Problem 85 - Chemistry: The Central Science in SI Units, Global Edition

Question

Hydrogen has two naturally occurring isotopes, \({ }^{1} \mathrm{H}\) and \({ }^{2} \mathrm{H}\). Chlorine also has two naturally occurring isotopes, \({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\). Thus, hydrogen chloride gas consists of four distinct types of molecules: \({ }^{1} \mathrm{H}^{35} \mathrm{Cl},{ }^{1} \mathrm{H}^{37} \mathrm{Cl},{ }^{2} \mathrm{H}^{35} \mathrm{Cl},\) and \({ }^{2} \mathrm{H}^{37} \mathrm{Cl}\). Place these four molecules in order of increasing rate of effusion.

Step-by-Step Solution

Answer

The order of effusion rates, from fastest to slowest, for these four hydrogen chloride molecules is: \({ }^{1} \mathrm{H}^{35} \mathrm{Cl} < { }^{2} \mathrm{H}^{35} \mathrm{Cl} < { }^{1} \mathrm{H}^{37} \mathrm{Cl} < { }^{2} \mathrm{H}^{37} \mathrm{Cl}\).

Step 1: Write down Graham's Law of Effusion Formula

Write down the formula for Graham's Law of Effusion: \[ Rate_1/Rate_2 = \sqrt{M_2/M_1} \] Where \(Rate_1\) and \(Rate_2\) are the effusion rates of two different gases and \(M_1\) and \(M_2\) are their respective molar masses.

Step 2: Calculate molar masses for each type of molecule

Next, calculate the molar masses of the four distinct types of hydrogen chloride molecules: 1. \({ }^{1} \mathrm{H}^{35} \mathrm{Cl}\): 1 (hydrogen) + 35 (chlorine) = 36 g/mol 2. \({ }^{1} \mathrm{H}^{37} \mathrm{Cl}\): 1 (hydrogen) + 37 (chlorine) = 38 g/mol 3. \({ }^{2} \mathrm{H}^{35} \mathrm{Cl}\): 2 (hydrogen) + 35 (chlorine) = 37 g/mol 4. \({ }^{2} \mathrm{H}^{37} \mathrm{Cl}\): 2 (hydrogen) + 37 (chlorine) = 39 g/mol

Step 3: Use Graham's Law of Effusion to determine the order of effusion rates

Since the effusion rate is inversely proportional to the square root of the molar mass, we can determine the order by comparing the square root of the molar masses: 1. \(\sqrt{36}\) = 6 2. \(\sqrt{37}\) ≈ 6.08 3. \(\sqrt{38}\) ≈ 6.16 4. \(\sqrt{39}\) ≈ 6.24 The lower the square root of the molar mass, the faster the effusion rate. Thus, the order of effusion rates, from fastest to slowest, is: \({ }^{1} \mathrm{H}^{35} \mathrm{Cl} ({ 6 }) < { }^{2} \mathrm{H}^{35} \mathrm{Cl} ({ 6.08 }) < { }^{1} \mathrm{H}^{37} \mathrm{Cl} ({ 6.16 }) < { }^{2} \mathrm{H}^{37} \mathrm{Cl} ({ 6.24 })\)

Key Concepts

Isotopes

Atoms of the same element that have different numbers of neutrons are called isotopes. For example, hydrogen has isotopes

• \( {}^1 \text{H} \) - also known as protium, which has one proton and no neutrons,
• \( {}^2 \text{H} \) - known as deuterium, with one proton and one neutron.

Chlorine, on the other hand, has isotopes

• \( {}^{35} \text{Cl} \) - with 18 neutrons, and
• \( {}^{37} \text{Cl} \) - with 20 neutrons.

Both hydrogen and chlorine isotopes occur naturally and participate in forming compounds, such as hydrogen chloride. Because the isotope variation alters the atomic mass while preserving chemical properties, isotopes play a significant role in phenomena such as effusion, as seen in Graham's Law.

Molar Mass

Molar mass is the weight of one mole of a substance, expressed in grams per mole (g/mol). It is crucial for understanding how different substances behave in chemical reactions and processes. When calculating the molar mass of a compound, you sum the isotopic masses of its constituent atoms based on their abundances.

For hydrogen chloride molecules in our exercise, considering the isotopes of hydrogen and chlorine, the different molar masses are as follows:

• \( {}^1\text{H}^{35}\text{Cl} \): 36 g/mol,
• \( {}^1\text{H}^{37}\text{Cl} \): 38 g/mol,
• \( {}^2\text{H}^{35}\text{Cl} \): 37 g/mol,
• \( {}^2\text{H}^{37}\text{Cl} \): 39 g/mol.

The knowledge of molar mass is pivotal when applying Graham's Law of Effusion, which requires comparing gases based on their molar mass properties.

Effusion Rates

Effusion is the process through which gas molecules escape through a tiny hole into a vacuum. The rate at which gases effuse is governed by Graham's Law of Effusion. According to this law, the rate of effusion is inversely proportional to the square root of the gas's molar mass:

\[\text{Rate}_1 / \text{Rate}_2 = \sqrt{M_2 / M_1}\]

This equation indicates that gases with smaller molar masses effuse faster because their particles move more quickly.
In the exercise, when comparing hydrogen chloride molecules with different isotopic compositions, we see varied effusion rates:

• \( {}^1\text{H}^{35}\text{Cl} \) effuses the fastest because it has the lowest molar mass, followed by
• \( {}^2\text{H}^{35}\text{Cl} \), \( {}^1\text{H}^{37}\text{Cl} \), and finally,
• \( {}^2\text{H}^{37}\text{Cl} \) with the highest molar mass effuses the slowest.

Understanding effusion rates is essential for fields such as chemistry and environmental science, where gas movement plays a crucial role.