Problem 130 - Chemistry

Question

The natural abundances of the two stable isotopes of hydrogen (hydrogen and deuterium) are \({ }_{1}^{1} \mathrm{H}\) : 99.985 percent and \({ }_{1}^{2} \mathrm{H}: 0.015\) percent. Assume that water exists as either \(\mathrm{H}_{2} \mathrm{O}\) or \(\mathrm{D}_{2} \mathrm{O} .\) Calculate the number of \(\mathrm{D}_{2} \mathrm{O}\) molecules in exactly \(400 \mathrm{~mL}\) of water. (Density \(=1.00 \mathrm{~g} / \mathrm{mL} .)\)

Step-by-Step Solution

Answer

The number of \(\mathrm{D}_{2}\mathrm{O}\) molecules in 400 mL of water is approximately \(9.033 \times 10^{22}\).

Step 1: Calculate mass of water

First, find the mass of the water. We know, density = mass/volume. so, mass = density * volume = \(1.00 \, \mathrm{g/mL} * 400 \, \mathrm{mL} = 400 \, \mathrm{g}\).

Step 2: Calculate mass of Deuterium

Next, find the mass of Deuterium. Assume the water exists only as \( \mathrm{H}_{2}\mathrm{O} \) and \( \mathrm{D}_{2}\mathrm{O} \). As such, 0.015 percent of the hydrogen atoms are Deuterium. So, the mass of deuterium = \(0.00015 * 400 \, \mathrm{g} = 0.06 \, \mathrm{g}\).

Step 3: Compute moles of Deuterium

Then, convert the mass of Deuterium to moles. The molar mass of Deuterium is 2.00 g/mol. So, moles of Deuterium = \(0.06 \, \mathrm{g} / 2.00 \, \mathrm{g/mol} = 0.03 \, \mathrm{mol}\).

Step 4: Calculate molecules of \( \mathrm{D}_{2}\mathrm{O} \)

Finally, calculate the number of \( \mathrm{D}_{2}\mathrm{O} \) molecules. As each \( \mathrm{D}_{2}\mathrm{O} \) molecule contains two Deuterium atoms, the moles of \( \mathrm{D}_{2}\mathrm{O} \) is half the moles of Deuterium. So, moles of \( \mathrm{D}_{2}\mathrm{O} \) = \(0.03 \, \mathrm{mol} / 2 = 0.015 \, \mathrm{mol}\). Then, to get the number of molecules, multiply by Avogadro's number \(6.022 \times 10^{23} \, \mathrm{mol}^{-1}\). Therefore, the number of \( \mathrm{D}_{2}\mathrm{O} \) molecules = \(0.015 \, \mathrm{mol} * 6.022 \times 10^{23} \, \mathrm{mol}^{-1} = 9.033 \times 10^{22}\).

Key Concepts

Understanding Deuterium

Deuterium is a fascinating component of chemistry, particularly when discussing isotopes. Isotopes are atoms of the same element that have different numbers of neutrons. In the case of hydrogen, its two stable isotopes are protium, which has no neutrons, and deuterium. Deuterium, often represented as \( ^2H \) or \( D \), has one neutron in addition to its single proton.Deuterium is uncommon in nature. It accounts for only about 0.015% of hydrogen found on Earth, which is why it's considered an isotope of interest when studying topics like heavy water. Heavy water (\( \mathrm{D}_2\mathrm{O} \)) is composed of deuterium rather than the typical hydrogen found in regular water. Scientists and chemists often study these isotopes to understand various physical, chemical, and nuclear reactions and to explore their potential applications. For example, deuterium's unique properties make it useful in nuclear reactors and certain types of lasers.

Role of Avogadro's Number

Avogadro's number is a key concept in chemistry that relates to the quantity of particles in a mole. Named after the scientist Amedeo Avogadro, this number is approximately \( 6.022 \times 10^{23} \). It represents the scale at which microscopic particles like atoms and molecules can be related to a macroscopic quantity of matter that we can measure in the laboratory.When calculating the number of molecules or atoms in a substance, Avogadro's number is essential. For example, in the exercise, this number helps determine how many molecules of heavy water (\( \mathrm{D}_2\mathrm{O} \)) are present in a given sample. Since \( 1 \text{ mole} \) of any substance contains \( 6.022 \times 10^{23} \) entities, multiplying the number of moles by Avogadro's number gives us the total count of molecules present.Understanding this concept is critical for students as it is foundational in chemistry, linking the atomic and molecular levels to measurable quantities.

Molar Mass Calculation Basics

Molar mass is a critical concept when performing calculations involving chemical reactions and stoichiometry. It helps convert between the mass of a substance and the number of moles. The molar mass is the mass of one mole of a given substance, measured in grams per mole (g/mol).To calculate molar mass, you typically sum the atomic masses of the elements present in a compound. For deuterium, the molar mass is approximately \( 2.00 \, \text{g/mol} \) since it contains one proton and one neutron.In the context of the exercise, knowing the molar mass of deuterium allows us to convert the calculated mass of deuterium into moles. This conversion is essential in finding any further relationships, such as determining the number of \( \mathrm{D}_2\mathrm{O} \) molecules.Grasping molar mass and its calculations is vital for students, as it forms the backbone of translating chemical formulas into real-world quantities.