Brownian Motion

Small particles suspended in a liquid or gas move in an irregular, unpredictable way. This movement is called Brownian motion. A suspended particle does not follow a smooth, straight path: its direction keeps changing.

The liquid or gas is made of molecules in continual random motion. These molecules collide with a suspended particle from different directions. Over a short interval, the impacts are uneven, so the particle experiences a resultant force in one direction. As the collisions continue, the resultant force fluctuates, causing the particle’s motion to change unpredictably.

The suspended particles are much larger than the molecules striking them. Their visible motion is therefore indirect evidence that matter is made of atoms and molecules in continual random motion: although the individual molecules cannot be seen, their uneven impacts can be observed through the motion they cause.

Gas Laws and the Molecular Model

A fixed amount of gas can change its pressure, volume and temperature. Changing one of these properties affects at least one of the others.

Temperature in the gas laws must be measured on the absolute temperature scale, in kelvin ($$\mathrm{K}$$), not in degrees Celsius. For a fixed amount of gas, measurements show three relationships:

• At constant temperature, pressure is inversely proportional to volume: halving the volume doubles the pressure.
• At constant volume, pressure is proportional to absolute temperature: doubling the temperature in kelvin doubles the pressure.
• At constant pressure, volume is proportional to absolute temperature: doubling the temperature in kelvin doubles the volume.

For a fixed amount of ideal gas, these relationships are combined in:

Here $$p$$ is pressure, $$V$$ is volume and $$T$$ is absolute temperature. The constant has the same value for different states of the same fixed amount of ideal gas.

Explaining gas laws with moving molecules

In the molecular model, gas molecules move continually and collide with the walls of their container. Each collision exerts a force on a wall. The combined effect of these collisions produces pressure: force per unit area.

If the gas is compressed at constant temperature, its molecules have less space to move through. They reach the walls more frequently, so the force exerted per unit area increases. The pressure therefore rises as the volume decreases.

Increasing the temperature increases the molecules’ average kinetic energy. They move faster, so their collisions with the walls become more frequent and more forceful. If the volume is fixed, the pressure rises.

If pressure is kept constant instead, the gas expands as it is heated. The larger volume reduces the frequency of wall collisions, allowing the pressure to remain constant despite the faster-moving molecules.

Gas laws as observations and explanations

Gas laws are empirical relationships: they were established from observations and measurements of gases. The molecular kinetic theory model uses assumptions about moving molecules to explain why those relationships occur. Understanding of gases thus developed from observed regularities to a model that accounts for their behaviour in terms of moving particles.

Kinetic Theory Derivation of Gas Pressure

Gas pressure arises because molecules collide with the walls of their container. Each collision transfers momentum to a wall; the combined rate of momentum transfer produces a force, and force per unit area is pressure.

The simple kinetic theory model treats a gas as a large number $$N$$ of molecules, each of mass $$m$$, moving randomly. The molecules are treated as point particles: their own volume is negligible compared with the container volume. They exert no forces on one another except during collisions. All collisions, including those with the walls, are elastic, and the molecules obey Newton’s laws.

Momentum transfer to a wall

Consider a cubic container of side length $$L$$. For one molecule, let $$c_x$$ be the component of its velocity perpendicular to a chosen wall. In an elastic collision with the wall, this component reverses while its magnitude stays the same. The molecule’s perpendicular momentum therefore changes by an amount of magnitude $$2m|c_x|$$. By conservation of momentum, the momentum transferred to the wall has the same magnitude.

For a molecule travelling freely between opposite walls, the distance between successive collisions with the same wall is $$2L$$. Its time between these collisions is therefore $$2L/|c_x|$$. Average force is momentum transferred per unit time, so the molecule’s average force on that wall is:

The calculation uses free travel between wall collisions. Collisions between molecules can change individual paths, but in an equilibrium gas their elastic collisions preserve the random distribution of motion used in the pressure result.

From molecular forces to gas pressure

The wall has area $$L^2$$ and the container has volume $$V=L^3$$. Adding the average contributions from all molecules and dividing by the wall area gives:

Here $$\langle c_x^2\rangle$$ is the mean of the squared velocity components perpendicular to the wall. The square matters: molecules moving towards either opposite wall contribute to pressure, even though their velocity components have opposite signs.

Random motion is isotropic, meaning that there is no preferred direction. The mean squared velocity components in the three perpendicular directions are therefore equal. Since each molecule’s squared speed is $$c^2=c_x^2+c_y^2+c_z^2$$, their means satisfy:

The root mean square speed is $$c_{\mathrm{rms}}=\sqrt{\langle c^2\rangle}$$. Substituting the isotropic relationship into the pressure equation gives the kinetic theory result:

This equation relates pressure $$p$$ and volume $$V$$ to the number of molecules, the mass of each molecule and their mean squared speed. It can be rearranged to calculate any one of these quantities when the others are known.

Molecular Kinetic Energy and Temperature

The molecules of an ideal gas move randomly. Their speeds are not all the same, so the kinetic energy associated with their motion from place to place—their translational kinetic energy—is described using an average.

The absolute temperature $$T$$, measured in kelvin, is proportional to the mean translational kinetic energy per molecule. A higher absolute temperature means a higher mean translational kinetic energy.

Here, $$m$$ is the mass of one molecule and $$c_{\mathrm{rms}}$$ is its root mean square speed: the square root of the mean of the squares of the molecular speeds. The constant $$k$$ is the Boltzmann constant, $$R$$ is the molar gas constant and $$N_A$$ is the Avogadro constant. They are related by $$k = R/N_A$$.

The expression uses the mean of the squares of the speeds, not the square of the mean speed. This is why it uses $$c_{\mathrm{rms}}$$ rather than the mean speed.

Gases at the same absolute temperature have the same mean translational kinetic energy per molecule, regardless of molecular mass. Their root mean square speeds can nevertheless differ: at the same temperature, heavier molecules have a lower root mean square speed than lighter molecules.

Internal energy of a monatomic ideal gas

A monatomic ideal gas consists of individual atoms. Its internal energy is the total random kinetic energy of those atoms. For $$N$$ atoms, the total is $$N$$ times the mean kinetic energy per atom:

For a fixed number of atoms, $$N$$ and $$k$$ are constant, so the internal energy $$U$$ is proportional to the absolute temperature $$T$$. For example, doubling the temperature in kelvin doubles the internal energy.