## What is Kinetic Theory of Gases?

**Kinetic theory of gases** and **kinetic gas equation** first-time developed by Bernoulli in 1738 to derive the molecular properties of gas molecules on the basis of the ideal gas law and mechanical energy. In the nineteenth century, the effort of Joule, Kronig, Clausius, Boltzmann, and Maxwell, gives the postulates and formula of the kinetic gas equation on the basis of root mean square velocity (RMS) and momentum of the gas molecule. Molecules in the gaseous state of matter move with very large speeds and the forces of attraction not sufficient to bind the molecules in one place which is given below the picture,

#### Solid liquid gas

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. Further increases the thermal energy, the vibrational motion rises, and the molecules break down to transform into a liquid state like ice to water. When the thermal energy much greater than the forces of attraction, then we have a gaseous state of matter.

### Postulates of Kinetic Theory of Gases

For study physics and chemistry, kinetic theory of gases consists following postulates or assumptions,

- Gas molecules are composed of very small discrete particles. For a gas, the mass and size of molecules are the same and different for different gases.
- The molecules are moving randomly in all directions of space with 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 collided with the walls of the container called wall collision but with themselves called an 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.
- 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 will be 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.
- Through the molecular velocity constantly changing 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 unit or single-molecule among N_{1} that has resultant velocity C_{1} and the component velocities are C_{x}, C_{y}, C_{z}. Therefore, C_{1}^{2} = C_{x}^{2} + C_{y}^{2} + C_{z}^{2}.

If the molecule will collide 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 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. Total rate change of momentum for the molecule given on 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, total change of momentum = 2mNC_{RMS}^{2}/l, where C_{RMS} = root means square velocity.

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

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

or, P × 3l^{3} = mNC_{RMS}^{2
}or, P × l^{3} = mNC_{RMS}^{2}/3

The kinetic gas equation derived from the kinetic theory of gases uses to calculate the root mean square velocity and density 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 gases, 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. Hence the RMS velocity of the hydrogen molecule four times greater than the oxygen molecule.

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

**Solution:** Numer of molecules (N) = 10^{23}, mass (m) = 10^{-22} gm = 10^{-25} Kg, volume (V) = 1 dm^{3} = 10^{-3} m^{3 }and 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.

### Kinetic Energy of Gas Molecules

#### Average kinetic energy formula

The average kinetic energy is defined as, E_{average} = mC_{RMS}^{2}/2. Again from the kinetic gas equation, PV = (2 × N × E_{average})/3. For ideal gas equation PV = RT and N = N_{0}. Therefore, 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 dependent on temperature only but independent of the nature of the gases.

#### Total kinetic energy formula

Again, the total kinetic energy for 1-mole gas = Avogadro number × average kinetic energy.

∴ 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 27⁰C.

**Answer:** From the kinetic theory of gases, the average kinetic energy per molecule or kinetic energy of 1-mole methane, KE_{methane} = (3 × 2 × 300)/2 = 900 calories.

### Kinetic Theory of Gases and Gas Law

From the average kinetic energy equation, PV = 2NE_{average}/3, but E_{average} = k’T. Therefore, PV = 2Nk’T/3, this energy equation calculates the necessary deductions of gas law like Boyle’s, Charles’s, and Avogadro’s law.

#### Charles law from kinetic theory of gases

From this equation, PV = 2Nk’T/3. When the pressure is kept constant, V ∝ T because N and k’ have constant values. Therefore, at constant pressure, the volume of the gas proportional to its kelvin temperature. This is Charles’s law.

#### Boyle’s law from kinetic theory of gases

When the temperature is kept constant, PV = constant. Therefore, the volume of the gas inversely proportional to its pressure. This is the Boyles law. The kinetic molecular theory or gas equation is also used to calculate the kinetic energy, density, diffusion, or effusion of gas molecules.