets less space to move around. Hence, the pressure exerted by the molecules also increases by $n$ times on the walls of the container. So, as $V1 > V2,\ P1 < P2$ ($n$ and $T$ are constant.)

When the temperature increases the molecules gains kinetic energy and thus starts moving with a higher speed. The volume being restricted, the pressure also increases accordingly. So, as $T1 > T2,\ P1 > P2$ ($n$ and $V$ being constant.)

## Kinetic Energy & Temperature of the Gas

The change in temperature affects the kinetic energy. They are directly proportional to each other. With the increase of temperature, the molecules gets agitated and starts vibrating away from its mean position travelling in higher speed than before. Thus, the kinetic energy of the particles increases as the temperature rises keeping the volume of the gas and the container in which the gas is kept constant so the number of molecules also remains constant.

## RMS Speed of Molecules

RMS speed for the root mean square speed. It is defined as the measure of the speed of the particles in a gas. The formula for RMS speed is as follows:

$v_{RMS}$ =  $\sqrt{(\frac{3RT}{M_{m}})}$

where,

$v_{RMS}$ is the root mean square of the speed in meters per second

$M_{m}$ is the molar mass of the gas in kilograms per mole

$R$ is the ideal gas constant = $8.314 kg\ \times\ m2/s2\ \times\ mol\ \times\ K$, and

$T$ is the temperature in Kelvin

Maxwell – Boltzmann Distribution:

$v_{RMS}$ =  $\sqrt{(\frac{3kT}{m})}$

where, $k$ is the Boltzmann constant and $m$ is the mass of one molecule of gas

Example: Calculate the Root Mean Square speed, $v_{RMS}$ in $m/s$ of Helium at $50^{\circ}C$

Solution: We need to convert the molar mass for Helium from $gm/mol$ to $kg/mol$

$M$ = $4.00\ gm/mol\ \times$ $\frac{1kg}{1000gm}$

$M$ = $4.00\ \times\ 10^{-3}$ $kg/mol$

We know that,

$v_{RMS}$ = $\sqrt{(\frac{3kT}{m})}$

$v_{RMS}$ = $\sqrt{\frac{(3 \times 8.314\ kg m^2/sec^2 \times mol \times K)(323 K)}{4.00 \times 10^{-3}}}$

$v_{RMS}$ = $1.42 \times 10^{3} m/s$

## Mean Free Path of the Gas Molecules

Mean free path of the gas molecules is the measure of how far the molecules travel between collisions. It is the average distance covered by the molecules in motion between collisions during diffusion when they collide with the walls of the container.

Formula for mean free path of the gas molecules is given by:

Mean free path $(\mu)$ = $\frac{1}{\sqrt{2 \pi d^{2}}}$ $(\frac{N}{V})$

Where, $d$ = diameter of the molecule

$(\frac{N}{V})$ = $density$

Factors Affecting Mean Free Path:

a) Density – Density and mean free path are inversely proportional to each other. As the density increases the molecules gets tightly packed such that the collision increases and the mean free path decreases accordingly

b) Radius of Molecule – Radius and the mean free path are inversely proportional to each other. As the radius keeps on increases the size of the molecules increases decreasing the intermolecular space between them and so the collision increases resulting in the decrease of the mean free path.
Example:

A gas has a density of $100$ particles $m^{-3}$ and a molecular diameter of $0.1\ m$. What is the mean free path?

Solution:

Mean free path of the gas molecules = $\frac{1}{\sqrt{2 \pi d^{2}}}$ $(\frac{N}{V})$

Where, $d$ = diameter of the molecule = $0.1\ m$

$(\frac{N}{V})$ = density = $100$ particles per meter cube

Therefore, mean free path of the gas molecules = $\frac{1}{\sqrt{2 \pi}}$ $(0.1)^{2}\ (100)$ = $0.225\ m$

## Ideal Gas Law

Ideal gas law is expressed as

$PV$ = $nRT$

Where, $n$ = number of moles

$R$ = Universal gas constant $(8.3145\ J/mol\ K)$

$P$ = Absolute pressure

$V$ = Volume

$T$ = Absolute temperature

Ideal gas law states that the collision between the molecules and atoms in a gas are perfectly elastic and the intermolecular forces of attraction among the molecules of the gas are absent. The volume occupied by the molecules is almost negligible as compared to the volume of the container in which the ideal gas is kept. The molecules behave as rigid spheres and the molecules move in straight line direction. In an ideal gas the total internal energy is in the form of kinetic energy and the change in internal energy is expressed as the the change in temperature
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