Derive the expression for molar kinetic energy of gas molecules from kinetic gas equation
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In order to explain the observed behavior of gases, a model was proposed based on the molecular and kinetic concepts of gas molecules.
This model takes into account the particulate nature of matter and the constant movement of particles. The various gas laws like Boyle’s law, Charles’s law, Dalton’s law of partial pressure, etc have been arrived at by experiment. There was no theoretical explanation for the behavior of gases. To account for the behavior of gases, a model was proposed which takes into account the absorbed behavior of gases.
A model was proposed which takes into account the molecular concept as well as the kinetic concept of gas molecules. The kinetic theory of gases has been devised to rationalize the behavior of gases. Bernoulli proposed this kinetic theory in 1738 and it was elaborated and developed by Clausius, Maxwell, Boltzmann, van der Waal and others.
The theory proposed to explain the particle nature of a gas is called kinetic theory of gases. The kinetic gas theory takes into account the particle nature of matter as well as their constant motion. Since its assumption relates to microscopic particles, this theory is also known as the ‘microscopic model
Kinetic Theory of Gases Definition
The kinetic molecular theory was proposed by Kronig, Clausius, Maxwell and Boltzmann in the nineteenth century. This theory proved to be a sound theoretical base to the various gas laws like Boyle's law, Charles Law, Grahams law, Dalton's law, etc. Kinetic theory is basically a theoretical proof for all the gas laws. Kinetic theory is defined as "a theory that gases consist of small particles in random motion."
Expression for the Pressure Exerted by the Gas
From the above postulates or assumptions of kinetic molecular theory of gases, it is possible, by applying the laws of classical mechanics, to derive an expression for the pressure of a gas.
Consider a gas enclosed in a cube, the sides of which are I meters long.
Let the number of molecules in the cube be c and let the mass of each molecule be m kg. Let the square mean square velocity of each molecules be I meters.
The molecules present in the cube will be moving in all possible directions. But since they do not prefer any particular direction and also because the pressure exerted on all sides is the same, it may be imagined that at any time one- third of the molecules move perpendicular to each pair of the faces, i.e., n / 3 molecules along x- axis, n / 3 along y-axis and n / 3 along the z-axis.
Now consider the motion of a single molecule striking against face A of the cube. Its momentum = mc. After striking the cube it will rebound with the same speed in the opposite direction. Therefore,
Change in momentum of the molecule colliding
= Initial momentum - Final momentum
= mc - (-mc) = 2mc.
Before another collision can occur with this small wall, the molecule must travel a distance of 2l. Number of collisions per second by a molecule on one face = c/2l.
So, Total change in momentum per second = 2mc x c/2l
Molecule at the face A = mc2/l
For n/3 molecules striking against the face A, the total change of momentum per second
= n / 3 x mc2/l = 1/3 mnc2 / l.
But change of momentum per second = force.
Therefore, total force on the face A of area l2 = 1/3 mnc2 / l.
Pressure = Force /Area
P = 1/3 mnc2 / l.
= 1/3. mnc2 / l3
l3 is the volume. So,
P = 1/3 mnc2 / V
This is called as the kinetic gas equation. This is applicable for ideal gases only.
Types of Molecular Velocities
The three types of molecular velocities are
Most probable velocityAverage velocityRoot mean square velocity
1. Most probable velocity (Cp)
This is defined as M the velocity possessed by maximum fraction of the total number of molecules of the gas at a given temperature.
Cp =2RTM−−−−√2RTM
Where
R = gas constant
T= temperature and
M is the molecular weight
2. Average velocity
It is expressed as C. Average velocity is the mean of the velocities of all the molecules. It is calculated using the formula.
C¯=8RTπM−−−−√C¯=8RTπM
3. Root mean square velocity
It is defined as the square root of the mean of the square of the velocities of all the molecules of the gas.
Root mean square velocity = 1.085 x Average velocity
Expression for Kinetic energy
According to kinetic gas equation,
PV = 1313mnc2c2
Where M = mn = molar mass of the gas.
Therefore,
PV = 2323 x 1212 Mc2c2
½ Mc2 is the kinetic energy of the gas.
So, Kinetic energy,
K.E = 3RT23RT2s−1s−1
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Explanation:
It is calculated using the formula. It is defined as the square root of the mean of the square of the velocities of all the molecules of the gas. Where M = mn = molar mass of the gas. ½ Mc2 is the kinetic energy of the gas
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