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26th May 2019 @ 9 min read
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.
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:
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.
Consider a single molecule moving with velocity v⃗. Let v⃗x, v⃗y, and v⃗z are the velocity components in the x, y, and z directions respectively.
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 v⃗x. 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.
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:
Here v1, v2, v3, v4… are the velocities for 1st, 2nd, 3rd, 4th,… collisions, and they all are equal to vx.
Therefore, the time intervals for respective collisions are
The average time taken t in k collisions is
When k tends to infinity, we get
This is the time period for a collision, and the reciprocal of it is frequency (f) of collisions.
The force F is defined as the rate of change of momentum ∆p.
The change in momentum is difference in momentum after a collision (−mvx) minus before a collision (mvx).
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:
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.
Using equation (1) and (2),
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,
Root mean square velocity v̄2 is given as
Substituting the above equation in (3),
Note: This force is a force exerted by a molecule on one wall of the cube.
By knowing the force on the wall, we can determine pressure (P), which is force per unit 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
y, and v̄2
z and v̄2 is given as:
Also, each of these velocities is equal.
From equation (5) and (6),
Substituting equation (7) in (4),
The equation (8) relates macroscopic pressure to microscopic velocity.
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.
Also, from statistical mechanics, the kinetic energy for monatomic gas is given as:
Here, k is the Boltzmann constant and its approximate value is 1.380 × 10−23 J K−1.
Using equation (9) and (10)
From equation (8) and (11),
The Boltzmann constant (k) is also ratio of the Gas constant (R) to the Avogadro’s constant (NA). Substituting in equation the above equation,
In the above equation, the ratio is the moles of gas n.
Finally, the ideal gas equation, PV = nRT, is derived using the kinetic theory of gases.
Root mean square velocity from equation (8) is given below.
Here, M is the mass of the gas (M = mN).
Inserting the value of PV from the ideal gas equation,
In the above equation Mw is the molecular weight of the ideal gas.
Finally, the root mean square velocity is given as:
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