Question

Based on your knowledge of the kinetic theory of gases, derive Graham's law of diffusion [Equation ( 5.17 )]

Short answer

Given the average speed of a gas is \( v_{avg} = \sqrt{\frac{8kT}{\pi M}} \), the ratio of the rates of diffusion of two gases A and B is given by Graham's law \( \frac{r_A}{r_B} = \frac{v_{avg,A}}{v_{avg,B}} = \sqrt{\frac{M_B}{M_A}} \).

Step-by-step solution

Understanding the Kinetic Theory of Gases

The kinetic theory of gases assumes that gases consist of a large number of particles, all of which are in constant, random motion. The theory holds that the speed of these particles depends on their temperature, with higher temperatures resulting in faster speeds. The faster the particles move, the more frequently and forcefully they collide with each other, leading to a higher pressure. This relates to the rate of diffusion, since the faster particles move, the more quickly a gas will spread out and diffuse.

Defining Rate of Diffusion

The rate of diffusion or effusion of a gas is defined as the volume of the gas which passes through a small hole per unit time. It is given by the formula \( rate = \frac{Volume}{Time} \). This rate of diffusion is proportional to the average speed of the gas particles.

Relating Speed to Molar Mass

The average speed of gas particles can be given by the formula \( v_{avg} = \sqrt{\frac{8kT}{\pi M}} \) where \( v_{avg} \) is the average speed, \( k \) is Boltzmann constant and \( T \) is temperature in Kelvin, \( M \) is the molar mass of the gas in kg.

Derive Graham's Law

Consider two gases A and B with molar masses \( M_A \) and \( M_B \) respectively. The rates of diffusion \( r_A \) and \( r_B \) are given by the average speeds of the gas particles, so \( r_A = v_{avg,A} \) and \( r_B = v_{avg,B} \). Therefore, as \( r_A = \sqrt{\frac{8kT}{\pi M_A}} \) and \( r_B = \sqrt{\frac{8kT}{\pi M_B}} \), the ratio \( \frac{r_A}{r_B} = \frac{v_{avg,A}}{v_{avg,B}} = \sqrt{\frac{M_B}{M_A}} \) .This equation is Graham's law of diffusion.

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases provides a foundational understanding of gas behavior, essential for deriving concepts such as Graham's Law of Diffusion. This theory posits that gases are made up of a large number of small particles (atoms or molecules) that are in constant, random motion. These particles move freely and have perfectly elastic collisions.

• Temperature and Particle Motion: The speed at which these particles move is directly linked to the temperature of the gas. Higher temperatures mean the particles move faster.
• Pressure and Collisions: As particles move, they collide with each other and the walls of their container, creating pressure.
• Speed and Diffusion: Faster particle speeds mean more rapid diffusion, or spreading out, of the gas.

Understanding these dynamics explains why gases behave as they do under various conditions, and why they rapidly fill the space available to them.

Rate of Diffusion

The rate of diffusion describes how quickly a gas moves through a space or passes through a barrier. In practical terms, it's how fast one gas can mix with another in the air.
The rate of diffusion or effusion (when gases escape through a tiny hole) is tied to the volume of gas moving per unit of time. Mathematically, this is expressed as:\[rate = \frac{Volume}{Time}\]Several factors influence this rate:

• Particle Speed: The speed at which particles move significantly affects the rate. Faster particles mean quicker diffusion.
• Concentration Gradient: A higher difference in concentrations between two areas results in faster diffusion.

By understanding these elements, we can predict how and why certain gases diffuse at the rates they do, which is crucial in applying Graham's Law.

Molar Mass

Molar mass is the mass of one mole of a substance, measured in grams per mole (g/mol) or kilograms per mol (kg/mol). It is a key factor in predicting the behavior of gases, particularly concerning the rate of diffusion.
The influence of molar mass on the speed of gas particles is demonstrated using the formula:\[v_{avg} = \sqrt{\frac{8kT}{\pi M}}\]Where:

• \(v_{avg}\): average speed of the gas particles
• \(k\): Boltzmann constant
• \(T\): temperature in Kelvin
• \(M\): molar mass of the gas in kilograms

A gas with a lower molar mass will move faster, resulting in a higher diffusion rate. Conversely, heavier gases diffuse more slowly. This relationship is pivotal in Graham's Law, where the diffusion rate of different gases is compared by taking the square root of the inverse of their molar masses.