B.3.2 Microscopic basis for macroscopic behavior

Kinetic Theory and Pressure

The kinetic theory of gases provides a microscopic explanation for the macroscopic behavior of gases, such as pressure and temperature.

Origin of Pressure in a Gas

1. Pressure arises from the collisions of gas molecules with the walls of their container.
2. Each collision exerts a force on the wall, and the cumulative effect of countless collisions results in pressure.

Deriving the Pressure Equation

To relate pressure to molecular motion, consider a cube of side $L$ containing $N$ molecules, each of mass $m$.

Assume the molecules move randomly with an average speed $v$.

• Momentum Change in a Collision:
  • A molecule moving with velocity $v_x$ along the x-axis collides elastically with a wall.
  • Before the collision, its momentum is $m{v}_x$; after the collision, it is $-m{v}_x$.
  • The change in momentum is $2m{v}_x$.
• Time Between Collisions:
  • The molecule travels a distance $2L$ (to the wall and back) in time $t=\frac{2L}{v_x}$.
• Force Exerted by the Molecule:
  • The average force exerted on the wall is given by the rate of change of momentum: $$F=\frac{2m{v}_x}{\frac{2L}{v_x}}=\frac{m{v}_x^2}{L}$$
• Total Pressure from All Molecules:
  • For $N$ molecules, the total pressure is the sum of the forces exerted by each molecule.
  • Using the root mean square speed $v_{\text{rms}}$, where $v_{\text{rms}}^2=\frac{v_1^2+v_2^2+\dots +v_N^2}{N}$, the pressure is: $$P=\frac{1}{3}\rho v_{\text{rms}}^2$$
• Here, $\rho =\frac{Nm}{V}$ is the density of the gas.

Internal Energy of an Ideal Gas

The internal energy of an ideal gas is the total kinetic energy of its molecules.

Expression for Internal Energy

The internal energy $U$ of an ideal gas is given by:

$$U=\frac{3}{2}N{k}_{B}T\text{or}U=\frac{3}{2}nRT$$

where:

• $N$ is the number of molecules.
• $n$ is the number of moles.
• $k_B$ is the Boltzmann constant ($1.38\times {10}^{-23}{\text{J K}}^{-1}$).
• $R$ is the universal gas constant ($8.31{\text{J K}}^{-1}{\text{mol}}^{-1}$).
• $T$ is the absolute temperature in kelvin.

Real vs. Ideal Gases

While the ideal gas model is useful, real gases deviate from this behavior under certain conditions.

Conditions for Ideal Gas Behavior

Ideal gases follow the assumptions of the kinetic theory, which include:

• Molecules are point particles with negligible volume.
• No intermolecular forces exist except during collisions.
• Collisions are perfectly elastic.

Deviations from Ideal Behavior

Real gases deviate from ideal behavior under high pressure or low temperature.

• High Pressure:
  • Molecules are closer together, so their volume becomes significant.
  • Intermolecular forces (e.g., van der Waals forces) become more pronounced.
• Low Temperature:
  • Molecules move more slowly, increasing the influence of attractive forces.
  • Gases may liquefy, violating the ideal gas assumption of no intermolecular forces.

Van der Waals Equation

To account for real gas behavior, the Van der Waals equation modifies the ideal gas law:

$$(P+\frac{a}{V^2})(V-b)=nRT$$

where:

• $a$ accounts for intermolecular attractions.
• $b$ accounts for the finite volume of molecules.

Summary

The kinetic theory of gases provides a framework for understanding gas behavior at both microscopic and macroscopic levels.