## Kinetic Theory and Kinetic Energy

**Kinetic theory** or kinetic molecular equation of gases first-time developed by Bernoulli in 1738 to calculate the molecular properties of gas molecules on the basis of the law of mechanical energy in learning chemistry or physics. This expression is the root law to explain the **kinetic** **energy** and molecular motion of gases. In the nineteenth century, the effort of Joule, Kronig, Clausius, Boltzmann, and Maxwell, gives the postulates and equation formula of the kinetic theory of gases on the basis of root mean square velocity (RMS) and momentum of the gas molecule. Solid molecules or particles are held very closely together and are entirely devoid of any translatory motion. When a specific heat supplied to 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 the liquid state like ice to water.

### Postulates of Kinetic Theory of Gas Molecules

The kinetic theory of gases postulates was developed for the formulation of the kinetic gas equation for study physics and chemistry.

- 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 dimensions 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. But in real or Van der Waals gas molecules has definite mass and volume.
- 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.
- This explains Boyle’s law since 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.

### Kinetic Gas Equation Formula Derivation

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}

Therefore, 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 = 2mC_{RMS}^{2}/l

where C_{RMS} = root means square velocity

But 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.

∴ P × 6l^{2} = (2mNC_{RMS}^{2})/l

or, P × 3l^{3} = mNC_{RMS}^{2}

These two gas formulas derived from the kinetic theory of gases uses to calculate the root mean square velocity and density of the gas molecules and valid for any shape of the container of our environment.

### RMS Velocity or Motion in Gas Molecules

RMS or root mean square speed 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. From the kinetic gas formula, 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

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 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 Formula and Kinetic Gas Equation

The average kinetic energy 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

or, 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. 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, calculate average kinetic energy per molecules means kinetic energy of 1-mole methane. Therefore, KE_{methane} = (3 × 2 × 300)/2 = 900 calories.

### Kinetic Molecular Theory 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 calculate the necessary deductions of gas law like Boyle’s, Charles, and Avogadro law.

- From this equation, when the pressure kept constant, the volume of the gas proportional to its kelvin temperature. This is Charles’s law.
- When the temperature kept constant, the volume of the gas inversely proportional to its pressure. This is the Boyles law.

Therefore, the kinetic molecular theory and equation of gases used to calculate the kinetic energy, density, diffusion or effusion of gas molecules based on certain postulates are applicable to derive the ideal gas law in chemistry and physics.