Question

A portion of the Sun's energy comes from the reaction $$ 4_{1}^{1} \mathrm{H} \longrightarrow{ }_{2}^{4} \mathrm{He}+2_{1}^{0} \mathrm{e} $$ which requires a temperature of \(10^{6}\) to \(10^{7} \mathrm{~K}\). (a) Use the mass of the helium- 4 nucleus given in Table 21.7 to determine how much energy is released when the reaction is run with \(1 \mathrm{~mol}\) of hydrogen atoms. (b) Why is such a high temperature required?

Short answer

(a) The energy released when the reaction is run with 1 mole of hydrogen atoms is \(2.58 \times 10^{12}\,\mathrm{J}\). (b) The high temperature is required for this reaction because, at such high temperatures, hydrogen nuclei (protons) obtain sufficient kinetic energy to overcome the electrostatic repulsion between them. This allows the protons to come close enough to each other for the strong nuclear force to act, which leads to nuclear fusion. The temperature should be in the range of \(10^6\) to \(10^7\) K, to provide the required kinetic energy for this reaction to occur.

Step-by-step solution

Determine the mass difference of the reaction

To find the energy released, we first need to determine the mass difference between the reactants and products. For this reaction, 4 hydrogen atoms combine to form a helium-4 nucleus and 2 positrons (e+). Therefore, the mass difference can be calculated using the equation: \[Δm = m_{reactants} - m_{products} \] From Table 21.7, we know the mass of helium-4 nucleus: \(4_{1}^{1} \mathrm{H} = 1.007825\,\mathrm{amu}\) \(_{2}^{4} \mathrm{He} = 4.001506\,\mathrm{amu}\) \(_{1}^{0} \mathrm{e} = 0.000548\,\mathrm{amu}\)

Calculate the mass difference

Now, we can calculate the mass difference using the masses of the reactants and products: \(Δm = 4(1.007825\,\mathrm{amu}) - (4.001506\,\mathrm{amu} + 2(0.000548\,\mathrm{amu}))\) \(Δm = 4.0313\,\mathrm{amu} - 4.002602\,\mathrm{amu}\) \(Δm = 0.028698\,\mathrm{amu}\)

Convert mass difference to energy

Now, we can convert the mass difference to energy using Einstein's mass-energy equivalence equation: \(E = Δm \times c^2\) where \(c\) is the speed of light (\(2.9979 \times 10^8\,\mathrm{m/s}\)). First, we need to convert the mass difference from amu to kg. We know that 1 amu = \(1.6605 \times 10^{-27}\,\mathrm{kg}\): \(Δm = 0.028698\,\mathrm{amu} \times\frac{1.6605 \times 10^{-27}\,\mathrm{kg}}{1\,\mathrm{amu}} = 4.766 \times 10^{-29}\,\mathrm{kg}\) Now we can calculate the energy: \(E = (4.766 \times 10^{-29}\,\mathrm{kg}) \times (2.9979 \times 10^8\,\mathrm{m/s})^2\) \(E = 4.29 \times 10^{-12}\,\mathrm{J}\)

Find the energy per mole of hydrogen atoms

Now, we have found the energy released for the reaction on a per atom basis. We need to find the energy released for 1 mole of hydrogen atoms. We know that 1 mole has \(6.022 \times 10^{23}\) atoms. Therefore, we can find the energy for 1 mol of hydrogen atoms as follows: \(E_{mol} = E \times 6.022 \times 10^{23}\,\mathrm{atoms}\) \(E_{mol} = (4.29 \times 10^{-12}\,\mathrm{J}) \times (6.022 \times 10^{23}\,\mathrm{atoms})\) \(E_{mol} = 2.58 \times 10^{12}\,\mathrm{J}\) (a) The energy released when the reaction is run with 1 mole of hydrogen atoms is \(2.58 \times 10^{12}\,\mathrm{J}\).

Explain the high temperature requirement

(b) The high temperature is required for this reaction because, at such high temperatures, hydrogen nuclei (protons) obtain sufficient kinetic energy to overcome the electrostatic repulsion between them. This allows the protons to come close enough to each other for the strong nuclear force to act, which leads to nuclear fusion. The temperature should be in the range of \(10^6\) to \(10^7\) K, to provide the required kinetic energy for this reaction to occur.

Key concepts

Understanding Mass-Energy Equivalence

The notion behind mass-energy equivalence is elegantly summarized by Albert Einstein's famous equation,\(E = mc^2\). This tells us that mass (\