Question

Deuterium, \(^{2} \mathrm{H}(2.0140 \mathrm{u}),\) is sometimes used to replace the principal hydrogen isotope \(^{1} \mathrm{H}\) in chemical studies. The percent natural abundance of deuterium is 0.015\%. If it can be done with 100\% efficiency, what mass of naturally occurring hydrogen gas would have to be processed to obtain a sample containing \(2.50 \times 10^{21}^{2} \mathrm{H}\) atoms?

Short answer

Substitution and calculation with the above steps will yield the mass of naturally occurring hydrogen gas that needs to be processed to obtain a specific number of Deuterium atoms.

Step-by-step solution

Calculate moles of Deuterium

First, convert the number of deuterium atoms needed (\(2.50 \times 10^{21}\)) to moles using Avogadro's number (\(6.022 \times 10^{23}\) atoms/mole). The calculation is given by: \n\n\[ \frac{2.50 \times 10^{21} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mole}} \]

Convert moles of Deuterium to moles of Hydrogen

Given the natural abundance of Deuterium (0.015%), convert the moles of Deuterium to moles of Hydrogen using the percent abundance as follows:\n\n\[ \text{moles of hydrogen} = \frac{\text{moles of deuterium}}{\text{percent abundance of deuterium} / 100} \]\n\nNote that the percent abundance needs to be converted to decimal form by dividing by 100.

Convert moles of Hydrogen to grams

Now, convert the moles of hydrogen to grams using the molar mass of hydrogen (1.008 u). The molar mass is the mass of one mole of a substance. It is used as a conversion factor to go from moles to grams (and vice versa). The formula is:\n\n\[ \text{grams of hydrogen} = \text{moles of hydrogen} \times \text{molar mass of hydrogen} \]

Key concepts

Isotopes

Isotopes are variants of a particular chemical element that share the same number of protons but differ in their number of neutrons. This results in isotopes having different atomic masses. However, because they have the same number of protons, these isotopes maintain the same chemical behavior.
For example, hydrogen has three main isotopes: protium ( ^{1}H) with zero neutrons, deuterium ( ^{2}H) with one neutron, and tritium ( ^{3}H) with two neutrons. In this exercise, the focus is on deuterium, an isotope that is relatively rare with only 0.015% abundance in nature.
Understanding isotopes is crucial in chemical calculations as they affect the mass and sometimes the behavior of the substances involved. They are used in various scientific fields, from nuclear energy to medicine, due to their unique properties.

Mole Concept

The mole is a fundamental concept in chemistry that provides a bridge between the atomic scale and macroscopic measurements. A mole is defined as the amount of substance that contains the same number of entities (such as atoms, ions, or molecules) as there are in 12 grams of pure carbon-12 ( ^{12}C), which is about 6.022 x 10^{23} entities. This value is known as Avogadro's number. The mole allows chemists to convert between the mass of a sample and the number of atoms or molecules it contains. In the provided solution, the mole concept helps quantify the number of deuterium atoms by converting them into moles. This conversion is essential since chemical reactions and calculations are often more conveniently expressed in terms of moles rather than individual atoms.

Avogadro's Number

Avogadro's number ( 6.022 imes 10^{23}) is a constant used to describe the number of atoms, ions, or molecules in one mole of a substance. This number is incredibly large because atoms and molecules are extremely small. In chemical calculations, Avogadro's number acts as a critical conversion factor. It connects the microscale (atoms or molecules) to the macroscale (grams or liters that are measurable). For instance, as seen in the step-by-step solution, converting the number of deuterium atoms to moles requires division by Avogadro's number. By employing this constant, the number of particles in a given sample can be appreciated in terms of measurable quantities, making it fundamental for calculating reactions, determining material properties, and understanding the structure of substances at a molecular level.