Problem 9 - Chemistry in 30 days

Question

Which of the following gases will have the highest rate of diffusion ? (a) \(\mathrm{O}_{2}\) (b) \(\mathrm{CO}_{2}\) (c) \(\mathrm{NH}_{3}\) (d) \(\mathrm{N}_{2}\)

Step-by-Step Solution

Answer

\( \mathrm{NH}_3 \) will have the highest rate of diffusion.

Step 1: Understanding Rate of Diffusion Formula

The rate of diffusion of a gas is inversely proportional to the square root of its molar mass, according to Graham's law of diffusion. The formula is: \[ r \propto \frac{1}{\sqrt{M}} \] where \( r \) is the rate of diffusion and \( M \) is the molar mass.

Step 2: Identify the Molar Masses

To determine which gas has the highest rate of diffusion, we first need to find the molar masses of the gases:- \( \mathrm{O}_2 \): 32 g/mol - \( \mathrm{CO}_2 \): 44 g/mol - \( \mathrm{NH}_3 \): 17 g/mol - \( \mathrm{N}_2 \): 28 g/mol.

Step 3: Apply Graham's Law

We apply Graham's law by comparing the inverse of the square root of the molar masses. The gas with the smallest molar mass will diffuse the fastest:- \( \sqrt{32} \): 5.66- \( \sqrt{44} \): 6.63- \( \sqrt{17} \): 4.12- \( \sqrt{28} \): 5.29The gas with the lowest square root value, hence the lowest molar mass, will have the highest rate of diffusion.

Step 4: Determine the Gas with Highest Diffusion Rate

Comparing the values, the smallest molar mass and square root value belong to \( \mathrm{NH}_3 \), which means it will have the highest rate of diffusion according to Graham's law.

Key Concepts

Molar Mass Calculations

Molar mass is a basic concept in chemistry and is used to measure the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It helps us understand how heavy or light a substance is on a molecular level. To calculate the molar mass of a compound, you sum the atomic masses of all the atoms present in the chemical formula. For example:

• For \( \mathrm{O}_2 \), each oxygen atom has a mass of approximately 16 g/mol. Thus, the molar mass of \( \mathrm{O}_2 \) is \( 16 \times 2 = 32 \) g/mol.
• Similarly, \( \mathrm{CO}_2 \) includes one carbon atom (12 g/mol) and two oxygen atoms (16 g/mol each), giving it a molar mass of \( 12 + 32 = 44 \) g/mol.
• \( \mathrm{NH}_3 \) consists of one nitrogen atom (14 g/mol) and three hydrogen atoms (1 g/mol each), resulting in a molar mass of \( 14 + 3 = 17 \) g/mol.
• Finally, \( \mathrm{N}_2 \) contains two nitrogen atoms, so its molar mass is \( 14 \times 2 = 28 \) g/mol.

Understanding molar mass is essential for many calculations in chemistry, including reactions and diffusion processes. It provides the necessary parameters to apply laws such as Graham's Law of Diffusion.

Rate of Diffusion in Gases

The rate of diffusion describes how quickly a gas spreads out through another substance or space. In the context of gases, Graham's Law provides a simple relationship to predict diffusion behavior.According to Graham's Law, the rate at which a gas diffuses is inversely proportional to the square root of its molar mass. This means:

• A lighter gas will diffuse more quickly than a heavier one.
• This is because lighter gas molecules have higher average speeds at a given temperature.
• The relationship can be expressed mathematically as \[ r_1/r_2 = \sqrt{M_2/M_1}\]where \( r_1 \) and \( r_2 \) are the diffusion rates, and \( M_1 \) and \( M_2 \) are the molar masses of the two gases.

In practice, this relationship helps chemists predict and compare how different gases will behave in various conditions, allowing them to account for variables like temperature and pressure while focusing on molecular mass differences.

Application of Gas Laws

Gas laws are fundamental principles in chemistry that predict how gases will behave under different conditions. Graham's Law of Diffusion, along with other gas laws such as Boyle's Law and Charles's Law, make up the basic toolkit for studying gaseous behavior. Graham's Law, specifically, has practical applications:

• In industrial settings, it can optimize processes that require specific gas mixtures.
• In science, it assists in separating gases, a critical application in both research and industry.
• In medicine, understanding gas diffusion is central to explaining oxygen delivery in the lungs and other respiratory processes.

Through these applications, gas laws form the foundation for deeper exploration into thermodynamics and other advanced areas. Understanding these principles enables anyone dealing with gases to manipulate and predict outcomes with confidence, fostering advancements in technology, research, and various practical fields.