KAHIBARO
Table of Contents
Microscopic Picture of Pressure
Pressure in a gas can be understood as the result of countless tiny collisions between gas molecules and the walls of a container. On the macroscopic level, pressure is a measurable quantity, defined as force per unit area. On the microscopic level, that force comes from molecules striking a surface and bouncing away.
A gas consists of a huge number of molecules moving in random directions. Each molecule has mass and velocity, so it carries momentum. When a molecule hits a wall, its momentum changes. A change in momentum means a force has acted. The wall exerts a force on the molecule, and the molecule exerts an equal and opposite force on the wall. The total effect of many such impacts is the pressure of the gas.
Pressure arises from molecular collisions with surfaces.
Macroscopic definition:
$$
P = \frac{F}{A}
$$
Microscopic origin:
pressure is the cumulative effect of momentum transfer from many molecules to the wall.
A Single Molecule Hitting a Wall
To see how this works, imagine a molecule of mass $m$ moving inside a cubic box of side length $L$. Suppose its velocity component perpendicular to one wall is $v_x$. When it strikes that wall elastically, the perpendicular component reverses direction. Before the collision, its momentum in the $x$ direction is
$$
p_x = mv_x
$$
After the collision, it becomes
$$
p_x' = -mv_x
$$
So the change in the molecule's momentum is
$$
\Delta p_x = p_x' - p_x = -mv_x - mv_x = -2mv_x
$$
The wall receives the opposite momentum change, so the momentum transferred to the wall has magnitude
$$
2mv_x
$$
This transfer happens every time the molecule returns to that same wall.
Time Between Collisions
After bouncing from one wall, the molecule travels to the opposite wall and comes back before striking the first wall again. The total distance for this round trip in the $x$ direction is $2L$. If the $x$ component of its speed is $v_x$, then the time between two successive hits on the same wall is
$$
\Delta t = \frac{2L}{v_x}
$$
The average force exerted by this one molecule on the wall is the momentum transferred per unit time:
$$
F = \frac{\Delta p}{\Delta t}
= \frac{2mv_x}{2L/v_x}
= \frac{mv_x^2}{L}
$$
So even one molecule contributes a small average force to the wall.
Many Molecules in the Container
A real gas contains a huge number of molecules. If there are $N$ molecules, each with possibly different velocity components, then the total force on the wall is
$$
F = \frac{m}{L}\left(v_{1x}^2 + v_{2x}^2 + \cdots + v_{Nx}^2\right)
$$
This can be written as
$$
F = \frac{m}{L}\sum_{i=1}^{N} v_{ix}^2
$$
The area of the wall is $A = L^2$, so the pressure is
$$
P = \frac{F}{A}
= \frac{m}{L^3}\sum_{i=1}^{N} v_{ix}^2
$$
Since $L^3 = V$, the volume of the cube,
$$
P = \frac{m}{V}\sum_{i=1}^{N} v_{ix}^2
$$
It is useful to introduce the average value of $v_x^2$:
$$
\overline{v_x^2} = \frac{1}{N}\sum_{i=1}^{N} v_{ix}^2
$$
Then
$$
P = \frac{Nm\overline{v_x^2}}{V}
$$
Why All Directions Matter
Gas molecules move in three dimensions, not just in the $x$ direction. Because the motion is random and there is no preferred direction in an ordinary gas, the average squared velocity is shared equally among the three coordinate directions:
$$
\overline{v_x^2} = \overline{v_y^2} = \overline{v_z^2}
$$
Also, the total squared speed is
$$
v^2 = v_x^2 + v_y^2 + v_z^2
$$
Taking averages gives
$$
\overline{v^2} = \overline{v_x^2} + \overline{v_y^2} + \overline{v_z^2}
$$
Since the three averages are equal,
$$
\overline{v^2} = 3\overline{v_x^2}
$$
So
$$
\overline{v_x^2} = \frac{1}{3}\overline{v^2}
$$
Substituting into the pressure formula gives
$$
P = \frac{Nm}{V}\cdot \frac{1}{3}\overline{v^2}
$$
Therefore,
$$
P = \frac{1}{3}\frac{Nm\overline{v^2}}{V}
$$
This is one of the central results of kinetic theory.
For an ideal gas, the microscopic expression for pressure is
$$
P = \frac{1}{3}\frac{Nm\overline{v^2}}{V}
$$
where $N$ is the number of molecules, $m$ is the mass of one molecule, $V$ is the volume, and $\overline{v^2}$ is the average of the square of the molecular speed.
Meaning of the Formula
This equation tells us what controls gas pressure microscopically. Pressure becomes larger if there are more molecules in a given volume, because more collisions occur. Pressure also becomes larger if the molecules move faster, because each collision transfers more momentum and collisions happen more often.
The formula shows that pressure depends on both particle density and molecular motion. If we define the number density as
$$
n = \frac{N}{V}
$$
then the pressure can be written as
$$
P = \frac{1}{3}nm\overline{v^2}
$$
This form makes the interpretation especially clear. Pressure is proportional to how many molecules are packed into space and to the average kinetic effect of their motion.
Assumptions Behind the Model
This molecular interpretation uses the ideal gas model. The derivation assumes that molecules are tiny compared with the container size, that they move randomly, that collisions with the walls are elastic, and that intermolecular forces are negligible except during brief collisions.
These assumptions are not exactly true for every real gas, but they work very well in many ordinary situations, especially at low density and moderate temperature.
Connection to Kinetic Energy
The average translational kinetic energy of one molecule is
$$
\overline{K} = \frac{1}{2}m\overline{v^2}
$$
Using the pressure formula,
$$
P = \frac{1}{3}\frac{Nm\overline{v^2}}{V}
$$
we can rewrite it as
$$
PV = \frac{1}{3}Nm\overline{v^2}
$$
Since
$$
\frac{1}{2}m\overline{v^2} = \overline{K}
$$
it follows that
$$
PV = \frac{2}{3}N\overline{K}
$$
This shows that pressure is directly linked to the average kinetic energy of the molecules.
A very important kinetic theory result is
$$
PV = \frac{2}{3}N\overline{K}
$$
This means that gas pressure is a macroscopic consequence of the microscopic kinetic energy of molecular motion.
Visualizing Molecular Impacts
The arrows represent molecular velocities. Molecules that hit the wall reverse their perpendicular velocity component. Repeated impacts from many molecules create a steady force on the wall, and therefore a steady pressure.
Summary Table
| Quantity | Microscopic meaning | Formula |
|---|---|---|
| Pressure | Force per unit area from molecular impacts | $P = \frac{F}{A}$ |
| Momentum change in one collision | Reversal of perpendicular motion | $\Delta p = 2mv_x$ |
| Average force from one molecule | Momentum transfer per time | $F = \frac{mv_x^2}{L}$ |
| Pressure of ideal gas | Result of all molecular impacts | $P = \frac{1}{3}\frac{Nm\overline{v^2}}{V}$ |
| Pressure and kinetic energy | Link between macroscopic and microscopic views | $PV = \frac{2}{3}N\overline{K}$ |
Physical Insight
The molecular interpretation of pressure is one of the most important bridges between microscopic physics and everyday experience. When you pump air into a tire, the pressure rises because more molecules are placed into the same volume. When a gas is heated, the pressure tends to rise because the molecules move faster and hit the walls harder and more often.
So pressure is not a mysterious continuous substance. It is the averaged effect of an enormous number of tiny, rapid molecular collisions.