Question

The natural abundances of the two stable isotopes of hydrogen (hydrogen and deuterium) are 99.99 percent \({ }_{1}^{1} \mathrm{H}\) and 0.01 percent \({ }_{1}^{2} \mathrm{H}\). 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 \(1.00 \mathrm{~g} / \mathrm{mL})\).

Short answer

There are approximately \(1.336 \times 10^{17}\) \(\text{D}_2\text{O}\) molecules in 400 mL of water.

Step-by-step solution

Calculate the mass of water

The density of water is given as \(1.00 \, \text{g/mL}\). Therefore, the mass of \(400 \, \text{mL}\) of water is calculated as follows:\[\text{Mass} = \text{Volume} \times \text{Density} = 400 \, \text{mL} \times 1.00 \, \text{g/mL} = 400 \, \text{g}\]

Determine moles of water

The molar mass of water (\(\text{H}_2\text{O}\)) is calculated as follows:\[\text{Molar Mass of } \text{H}_2\text{O} = 2 \times 1.01 + 16.00 = 18.02 \, \text{g/mol} \]Now, calculate the number of moles of \(\text{H}_2\text{O}\) using its mass:\[\text{Moles of } \text{H}_2\text{O} = \frac{400 \, \text{g}}{18.02 \, \text{g/mol}} \approx 22.2 \, \text{mol}\]

Calculate the number of water molecules

Using Avogadro's number \((6.022 \times 10^{23} \, \text{molecules/mol})\), calculate the number of water molecules:\[\text{Number of water molecules} = 22.2 \, \text{mol} \times 6.022 \times 10^{23} \, \text{molecules/mol} \approx 1.336 \times 10^{25} \, \text{molecules}\]

Calculate the fraction of \(\text{D}_2\text{O}\)

Since the natural abundance of deuterium \((\text{D})\) is 0.01%, the probability of a water molecule being \(\text{D}_2\text{O}\) is:\[\left(\frac{0.01}{100}\right)^2 = 1 \times 10^{-8}\]This is because both hydrogen atoms in \(\text{D}_2\text{O}\) must be deuterium atoms.

Calculate the number of \(\text{D}_2\text{O}\) molecules

Now, calculate the number of \(\text{D}_2\text{O}\) molecules by multiplying the total number of water molecules by the probability of a molecule being \(\text{D}_2\text{O}\):\[\text{Number of } \text{D}_2\text{O} \text{ molecules} = 1.336 \times 10^{25} \times 1 \times 10^{-8} = 1.336 \times 10^{17}\]

Key concepts

Understanding Natural Abundance

Natural abundance explains the proportion of each isotope present in the environment. Isotopes are atoms of the same element that vary in the number of neutrons they contain. For hydrogen, the two stable isotopes are protium

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Molar Mass Calculation Simplified

Molar mass is essentially the weight of one mole of a substance, expressed in grams per mole (g/mol). This is calculated by adding the atomic masses of all the atoms in a molecule. For a water molecule (\(\text{H}_2\text{O}\)), these atomic masses are for two hydrogen atoms and one oxygen atom.

The atomic mass of hydrogen is approximately \(1.01 \text{ g/mol}\).

The atomic mass of oxygen is approximately \(16.00 \text{ g/mol}\).

Thus, the molar mass of water is calculated as:\[\text{Molar mass of H}_2\text{O} = 2 \times 1.01 + 16.00 = 18.02 \text{ g/mol}\]
Once you know the molar mass, you can calculate the number of moles in any given mass of water by using the formula:\[\text{Moles of water} = \frac{\text{mass}}{\text{molar mass}}\]This is a crucial step in determining how many molecules comprise that specific mass.

Demystifying Avogadro's Number

Avogadro's number is a fundamental constant in chemistry that represents the number of atoms or molecules in one mole of any substance. This number is approximately \(6.022 \times 10^{23}\).

• It allows us to convert between atoms/molecules and moles, making it possible to move from the atomic scale to the macroscopic scale.
• When you know the number of moles, you simply multiply it by Avogadro's number to find the total number of molecules. For instance:\[\text{Total molecules} = \text{moles} \times 6.022 \times 10^{23}\]Understanding this conversion is key to performing calculations involving chemical reactions and compositions.

Essentials of Water Density

Water density is the measure of mass per unit volume. Typically, water has a density of \(1.00 \text{ g/mL}\) at its maximum density, which occurs at 4 degrees Celsius. This density means that:

• For every milliliter of water, it contains about 1 gram.
• This property allows easy calculations for converting between mass and volume for water-related calculations.
  Given the density, you can determine the mass from the volume using the formula:\[\text{Mass} = \text{Volume} \times \text{Density}\]This conversion is simple yet powerful for various practical applications, notably in calculating how much of a particular substance is found in a given body of water.