Kinetic Theory of Gases
Bridging the gap between microscopic molecular motion and macroscopic gas laws.
Gas laws like Boyle's and Charles's describe how gases behave (macroscopically). To explain why they behave this way, scientists developed a theoretical model based on the microscopic motion of individual molecules. This is the Kinetic Molecular Theory of Gases (KTG).
1. Postulates of KTG
The entire theory rests on a set of idealized assumptions. Memorize these, as they define what an "Ideal Gas" is:
- Particle Nature: Gases consist of large numbers of tiny particles (atoms or molecules) that are far apart relative to their size.
- Negligible Volume: The actual volume occupied by the gas molecules themselves is negligibly small compared to the total empty space of the container. (Explains high compressibility).
- No Intermolecular Forces: There are no forces of attraction or repulsion between the gas molecules.
- Random Motion: Molecules are in constant, rapid, random, straight-line motion.
- Perfectly Elastic Collisions: When molecules collide with each other or the walls of the container, there is no loss of total kinetic energy. (Explains why gases don't settle down over time).
- Kinetic Energy & Temperature: The average kinetic energy of the gas particles is directly proportional to the absolute temperature (in Kelvin).
2. The Kinetic Gas Equation
Using the postulates above and classical mechanics, physicists derived a fundamental equation that calculates the pressure exerted by a gas based on molecular collisions:
- $P$ = Pressure of the gas
- $V$ = Volume of the container
- $m$ = Mass of a single gas molecule
- $N$ = Total number of gas molecules
- $u_{rms}$ = Root Mean Square speed of the molecules
3. Average Kinetic Energy
By combining the KTG equation ($PV = \frac{1}{3} m N u^2$) with the Ideal Gas Equation ($PV = nRT$), we can derive the formulas for Kinetic Energy.
For 1 Mole of Gas:
For 1 Single Molecule:
Divide the molar kinetic energy by Avogadro's number ($N_A$). The term $\frac{R}{N_A}$ becomes the Boltzmann Constant ($k_B$).
4. Molecular Speeds (Maxwell-Boltzmann)
Not all gas molecules move at the same speed. They have a distribution of speeds. We define three important types of speeds:
| Speed Type | Symbol | Formula | Definition |
|---|---|---|---|
| Root Mean Square Speed | $u_{rms}$ | $\sqrt{\frac{3RT}{M}}$ | The square root of the average of the squares of speeds. Used in KE calculations. |
| Average Speed | $u_{avg}$ | $\sqrt{\frac{8RT}{\pi M}}$ | The simple arithmetic mean of all molecular speeds. |
| Most Probable Speed | $u_{mp}$ | $\sqrt{\frac{2RT}{M}}$ | The speed possessed by the maximum fraction of molecules in the gas. |
The Golden Ratio of Speeds:
$$u_{mp} : u_{avg} : u_{rms}$$ $$\sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3}$$ $$1 : 1.128 : 1.224$$
Therefore, $u_{rms} \gt u_{avg} \gt u_{mp}$.
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