Question

What is the ratio of rates of effusion of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) under the same conditions?

Short answer

The ratio of the rates of effusion of \(\mathrm{N}_2\) to \(\mathrm{O}_2\) is approximately 1.069.

Step-by-step solution

Understanding the Problem

We need to find the ratio of the rates of effusion for nitrogen gas (\(\mathrm{N}_2\)) and oxygen gas (\(\mathrm{O}_2\)) under the same conditions. Effusion is the process by which gas molecules escape through a tiny hole into a vacuum. We can use Graham's law of effusion which states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

Graham's Law of Effusion

Graham's Law of Effusion states that \( \frac{\text{Rate of effusion of gas 1}}{\text{Rate of effusion of gas 2}} = \sqrt{\frac{M_2}{M_1}} \), where \( M_1 \) and \( M_2 \) are the molar masses of the gases. Therefore, to find the ratio of the effusion rates of \( \mathrm{N}_2 \) and \( \mathrm{O}_2 \), we substitute their molar masses into this formula.

Identify Molar Masses

The molar mass of \(\mathrm{N}_2\) is 28.02 g/mol and the molar mass of \(\mathrm{O}_2\) is 32.00 g/mol. These values will be used in the square root equation provided by Graham's Law.

Calculate the Ratio using Molar Masses

Substitute the molar masses into Graham's Law: \[ \frac{\text{Rate of \(\mathrm{N}_2\)}}{\text{Rate of \(\mathrm{O}_2\)}} = \sqrt{\frac{M_{O_2}}{M_{N_2}}} = \sqrt{\frac{32.00}{28.02}} \]. Calculate this expression to find the ratio.

Calculate the Square Root

Perform the calculation: \[ \sqrt{\frac{32.00}{28.02}} \approx \sqrt{1.1428} \approx 1.069 \]. So the ratio of the rate of effusion of \(\mathrm{N}_2\) to \(\mathrm{O}_2\) is approximately 1.069.

Key concepts

Effusion

Effusion is a unique and intriguing process where gas molecules escape through a teeny tiny hole into an empty space or vacuum. Imagine it like air slowly leaking from a punctured balloon. This phenomenon occurs because gas molecules are always in motion and, when given an opportunity, slip through openings.
Graham's Law of Effusion helps us understand how fast different gases escape. It tells us that lighter molecules effuse faster than heavier ones due to their speed. Through this understanding, we can predict and calculate important outcomes in chemical processes.

• Describes gas molecule movement from a container to an empty space.
• Occurs due to random and constant motion of gas molecules.
• Utilizes Graham's Law for calculating effusion rates.

Molar Mass

Molar mass is simply the mass of one mole of a substance. It's like saying how heavy a bunch of the same kind of tiny particles is. In this case, we're focusing on gases, which are made up of various types of molecules.
When using Graham's Law of Effusion, molar mass plays a key role because the rate of effusion is inversely related to the square root of the gas's molar mass. Simply put, the lighter the gas (lower molar mass), the faster it will effuse.
For instance, by comparing the molar masses of nitrogen and oxygen, we see that nitrogen's lower mass indicates it will effuse more rapidly.

• Molar mass determines the heaviness of a gas molecule.
• In effusion, lighter molar mass equates to quicker movement.
• Critical for calculating effusion rate ratios using Graham's Law.

Gas Molecules

Gas molecules are the tiny pieces that make up gases like air, nitrogen, and oxygen. They're in constant, random motion, moving about easily and colliding with each other and their container walls.
Their quick motion and small size allow them to take up the shape and volume of their containers. In a process like effusion, their random movement makes it possible for some molecules to escape through an opening.
Understanding their movement and behavior is crucial for explaining how and why effusion happens.

• They're constantly moving and colliding.
• Small and able to fill any container shape.
• Movement explains processes like effusion.

Nitrogen Gas

Nitrogen gas, or \(\mathrm{N}_{2}\), composes a major part of Earth's atmosphere, taking up about 78%. It is a diatomic molecule, meaning it's composed of two nitrogen atoms bonded together.
In terms of effusion and chemical behavior, nitrogen is lighter compared to some other gases like oxygen, due to its lower molar mass of 28.02 g/mol.
Because of its weight, nitrogen effuses more quickly than heavier gases like oxygen under the same conditions. This property is well understood using Graham's Law of Effusion.

• Main component of Earth's atmosphere (78%).
• Diatomic molecule with a molar mass of 28.02 g/mol.
• Effuses more quickly than oxygen under similar conditions.

Oxygen Gas

Oxygen gas, represented as \(\mathrm{O}_{2}\), is vital for life, making up about 21% of the Earth's atmosphere. It's also a diatomic molecule, consisting of two oxygen atoms joined together.
With a molar mass of 32.00 g/mol, it's heavier than nitrogen, necessitating special calculations when predicting effusion rates.
Though essential for respiration and combustion, its effusion rate is slower than lighter gases like nitrogen, which is a fascinating insight using Graham's Law.

• Essential for life and makes up 21% of the atmosphere.
• Diatomic with a molar mass of 32.00 g/mol.
• Effuses slower than lighter gases like nitrogen.