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 J $\cdot \mathrm{m}/{\mathrm{C}}^2]$

Short answer

To estimate the temperature needed for the fusion of deuterium to make an α particle (helium nucleus), we can use Coulomb's law to first calculate the energy required for the process. Using the given values, we find the energy to be approximately $E\approx 7.68\times {10}^{-13}\mathrm{J}$. To relate this energy to temperature, we can use the equipartition theorem, which gives us the temperature value as approximately $T\approx 3.71\times {10}^7\mathrm{K}$. Therefore, the estimated temperature needed for the fusion of deuterium to make an α particle is approximately $3.71\times {10}^7\mathrm{K}$.

Step-by-step solution

Understand the given data

We are given the form of Coulomb's law as: $$E=9.0\times {10}^9\frac{Q_1{Q}_2}{r}$$ where $Q_1$ and $Q_2$ are the charges of the two particles (in this case, deuterium nuclei) with unit charge, i.e., the charge of a proton, $Q=1.6\times {10}^{-19}\mathrm{C}$, $r=2\times {10}^{-15}\mathrm{m}$ is the separation between the protons in the helium nucleus.

Calculate the energy using Coulomb's law

Using the given values, we can calculate the energy required for the fusion process: $$E=9.0\times {10}^9\frac{(1.6\times {10}^{-19}{)}^2}{2\times {10}^{-15}}$$ Calculate this expression to find the energy value: $$E\approx 7.68\times {10}^{-13}\mathrm{J}$$

Estimate the temperature needed

Now, we need to estimate the temperature needed for the fusion process. We can relate energy and temperature using the equipartition theorem, which states that the average kinetic energy per particle is given by: $$\frac{3}{2}{k}_{B}T=E$$ where $k_B=1.38\times {10}^{-23}\mathrm{J}/\mathrm{K}$ is the Boltzmann constant and $T$ is the temperature in Kelvin. Plugging in the energy value we found in step 2, we can solve for temperature: $$T=\frac{2E}{3k_B}=\frac{2(7.68\times {10}^{-13}\mathrm{J})}{3(1.38\times {10}^{-23}\mathrm{J}/\mathrm{K})}$$ Calculate this expression to find the temperature value: $$T\approx 3.71\times {10}^7\mathrm{K}$$ So, the estimated temperature needed to achieve the fusion of deuterium to make an α particle is approximately $3.71\times {10}^7\mathrm{K}$.

Key concepts

Deuterium

Deuterium is an isotope of hydrogen. Unlike the more common hydrogen atom, which contains just a single proton, deuterium has both a proton and a neutron in its nucleus. This makes deuterium twice as heavy as ordinary hydrogen. It is often referred to as "heavy hydrogen" due to its extra neutron.
Deuterium plays a crucial role in nuclear fusion processes:

• Nuclear fusion reactions involving deuterium can produce significant amounts of energy.
• In the case of deuterium-deuterium (D-D) fusion, two deuterium nuclei combine to form either helium-3 and a neutron or tritium and a proton.

These reactions are important for energy generation in stars and are being investigated for their potential in clean energy production on Earth.

α Particle

The α particle, or alpha particle, is a type of ionizing radiation. It consists of two protons and two neutrons bound together, which is why it is identical to the nucleus of a helium atom. Alpha particles are commonly emitted during radioactive decay:

• They typically come from the decay of heavy elements such as uranium, radium, and thorium.
• Because α particles have a +2 charge, they interact strongly with matter and can be stopped by a sheet of paper or human skin.

In the context of nuclear fusion, the creation of an α particle signifies the conversion of lighter nuclei, like deuterium, into heavier elements such as helium, releasing energy in the process.

Coulomb's Law

Coulomb's Law describes the electrostatic interaction between charged particles. It states that the force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The formula is often written as:$$F=k\frac{Q_1{Q}_2}{r^2}$$where:

• $F$ is the electrostatic force between the charges.
• $k=9.0\times {10}^9{\text{ Nm}}^2/{\text{C}}^2$ is the electrostatic constant.
• $Q_1$ and $Q_2$ are the charges.
• $r$ is the separation distance between the charges.

Coulomb's law helps calculate the energy needed to overcome the electrostatic repulsion between nuclei during fusion. When two positively charged nuclei like deuterium come together, the energy calculated using Coulomb's law shows how much kinetic energy is needed to bring the nuclei close enough for the strong nuclear force to cause them to fuse.

Equipartition Theorem

The Equipartition Theorem is a principle of statistical mechanics. It states that energy is equally distributed among all available degrees of freedom of a system, each contributing $\frac{1}{2}{k}_{B}T$ to the system's total energy, where $k_B$ is the Boltzmann constant and $T$ is the temperature.In the case of gases and other systems in thermal equilibrium:

• The translational kinetic energy for each degree of freedom is $\frac{1}{2}{k}_{B}T$.
• For three-dimensional motion, as with gas particles, the total translational energy is therefore $\frac{3}{2}{k}_{B}T$.

This theorem helps establish a relationship between temperature and kinetic energy, crucial in estimating the temperature necessary for particles to overcome the energy barriers posed by Coulomb's law in fusion reactions. By equating the kinetic energy to the potential energy needed for fusion, one can calculate the required temperature for processes like deuterium fusion to form α particles.