Question

It is possible to estimate the activation energy for fusion by calculating the energy required to bring two deuterons close enough to one another to form an alpha particle. This energy can be obtained by using Coulomb's law in the form \(E=8.99 \times 10^{9} q_{1} q_{2} / r\), where \(q_{1}\) and \(q_{2}\) are the charges of the deuterons \(\left(1.60 \times 10^{-19} \mathrm{C}\right), r\) is the radius of the He nucleus, about \(2 \times 10^{-15} \mathrm{~m}\), and \(E\) is the energy in joules. (a) Estimate \(E\) in joules per alpha particle. (b) Using the equation \(E=m v^{2} / 2\), estimate the velocity (meters per second) each deuteron must have if a collision between the two of them is to supply the activation energy for fusion \((m\) is the mass of the deuteron in kilograms).

Short answer

Answer: The activation energy for fusion is 1.15 x 10^-11 joules per alpha particle, and the required velocity for each deuteron is approximately 2.98 x 10^7 meters per second.

Step-by-step solution

Calculate the energy using Coulomb's law

We are given Coulomb's law in the form \(E = 8.99 \times 10^9 \frac{q_1 q_2}{r}\). We will substitute the given values to calculate the energy E. $$ E = 8.99 \times 10^9 \frac{1.60 \times 10^{-19} \mathrm{C} \times 1.60 \times 10^{-19} \mathrm{C}}{2 \times 10^{-15} \mathrm{m}} $$ $$ E = \frac{2.30 \times 10^{-26}}{2 \times 10^{-15}} \mathrm{J} $$ $$ E = 1.15 \times 10^{-11} \mathrm{J} $$ The energy E per alpha particle is approximately \(1.15 \times 10^{-11}\) joules.

Calculate the velocity for fusion

Now, we will use the equation \(E = \frac{1}{2}mv^2\) to estimate the velocity each deuteron must have for fusion. We can rewrite the equation to solve for v as follows: $$ v = \sqrt{\frac{2E}{m}} $$ The mass of a deuteron is approximately \(3.34 \times 10^{-27}\) kg. We will substitute the values for E and m to find the velocity. $$ v = \sqrt{\frac{2 \times 1.15 \times 10^{-11} \mathrm{J}}{3.34 \times 10^{-27} \mathrm{kg}}} $$ $$ v \approx 2.98 \times 10^7 \mathrm{m/s} $$ Each deuteron must have a velocity of approximately \(2.98 \times 10^7\) meters per second for fusion to occur. In conclusion, the energy in joules per alpha particle is \(1.15 \times 10^{-11}\) joules and the required velocity for fusion is approximately \(2.98 \times 10^7\) meters per second.

Key concepts

Coulomb's Law

Coulomb's Law helps us understand the forces between charged particles. It is used in calculating how much electrical energy is needed to make two charged objects come close enough to interact. In the context of deuterons—particles of interest in nuclear reactions—Coulomb's Law gives us a way to estimate the activation energy for fusion. The formula provided, \(E = 8.99 \times 10^9 \frac{q_1 q_2}{r}\), calculates the energy necessary to move two deuterons into the proximity needed for them to form an alpha particle. Here, \(q_1\) and \(q_2\) represent the electric charges of the deuterons, while \(r\) represents the distance that should separate them; this distance relates to the radius of a Helium nucleus, around \(2 \times 10^{-15}\) meters.

To do this calculation, we need to know the charge of a deuteron, which is about \(1.60 \times 10^{-19}\) C (Coulombs). Plugging these values into the formula allows us to compute the energy, providing a necessary value to understand and compare with other parts of the nuclear fusion process.

Deuterons

Deuterons are nuclei of deuterium, an isotope of hydrogen. They consist of one proton and one neutron and play a significant role in nuclear physics, especially in fusion reactions.

Nuclear fusion often involves deuterons due to their simplicity and relative abundance. As a hydrogen isotope, deuterons have a single positive charge of \(1.60 \times 10^{-19}\) C. Fusion involving deuterons is a topic of considerable interest because it's a potential source of clean energy. However, achieving the conditions necessary for deuterons to overcome their natural repulsion and fuse requires a specific amount of energy—often represented by the activation energy computed earlier.

This activation energy is important for determining the velocity each deuteron must achieve in order to close the gap between them and allow nuclear forces to bring them together into an alpha particle.

Alpha Particle

An alpha particle is a fundamental constituent of nuclear physics. Comprising two protons and two neutrons, it essentially forms the same structure as a Helium nucleus. When deuterons collide with enough energy, they can overcome their repulsion and fuse to form an alpha particle.

Alpha particles are characterized by stability due to the strong nuclear forces binding the protons and neutrons together. Recognizing the structural change from two independent deuterons to a single alpha particle is crucial for comprehending nuclear reactions. This reaction is part of the energy balance that makes fusion a source of potential energy—a perspective supported by modern research into fusion energy.

The transition from deuterons to alpha particles involves overcoming the electrical repulsion calculated by Coulomb's Law and is the reason activation energy is important in achieving such nuclear transformations.

Energy Calculation

The energy calculation is a primary focus when considering nuclear reactions, especially fusion. Energy isn't just about forming an alpha particle; it's also about ensuring that the conditions—such as the velocity of deuterons—are right for the fusion to take place.

After calculating activation energy using Coulomb's Law, we next compute the kinetic energy needed for each deuteron to ensure fusion can occur. We use the formula \(E = \frac{1}{2}mv^2\), which models how kinetic energy is associated with mass and velocity. By rearranging to find velocity, \(v = \sqrt{\frac{2E}{m}}\), where \(m\) is the mass of a deuteron and \(E\) is the required energy, we establish how fast the deuterons should travel for successful fusion.

From the earlier step, with \(E\) being \(1.15 \times 10^{-11}\) joules and \(m\) being \(3.34 \times 10^{-27}\) kg, the velocity calculation provides an approximation of \(2.98 \times 10^7\) meters per second, illustrating the intense conditions necessary for nuclear fusion.