Question

Describe how the mass of a gas particle affects its rate of effusion and diffusion.

Short answer

The mass of a gas particle affects its rate of effusion and diffusion as described by Graham's law, which states that the rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass: \( r \propto \frac{1}{\sqrt{M}} \). As the molar mass of a gas particle increases, its rate of effusion or diffusion decreases, meaning that heavier gas particles effuse and diffuse more slowly than lighter gas particles. Conversely, lighter gas particles effuse and diffuse more quickly than heavier gas particles.

Step-by-step solution

Define Effusion and Diffusion

Effusion is the process by which a gas escapes through a small opening, such as a pinhole, into a vacuum or lower-pressure environment. Diffusion is the process by which gas particles spread, mix, and distribute themselves evenly throughout a container due to their random motion.

Introduce Graham's Law

Graham's law relates the rate of effusion or diffusion of two ideal gases to their molar masses. The law states that the rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass: \[ r \propto \frac{1}{\sqrt{M}} \] where \(r\) is the rate of effusion or diffusion and \(M\) is the molar mass.

Derive the Relationship between Rate and Mass

To compare the rates of effusion or diffusion of two gases, we can use Graham's law in the form: \[ \frac{r_1}{r_2} = \frac{\sqrt{M_2}}{\sqrt{M_1}} \] Here, \(r_1\) and \(M_1\) represent the rate and molar mass of gas 1, while \(r_2\) and \(M_2\) represent the rate and molar mass of gas 2.

Discuss the Effects of Mass on Effusion and Diffusion

According to the derived relationship, as the molar mass of a gas particle increases, its rate of effusion or diffusion decreases. Conversely, as the molar mass of a gas particle decreases, its rate of effusion or diffusion increases. This means that lighter gas particles effuse and diffuse more quickly than heavier gas particles.

Key concepts

Effusion and Diffusion

Understanding the behavior of gases is essential in many scientific and industrial processes. Two key terms when discussing gas behavior are effusion and diffusion. Effusion is the process in which gas molecules pass through a tiny opening, such as a puncture, into a lower-pressure area. This happens because gas molecules are in constant motion and when they encounter the opening, some escape. Picture a balloon slowly losing air through a small hole. That's effusion in action.

Diffusion, on the other hand, is when gas molecules spread out across a space until they are uniformly distributed. It's like opening a perfume bottle in a room; gradually, the scent spreads until it reaches every corner. Diffusion occurs due to the random, haphazard movement of gas molecules colliding and spreading out. Both effusion and diffusion are influenced by the properties of the gas particles, including their molar mass.

Molar Mass

Molar mass is a property of a substance that is critical to understanding the behavior of gases. It is defined as the mass of one mole of a given substance and is usually expressed in grams per mole (\text{g/mol}). A mole is a specific number of particles, roughly 6.022\(\times\)10²³, known as Avogadro's number.

The molar mass tells us about the weight of gas molecules, which in turn influences how they behave when it comes to effusion and diffusion. Generally, the lighter a gas molecule (lower molar mass), the faster it is in moving and spreading. This is intuitive if you imagine how a feather (light) may fly away faster than a stone (heavy) when a gust of wind hits them.

Rates of Effusion

When we delve into the rates of effusion, we apply Graham's law to explore the quantitative aspects of how fast gases escape through a pinhole. Graham's law provides a direct formula for the relationship between the rate of effusion and the molar mass of a gas:
\[ r \propto \frac{1}{\sqrt{M}} \]
Here, \(r\) represents the rate of effusion, and \(M\) denotes the molar mass. The law implies that a gas with a lower molar mass will effuse faster than a gas with a higher molar mass. Interestingly, this means that under the same conditions, helium gas will effuse much quicker than oxygen gas, due to its much smaller molar mass.

Chemical Kinetics

Chemical kinetics is the field of chemistry that studies the rates of chemical reactions and the factors that affect these rates. It is directly related to the concepts of effusion and diffusion in gases because the rate at which molecules move can influence how quickly reactions happen.

In chemical reactions, molecules must collide with enough energy to react. Faster-moving molecules (like those in a lower molar mass gas diffusing rapidly) can lead to more frequent and energetic collisions and thus, potentially, faster reactions. In the grand scheme of things, the principles of chemical kinetics provide a broader context for understanding how gas behavior fits into more complex chemical processes.