Question

# Derive Boyle’s law using kinetic theory of gases.

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Hint: Use the expression for pressure in kinetic theory of gases to derive Boyle’s law at a constant temperature.

Complete step by step solution:
The pressure exerted by a gas according to kinetic theory of gases is,
$P=\dfrac{1}{3}\rho {{v}^{2}}$
$\rho$ is the density and $v$ is the root mean square velocity of the gas molecules
\begin{align} & \rho =M/V \\ & P=\dfrac{M}{3V}{{v}^{2}} \\ & PV=\dfrac{1}{3}M{{v}^{2}}=\dfrac{2}{3}{{E}_{k}} \\ \end{align}

Where M is mass, $V$ is volume and ${{E}_{k}}$ is the kinetic energy of the gas.
At a constant temperature, the velocity will be constant and hence the kinetic energy will be constant.
$PV=$ constant

Thus, at a constant temperature the pressure of a gas is inversely proportional to the volume occupied by it. This is Boyle’s law.

Additional information: The kinetic theory of gases assumes that a gas is made up of a large number of identical particles (atoms or molecules) which are in rapid random thermal motion. They undergo elastic collisions between themselves and with the walls of the container in which they are enclosed. The size of the constituent atom or molecule is assumed to be much smaller than the mean distance between them. All other types of interactions are neglected. Although this theory is useful in explaining properties such as viscosity, thermal conductivity, mass diffusivity and various macroscopic properties of gases it has several drawbacks. The assumption that intermolecular forces and volume of the particles is negligible is not always valid. It cannot be used to determine properties of condensed states of matter.

Note: Boyle’s law along with Charles’s law, Gay-Lussac’s law and Avogadro’s hypothesis gives the ideal gas law.