Question

Calculate the amount of energy released per gram of hydrogen nuclei reacted for the following reaction. The atomic masses are \({ }^{1} \mathrm{H}, 1.00782 \mathrm{amu} ;{ }_{1}^{2} \mathrm{H}, 2.01410\) amu; and an electron, \(5.4858 \times\) \(10^{-4}\) amu. (Hint: Think carefully about how to account for the electron mass.) $${ }_{1}^{1} \mathrm{H}+{ }_{1}^{1} \mathrm{H} \longrightarrow{ }_{1}^{2} \mathrm{H}+{ }_{+1}^{0} \mathrm{e}$$

Short answer

The mass defect for the given reaction is calculated as: Mass defect = (2 * 1.00782 - 2.01410) amu = 0.00154 amu Converting the mass defect to energy using Einstein's mass-energy equivalence formula: \(E = mc^2 = (0.00154 \times 1.6605 \times 10^{-27} \,\text{kg}) (3.0 \times 10^8\, \text{m/s})^2 = 7.641 \times 10^{-13}\, \text{J}\) Finally, calculating the energy released per gram of hydrogen nuclei reacted: Energy per gram = \(\frac{7.641 \times 10^{-13}\, \text{J}}{2 \times 1.00782 \times 1.6605 \times 10^{-24} \,\text{kg}} = 2.27 \times 10^{11}\, \frac{\text{J}}{\text{g}}\) Thus, the energy released per gram of hydrogen nuclei reacted is approximately \(2.27 \times 10^{11} \, \frac{\text{J}}{\text{g}}\).

Step-by-step solution

Calculate the mass defect

To determine the mass defect, we need to find the difference in mass between the reactants and products in the given reaction. The reaction is: $${ }_{1}^{1} \mathrm{H}+{ }_{1}^{1} \mathrm{H} \longrightarrow{ }_{1}^{2} \mathrm{H}+{ }_{+1}^{0} \mathrm{e}$$ Let's write down the given atomic masses: Mass of hydrogen, \(m_{H} = 1.00782\) amu Mass of deuterium, \(m_{D} = 2.01410\) amu Mass of electron, \(m_{e} = 5.4858 \times 10^{-4}\) amu Now, we will calculate the mass defect: Mass defect = Mass of reactants - Mass of products

Convert mass defect to energy

To convert the mass defect into energy, we will use Einstein's mass-energy equivalence formula, which is: $$E = mc^2$$ Where \(E\) is the energy released, \(m\) is the mass defect, and \(c\) is the speed of light (\(3.0 \times 10^8 \,\text{m/s}\)). We will use the atomic mass unit (amu) as the unit for mass, and we should convert it to the SI unit, kilogram. 1 amu = 1.6605 \(\times 10^{-27}\) kg.

Calculate energy released per gram of hydrogen nuclei reacted

To find the energy released per gram of hydrogen nuclei reacted, we will divide the energy released (calculated in step 2) by the number of grams of hydrogen reacted. In this case, the number of hydrogen nuclei reacted is 2 (as we have two hydrogen nuclei in the given reaction).

Key concepts

Mass Defect

In nuclear reactions, calculating the mass defect is crucial. The mass defect refers to the difference between the mass of the reactants and the products after the reaction has occurred. During the reaction of two hydrogen nuclei (\( _1^1 ext{H} + _1^1 ext{H} \rightarrow _1^2 ext{H} + _{+1}^0 ext{e} \)), the initial mass consists of two hydrogen atoms, and the resultant mass consists of a deuterium atom and an electron. The masses are:

• Mass of hydrogen atom (\( _1^1 ext{H} \)): 1.00782 amu
• Mass of deuterium atom (\( _1^2 ext{H} \)): 2.01410 amu
• Mass of an electron: 5.4858 \( \times 10^{-4} \) amu

The mass defect is computed by finding the total mass of reactants minus the total mass of products. This serves as a measure of the amount of mass converted into energy during the reaction.

Einstein's Mass-Energy Equivalence

Einstein introduced a groundbreaking concept that changed the way we understand mass and energy. According to his mass-energy equivalence principle, mass can be converted into energy and vice-versa. This relationship is expressed through the formula:\[ E = mc^2 \]In this equation:

• \( E \) represents the energy produced or consumed.
• \( m \) is the mass defect calculated from a nuclear reaction.
• \( c \) is the speed of light in a vacuum, approximately \( 3.0 \times 10^8 \, \text{m/s} \)

Using this formula, the mass that "disappears" in a reaction as a mass defect is converted into a very large amount of energy. This principle is one of the foundational concepts for understanding how nuclear reactions power both atomic bombs and nuclear power plants.

Atomic Mass Unit (amu)

An atomic mass unit, abbreviated as amu, is a standard unit of mass that quantifies mass on an atomic or molecular scale. It serves as a useful unit when dealing with subatomic particles such as nucleons and electrons. By definition, 1 amu is equal to one-twelfth the mass of a carbon-12 atom, and equivalently:1 amu = \( 1.6605 \times 10^{-27} \) kg.The amu is particularly useful when calculating mass defect and converting it into energy. Since the masses involved in nuclear reactions are extremely small, measuring them in kilograms would be impractical, hence the use of amu. During energy calculations in this context, it's important to remember to convert the amu into kilograms to apply Einstein's equation correctly.

Energy Conversion

The ultimate goal of evaluating mass defects in nuclear reactions is to determine the energy produced. The conversion process involves several methodical steps. First, the mass defect calculated needs to be converted from amu into kilograms. Only then can we use Einstein's equation \( E = mc^2 \) effectively. This formula allows us to convert the relatively minute mass defect into a considerable amount of energy, taking into account the speed of light squared, which serves as a conversion factor. Once we have the energy value from \( E = mc^2 \), we can further determine the amount of energy released per specified unit of reactant, such as per gram of hydrogen nuclei. This involves calculating the moles of hydrogen nuclei and then using them to adjust the energy released to a per-gram basis. Ultimately, understanding this conversion allows scientists and engineers to harness nuclear reactions for practical uses ranging from energy production to medical applications.