Question

Calculate the amount of energy released per gram of hydrogen nuclei reacted for the following reaction. The atomic masses are \(_{1}^{1} \mathrm{H}, \quad 1.00782\) u; \(_{1}^{2} \mathrm{H}, \quad 2.01410 \quad \mathrm{u} ;\) and an electron, \(5.4858 \times 10^{-4}\) u. (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 energy released per gram of hydrogen nuclei reacted for the given nuclear reaction can be found by: 1. Calculate the mass of reactants (\(2 \times 1.00782\) u) and products (\(2.01410 + 5.4858 \times 10^{-4}\) u). 2. Find the mass difference between reactants and products (∆m) (\(2 \times 1.00782 - (2.01410 + 5.4858 \times 10^{-4})\)). 3. Convert ∆m from atomic mass units (u) to kilograms (kg) using the conversion factor 1 u = \(1.660539 \times 10^{-27}\) kg. 4. Calculate the energy released (E) from the mass difference using Einstein's mass-energy equivalence formula: \(E = ∆m \times c^2\), where \(c = 3 \times 10^8 m/s\) is the speed of light. 5. Convert the mass of hydrogen nucleus to grams using the conversion factor 1 u = \(1.660539 \times 10^{-24}\) g. 6. Calculate the energy released per gram of hydrogen nuclei reacted by dividing E by the total mass of 2 hydrogen nuclei in grams: \(\text{Energy per gram} = \frac{E}{2 \times m_{\text{H}} \text{(in g)}}\). This process will give you the energy released per gram of hydrogen nuclei reacted for the given nuclear reaction.

Step-by-step solution

Mass of reactants

We are given the mass of the reactants: Mass of hydrogen nucleus \(_1^1\text{H}\), \(m_{\text{H}} = 1.00782\) atomic mass units (u) We have 2 of these nuclei in the reaction: Total mass of hydrogen nuclei = \(2 \times m_{\text{H}} = 2 \times 1.00782\) u

Mass of products

We are given the mass of the products: Mass of deuteron \(_1^2\text{D}\), \(m_{\text{D}} = 2.01410\) u Mass of an electron, \(m_{\text{e}} = 5.4858 \times 10^{-4}\) u Now, let's calculate the total mass of the reactants and products. Reactants total mass: $$M_{\text{reactants}} = 2 \times 1.00782 \quad \mathrm{u}$$ Products total mass: $$M_{\text{products}} = 2.01410 + 5.4858 \times 10^{-4} \quad \mathrm{u}$$ Step 2: Calculate the mass difference and convert it to energy

Mass difference

The mass difference between reactants and products ∆m is given by: ∆m = \(M_{\text{reactants}} - M_{\text{products}}\) Calculate the mass difference: $$ \Delta m = (2 \times 1.00782) - (2.01410 + 5.4858 \times 10^{-4}) $$

Convert mass difference to energy

To convert the mass difference to energy, we use Einstein's mass-energy equivalence formula \(E = ∆m c^2\), where \(c = 3 \times 10^8 m/s\) is the speed of light. We must first convert the mass difference (∆m) from atomic mass units (u) to kilograms (kg). The conversion factor is 1 u = \(1.660539 \times 10^{-27}\) kg. ∆m (in kg) = ∆m (in u) × \(1.660539 \times 10^{-27} kg/u\) Now, calculate the energy released (E): $$ E = \Delta m \times c^2 $$ Step 3: Calculate energy released per gram of hydrogen nuclei reacted

Energy per gram

To find the energy released per gram of hydrogen nuclei reacted, we will divide the energy released (E) by the mass of one gram of hydrogen nuclei. First, convert the mass of hydrogen nucleus to grams: Mass of hydrogen nucleus in grams = \(m_{\text{H}}\) (in u) × \(1.660539 \times 10^{-24}\) g/u We have 2 hydrogen nuclei in the reaction, so the total mass of 2 hydrogen nuclei in grams: $$2 \times m_{\text{H}} \text{(in g)}$$ Next, calculate the energy released per gram of hydrogen nuclei reacted: $$ \text{Energy per gram} = \frac{E}{2 \times m_{\text{H}} \text{(in g)}} $$ Following these steps will give you the amount of energy released per gram of hydrogen nuclei reacted for the given nuclear reaction.

Key concepts

Mass-Energy Equivalence

Mass-energy equivalence is a concept introduced by Albert Einstein. It states that mass and energy are two sides of the same coin. This concept is beautifully captured in the famous formula: \(E=mc^2\). Here, \(E\) represents energy, \(m\) stands for mass, and \(c\) denotes the speed of light, which is approximately \(3 \times 10^8 \text{ m/s}\).

This equation tells us that a small amount of mass can be converted into a huge amount of energy, due to the \(c^2\) factor, which is an enormous number. In nuclear reactions, like the fusion of hydrogen nuclei, a minute mass difference due to nuclear binding can result in an incredible release of energy.

Understanding mass-energy equivalence is crucial for grasping how energy is produced in processes like nuclear fusion and fission.

Atomic Mass Units

Atomic mass units (abbreviated as u) are a standard unit of mass that quantifies the mass of atoms or subatomic particles. One atomic mass unit is approximately \(1.660539 \times 10^{-27} \text{ kg}\).

Instead of dealing with tiny and inconveniently small numbers when referring to particles, we use amu because it makes calculations and comparisons much more manageable. For instance, the hydrogen nucleus or proton has an approximate mass of \(1.00782 \text{ u}\).

When you're calculating the energy in a nuclear reaction, you'll often start by using amu to find the mass difference, then convert it to kilograms. This is essential because the mass-energy equivalence \(E=mc^2\) formula requires mass in kilograms to calculate energy in joules.

This makes the concept of atomic mass units an important tool in the study of nuclear physics, allowing physicists to easily manage and calculate incredibly small masses and resultant energies in reactions.

Hydrogen Nuclei

Hydrogen nuclei, also known as protons when isolated, are the simplest atomic nuclei, consisting of a single proton. They play a crucial role in nuclear reactions, particularly in the field of nuclear fusion — the process powering the Sun.

In the given reaction, two hydrogen nuclei (protons) fuse to form a deuterium nucleus (a hydrogen isotope with one proton and one neutron) and release a positron.

This kind of reaction is central to understanding how stars convert hydrogen into helium and release energy as part of the stellar lifecycle.

• Proton mass: \(1.00782 \text{ u}\)
• Deuterium (hydrogen with an extra neutron) results in a more stable configuration with slightly less mass, releasing energy in the process.

Understanding hydrogen nuclei and their interactions is vital for fields like astrophysics and energy research, where fusion has promising applications for sustainable energy production.

Energy Conversion

Energy conversion in nuclear reactions involves transforming mass into energy as per mass-energy equivalence. When hydrogen nuclei undergo fusion, there is a slight loss of mass, manifesting as enormous energy output.

This energy can be quantified using Einstein’s equation \(E=mc^2\). The first step is determining the mass difference between reactants and products in atomic mass units and then converting it to energy.

Here’s how you do it:

• Calculate the total mass of reactants and products.
• Determine the mass difference, \(\Delta m\) — the critical factor for energy release.
• Convert \(\Delta m\) from amu to kilograms (1 amu = \(1.660539 \times 10^{-27} \text{ kg}\)).
• Apply \(E=mc^2\) to find energy in joules.
• To find energy per gram of reactant, calculate how much energy the reaction releases for the mass of hydrogen nuclei involved.

This process makes it possible to harness the massive power of nuclear reactions for applications like nuclear power and potential fusion power plants.