Derive the formula for the pressure exerted by an ideal gas on the walls of its container.

Pressure ($P$) in a gas is the force exerted by the gas molecules per unit area on the walls of its container. This force arises from the continuous collisions of the gas molecules with the container walls.

Consider an ideal gas consisting of $N$ identical molecules, each of mass $m$, confined within a cubic container of side length $L$. The molecules move randomly and undergo elastic collisions with each other and with the container walls.

Let's consider a single molecule moving with velocity $\vec{v} = (v_x, v_y, v_z)$. When this molecule collides elastically with a wall perpendicular to the x-axis, its x-component of velocity reverses, while $v_y$ and $v_z$ remain unchanged.
The initial momentum in the x-direction is $p_{xi} = m v_x$.
The final momentum in the x-direction is $p_{xf} = -m v_x$.
The change in momentum for the molecule is $\Delta p_{molecule} = p_{xf} - p_{xi} = -2m v_x$.
By Newton's third law, the momentum imparted to the wall by this collision is $\Delta p_{wall} = +2m v_x$.

The molecule travels a distance $L$ to hit the opposite wall and another $L$ to return to the original wall. The time taken for the molecule to complete this round trip and collide with the same wall again is $\Delta t = \frac{2L}{v_x}$.

The average force exerted by this single molecule on the wall is the rate of change of momentum:

For $N$ molecules in the container, the total force on the wall is the sum of the forces due to each molecule:

We can express the sum using the average of the square of the x-component of velocity, $\overline{v_x^2}$:

So, the total force is $F_{total} = \frac{N m \overline{v_x^2}}{L}$.

Since the motion of molecules is random and isotropic (uniform in all directions), the average squared velocity components are equal:

The total average squared velocity is $\overline{v^2} = \overline{v_x^2} + \overline{v_y^2} + \overline{v_z^2} = 3 \overline{v_x^2}$.
Therefore, $\overline{v_x^2} = \frac{1}{3} \overline{v^2}$.

Substitute this into the total force equation:

Pressure $P$ is defined as force per unit area. The area of one wall of the cubic container is $A = L^2$.

Since $L^3$ is the volume $V$ of the container:

Rearranging this gives the fundamental equation for pressure from the kinetic theory of gases:

This equation relates the macroscopic property of pressure to the microscopic properties of the gas molecules (mass, number, and average squared speed).

We know that the average translational kinetic energy of a single molecule is $K.E = \frac{1}{2} m \overline{v^2}$.
From the previous question, we also know that $K.E = \frac{3}{2} k_B T$.
Equating these two expressions for kinetic energy:

Substitute this into the pressure equation from Step 8:

This is the ideal gas law, where $N$ is the total number of molecules and $k_B$ is the Boltzmann constant. If we express $N$ in terms of the number of moles $n$ and Avogadro's number $N_A$ ($N = n N_A$), and use the universal gas constant $R = N_A k_B$, we get the more common form:

This derivation shows how the kinetic theory of gases provides a microscopic foundation for the ideal gas law.