Problem 13 - An Introduction to Modern Astrophysics

Question

Calculate the amount of energy released or absorbed in the following reactions (express your answers in \(\mathrm{MeV}\) ): (a) \(^{12} \mathrm{C}+_{6}^{12} \mathrm{C} \rightarrow_{12}^{24} \mathrm{Mg}+\gamma\) (b) \(^{12} \mathrm{C}+^{12} \mathrm{C} \rightarrow_{8}^{16} \mathrm{O}+2_{2}^{4} \mathrm{He}\) (c) \(_{9}^{19} \mathrm{F}+_{1}^{1} \mathrm{H} \rightarrow_{8}^{16} \mathrm{O}+_{2}^{4} \mathrm{He}\) The mass of \(^{12} \mathrm{C}\) is \(12.0000 \mathrm{u}\), by definition, and the masses of \(_{8}^{16} \mathrm{O},_{9}^{19} \mathrm{F},\) and \(_{12}^{24} \mathrm{Mg}\) are \(15.99491 \mathrm{u}\) \(18.99840 \mathrm{u},\) and \(23.98504 \mathrm{u}\), respectively. Are these reactions exothermic or endothermic?

Step-by-Step Solution

Answer

Energy released: (a) 13.94 MeV, (b) 0.83 MeV, (c) 8.12 MeV. All reactions are exothermic.

Step 1: Understand the Problem

We need to calculate the energy released or absorbed for each nuclear reaction given in the problem. For each reaction, we are comparing the binding energy of the reactants with that of the products, to ascertain if energy is released or absorbed.

Step 2: Calculate Mass Defect for Reaction (a)

For the reaction \(^{12} \mathrm{C}+_{6}^{12} \mathrm{C} \rightarrow_{12}^{24} \mathrm{Mg}+\gamma\), calculate the initial mass: \(2 \times 12.0000 = 24.0000\, \text{u}.\) Calculate the final mass: \(23.98504\, \text{u}\). Mass defect \( = 24.0000 - 23.98504 = 0.01496\, \text{u}\).

Step 3: Convert Mass Defect to Energy for Reaction (a)

Use the conversion factor \(1\, \text{u} = 931.5\, \mathrm{MeV/c^2}\). Energy released \( = 0.01496\, \text{u} \times 931.5\, \text{MeV/u} = 13.94\, \text{MeV}\). Reaction (a) is exothermic since energy is released.

Step 4: Calculate Mass Defect for Reaction (b)

For the reaction \(^{12} \mathrm{C}+^{12} \mathrm{C} \rightarrow_{8}^{16} \mathrm{O}+2_{2}^{4} \mathrm{He}\), initial mass \(= 24.0000\, \text{u}\). Final mass: \(15.99491 + 2 \times 4.00260 = 23.99911\, \text{u}\). Mass defect \( = 24.0000 - 23.99911 = 0.00089\, \text{u}\).

Step 5: Convert Mass Defect to Energy for Reaction (b)

Energy released \(= 0.00089\, \text{u} \times 931.5\, \text{MeV/u} = 0.83\, \text{MeV}\). Reaction (b) is exothermic.

Step 6: Calculate Mass Defect for Reaction (c)

For the reaction \(_{9}^{19} \mathrm{F}+_{1}^{1} \mathrm{H} \rightarrow_{8}^{16} \mathrm{O}+_{2}^{4} \mathrm{He}\), initial mass \(= 18.99840 + 1.007825 = 20.006225\, \text{u}\). Final mass \(= 15.99491 + 4.00260 = 19.99751\, \text{u}\). Mass defect \(= 20.006225 - 19.99751 = 0.008715\, \text{u}\).

Step 7: Convert Mass Defect to Energy for Reaction (c)

Energy released \(= 0.008715\, \text{u} \times 931.5\, \text{MeV/u} = 8.12\, \text{MeV}\). Reaction (c) is exothermic.

Key Concepts

Mass Defect

In nuclear reactions, understanding mass defect is crucial. Mass defect refers to the difference between the total mass of the individual nucleons (protons and neutrons) that make up the nucleus, and the actual mass of the nucleus. This discrepancy arises because some of the mass is converted into binding energy, which holds the nucleus together. Let's break it down:

• The measured mass of a nucleus is always less than the sum of its individual components.
• This mass difference is known as the mass defect: \[ \text{Mass defect} = \text{Sum of masses of individual nucleons} - \text{Mass of nucleus}. \]
• The mass defect is often measured in atomic mass units (u).

When we calculate the mass defect during a nuclear reaction, we identify how much mass was "lost," as this loss equates to a release of energy, which we can track using Einstein's famous equation: \[ E = mc^2, \] where **c** is the speed of light. For practical calculations in nuclear physics, we often use: \[ 1\, \text{u} = 931.5\, \text{MeV/c}^2. \] This equivalence helps us convert the mass defect directly into an energy unit, typically MeV (Mega electron Volts), allowing us to understand the energy changes in nuclear reactions.

Binding Energy

Binding energy is the energy required to break a nucleus into its separate nucleons (protons and neutrons). It's a measure of the nucleus's stability: the greater the binding energy, the more stable the nucleus. In a nuclear reaction, we compare the binding energies of the reactants with the products to determine whether energy is absorbed or released. Here's a breakdown of binding energy key points:

• Binding energy helps explain why certain nuclei are more stable than others.
• In nuclear reactions, comparing binding energy before and after the reaction provides insight into energy release or absorption.\[ \text{Binding energy} = \text{(Mass defect)} \times c^2. \]
• The unit of binding energy is typically MeV.

In our given reactions, binding energy is central to determining the energetics:

• If the total binding energy of the products is higher than that of the reactants, the reaction releases energy (exothermic); if lower, energy is absorbed (endothermic).
• Nuclear reactions involving large increases in binding energy tend to be highly exothermic, making them of particular interest in power generation, such as nuclear reactors and fusion processes.

Exothermic Reactions

Exothermic reactions are processes that release energy into the surroundings, often in the form of heat or light. In nuclear chemistry, when a reaction is exothermic, it means that the total energy of the products is less than the total energy of the reactants. Let's understand this with a few details:

• During an exothermic nuclear reaction, the energy is released because of a net change in binding energy.
• The difference in binding energies between reactants and products appears as energy, mostly released as kinetic energy of the products or as photons called gamma rays.
• This energy release makes exothermic reactions crucial for applications like nuclear power generation.

In the context of the given reactions:

• All three reactions (a, b, and c) are exothermic, as they demonstrate energy release upon completion. This is seen in the calculated positive release of MeV energy values.
• The released energy can be used to power turbines in a nuclear reactor or fuse nuclei in a fusion reaction.
• Recognizing exothermic reactions helps us understand processes like stellar nucleosynthesis and the potential of nuclear reactions for energy production on Earth.