Question

Give the mathematical expression for Boyle’s law.

Answer 1

Hint: The explanation for the macroscopic properties of the gases such as volume, temperature, and pressure is called the Kinetic theory of the gases. For mathematical expression, use the pressure expression in the kinetic theory of the gases to derive Boyle’s law.

Complete step by step answer:
According to the kinetic theory of the gases, the pressure expressed by gas is,
$ \Rightarrow P = \dfrac{1}{3}\rho {v^2}$
Where,
$\rho $ is the density and $v$ is the velocity of the gas molecules.
The value of the density can be derived by dividing the values of the mass and volume. That is,
$ \Rightarrow \rho = \dfrac{M}{V}$
Where,
$M$is the mass and $V$is the volume.
Substitute the density in the pressure equation.
$ \Rightarrow P = \dfrac{1}{3}\dfrac{M}{V}{v^2}$
$ \Rightarrow P = \dfrac{M}{{3V}}{v^2}$
Take the volume to the left-hand side we get,
$ \Rightarrow PV = \dfrac{M}{3}{v^2}$
$ \Rightarrow PV = \dfrac{1}{3}M{v^2}$
Equating the pressure-volume equation to the kinetic energy equation,
$ \Rightarrow PV = \dfrac{1}{3}M{v^2} = \dfrac{2}{3}KE$
Where $KE$ is the kinetic energy.
At the constant temperature, the volume will also be constant and hence the kinetic energy will also be constant.$PV$is equal to constant.
Therefore, the pressure of the gas is inversely proportional to the volume of the gas occupied. This is known as Boyle’s law.

Additional information:
The kinetic theory of the gases assumes that they are made up of large identical particles that move around a fast random thermal motion. Inside the walls of an enclosed container, they undergo elastic collisions between themselves. The size of every gas molecule is much smaller than the mean distance between the gas molecules. Other kinds of interactions can be neglected.

Note:
The kinetic gas theory is useful in determining the thermal conductivity, mass diffusivity it also has several other drawbacks. The volume of the gas molecules and the intermolecular distances are always negligible and they are not valid. Along with Boyle’s law, Charles law, Gay-lussac’s law, and Avagadro’s hypothesis also give the ideal gas law.