Molecular Speeds
The Maxwell-Boltzmann Distribution and the Three Speeds.
Gas molecules are in constant, chaotic motion, colliding millions of times per second. Because of these collisions, molecules are constantly speeding up and slowing down. Therefore, we cannot talk about the "speed" of a gas; we must use statistical averages to describe the distribution of molecular speeds.
1. Maxwell-Boltzmann Distribution Curve
James Clerk Maxwell and Ludwig Boltzmann derived a statistical formula showing how molecular speeds are distributed at a given temperature. The curve plots the Fraction of Molecules against their Speed.
Key Feature: The curve is asymmetrical (skewed to the right). A very small fraction of molecules have extremely low or extremely high speeds. Most are clustered around the peak.
2. The Three Types of Speeds
Because the distribution curve is skewed, the most probable, average, and RMS speeds are all different values.
| Speed Type | Symbol | Definition | Formula |
|---|---|---|---|
| Most Probable Speed | $u_{mp}$ | The speed possessed by the maximum fraction of gas molecules (the peak of the curve). | $\sqrt{\frac{2RT}{M}}$ |
| Average Speed | $u_{avg}$ | The simple arithmetic mean of the speeds of all molecules in the gas. | $\sqrt{\frac{8RT}{\pi M}}$ |
| Root Mean Square Speed | $u_{rms}$ | The square root of the mean of the squares of the speeds. Directly defines the Kinetic Energy. | $\sqrt{\frac{3RT}{M}}$ |
3. The "Golden Ratio" of Speeds
Because $\frac{RT}{M}$ is common to all three formulas, we can determine the exact ratio between them. This ratio is a staple of objective chemistry exams.
Crucial Inequality:
$u_{rms} \gt u_{avg} \gt u_{mp}$
The RMS speed is always the highest, sitting furthest to the right on the distribution curve.
4. Effects of Temperature ($T$) and Molar Mass ($M$)
How does the Maxwell-Boltzmann curve change when we alter the physical conditions?
1. Effect of Increasing Temperature ($T_2 > T_1$)
- As temperature increases, all molecular speeds increase.
- The peak of the curve shifts to the right (higher speeds).
- The curve broadens and flattens. The fraction of molecules moving at the most probable speed actually decreases, but the distribution of speeds becomes much wider.
- Exam Trap: The total area under the curve represents the total number of molecules. This area remains constant regardless of temperature!
2. Effect of Molar Mass ($M$)
- Speed is inversely proportional to the square root of Molar Mass ($u \propto 1/\sqrt{M}$).
- Lighter gases (like $H_2$ or $He$) move much faster. Their curves are broad, flat, and shifted far to the right.
- Heavier gases (like $CO_2$ or $Cl_2$) move sluggishly. Their curves are sharp, tall, and shifted to the left.
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