Problem 84 - Chemistry The Science in Context

Question

It has been proposed that electrical power production in the future might be based on the fusion of deuterium to helium-4. a. Write a radiochemical equation describing the reaction (assume that \(^{4} \mathrm{He}\) is the only product). b. Calculate how much energy is released during the formation of 1 mole of \(^{4} \mathrm{He}\).

Step-by-Step Solution

Answer

Answer: The radiochemical equation for the fusion of deuterium to helium-4 is: $$ ^2\mathrm{H} + ^2\mathrm{H} \rightarrow ^4\mathrm{He} $$ The energy released during the formation of 1 mole of helium-4 is approximately \(3.8857\times10^{-12}\,\mathrm{J/mol}\).

Step 1: Write a radiochemical equation for deuterium fusion to helium-4

The fusion of deuterium (\(^2\mathrm{H}\)) to form helium-4 (\(^4\mathrm{He}\)) involves the combination of two deuterium nuclei. We write the radiochemical equation as follows: $$ ^2\mathrm{H} + ^2\mathrm{H} \rightarrow ^4\mathrm{He} $$

Step 2: Calculate the atomic masses

Next, we need to calculate the atomic masses of deuterium and helium-4. We have: 1. Deuterium atomic mass: \(M(^2\mathrm{H}) = 2.014\,\mathrm{amu}\) 2. Helium-4 atomic mass: \(M(^4\mathrm{He}) = 4.002\,\mathrm{amu}\) where \(\mathrm{amu}\) stands for atomic mass units.

Step 3: Calculate the mass difference

Now we need to calculate the mass difference between the reactants and products in the radiochemical equation. This mass difference will be converted to energy using Einstein's equation (\(E=mc^2\)). The mass difference is: $$ \Delta m = [2M(^2\mathrm{H}) - M(^4\mathrm{He})] \times 1\,\mathrm{mol} $$ Plugging in the values from Step 2, we get: $$ \Delta m = [(2\times2.014) - 4.002]\,\mathrm{amu/mol} = 0.026\,\mathrm{amu/mol} $$

Step 4: Convert mass difference to energy

To convert the mass difference to energy, we use Einstein's equation and the conversion factor between amu and kg: 1 amu = 1.66054 × 10\(^{-27}\) kg Einstein's equation: $$ E = \Delta m \times c^2 $$ where \(E\) is the energy released, \(\Delta m\) is the mass difference, and \(c\) is the speed of light (\(3×10^8 m/s\)). First, we convert the mass difference to kg: $$ \Delta m = 0.026\,\mathrm{amu/mol} \times 1.66054\times10^{-27}\,\mathrm{kg/amu} = 4.3174\times10^{-29}\,\mathrm{kg/mol} $$ Now we apply Einstein's equation: $$ E = 4.3174\times10^{-29}\,\mathrm{kg/mol} \times (3\times10^8\,\mathrm{m/s})^2 = 3.8857\times10^{-12}\,\mathrm{J/mol} $$ So, the energy released during the formation of 1 mole of helium-4 is approximately \(3.8857\times10^{-12}\,\mathrm{J/mol}\).

Key Concepts

Understanding Deuterium Fusion

Deuterium fusion is a nuclear reaction in which two deuterium nuclei, which are isotopes of hydrogen, combine to form a heavier atom. In this case, the product is helium-4. Fusion reactions require very high temperatures and pressures to bring the positively charged nuclei close enough together for the strong nuclear force to bind them. In the fusion process, energy is released because the mass of the resulting helium-4 nucleus is less than the combined mass of the original deuterium nuclei. This mass difference is due to the conversion of some mass to energy, as described by Einstein’s famous equation, \( E=mc^2 \). The equation for the fusion of two deuterium atoms to form helium-4 can be written as:

• \( ^2\mathrm{H} + ^2\mathrm{H} \rightarrow ^4\mathrm{He} \)

This equation shows that two deuterium atoms combine to produce one helium-4 atom, releasing energy in the process. Deuterium is naturally found in water and can be easily extracted, which is why scientists are looking into deuterium fusion as a potential clean energy source for the future.

Explaining Radiochemical Equations

To understand radiochemical equations, it's important to remember that they represent nuclear reactions where atomic nuclei change. These equations are similar to chemical equations but involve nuclear particles like protons, neutrons, or entire nuclei instead of atoms.Radiochemical equations help track the changes in the nucleus of atoms during a nuclear reaction. Unlike chemical reactions, which only involve the electron cloud around an atom, nuclear reactions can change an atom's core identity, creating entirely new elements or isotopes.In the case of deuterium fusion, the radiochemical equation is:

• \(^2\mathrm{H} + ^2\mathrm{H} \rightarrow ^4\mathrm{He} \)

This equation indicates that the two hydrogen nuclei (deuterium) combine to form a helium nucleus. Understanding this helps to visualize the process. The process of writing these equations involves ensuring that both the mass and charge are balanced across the reaction, just like in standard chemical equations.

Calculating the Energy from the Reaction

When deuterium nuclei fuse to form helium, a small amount of mass is lost. This mass loss is converted into energy, which can be calculated using the formula \( E=mc^2 \), where \( E \) is the energy, \( m \) is the mass difference, and \( c \) is the speed of light.The first step is to determine the mass difference \( \Delta m \) between the reactants and products. According to the given data:

• Mass of deuterium = \( 2.014 \, \mathrm{amu} \)
• Mass of helium-4 = \( 4.002 \, \mathrm{amu} \)

The mass difference is calculated as:

• \( \Delta m = (2 \times 2.014) - 4.002 \, \mathrm{amu} = 0.026 \, \mathrm{amu} \)

After finding the mass difference, this value is converted to kilograms for the energy calculation:

• \( \Delta m = 0.026 \, \mathrm{amu/mol} \times 1.66054 \times 10^{-27} \, \mathrm{kg/amu} = 4.3174 \times 10^{-29} \, \mathrm{kg/mol} \)

The energy released is calculated by plugging this mass difference into Einstein's equation:

• \( E = 4.3174 \times 10^{-29} \, \mathrm{kg/mol} \times (3 \times 10^8 \, \mathrm{m/s})^2 = 3.8857 \times 10^{-12} \, \mathrm{J/mol} \)

This energy, though small for a single reaction, illustrates the enormous potential of deuterium fusion as a powerful energy source when considered at large scales.