Question

Calculate the amount of energy released in \(\mathrm{kJ} / \mathrm{mol}\) for the fusion reaction of \({ }^{1} \mathrm{H}\) and \({ }^{2} \mathrm{H}\) atoms to yield a \({ }^{3} \mathrm{He}\) atom: $$ { }_{1}^{1} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \longrightarrow{ }_{2}^{3} \mathrm{He} $$ The atomic masses are \({ }^{1} \mathrm{H}(1.00783 \mathrm{u}),{ }^{2} \mathrm{H}(2.01410 \mathrm{u})\), and \({ }^{3} \mathrm{He}(3.01603 \mathrm{u})\).

Short answer

The energy released is approximately \(5.308 \times 10^8\,\text{kJ/mol}\).

Step-by-step solution

Write the reaction equation

The fusion reaction we are examining is:\[ {}_{1}^{1} \mathrm{H} + {}_{1}^{2} \mathrm{H} \rightarrow {}_{2}^{3} \mathrm{He} \]This equation represents the fusion of a hydrogen-1 isotope with a hydrogen-2 isotope to form a helium-3 isotope.

Calculate the initial mass of reactants

To find the total mass of the reactants, we sum the atomic masses of \(^1\mathrm{H}\) and \(^2\mathrm{H}\):\[\text{Initial mass} = 1.00783\,\text{u} + 2.01410\,\text{u} = 3.02193\,\text{u}\]

Record the mass of the product

The mass of the product \(^3\mathrm{He}\) is given as:\[\text{Mass of } ^3\mathrm{He} = 3.01603\,\text{u}\]This is the mass of helium-3 produced by the reaction.

Calculate mass defect

The mass defect is the difference between the initial mass of the reactants and the mass of the product:\[\text{Mass defect} = 3.02193\,\text{u} - 3.01603\,\text{u} = 0.00590\,\text{u}\]

Calculate energy released using mass-energy equivalence

Use Einstein’s mass-energy equivalence equation \(E = mc^2\), where \(m\) is the mass defect and \(c\) is the speed of light \(3.00 \times 10^8\,\text{m/s}\). First, convert the mass defect from atomic mass units (u) to kilograms. Given that \(1\,\text{u} = 1.66054 \times 10^{-27}\,\text{kg}\):\[m = 0.00590\,\text{u} \times 1.66054 \times 10^{-27}\,\text{kg/u} = 9.79719 \times 10^{-30}\,\text{kg}\]Now calculate the energy:\[E = (9.79719 \times 10^{-30}\,\text{kg})(3.00 \times 10^8\,\text{m/s})^2 = 8.81747 \times 10^{-13}\,\text{J}\]

Convert energy from joules per mole to kilojoules per mole

One mole of particles corresponds to Avogadro's number \(6.022 \times 10^{23}\) particles. Convert energy per reaction obtained in joules to kilojoules per mole:\[E_{\text{per mole}} = 8.81747 \times 10^{-13}\,\text{J} \times 6.022 \times 10^{23} = 5.308\times 10^{11}\,\text{J/mol}\]Convert joules to kilojoules:\[E_{\text{in kJ/mol}} = \frac{5.308 \times 10^{11}\,\text{J/mol}}{1000} = 5.308 \times 10^{8}\,\text{kJ/mol}\]

Key concepts

Mass Defect

The concept of mass defect revolves around the peculiar phenomenon where the mass of an atomic nucleus is less than the sum of the individual masses of its constituent protons and neutrons. In nuclear fusion reactions, like the one involving Hydrogen-1 and Hydrogen-2 to form Helium-3, there is a small disparity in mass before and after the reaction.
This "missing" mass, known as the mass defect, occurs because part of the mass is transformed into energy during the process. In simpler terms, the mass defect is the manifestation of mass being converted into energy, rather than being destroyed or lost. The initial mass of hydrogen isotopes, which was calculated as 3.02193 u, is slightly higher than the final mass of Helium-3, which is 3.01603 u. This tiny discrepancy results in a mass defect of 0.00590 u, yet it heralds the release of a tremendous amount of energy.

Einstein's Mass-Energy Equivalence

Einstein's mass-energy equivalence principle, famously expressed in the formula \( E = mc^2 \), uncovers the profound relationship between mass and energy. It suggests that mass can be transformed into energy and vice versa, hence, they are essentially two forms of the same entity.
In the context of nuclear fusion, the mass defect turns into an incredible energetic payoff. Using \( E = mc^2 \), the mass defect calculated earlier is multiplied by the speed of light squared, \( 3.00 \times 10^8\,\text{m/s} \), to determine the energy released. For instance, for the fusion of Hydrogen-1 and Hydrogen-2 into Helium-3, a mass defect of 0.00590 u is equivalent to an energy release of about \( 8.81747 \times 10^{-13} \) J per reaction.

• This enormous energy potential is what propels the interest in nuclear fusion as an energy source, especially given its implications for clean and virtually limitless energy.

Helium-3 Production

Helium-3 is an isotope of helium with two protons and one neutron. Its production through fusion reactions, like the merging of Hydrogen-1 and Hydrogen-2 isotopes, is significant for a number of reasons.
Firstly, Helium-3 is rare on Earth, making its potential production through fusion a valuable resource for the future. Helium-3 has promising applications, including its use in advanced nuclear fusion reactors and in cryogenics because of its unique properties.

• The fusion reaction discussed converts simple isotopes of hydrogen into Helium-3, releasing substantial energy in the process.
• Potentially, Helium-3 could fuel a different kind of fusion reactor, known as a D-³He fusion reactor, which would produce highly efficient energy with fewer radioactive by-products than conventional methods.

Understanding Helium-3 production not only aids energy innovation but also shows how seemingly minor atomic processes could have monumental impacts on global energy strategies.