Kinetic Gas Equation Derivation

Kinetic Gas Equation Derivation

What Is Kinetic Theory of Gases?

The kinetic theory of gases describes a 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

\(\begin{array}{l}P=\frac{F}{A}\end{array} \)

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

\(\begin{array}{l}\Delta p=-mv_x-(mv_x)\end{array} \)



\(\begin{array}{l}\Delta p=-2mv_x\end{array} \)

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

\(\begin{array}{l}\Delta p=-2Nmv_x\end{array} \)

The force is given by the equation

\(\begin{array}{l}F=\frac{\Delta p}{t}\end{array} \)

Therefore,

\(\begin{array}{l}F=\frac{-2Nmv_{x}}{t}\end{array} \)

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

\(\begin{array}{l}t=\frac{2l}{v_x}\end{array} \)

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

\(\begin{array}{l}F=\frac{-2Nmv_x}{\frac{2l}{v_x}}\end{array} \)



\(\begin{array}{l}F_{molecules}=\frac{-2Nmv_x}{\frac{2l}{v_x}}=\frac{-Nmv_{x}^2}{l}\end{array} \)

Therefore, the force exerted on the wall is

\(\begin{array}{l}F_{wall}=\frac{Nmv_{x}^2}{l}\end{array} \)
.

Now, the pressure P is given by the equation

\(\begin{array}{l}P=\frac{Force\,on\,the\,wall}{Area}=\frac{\frac{Nmv_{x}^2}{l}}{l^2}=\frac{Nmv_{x}^2}{l^3}\end{array} \)

Hence,

\(\begin{array}{l}PV={Nmv_{x}^2}..(1)\end{array} \)

Since vx, vy and vZ are independent speeds in three directions and if we consider the gas molecules in bulk, then

\(\begin{array}{l}V_x^2=V_y^2=V_z^2\end{array} \)



We know, volume V = l3

Hence,

\(\begin{array}{l}v^2=3v_x^2\end{array} \)

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

\(\begin{array}{l}PV=\frac{Nmv^2}{3}\end{array} \)

Therefore,

\(\begin{array}{l}PV=\frac{1}{3}mNv^2\end{array} \)

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

\(\begin{array}{l}V_{rms}=\frac{\sqrt{v_1^2+v_2^2+v_3^2……..+v_n^2}}{N}\end{array} \)

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

Stay tuned to BYJU’S to learn more important physics derivation.

Also, Read

Derivation of Biot Savart’s Law

Derivation of Lens Formula

Derivation of Equation of Motion

Stoke’s Law Derivation

Frequently Asked Questions – FAQs

Q1

What is the kinetic theory of gases?

The kinetic theory of gases describes a 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.
Q2

What are the assumptions of kinetic theory?

Following are the assumptions of kinetic gas theory:
  • The molecules do not interact with each other.
  • The collision of molecules with themselves or walls will be an elastic collision.
  • The momentum is conserved.
  • Kinetic energy will be conserved.
Q3

What is the ideal gas law?

The ideal gas law states that the product of the pressure and the volume of one gram molecule of an ideal gas is equal to the product of the absolute temperature of the gas and the universal gas constant.
Q4

Why is the ideal gas inaccurate

The ideal gas only holds true when the conditions at consideration are ideal. Under high pressure and low temperature, the molecular size and the intermolecular forces become important to be considered and are no longer negligible, so essentially the ideal gas law won’t work.
Q5

Who derived the ideal gas equation?

Benoît Paul Émile Clapeyron derived the ideal gas.
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  1.  Joyce
    August 21, 2019 at 11:01 pm

    The lesson is great

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