Question

A portion of the Sun's energy comes from the reaction \begin{equation}4_{1}^{1} \mathrm{H} \longrightarrow_{2}^{4} \mathrm{He}+2_{1}^{0} \mathrm{e} \end{equation}which requires a temperature of \(10^{6}\) to \(10^{7} \mathrm{K}\) . Use the mass of the helium-4 nucleus given in Table 21.7 to determine how much energy is released per mol of hydrogen atoms.

Short answer

The energy released per mol of hydrogen atoms in the given nuclear reaction is determined by first calculating the mass deficit, \( Δm = 4 * m(H) - [ m(He-4) + 2 * (mass \, of \, electron) ] \), and then converting it to energy using Einstein's mass-energy equivalence equation, \( E = Δm * c^2 \). Finally, divide the total energy by the molar amount of hydrogen atoms (4 mol) to find the energy released per mol of hydrogen atoms, with the final answer in J/mol (joules per mol).

Step-by-step solution

Understand the Nuclear Reaction

The given nuclear reaction is: \begin{equation} 4_{1}^{1} \mathrm{H} \longrightarrow_{2}^{4} \mathrm{He} + 2_{1}^{0} \mathrm{e} \end{equation} In this reaction, four hydrogen atoms combine to form one helium nucleus and two electrons are released.

Use the Mass of Helium-4 Nucleus

Using the mass of helium-4 nucleus from Table 21.7, we have m(He-4) = 4.001506 * 10^{-3} kg/mol. In addition, we know that the mass of hydrogen atom is m(H) = 1.00794 * 10^{-3} kg/mol.

Calculate the Mass Deficit

The mass before the reaction is equal to the mass of four hydrogen atoms, and the mass after the reaction is the sum of the mass of the helium-4 nucleus and the mass of the two electrons. First, let's calculate the mass of two electrons, which is 2 * (9.109383 * 10^{-31}) kg (Refer to the electron mass in atomic mass units in the Table, and convert it to kg/mol). Now, let's calculate the mass deficit, which is the difference between the mass before the reaction and the mass after the reaction: \( Δm = 4 * m(H) - [ m(He-4) + 2 * (mass \, of \, electron) ] \)

Convert Mass Deficit to Energy

We can use Einstein's mass-energy equivalence equation to convert the mass deficit to energy: \( E = Δm * c^2 \) Here, c is the speed of light (3.0 * 10^8 m/s).

Determine the Energy Released Per Mol of Hydrogen Atoms

Now that we have calculated the total energy released during the reaction, divide this value by the molar amount (4 mol) of hydrogen atoms to find the energy released per mol of hydrogen atoms. Remember to state the final answer in J/mol (joules per mol), which is the standard unit of energy.

Key concepts

Nuclear Chemistry

Nuclear chemistry is a fascinating branch of chemistry that deals with the reactions of atomic nuclei. These reactions are different from chemical reactions, which involve electrons and the making or breaking of chemical bonds.

Nuclear reactions involve changes in an atom's nucleus and can release or absorb incredible amounts of energy. A familiar example of a nuclear reaction is the fusion that takes place in the Sun, where hydrogen atoms merge to form helium atoms. During these reactions, the principle of conservation of mass is apparently violated. That's because some of the mass is actually converted to energy, according to Einstein's mass-energy equivalence principle.

Mass-Energy Equivalence

The mass-energy equivalence is a groundbreaking concept introduced by Albert Einstein. This principle tells us that mass and energy are two forms of the same thing and can be converted into each other.

The famous equation, E=mc², where E stands for energy, m for mass, and c for the speed of light squared, quantifies this relationship. This means that a small amount of mass can be converted into a large amount of energy, and this is the fundamental principle that powers reactions like those in the Sun.

Einstein's Equation

Einstein's equation, written as E=mc², describes the conversion of mass (m) into energy (E), with c representing the speed of light in a vacuum. In the context of our exercise, when four hydrogen atoms combine to form a helium-4 nucleus, mass is lost.

This loss of mass, known as the mass deficit, is what's converted into energy. In the calculation process, we measure the mass of hydrogen and helium, find the difference, and then use Einstein's equation to determine the energy released.

Helium-4 Nucleus

A helium-4 nucleus, also known as an alpha particle, is a highly stable configuration of two protons and two neutrons. It's the end product in the fusion reaction of hydrogen atoms that takes place in the Sun and other stars.

The stability of the helium-4 nucleus comes from its strong nuclear forces that balance out electrostatic repulsion. When calculating the energy released during fusion, the mass of the helium-4 nucleus is crucial because the mass of helium-4 after the reaction is less than the mass of the original four hydrogen atoms. This difference in mass is key to calculating the energy produced.

Hydrogen Fusion

Hydrogen fusion is the process that fuels the Sun and stars, combining hydrogen nuclei (protons) to form heavier elements. In our context, four hydrogen atoms fuse to create one helium-4 nucleus.

It's important to note that this reaction doesn’t just happen anywhere; it requires extreme conditions of temperature and pressure, such as those found in the core of the Sun. The process of fusion produces a vast amount of energy, which is due to the mass defect we calculate in the exercise. This energy is what we observe as light and heat radiating from the Sun.