Kinetic Molecular Theory and Graham's Law

Kinetic Molecular Theory

Developed in the second half of the 19th century to explain gas behavior after gas laws were established. Gas laws describe gas behavior, while kinetic molecular theory explains it.

Key Points

• All gases exhibit similar physical characteristics and behaviors, irrespective of their chemical identity.
• Real gases deviate from ideal behavior, but corrections can be applied in calculations.
• Developed by James Maxwell, Ludwig Boltzmann, and others.

Assumptions of Kinetic Molecular Theory

1. Negligible Volume: Gas particles' volume is negligible compared to the container volume.
2. No Intermolecular Forces: Gas atoms or molecules exhibit no intermolecular attractions or repulsions.
3. Continuous Random Motion: Gas particles are in continuous, random motion, colliding with each other and the container walls.
4. Elastic Collisions: Collisions between gas particles or with the container walls are elastic, meaning conservation of momentum and kinetic energy.
5. Kinetic Energy and Temperature: The average kinetic energy of gas particles is proportional to the absolute temperature (in Kelvins) and is the same for all gases at a given temperature, regardless of chemical identity or atomic mass.

Applications and Visualization

• Gas particles can be visualized as rubber balls bouncing off each other and the container walls.
• Real gas particles have measurable mass and volume, and collisions are not perfectly elastic.

Average Molecular Speeds

Kinetic Energy Equation

$KE = \frac{1}{2}mv^2 = \frac{3}{2}k_bT$

Where:

• $KE$ = Kinetic Energy
• $m$ = mass
• $v$ = speed
• $k_b$ is the Boltzmann constant ($1.38 \times 10^{-23} J/K$)
• $T$ = Temperature in Kelvins

Root Mean Square Speed

The root mean square speed ($u_{rms}$) is a measure of average molecular speed.

$u_{rms} = \sqrt{\frac{3RT}{M}}$

Where:

• $R$ = Ideal gas constant
• $T$ = Temperature in Kelvins
• $M$ = Molar mass

Maxwell-Boltzmann Distribution

• Depicts the distribution of gas particle speeds at a given temperature.
• As temperature increases, the curve flattens and shifts to the right, indicating that more molecules are moving at higher speeds.

Example: Xenon Difluoride

What is the average speed of xenon difluoride molecules at 20°C?

• $R = 8.314 \frac{J}{K \cdot mol}$
• Molar mass of $XeF_2$ = 169.3 g/mol = 0.1693 kg/mol
• $T = 20°C = 293 K$

$u_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3 \times 8.314 \times 293}{0.1693}} \approx 208 \frac{m}{s}$

Graham's Law of Diffusion and Effusion

Diffusion

Diffusion is the movement of molecules from high concentration to low concentration through a medium.

The kinetic molecular theory predicts that heavier gases diffuse more slowly than lighter gases due to their different average speeds.

Graham's Law

In 1832, Thomas Graham stated that the rates at which two gases diffuse are inversely proportional to the square roots of their molar masses under isothermal and isobaric conditions.

$\frac{r<em>1}{r</em>2} = \sqrt{\frac{M<em>2}{M</em>1}}$

Where:

• $r<em>1$ and $r</em>2$ are the diffusion rates of gas 1 and gas 2, respectively.
• $M<em>1$ and $M</em>2$ are the molar masses of gas 1 and gas 2, respectively.

A gas with a molar mass four times that of another gas will travel half as fast.

Effusion

Effusion is the flow of gas particles under pressure from one compartment to another through a small opening.

For two gases at the same temperature, the rates of effusion are proportional to their average speeds.

Example

Oxygen molecules travel at an average speed of approximately 500 m/s at a given temperature. Calculate the average speed of hydrogen molecules at the same temperature.

• Molar mass of $O_2$ = 32 g/mol
• Molar mass of $H<em>2$ = 2 g/mol $\frac{r</em>1}{r<em>2} = \sqrt{\frac{M</em>2}{M_1}}$

$r<em>2 = r</em>1 \times \sqrt{\frac{M<em>1}{M</em>2}} = 500 \frac{m}{s} \times \sqrt{\frac{32 \frac{g}{mol}}{2 \frac{g}{mol}}} = 500 \frac{m}{s} \times \sqrt{16} = 500 \frac{m}{s} \times 4 = 2000 \frac{m}{s}$