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Let us understand the relation between kinetic energy and gas pressure. When gas is stored in a container, the molecules of the gas, which are in a state of random movement, continuously collide with each other and the walls of the container. Due to this continuous collision, a steady force is exerted on the wall of the container. The force exerted per unit area of the wall is called gas pressure. Since the gas molecules are in random motion, they possess certain kinetic energy. The kinetic energy of gas per unit volume is related to the pressure of gas by the following equation:
K.E = 3/2.P
K.E = kinetic energy of gas per unit volume; P = pressure of gas
This equation is derived on the basis of the Kinetic Theory of Gases.
Kinetic Theory of Gases
The Kinetic Theory of Gases states that the molecules of gases are in continuous and random motion. These rapidly moving molecules collide with each other and also with the walls of the container and exert a force on the walls. This force per unit area is called pressure of the gas. The five postulates of the Kinetic Theory of Gases are:
- The volume occupied by the molecules of gas is negligible compared to the volume of the gas itself.
This means that molecules of gas are separated by a distance that is much larger than the size (diameter) of the molecules themselves.
- The molecules of an ideal gas do not exert attractive forces on each other.
- The molecules of gas are in continuous and random motion. They follow Newton’s Law of Motion.
This means that the molecules move in straight lines until they collide with each other or with the walls of the container.
- Collisions of the molecules with each other are totally elastic.
This means when two molecules collide, they change their directions and kinetic energies, but the total kinetic energy is conserved.
- The average kinetic energy of the gas molecules is proportional to the absolute temperature.
What is Gas Pressure?
The force per unit area exerted by gas molecules on the walls of the container is called gas pressure. When molecules collide with the walls of the container, their momentum changes.
According to Newton’s Second Law of Motion, the rate of change in the momentum of gas molecules is equal to the force exerted by them on the walls of the container. The force exerted by the molecules per unit area of the wall of the container generates gas pressure.
According to the Kinetic Theory of Gases, gas pressure is represented by the following equation:
P= ⅓ .2
Where, P = gas pressure
= density of the gas molecules
= root mean square speed of gas molecules
Kinetic energy of gas molecules
The molecules of gas are in random motion. Due to this random and continuous motion, the gas molecules attain a certain energy. This energy is known as the kinetic energy of gas molecules.
The expression for kinetic energy is given by the following equation:
K.E = ½.mv2
Where, K.E = kinetic energy of gas molecules
m = mass of gas molecules
= root mean square speed of gas molecules
Root mean square velocity of molecules
The square root of the mean of squares of the velocity of each gas molecule is called root mean square velocity of gas molecules.
It is denoted by the symbol vrms
The following equation represents how root mean square velocity of gas molecules is calculated:
vrms = 3RT/M
Where, M = molar mass of the gas
R = gas constant
T = temperature
Derivation of the relation between kinetic energy per unit volume and gas pressure
The equation which establishes the relation between kinetic energy per unit volume and gas pressure is derived based on the Kinetic Theory of Gases.
Kinetic energy is denoted by the following equation:
K.E = ½.mv2
Where, K.E = kinetic energy of gas molecules
m = mass of gas molecules
= root mean square speed of gas molecules
Since density = mass/volume
mass = density x volume
Mass = v
Putting the value of mass (m) in the equation of kinetic energy we get:
K.E = ½.Vv2
This is the kinetic energy of gas.
Kinetic energy of gas per unit volume = K.E / volume (v)
Kinetic energy of gas per unit volume = ½.v2 ………….. (i)
According to the Kinetic Theory of Gases, gas pressure is represented by the following equation:
P = ⅓ .v2 ………….. (ii)
Where, P = gas pressure
= density of the gas molecules
= root mean square speed of gas molecules
The relation between kinetic energy per unit volume and gas pressure is obtained by dividing equation (i) by equation (ii):
K.E/P = ½ .2/ ⅓ .2
K.E/P = 3/2
K.E = 3/2 x p
Hence, the kinetic energy of gas per unit volume is related to pressure of gas by the following equation:
K.E = 3/2.P
(where, K.E = kinetic energy of gas per unit volume; P = pressure of gas)
Conclusion
According to the Kinetic Theory of Gases, the molecules of a gas are in random motion. As a consequence of this, they frequently collide with each other as well as with the walls of the container. The collision that takes place here is totally elastic in nature. So, there is a change in the momentum of the molecules, but the total momentum of the system is conserved. Due to changes in momentum, these molecules exert pressure on the walls of the container. The pressure exerted on the per unit area of the walls is called gas pressure. The kinetic energy of the gas molecules (which is due to the motion of the molecules) is, thus, related to gas pressure. The kinetic energy per unit volume of a given gas is 3/2 times the pressure of the gas.
