## Kinetic Theory of Gases Formula

**Kinetic theory of gases** and the **kinetic gas equation** was first developed by Bernoulli in 1738 to derive the molecular properties of gas molecules on the basis of the ideal gas law and mechanical energy formula. For the study of physics and chemistry, the kinetic theory of gases consists of many postulates or assumptions. In the nineteenth century, the effort of Joule, Kronig, Clausius, Boltzmann, and Maxwell, gives the postulates or assumptions of the kinetic theory of gases and kinetic gas equation or formula on the basis of root mean square velocity (RMS) and momentum of the gas molecule.

Molecules in the gaseous state of matter move at very large speeds and the forces of attraction are not sufficient to bind the molecules in one place.

## Solid Liquid and Gas Molecules

**Solid Molecules:**Solid molecules or particles are held very closely together and are entirely devoid of any translatory motion. When a specific heat is supplied to a crystalline solid, it takes the form of vibrational motion with the rise of temperature.**Liquid Molecules:**Further increases the thermal energy, the vibrational motion rises, and the molecules break down to transform into a liquid state like ice to water.**Gas Molecules:**When the thermal energy is much greater than the forces of attraction, then we have a gaseous state of matter. The gas molecules move randomly and collide with the walls of the container (wall collision) and with themselves (intermolecular collision). These collisions are perfectly elastic. Therefore, there occurs conservation of energy because of no loss of kinetic energy or momentum of the molecules by this collision.

## Postulates of Kinetic Theory of Gases

To derive the kinetic equation or formula, the kinetic theory of gases consists of the following postulates or assumptions,

- Gas molecules are composed of very small discrete particles. For the same gases, the mass and size of molecules are the same but the mass and size are different for different gases.
- The molecules are moving randomly in all directions of space at a variety of speeds. Some are very fast others are slow.
- Due to random motion, the gas molecules are executing two types of collisions. When it collides with the walls of the container called a wall collision but with themselves called an intermolecular collision.
- Gas molecules are assumed to be point masses. Hence the gas sizes are very small in comparison to the distance where they travel.
- Especially at low pressure, the gas molecules have no intermolecular attraction. Therefore, one molecule can exert pressure independent of the influence of other molecules.
- The pressure exerted by a gas is due to the uniform wall collisions. Hence higher the frequency of the wall collision greater the pressure of the gas. It explains Boyle’s law. When the volume is reduced at a constant temperature, wall collision becomes more frequent and pressure is increased.
- The molecular velocity constantly changes due to the intermolecular collision but the average kinetic energy of the gas molecules remains fixed at a given temperature.

## Derivation of Kinetic Gas Equation

Let us take a cubic container with edge length l containing N molecules of gas of molecular mass = m, and RMS speed = C_{RMS} at temperature T and pressure P.

Among these molecules, N_{1} has velocity C_{1}, N_{2} has velocity C_{2}, N_{3} has velocity C_{3} and so on.

Let us concentrate our calculation on a single molecule among N_{1} that has resultant velocity C_{1} and the component velocities are C_{x}, C_{y}, and C_{z}.

∴ C_{1}^{2} = C_{x}^{2} + C_{y}^{2} + C_{z}^{2}

If the molecule collides with walls A and B of the container with the component velocity C_{x} and other opposite faces by C_{y} and C_{z}. Therefore, the change of momentum along the X-direction for a single collision,

= m C_{x} − (− m C_{x})

= 2 m C_{x}

A rate change of momentum along X, Y, and Z-direction is 2mC_{x}^{2}/l, 2mC_{y}^{2}/l, and 2mC_{z}^{2}/l.

The total rate change of momentum for the molecule given above the picture,

= 2m(C_{x}^{2} + C_{y}^{2} + C_{z}^{2})/l

= 2mC_{1}^{2}/l

For N_{1} molecules,

Change of momentum = 2mN_{1}C_{1}^{2}/l

If we consider all the molecules of the gas present in this cubic container,

A total change of momentum = 2mNC_{RMS}^{2}/l

Where C_{RMS} = root means square velocity

## Kinetic Gas Equation

According to Newton’s second law of motion, the rate of change of momentum due to wall collision is equal to the force developed within the gases.

Therefore, P × 6l^{2} = (2mNC_{RMS2})/l

or, P × 3l_{3} = mNC_{RMS2}

or, P × l_{3} = mNC_{RMS2}/3

∴ **PV = mNC _{RMS2}/3**

The kinetic gas equation derived from the kinetic theory of gases is used to derive the root mean square velocity and density formula of the gas molecules. It is valid for any shape of the container of our environment.

## Root Mean Square Speed

Root mean square speed or RMS is defined as the square root of the average of the squares of speeds of the gases. The kinetic gas equation and ideal gas law may be used to formulate the RMS velocity of the gases.

### Root Mean Square Speed Formula

From the kinetic gas equation,

PV = (mN C_{RMS}^{2})/3

where mN = M = molecular mass of the gases.

The ideal gas equation for 1 mole gas,

PV = RT

Therefore, 3RT = M × C_{RMS}^{2}

or, C_{RMS}^{2} = 3RT/M

### Root Mean Square Velocity of Oxygen and Hydrogen

From the above equation, root means square velocity (RMS) depends on the molar mass and temperature of the gases. Therefore, at a given temperature RMS velocity decreases with the increasing molecular weight of gas molecules.

For oxygen, C_{RMS}^{2} = 3RT/16

For hydrogen, C_{RMS}^{2} = 3RT

∴ C_{RMS}^{2} for oxygen/C_{RMS}^{2} for hydrogen = 1/16

or, C_{RMS} for oxygen/C_{RMS} for hydrogen = 1/4

Hence the RMS velocity of the hydrogen molecule is four times greater than the oxygen molecule.

**Problem:** Calculate the pressure of 10^{23} gas molecules each of the molecules having mass = 10^{−22} g and a container of volume = 1 dm^{3}. Given C_{RMS} 10^{5} cm s^{−1}.

**Solution:** Numer of molecules (N) = 10^{23}

mass (m) = 10^{−22} g = 10^{−25} Kg

volume (V) = 1 dm^{3} = 10^{−3 }m^{3}

C_{RMS} = 10^{5} cm sec^{−1} = 10^{3} m sec^{−1}

From the kinetic gas equation,

pressure (P) = (10^{−25} × 10^{23} × 10^{−6})/(3 × 10^{3})

= 0.333 × 10^{7} Pascal (Pa)

## Kinetic Energy of Gases Formula

### Average Kinetic Energy Formula

The average kinetic energy is defined as,

E_{average} = mC_{RMS}^{2}/2

However, from the kinetic gas equation,

PV = (2 × N × E_{average})/3

The ideal gas equation is PV = RT and N = N_{0}.

∴ RT = (2 × N_{0} × E_{average})/3

Hence average kinetic energy,

= 3RT/2N_{0} = 3kT/2

Where k = R/N_{0} = Boltzmann constant

= 1.38 × 10^{−23} J K^{−1}

Therefore, the average kinetic energy is dependent on temperature only but independent of the nature of the gases.

### Total Kinetic Energy Formula

The total kinetic energy for 1 mole gas = Avogadro number × average kinetic energy.

The total kinetic energy for 1 mole gases,

E_{kinetic energy} = N_{0} × (3RT/2N_{0})

= 3RT/2

For n-mole gases,

E_{kinetic energy} = 3nRT/2

**Question:** Calculate the average kinetic energy for hydrocarbon like methane gas at a temperature of 27 ⁰C.

**Answer:** From the kinetic theory of gases, the formula for average kinetic energy per molecule or kinetic energy of 1 mole of methane,

KE_{methane} = (3 × 2 × 300)/2

= 900 calories

## Derivation of Gas Laws from Kinetic Gas Equation

From the average kinetic energy equation,

PV = 2NE_{average}/3, but E_{average} = k’T

From the above two equations,

PV = 2Nk’T/3

Such energy equation calculates the necessary deductions of gas law like Boyle’s, Charles’s, and Avogadro’s law.

### Charles Law from Kinetic Gas Equation

From the energy equation,

PV = 2Nk’T/3

When the pressure is kept constant, V ∝ T because N and k’ have constant values. Hence at constant pressure, the volume of the gas is proportional to its kelvin temperature. Therefore, Charles’s law can be derived from the kinetic gas equation.

### Boyle’s Law from Kinetic Gas Equation

From the energy equation, when the temperature is kept constant, PV = constant. Hence the volume of the gas is inversely proportional to its pressure. Therefore, Boyles’s law can be derived from the kinetic gas equation.

The postulates or assumptions of the kinetic molecular theory of gases and the kinetic gas equation or formula are also used to calculate the kinetic energy, density, diffusion, or effusion of gas molecules.