Problem 43 - Chemistry Matter and Change

Question

What is the ratio of effusion rates of krypton and neon at the same temperature and pressure?

Step-by-Step Solution

Answer

The ratio of effusion rates of krypton to neon at the same temperature and pressure is approximately \(0.489:1\).

Step 1: Write Graham's law of effusion equation

First, let's write down the equation for Graham's law of effusion: \(Rate_{1} / Rate_{2} = \sqrt{(M_{2} / M_{1})}\) where Rate₁ and Rate₂ are the effusion rates of the two gases, and M₁ and M₂ are their respective molar masses.

Step 2: Substitute molar masses of krypton and neon

Now, let's substitute the molar masses of krypton (M₁ = 83.8 g/mol) and neon (M₂ = 20.2 g/mol) into the equation: \(Rate_{Kr} / Rate_{Ne} = \sqrt {(M_{Ne} / M_{Kr})}\) \(Rate_{Kr} / Rate_{Ne} = \sqrt{(20.2 / 83.8)}\)

Step 3: Calculate the ratio of effusion rates

Lastly, let's calculate the ratio of the effusion rates: \(Rate_{Kr} / Rate_{Ne}= \sqrt{(20.2 / 83.8)}\approx 0.489\) So, the ratio of effusion rates of krypton to neon at the same temperature and pressure is approximately 0.489:1.

Key Concepts

Effusion Rates

Effusion is the process by which gas particles pass through a tiny hole into a vacuum. It's a fascinating phenomenon that occurs because gas molecules are constantly in motion, bouncing around and colliding with each other. When they encounter a small opening, some of these particles inevitably slip through. Effusion rates refer to the speed at which these gas particles escape through the hole.

Now, you might wonder how we measure that speed. Here, the effusion rate is often expressed as how much gas escapes per unit time. The rate is influenced by several factors: the size of the opening, the temperature of the gas, and the nature of the gas itself.

According to Graham's law, lighter gases tend to effuse more rapidly than heavier ones. This is because lighter molecules move faster at a given temperature, facilitating quicker passage through the opening.

Molar Mass

Molar mass is a key concept in chemistry that helps us understand the weight of a substance on a molecular level. Think of it as the mass of a given substance divided by the amount of substance, which is measured in moles. It’s typically expressed in grams per mole (g/mol).

Molar mass is critical when dealing with gases, especially in the context of effusion, because it directly affects the rate of effusion. Gases with lower molar masses tend to move more quickly because their molecules are lighter and can navigate through spaces more easily.

In the context of Graham’s law, the molar mass becomes an integral part of calculating the effusion rate ratio. By knowing the molar masses of the two gases involved, we can determine how their rates of effusion compare. For the gases krypton and neon, knowing their molar masses helps us calculate that neon effuses almost twice as fast as krypton, given that neon is lighter.

Graham's Law

Graham's law of effusion is a handy formula that relates the effusion rates of two gases to their molar masses. Named after Scottish chemist, Thomas Graham, this law provides a straightforward way to predict how gases will behave.

The fundamental principle behind Graham's law is that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it’s expressed as: \[Rate_{1} / Rate_{2} = \sqrt{(M_{2} / M_{1})}\] where \(Rate_1\) and \(Rate_2\) are the effusion rates of the two gases, and \(M_1\) and \(M_2\) are their respective molar masses.

This equation tells us that if one gas is lighter than another, it will effuse more quickly. Graham’s law is incredibly useful when comparing different gases. By applying this law, you can deduce that neon, with a lower molar mass, will effuse more rapidly than the heavier krypton under the same conditions of temperature and pressure.