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.
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
Let us consider a gas molecule moving in the xdirection towards face A. The molecule hits the wall with a velocity V_{x} and rebounds back with the same velocity V_{x}, and will experience a change of momentum which is equal to
For a total of N number of gas molecules in the container, the total change in momentum is given by
The force is given by the equation
Therefore,
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
Substituting the value of t in the force equation, we get the force on the molecules as
Therefore, the force exerted on the wall is
Now, the pressure P is given by the equation
Hence,
Since v_{x}, v_{y} and v_{z} are independent speeds in three directions and if we consider the gas molecules in bulk, then
We know, volume V = l^{3}
Hence,
Substituting the above condition in eq (1), we get
Therefore,
This equation above is known as the kinetic theory equation.
The velocity v in the kinetic gas equation is known as the rootmeansquare velocity and is given by the equation
We use this equation to calculate the rootmeansquare velocity of gas molecules at any given temperature and pressure.
Not all the air molecules surrounding us travel at the same speed. Some air molecules travel faster, some move at moderate speeds, and some air molecules will hardly move at all. Hence, instead of asking about the speed of any particular gas molecule, we ask about the distribution of speed in a gas at a certain temperature. James Maxwell and Ludwig Boltzmann came up with a theory to show how the speeds of the molecules are distributed for an ideal gas. The distribution is often represented using the following graph.
Consider a system that consists of identical yet distinguishable particles.
We now want to know the number of particles at a given energy level.
The energy levels available within the system are given by ε_{0}, ε_{1}, ε_{2} ………., ε_{r}. The energy levels are fixed for the system.
The number of particles sitting at each energy level is variable, and the number of particles is given by n_{1}, n_{2}, n_{3} …….., n_{r}.
The number of ways to attain a given microstate is given by the formula
We need to know the combination of n1,n2,n3 up to nr that maximizes the value of ω. This combination results in the most probable microstate. If it is the most probable state, then it can be considered the equilibrium state. To find the combination, let us alter eq (1) as follows
To maximise the value of ω, instead of dealing with ω let us deal with \lnω.
According to Sterling’s approximation, the log of factorial can be approximated as
Making use of the Stirling approximation in eq (3), we get
By taking the derivative of the above equation, we get
The above equation implies that
As we know that n is a constant, therefore we can say that
This means that
The sum of all the changes in the system should sum up to zero.
The total energy of the system should also be constant.
Differentiating the above equation, we get
To maximise the function (4) subjected to constraints (5) and (6), let us use Lagrange’s method of undetermined multipliers.
Multiplying (5) by ɑ and multiplying (6) by 𝛽 and then adding 4, 5 and 6, we get
From eq (7) to be zero, the term
Simplifying the above equation, we get
The above equation signifies the number of particles in i th level.
In eq (8), the only variable parameter is ω, therefore taking the sum, we get
Simplifying, we get
The above equation is the same as
where P is the partition function,
Substituting (9) in (8), we get
The value of 𝛽 in the above equation is
where k_{b} is the Boltzmann constant and T is the temperature.
Substituting the value of 𝛽 in eq (10), we get
The above expression helps find the number of particles in the most probable microstate. This expression is known as the Maxwell Boltzmann statistics expression.
The kinetic theory of gases relates the macroscopic property of the gas, like – Temperature, Pressure, and Volume to the microscopic property of the gas, like – speed, momentum, and position. In this model, the atoms and molecules are continually in random motion, constantly colliding with one another and the walls of the container within which the gas is enclosed. It is this motion that results in physical properties such as heat and pressure. In this article, let us delve deeper into the kinetic theory of gases.
The gases are made up of a large number of molecules and they are flying in a random direction at a certain speed. By knowing their speed or position, one can figure out the macroscopic properties. In other words, by knowing the value of velocity or the internal energy of gas molecules, one should be able to figure out the temperature or pressure.
Kinetic Theory Assumptions
Consider a cubic box of length l filled with the gas molecule of mass m, moving along the xaxis with velocity v_{x }Therefore its momentum is mv_{x}.
The gas molecules collide with the walls. At wall 1, it collides and gains momentum mv_{x}.
Similarly, the molecules collide wall 2, reversing the momentum i.e., mv_{x}.
Thus, the change in the momentum is given by
Δp = mv_{x}(mv_{x}) = 2mv_{x}—–(1)
After the collision, the molecule travels a distance of 2l before colliding again with wall 1.
Thus, the time taken is given by
These continuous collisions carry a force, given by
Thus, by substituting the values from equations (1) and (2), we get
The continuous collisions also exert pressure on the wall given by
We know that the gas is made of N numbers of molecules and moves in all possible directions. Thus, the pressure exerted on wall 1 by the collision of N number of gas molecules is given by
where,
is the average velocity (or velocity component) of all gas molecules colliding with wall 1 along x
direction.
And l^{3} = V
On extending the above equation to three dimensions, we get
For a large number of gas molecules:
Substituting equation (7) in equation (6), we get
On rearranging,
We know that PV=nRT —(9)
Thus equating equations (8) and (9), we get
Multiplying and Dividing by 2, we get
Rearranging equation (10), we get
Here, N is the total number of gas molecules in a given region of space
n is the number of moles of gas present in a given region of space.
Avagadro’s number in this context is the number of molecules present in the one mole of gas.
NA = 6.022140857 × 10^{23}.
Substituting N_{A} in equation (11),
Thus, Average Kinetic Energy of a gas molecule is given by
For monoatomic molecules, the total internal energy is given by
According to the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, T is the temperature, and R is the universal gas constant. The Van der Waals equation is also known as the Van der Waals equation of state for real gases which do not follow ideal gas law. The Van der Waals Equation derivation is explained below.
Following is the derivation of Van der Waals equation for one mole of gas that is composed of noninteracting point particles which satisfies ideal gas law:
(proportionality between particle surface and number density)
(substituting nV_{m} = V)
Compressible fluids like polymers have varying specific volume which can be written as follows:
Where,
(p + A)(V  B) = CT
p: pressure
V: specific volume
T: temperature
A,B,C: parameters
This was the derivation of Van der Waals equation. Stay tuned with BYJU’S and learn various other Physics related topics.
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