Kinetic Gas Equation Derivation

Bernoulli established the kinetic theory of gases and the kinetic gas equation for the first time in 1738 to obtain the molecular characteristics of gas molecules using the ideal gas law and mechanical energy. On the basis of root mean square velocity (RMS) and momentum of the gas molecule, Joule, Kronig, Clausius, Boltzmann, and Maxwell developed the postulates and formula of the kinetic gas equation in the nineteenth century. The kinetic theory of the gases model considers that molecules are very small in comparison to the distance between them. The molecules are in continual, random motion, colliding with each other and the container’s walls on a regular basis.

Kinetic Theory of Gas

The kinetic theory of gases is commonly used to explain gas molecule behaviour. It is mostly the study of gas molecules at the macroscopic level. The following are the five postulates of the kinetic theory of gases:

Gas is made up of a great number of molecules that move about at random all the time.

The volume of gas molecules is negligible since the distance between them is frequently higher than the size of the molecules.

There are no intermolecular interactions.

Molecules collide, and the container’s walls are always elastomeric.

The average kinetic energy of all molecules can be affected by temperature.

Factors Affecting the Behaviour of Gases

Temperature (T): The pressure of gas molecules increases as the temperature rises.

When the volume of a container is lowered, the gas molecules have less room to move around. As a result, they’ll hit the container’s walls more often, increasing the pressure.

At any given steady temperature, as the pressure of the gas increases, the volume of the gas decreases.

Quantity (n): As the number of gas molecules in a particular volume container grows, pressure rises.

The thermodynamic behaviour of gases is defined by the kinetic theory of gases. The theory explains a gas’s macroscopic quantities in terms of the microscopic nature of the atoms and molecules that make up the gas’s composition. The physical nature of solids and liquids can be characterised by their shape, size, mass, and volume, but the kinetic theory can be applied to gases because they have no distinct shape or size and their volume or mass cannot be observed directly.

Kinetic Molecular Theory of Gases

It explains some transfer qualities like viscosity, thermal conductivity, and mass diffusivity, as well as the three properties of gases: volume, pressure, and temperature. This theory is based on attempts to construct a relationship between macroscopic qualities and microscopic phenomena, as well as the molecular structure of the gas in terms of a large number of sub-microscopic particles. The molecules of the gases are said to be in constant random motion, and particles have random elastic collisions with each other and the container’s enclosing walls. The theory presupposes a large inter-particle distance and that the particles are relatively small in size.

Derivation of Kinetic Gas Equation

Consider a cubical container with a length of l‘ filled with gas molecules, each with a mass of m and a total number of gas molecules in the container of N. The gas molecules travel in random directions at a velocity of V due to the influence of temperature.

The pressure of the gas molecules is defined as the force exerted by each gas molecule per unit area of the container’s wall, and is calculated using the equation.

 P=F/A

Consider a gas molecule travelling towards face A in the X-direction. The molecule collides with the wall with a velocity VXand returns with the same velocity VX resulting in a shift of momentum equal to

 Δp= -2m VX

For a total no of N gas molecules change in momentum will be given as:

 Δp= -2Nm VX

Here the force is given as:

 F= p/t

Hence F= -2Nm VX/t

After hitting wall A, gas molecules will flow back across the box, collide with the opposite face, and then hit wall A again after a time t determined by the equation.

 t=2l/VX

Now substituting the value of t in force equation we get:

 F= -2NmVX /2l/VX

 Fmolecules= –2NmVX 2lVX=-NmVX2/l

Hence, the force exerted on will be:

 Fwall=NmVX2/l

Now the pressure will be given as:

 P=FA=NmVX2l3

Hence PV=NmVX2

Because VX , VY and Vzare three independent speeds in three directions, and we examine gas molecules in bulk:

 VX2= Vy2= Vz2

Hence V2=3  VX2

Now substituting the above equation we get :

 PV=NmV2/3

Therefore,

  PV=13 NmV2

Hence, the above equation is known as the kinetic theory equation.

Conclusion

The kinetic theory of gases is commonly used to explain gas molecule behaviour. The kinetic theory of the gases model considers that molecules are very small in comparison to the distance between them. Bernoulli established the kinetic theory of gases and the kinetic gas equation for the first time in 1738 to obtain the molecular characteristics of gas molecules using the ideal gas law. There are no intermolecular interactions in the kinetic theory of gases. Kinetic molecular theory explains some transfer qualities like viscosity, thermal conductivity, and mass diffusivity, as well as the three properties of gases: volume, pressure, and temperature. The physical nature of solids and liquids can be characterised by their shape, size, mass, and volume.