Problem 158 - Chemistry The Science in Context

Question

Which noble gas effuses at about half the effusion rate of \(\mathrm{O}_{2} ?\)

Step-by-Step Solution

Answer

Answer: Helium (He)

Step 1: Find the molar mass of \(\mathrm{O}_{2}\)

To find the molar mass of \(\mathrm{O}_{2}\), we just need to sum the molar masses of the individual oxygen atoms. The molar mass of a single oxygen atom is approximately 16 g/mol. Since there are 2 oxygen atoms in \(\mathrm{O}_{2}\), the molar mass of \(\mathrm{O}_{2}\) is: $$M_{O_2} = 2 \cdot 16 \,\text{g/mol} = 32 \,\text{g/mol}$$

Step 2: Use Graham's Law of Effusion

Graham's law of effusion states that the rates of effusion of two gases are inversely proportional to the square roots of their molar masses: $$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$$ Here, \(r_1\) and \(M_1\) are the effusion rate and molar mass of the first gas (\(\mathrm{O}_{2}\)), and \(r_2\) and \(M_2\) are the effusion rate and molar mass of the second gas (the noble gas we want to identify). From the problem statement, we know that the noble gas effuses at about half the rate of \(\mathrm{O}_{2}\), so we can write: $$\frac{r_1}{r_2} = \frac{1}{2} = \sqrt{\frac{M_2}{32 \,\text{g/mol}}}$$ Next, we need to solve for the molar mass \(M_2\) of the noble gas.

Step 3: Solve for \(M_2\)

To solve for \(M_2\), we can rearrange the equation from Step 2 and square both sides: $$M_2 = 32 \,\text{g/mol}\cdot \left(\frac{r_1}{r_2}\right)^2 = 32 \,\text{g/mol}\cdot \left(\frac{1}{2}\right)^2 = 32 \,\text{g/mol}\cdot \frac{1}{4} = 8 \,\text{g/mol}$$

Step 4: Identify the noble gas

Now that we have found the molar mass of the noble gas, we can look at the list of noble gases and their molar masses to identify which gas has a molar mass of 8 g/mol: - Helium (He): ~4 g/mol - Neon (Ne): ~20 g/mol - Argon (Ar): ~40 g/mol - Krypton (Kr): ~84 g/mol - Xenon (Xe): ~131 g/mol - Radon (Rn): ~222 g/mol From this list, no noble gas has an exact molar mass of 8 g/mol. However, the closest gas is helium with a molar mass of ~4 g/mol. Thus, helium is the noble gas that effuses at about half the effusion rate of \(\mathrm{O}_{2}\).

Key Concepts

Molar Mass

Molar mass is an essential concept in chemistry that helps us understand how much one mole of a substance weighs. It's the sum of the masses of all the atoms in a molecule, usually expressed in grams per mole (g/mol). For example, the molar mass of an oxygen molecule (\(\mathrm{O}_2\)) is calculated by adding the molar masses of two oxygen atoms, each weighing about 16 g/mol. Hence, the molar mass of \(\mathrm{O}_2\) is \(2 \times 16 \, \text{g/mol} = 32 \, \text{g/mol}\).
Knowing the molar mass of a substance allows you to predict how it behaves under certain conditions, such as during a reaction or when it effuses through a membrane. Understanding Graham's Law of Effusion, which involves molar masses, can predict how quickly gases will escape through small openings. This highlights the importance of molar mass in determining the properties and behavior of gases, illustrating how interconnected these principles are in chemistry.

Noble Gases

Noble gases are a group of elements in the periodic table characterized by their complete valence electron shells. This property makes them very stable and unreactive under normal conditions. They are known for their unique traits:

• They exist as single atoms, not molecules.
• They are odorless, colorless, and tasteless.
• They have low chemical reactivity.

Common noble gases include helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn). With diverse applications due to their stability, such as neon lighting or helium being used for cooling superconductors, they are vital in both scientific and everyday contexts. Regarding molar masses, noble gases have distinct values, with helium having the lowest molar mass of approximately 4 g/mol. They don't react easily, making them suitable for various uses where non-reactivity is crucial. Understanding these gases provides context for why helium effuses more readily compared to heavier noble gases like xenon.

Oxygen

Oxygen is a diatomic molecule represented as \(\mathrm{O}_2\), meaning it consists of two oxygen atoms bonded together. It is a vital element for life, essential for respiration in most organisms. Besides being a significant component of the Earth's atmosphere, it is involved in combustion processes and the formation of water and carbon dioxide.
Oxygen's molar mass is approximately 32 g/mol, which affects its behavior in various scenarios, including the rate at which it effuses. Effusion is the process by which a gas escapes through a tiny hole into a vacuum, and oxygen's moderate molar mass allows it to effuse at a specific rate. Understanding the molar mass of oxygen enables us to apply Graham's Law of Effusion, which compares the effusion rates of different gases and highlights the comparative effusion rate differences between \(\mathrm{O}_2\) and other gases like helium. Such comparisons help in predicting the movement and speed of different gases in mixtures and are useful in industries where gas separation and purification are necessary.