Question

What is the ratio of the rate of effusion of the most abundant gas, nitrogen, to the lightest gas, hydrogen?

Short answer

Answer: The ratio of the rate of effusion of nitrogen to hydrogen is approximately 1:3.74.

Step-by-step solution

Understanding Graham's Law of Effusion

Graham's law of effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it can be written as: Rate_1 / Rate_2 = sqrt(Molar_mass_2 / Molar_mass_1) Where Rate_1 and Rate_2 are the rates of effusion of two different gases, and Molar_mass_1 and Molar_mass_2 are their respective molar masses. We will use this formula to calculate the ratio of the rate of effusion of nitrogen (N_2) to hydrogen (H_2).

Find the molar masses of nitrogen and hydrogen

First, we need to find the molar masses of nitrogen and hydrogen. The molar mass of a molecule can be found by adding up the atomic masses of all the atoms present in the molecule. For nitrogen (N_2), it consists of 2 nitrogen atoms, and each nitrogen atom has a molar mass of 14.01 g/mol. The molar mass of a nitrogen molecule is: Molar_mass_N2 = 2 * 14.01 g/mol = 28.02 g/mol For a hydrogen molecule (H_2), it consists of two hydrogen atoms, and each hydrogen atom has a molar mass of approximately 1 g/mol. So, the molar mass of a hydrogen molecule is: Molar_mass_H2 = 2 * 1 g/mol = 2 g/mol

Apply Graham's Law to find the rate of effusion ratio

Now, we can use Graham's Law to calculate the ratio of the rate of effusion of nitrogen (N_2) to hydrogen (H_2): Rate_N2 / Rate_H2 = sqrt(Molar_mass_H2 / Molar_mass_N2) Plugging in the values that we calculated earlier, we get: Rate_N2 / Rate_H2 = sqrt(2 g/mol / 28.02 g/mol)

Simplify the fraction and calculate the ratio

Now, we will simplify the fraction within the square root: Rate_N2 / Rate_H2 = sqrt(1 / 14.01) To find the ratio, calculate the square root: Rate_N2 / Rate_H2 ≈ sqrt(1/14.01) ≈ \dfrac{1}{3.74} So, the ratio of the rate of effusion of nitrogen to hydrogen is approximately 1:3.74, or in other words, hydrogen effuses approximately 3.74 times faster than nitrogen.

Key concepts

Rate of Effusion

The rate of effusion refers to how quickly a gas escapes through a tiny hole into a vacuum. This concept is an essential part of understanding gas behavior and dynamics. According to Graham's Law of Effusion, the rate at which a gas effuses is inversely proportional to the square root of its molar mass.
This means that lighter gases effuse more quickly than heavier gases.
The mathematical formula is expressed as:

• \( \frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{\text{Molar mass}_2}{\text{Molar mass}_1}} \)

Where \( \text{Rate}_1 \) and \( \text{Rate}_2 \) are the effusion rates of two different gases, and \( \text{Molar mass}_1 \) and \( \text{Molar mass}_2 \) are their molar masses.
This principle is widely used to compare how different gases diffuse in various conditions.

Molar Mass

Molar mass is the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). It is the sum of the atomic masses of all the atoms in a molecule.
This measurement is crucial in chemistry as it allows us to relate a substance's mass to its amount in moles, providing a bridge between the atomic scale and the macroscopic scale that we can measure and observe.
In the context of effusion, a gas with a smaller molar mass will effuse more rapidly than a gas with a larger molar mass.

• For example, hydrogen, with a low molar mass of about 2 g/mol, effuses faster than nitrogen, which has a molar mass of about 28.02 g/mol.

Understanding molar mass provides insight into how different gases behave under various circumstances, such as how they will act when they effuse.

Nitrogen (N2) Gas

Nitrogen gas makes up about 78% of the Earth's atmosphere, making it the most abundant gas. It exists as a diatomic molecule, \( N_2 \), meaning each molecule contains two nitrogen atoms.
The molar mass of nitrogen gas is about 28.02 g/mol, calculated by adding the atomic mass of two nitrogen atoms (each approximately 14.01 g/mol).
Nitrogen is a diatomic molecule because it readily forms triple bonds between its atoms, which makes it very stable.

• This stability is why nitrogen is unreactive under standard conditions.
• It plays a fundamental role in the chemical processes of plants and animals, mainly as a source of nitrogen in proteins and nucleic acids.

Understanding nitrogen's characteristics helps in predicting its behavior in various applications, such as its rate of effusion compared to other gases.

Hydrogen (H2) Gas

Hydrogen gas is the lightest and most abundant chemical element in the universe. Its molecular form, \( H_2 \), consists of two hydrogen atoms bonded together.
It has a notably low molar mass of about 2 g/mol, making it much lighter than many other gases.

• This low molar mass means hydrogen effuses much faster than heavier gases like nitrogen.
• Due to its lightweight, hydrogen is often used in applications like fuel cells and as a lifting gas in balloons.

Because it's so light, hydrogen gas tends to rise and disperse rapidly in air, a behavior that directly results from its low molar mass.
This characteristic makes hydrogen a useful reference gas in studies of effusion and diffusion. Understanding hydrogen's properties is key to grasping its rate of effusion compared to other gases like nitrogen.