Kinetic Gas Theory

The kinetic theory of gases explains the behavior of gases based on the assumption that gas molecules are in constant random motion and follow the principles of classical mechanics. This model treats gases as composed of particles that collide elastically with each other and with the walls of their container.

  • Molecules move in straight lines between collisions.
  • Collisions are elastic, meaning no kinetic energy is lost.
  • The volume of the gas molecules themselves is negligible compared to the volume of the container.
  • No long-range forces exist between gas molecules except during collisions.

Classical Mechanics Review

Several key concepts from classical mechanics help explain the kinetic theory of gases and the origin of pressure:

  • Velocity, \( \mathbf{v} \):
    • Rate of change of position (motion).
    • Defines the direction of travel.
    • Mathematically, velocity is given by:

      \[ \mathbf{v} = \frac{\Delta \mathbf{x}}{\Delta t} \]

  • Linear Momentum, \( \mathbf{p} \):
    • The momentum of an object is the product of its mass and velocity.
    • Mathematically, momentum is given by:

      \[ \mathbf{p} = m \mathbf{v} \]

  • Linear Acceleration, \( \mathbf{a} \):
    • Rate of change of velocity.
    • Mathematically, acceleration is given by:

      \[ \mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t} \]

  • Newton's Second Law:
    • The acceleration of a body with mass \(m\) is proportional to the applied force \( \mathbf{F} \).
    • Acceleration occurs in the direction of the applied force.
    • Newton's second law is mathematically expressed as:

      \[ \mathbf{F} = m \mathbf{a} \]

  • Newton's Third Law:
    • The law of conservation of momentum.
    • Momentum is constant in the absence of a force acting on an object.
    • This is expressed as:

      \[ \mathbf{p} = \text{constant} \]

Derivation of Pressure

In the kinetic theory of gases, pressure is explained as the result of gas molecules colliding with the walls of the container. Each collision exerts a force on the walls, and the cumulative effect of all collisions determines the pressure. The derivation starts by considering the velocity components of a single gas molecule in the $x$, $y$, and $z$ directions. The velocity of a molecule is denoted by $v$, which has three components: $v_x$, $v_y$, and $v_z$.

The total velocity of the molecule is:

\[v^2 = v_x^2 + v_y^2 + v_z^2\]
.

;Assuming that the gas molecules are moving randomly in all directions, the average velocity squared in each direction is the same, which allows us to write:

\[\langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle = \frac{1}{3} \langle v^2 \rangle\]
.

The momentum change for a molecule colliding elastically with the wall in the x-direction is given by:

\[ \Delta p_x = mv_x - (-mv_x) = 2mv_x\]

where $m$ is the mass of the molecule and $v_x$ is its velocity in the $x$-direction. The time between collisions with the same wall is the time it takes for the molecule to travel a distance of $2L$ (where $L$ is the length of the container) and return to the wall:

\[ \Delta t = \frac{2L}{v_x} \]

The force exerted by the molecule on the wall is given by the change in momentum per unit time:

\[ F = \frac{\Delta p_x}{\Delta t} = \frac{2mv_x}{\frac{2L}{v_x}} = \frac{mv_x^2}{L}\]
.

To find the total force exerted by all molecules, we need to sum over all the molecules in the system. The total pressure $p$ is the force exerted by the molecules on the walls per unit area $A$ of the wall. Assuming there are $N$ molecules, the total pressure can be written as:

\[ p = \frac{1}{3} \frac{N m \langle v^2 \rangle}{V} \]

where:

  • $N$ is the total number of molecules in the system,
  • $m$ is the mass of a single molecule,
  • $V$ is the volume of the container, and
  • $\langle v^2 \rangle$ is the average squared velocity of the molecules.

This equation shows that the pressure is directly proportional to the number of molecules, the mass of the molecules, and the average velocity squared. This relationship is central to the kinetic theory of gases, as it connects the microscopic motion of gas molecules to the macroscopic observable property of pressure.