Molecular Motion

Overview

Molecular Motion studies how gas molecules move and how this motion relates to temperature, pressure, and internal energy.

Definition

Gas molecules are in continuous random motion. They move in straight lines between collisions and undergo elastic collisions with other molecules and container walls.

Because molecules have a range of speeds, we use averages such as mean speed, mean square speed, and root mean square speed.

Why It Matters

Molecular motion explains why hot gases exert higher pressure, why lighter gases move faster, why temperature is linked to kinetic energy, why gases diffuse, and why internal energy rises when temperature rises.

Key Representations

At any instant, some molecules move fast and some slowly. Directions vary continuously. The average velocity of all molecules in a container may be zero, but the average speed is not zero.

For one ideal gas molecule:

$$⟨E_k⟩=\frac{3}{2}kT$$

For $N$ molecules:

$$U=\frac{3}{2}NkT$$

Using $N=n{N}_A$:

$$U=\frac{3}{2}nRT$$

The root mean square speed is:

$$v_{\text{rms}}=\sqrt{⟨v^2⟩}$$

where:

$$⟨v^2⟩=\frac{v_1^2+v_2^2+v_3^2+\cdots }{N}$$

For an ideal gas:

$$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$$

where $M$ is molar mass in ${\text{kg mol}}^{-1}$.

Since:

$$v_{\text{rms}}\propto \sqrt{T}$$

if $T$ becomes $4T$, $v_{\text{rms}}$ doubles.

Since:

$$v_{\text{rms}}\propto \frac{1}{\sqrt{M}}$$

lighter gases move faster at the same temperature.

Gas pressure arises because molecules collide with container walls and change momentum. Higher pressure occurs when molecules move faster, collisions are more frequent, or more molecules are present.

Common Exam Traps

Average velocity is not average speed. Because directions cancel, average velocity may be zero while average speed is positive.

Use kelvin in $v_{\text{rms}}\propto \sqrt{T}$.

Doubling $T$ gives $v_{\text{rms}}$ multiplied by $\sqrt{2}$, not 2.

At the same temperature, heavier molecules move more slowly because $v_{\text{rms}}\propto 1/\sqrt{M}$.

Temperature relates to average kinetic energy per molecule, not total energy of all molecules.

For a fixed amount of ideal gas, $U\propto T$.

Links

• Thermal Physics B
• Kinetic Theory and Ideal Gases
• Kinetic Theory of Matter
• Ideal Gases
• Temperature and Heat
• First Law and Thermodynamic Processes
• Thermodynamic Processes