Question

Estimate the temperature needed to achieve the fusion of deuterium to make an \(\alpha\) particle. The energy required can be estimated from Coulomb's law [use the form \(E=9.0 \times 10^{9}\) \(\left(Q_{1} Q_{2} / r\right)\), using \(Q=1.6 \times 10^{-19} \mathrm{C}\) for a proton, and \(r=2 \times\) \(10^{-15} \mathrm{~m}\) for the helium nucleus; the unit for the proportionality constant in Coloumb's law is \(\left.\mathrm{J} \cdot \mathrm{m} / \mathrm{C}^{2}\right]\).

Short answer

The estimated temperature needed to achieve the fusion of deuterium to make an alpha particle, using Coulomb's law and the provided values, is approximately \( 8.35 \times 10^9 K \).

Step-by-step solution

Identify the variables and formulas we will use

In this problem, we are given the following values and formulas: 1. Coulomb's Law constant: \( E = 9.0 \times 10^9 \frac{J \cdot m}{C^2} \) 2. Charge of proton: \( Q = 1.6 \times 10^{-19} C \) for a proton 3. Distance between protons in helium nuclei: \( r = 2 \times 10^{-15} m \) We need to find the energy required for the fusion using Coulomb's Law: \( E = \frac{9.0 \times 10^9 Q_1 Q_2}{r} \)

Calculate the energy required for the fusion of deuterium

We now need to find Q_1 and Q_2, the charge of the two deuterium nuclei. Since each deuterium nucleus contains one proton, their charges are equal and given as: \( Q_1 = Q_2 = 1.6 \times 10^{-19} C \). Now we can calculate the energy using the formula: \( E = \frac{9.0 \times 10^9 Q_1 Q_2}{r} = \frac{9.0 \times 10^9 (1.6 \times 10^{-19} C)^2}{2 \times 10^{-15} m} \)

Simplify the expression to find the energy

Calculate the energy: \( E = \frac{9.0 \times 10^9 (1.6 \times 10^{-19} C)^2}{2 \times 10^{-15} m} = \frac{9.0 \times 10^9 \times (2.56 \times 10^{-38} C^2)}{2 \times 10^{-15} m} \) \( E = \frac{2.304 \times 10^{-28} \frac{J \cdot m \cdot C^2}{m}}{2 \times 10^{-15}} \) \( E = 1.152 \times 10^{-13} J \) So, the energy required for the fusion of deuterium is \( 1.152 \times 10^{-13} J \).

Calculate the temperature required for the fusion

Now that we have the energy required for the fusion, we can calculate the temperature needed. We will use the relationship between energy and temperature, which is given by the equation: \( E = k_B T \) Where \( E \) is the energy, \( k_B \) is the Boltzmann constant (\( 1.38 \times 10^{-23} \frac{J}{K} \)), and \( T \) is the temperature in Kelvin. Solving for the temperature: \( T = \frac{E}{k_B} = \frac{1.152 \times 10^{-13} J}{1.38 \times 10^{-23} \frac{J}{K}} \)

Calculate the temperature

Calculate the temperature: \( T = \frac{1.152 \times 10^{-13} J}{1.38 \times 10^{-23} \frac{J}{K}} \approx 8.35 \times 10^9 K \) So, the estimated temperature needed to achieve the fusion of deuterium to make an alpha particle is approximately \( 8.35 \times 10^9 K \).

Key concepts

Coulomb's Law

Coulomb's Law is a principle of physics that describes the force between two charged particles. In the context of nuclear fusion, it helps us understand the repulsive force that two positively charged nuclei experience when they come close to each other. Coulomb's Law can be expressed as \( E = \frac{k Q_1 Q_2}{r} \), where \( E \) is the magnitude of the electrostatic force, \( k \) is Coulomb's constant (\( 9.0 \times 10^9 \frac{J \cdot m}{C^2} \)), \( Q_1 \) and \( Q_2 \) are the charges of the particles, and \( r \) is the distance between them.

In our exercise, Coulomb’s Law was used to calculate the energy required to overcome the electrostatic repulsion between two deuterium nuclei, which is the first step in fusing them to form a helium nucleus, or an alpha particle. The ease with which learners can understand this concept is improved by visualizing the particles and the forces acting upon them, picturing the particles drawing closer together as they prepare to fuse.

Nuclear Fusion

Nuclear fusion is the process by which two light atomic nuclei combine to form a heavier nucleus, releasing a significant amount of energy in the process. This is the same mechanism that powers the sun and other stars, where hydrogen atoms fuse to form helium under high temperatures and pressures. For fusion to occur, the kinetic energy of the particles must be sufficient to overcome their electrostatic repulsion, as described by Coulomb's Law.

The deuterium fusion process specifically involves two deuterium nuclei fusing to form helium. Students benefit from understanding that the high temperatures required for fusion give the nuclei enough energy to overcome the electrostatic forces of repulsion—essentially, the thermal energy transforms into the kinetic energy needed for fusion.

Boltzmann Constant

The Boltzmann constant (\( k_B \)), named after physicist Ludwig Boltzmann, relates the average kinetic energy of particles in a gas with the temperature of the gas. This constant is crucial when linking microscopic physics to macroscopic phenomena, playing a pivotal role in the field of statistical mechanics.

In the context of our problem, the Boltzmann constant was used to calculate the temperature necessary for nuclear fusion. It acts as a bridge between the microscopic energy (calculated using Coulomb’s Law) and the macroscopic world where temperature is measured. To simplify the connection between these scales, students should appreciate that the Boltzmann constant enables us to scale up the subatomic processes to thermodynamic temperatures that we can conceptualize and measure.

Energy Calculation

Energy calculation is fundamental in physics, allowing us to quantify the work, heat, and other forms of energy involved in processes. In the deuterium fusion scenario, we calculate the energy needed to overcome Coulomb repulsion to fuse nuclei. The calculated energy gives us insight into the conditions necessary for fusion to occur, such as the extremely high temperatures found within stars.

To make this concrete for students, consider explaining it as follows: just as a car needs a certain amount of fuel to start and run, deuterium nuclei require a specific amount of energy to initiate fusion. The step-by-step calculation of this energy and subsequent translation into temperature are pivotal for learners to grasp the scale of energy involved in nuclear processes as compared to those familiar in everyday life.