Question

drive gas equation by kinetic equation​

Answer 1

What Is Kinetic Theory Of Gases?

The kinetic theory of gases describes gas as a large number of submicroscopic particles such as atoms and molecules, all of which are in random and constant motion. The randomness arises from the particles collisions with each other and with the walls of the container.

Derivation Of Kinetic Gas Equation

Derivation Of Kinetic Gas Equation

Consider a cubical container of length ‘l ’ filled with gas molecules each having mass ‘m’ and let N be the total number of gas molecules in the container. Due to the influence of temperature, the gas molecules move in random directions with a velocity ‘v.’

The pressure of the gas molecules is the force exerted by the gas molecule per unit area of the wall of the container and is given by the equation

P=FA

Let us consider a gas molecule moving in the x-direction towards face A. The molecule hits the wall with a velocity Vx and rebounds back with the same velocity Vx, and will experience a change of momentum which is equal to Δp=−2mVx.

For a total of N number of gas molecules in the container, all such change in momentum is given by

Δp=−2NmVx

The force is given by the equation

F=Δpt

Therefore,

F=−2NmVxt

Gas molecules will hit the wall A and will travel back across the box, collide with the opposite face and hit face A again after a time t which is given by the equation

t=2lVx

Substituting the value of t in the force equation, we get the force on the molecules as

F=−2NmVx2lVx

Fmolecules=−2NmVx2lVx=−NmV2xl

Therefore, the force exerted on the wall is Fwall=NmV2xl.

Now, the pressure P is given by the equation

P=ForceonthewallArea=NmV2xll2=NmV2xl3

Hence, PV=NmV2x (1)

Since Vx, Vy and VZ are independent speed in three directions and if we consider the gas molecules in bulk, then

V2x=V2y=V2z

Hence,

V2=3V2x

Substituting the above condition in eq (1), we get

PV=NmV23

Therefore,

PV=13mNV2

This equation above is known as the kinetic theory equation.

The velocity V in the kinetic gas equation is known as the root-mean-square velocity and is given by the equation

Vrms=V21+V22+V23……..+V2n√N

We use this equation to calculate the root-mean-square velocity of gas molecules at any given temperature and pressure

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