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Derivation of Ideal Gas Equation from Kinetic Theory of Gases

26th May 2019 @ 9 min read

Physical Chemistry

Ideal gas equation is PV = nRT. This equation can easily be derived from the combination of Boyle’s law, Charles’s law, and Avogadro’s law. But here, we will derive the equation from the kinetic theory of gases. The kinetic theory of gases is a very important theory which relates macroscopic quantities like pressure to microscopic quantities like the velocity of gas molecules. This equation is applicable only for ideal gases, but be approximated for real gas under some conditions.

In 1856 German chemist August Kronig had developed the simple gas model. He had only considered the translational motion of gas particles in his model. Later, in 1857 Rudolf Clausius independent of Kronig developed a sophisticated version of the kinetic theory of gases. Clausius had not only considered translational motion but also the rotational and vibrational motion of gas molecules.

Assumptions in Ideal Gas Equation

The behaviour of gases is simplified significantly by making many assumptions. This is necessary to avoid complexity. The assumptions in the theory are as follows:

Derivation of Ideal Gas Equation

Visualise a cube in space as shown in the figure below. L be the length of the cube and Area, A. V be the volume of the cube.

The cube is filled with an ideal gas of pressure P, at temperature T. Let n and N be the moles and the number of molecules of the gas in the cube. m and M are the mass of a molecule and the gas.

A cube with the length L and the surface area A
Figure 1: A cube in space with length L and area A

Consider a single molecule moving with velocity v. Let vx, vy, and vz are the velocity components in the x, y, and z directions respectively.

Step 1: To Determine the Frequency of Collisions.

For simplicity, we will start with the x-direction as depicted in the below figure. Initially, at x = 0 a molecule with mass m moves with velocity vx. It strikes the wall at x = L. This is the first collision. The total distance travel in the first collision is L0 and the time taken is t1. The second collision takes place at t1 + t, and the distance travel between the 1st collision and 2nd collision is 2L. The time interval between the 1st and 2nd collision is ∆t = (t1 + t) −t1 = t. This is also shown in the figure below.

The figure depicts various instances of molecular collision with time.
Figure 2: Location of a molecule at different instant of time

For the 3rd collision, the time taken is t1 + 2t, but the time interval between preceding collision is the same ∆t = (t1 + 2t) − (t1 + t) = t. Also, the distance travel by the molecule between the 2nd and 3rd collision is the same 2L. This two quantities, time interval and distance travel, will remain the same for all successive collisions. So, we can calculate the speed vx which is distance travel divided time interval for all collisions as follows:

velocity of a molecule, v = lenght, 2L∕timw, t

Here v1, v2, v3, v4… are the velocities for 1st, 2nd, 3rd, 4th,… collisions, and they all are equal to vx.

Velocities are equal: v1=v2=v3…

Therefore, the time intervals for respective collisions are

time=distance∕velocity: t=2L/v

The average time taken t in k collisions is

The average time for collision = 2L∕v

When k tends to infinity, we get

The average time for collision, t = 2L∕v

This is the time period for a collision, and the reciprocal of it is frequency (f) of collisions.

Frequency of collision, f=1∕t=v∕2L

Step 2: To Derive Force

The force F is defined as the rate of change of momentum ∆p.

force is the rate of change of momentum.

The change in momentum is difference in momentum after a collision (−mvx) minus before a collision (mvx).

Change of momentum = final momentum − initial momentum

Note: The negative momentum means the molecule has lost the momentum after the collision.

The force exerted by wall Fwm on the molecule is given as:

Force, F=momentum, p × frequency, f

The Newton’s 3rd law says at every action there is an equal and opposite reaction. So, the force exerted by the molecule on the wall Fmw is opposite of the force exerted by the wall on the molecule Fwm.

force by wall = − force on wall

Using equation (1) and (2),

Force by molecule on wall, Fmw = mv2∕L

In the beginning, we have considered only one molecule; So, the above force equation is for only one molecule. The force by N molecules moving in x-direction is,

Force by molecules in moving in x-direction

Root mean square velocity v̄2 is given as

Root mean sqaure velocity, v rms sqaure

Substituting the above equation in (3),

Force in terms of root mean sqaure velocity

Note: This force is a force exerted by a molecule on one wall of the cube.

Step 3: To Determine the Pressure of the Gas

By knowing the force on the wall, we can determine pressure (P), which is force per unit area.

Pressure on the wall is force divided by area.

Here, V is the volume of the cube.

In equation (4), we have only incorporated the x direction. We need to replace v̄2
x
by v̄2.

From mathematics, the relation between v̄2
x
, v̄2
y
, and v̄2
z
and v̄2 is given as:

Vector addition: v^2=vx^2+vy^2+vz^2

Also, each of these velocities is equal.

Vectors in x, y, z-directions are equal in magnitude: vx^2=vy^2=vz^2.

From equation (5) and (6),

 vector v sqaure = vector vx sqaure+vector vy sqaure+vector vz sqaure Vector v sqaure is thrice vector vx square.

Substituting equation (7) in (4),

Pressure of ideal gas from the kinetic theory of gases, P = Nmv2∕3V

The equation (8) relates macroscopic pressure to microscopic velocity.

Step 4: To Determine Energy associated with Ideal Gas

The problem with equation (8) is that the rms velocity v̄2 is not a convenient quantity to measure. We will replace it with a more convenient quantity: temperature which can easily measure by a thermometer.

The kinetic energy equation from classical mechanics is half of the mass times square of velocity.

Kinetic energy, KE=1∕2 mv2

Also, from statistical mechanics, the kinetic energy for monatomic gas is given as:

Kinetic energy, KE = 3∕2 kT

Here, k is the Boltzmann constant and its approximate value is 1.380 × 10−23 J K−1.

Using equation (9) and (10)

Kinetic energy, KE = 3∕2 kT = 1∕2 mv2 mv2=3kT

Step 5: To Determine Ideal Gas Equation

From equation (8) and (11),

Pressure of gas, P =NkT∕V

The Boltzmann constant (k) is also ratio of the Gas constant (R) to the Avogadro’s constant (NA). Substituting Boltzmann's constant, kB = gas constant, R∕ Avogadro's constant, NA in equation the above equation,

Pressure of gas, P=NRT∕NA∕V

In the above equation, the ratio Ratio of N to NA is the moles of gas n.

Ideal gas eqaution, P=nRT∕V

Finally, the ideal gas equation, PV = nRT, is derived using the kinetic theory of gases.

Step 6: To Determine Root Mean Square Velocity from Ideal Gas Equation.

Root mean square velocity from equation (8) is given below.

Root mean square velocity, vector v2=3PV∕M

Here, M is the mass of the gas (M = mN).

Inserting the value of PV from the ideal gas equation,

v2=3RT∕Mw

In the above equation Mw is the molecular weight of the ideal gas.

Finally, the root mean square velocity is given as:

Root mean square velocity, v rms = square root (3RT∕Mw)

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