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 \(\left.\mathbf{J} \cdot \mathbf{m} / \mathbf{C}^{2}\right]\).

Short answer

The estimated temperature required to achieve the fusion of deuterium to make an alpha particle can be calculated using Coulomb's law. By substituting the given values and solving for energy, we find that the energy required is approximately \(1.152 \times 10^{-13} \mathrm{J}\). Using the Boltzmann constant, we can convert this energy to temperature, which results in an estimated temperature of \(8.35 \times 10^9 \mathrm{K}\).

Step-by-step solution

Identify the Charges of Particles

In the fusion process, we are dealing with two deuterium nuclei, i.e. two protons (since each deuterium nucleus has only one proton). Also, the charge of a proton is given as \(Q = 1.6 \times 10^{-19} \mathrm{C}\). So, for the two protons, we have \(Q_1 = Q_2 = 1.6 \times 10^{-19} \mathrm{C}\).

Calculate the Energy Required Using Coulomb's Law

Now, we will use Coulomb's law to calculate the energy required for the fusion process. Coulomb's law is expressed as: \(E=9.0 \times 10^{9} \left(\frac{Q_{1} Q_{2}}{r}\right)\) Where \(Q_1\) and \(Q_2\) are the charges of the two particles, and \(r\) is the distance between them, which is given as \(2 \times 10^{-15} \mathrm{m}\) for the helium nucleus. Plug in the values and solve for \(E\): \(E = 9.0 \times 10^9 \left(\frac{(1.6 \times 10^{-19} \mathrm{C})(1.6 \times 10^{-19} \mathrm{C})}{2 \times 10^{-15} \mathrm{m}}\right)\)

Simplify the Expression and Calculate Energy

Solve the equation for \(E\): \(E = 9.0 \times 10^9 \left(\frac{2.56 \times 10^{-38} \mathrm{C}^2}{2 \times 10^{-15} \mathrm{m}}\right)\) \(E = 9.0 \times 10^9 \times 1.28 \times 10^{-23} \mathrm{J} = 1.152 \times 10^{-13} \mathrm{J}\) So, the energy required for the fusion process is approximately \(1.152 \times 10^{-13} \mathrm{J}\).

Convert Energy to Temperature

Now, we will convert the energy we found in the previous step to temperature. Temperature (\(T\)) can be related to energy (\(E\)) through the Boltzmann constant (\(k_B = 1.38 \times 10^{-23} \mathrm{J/K}\)) with the following equation: \(T = \frac{E}{k_B}\) Plug in the values and solve for temperature: \(T = \frac{1.152 \times 10^{-13} \mathrm{J}}{1.38 \times 10^{-23} \mathrm{J/K}} = 8.35 \times 10^9 \mathrm{K}\) So, the estimated temperature needed to achieve the fusion of deuterium to make an alpha particle is approximately \(8.35 \times 10^9 \mathrm{K}\).

Key concepts

Coulomb's Law

When we talk about nuclear fusion, and specifically the fusion of deuterium to form helium, Coulomb's law plays a crucial role in understanding the forces at play. Coulomb's law quantifies the amount of electrostatic force between two charged particles. In mathematical terms, it's represented as:

\(E= k_e \cdot \frac{Q_{1}Q_{2}}{r}\),

where \(E\) is the electrostatic energy, \(Q_1\) and \(Q_2\) are the charges of the particles, \(r\) is the distance between the centers of the two charges, and \(k_e\) is Coulomb's constant, which is approximately \(9.0 \times 10^9 \frac{Nm^2}{C^2}\). For the deuterium fusion problem, understanding how much energy is needed to overcome the electrostatic repulsion between the positively charged deuterium nuclei (protons) becomes vital. The energy calculated gives us an insight into how 'strong' the repulsion is and therefore how much energy we need to add (in the form of heat/temperature) to force these nuclei close enough together to overcome this repulsion and allow fusion to occur.

Deuterium Fusion

Deuterium fusion is a key process in the quest for achieving nuclear fusion power. Deuterium, an isotope of hydrogen with one proton and one neutron, can fuse under extreme conditions of temperature and pressure, similar to the conditions found in the sun's core. The process involves two deuterium nuclei coming so close that nuclear forces overcome the electrostatic repulsion between them, allowing for the formation of a helium nucleus, along with the release of a neutron and a significant amount of energy.

The temperatures required for deuterium fusion are in the range of millions of degrees; this high temperature is necessary because it provides the deuterium nuclei with enough kinetic energy to overcome their mutual electrostatic repulsion, as described by Coulomb's law. This also serves to illustrate why studying fusion reactions requires a good understanding of multiple physical concepts – from electrostatics (to describe the repulsion) to thermodynamics (to describe the environment needed for fusion to occur).

Boltzmann Constant

Understanding the role of the Boltzmann constant is also critical in grasping the relationship between energy and temperature. The Boltzmann constant (\(k_B\)) connects the average kinetic energy of particles in a gas with the temperature of the gas. It's valued at approximately \(1.38 \times 10^{-23} \frac{J}{K}\), and it gives us a bridge between the microcosmic world of atoms and molecules and the macroscopic world that we can measure with thermometers.

When considering the temperatures necessary for deuterium fusion, the Boltzmann constant allows us to convert the calculated energy required to overcome Coulomb's law to an equivalent temperature. This helps us understand that to initiate deuterium fusion, we need to reach temperatures roughly over 8 billion Kelvin, which underlines why achieving controlled fusion on Earth is such a monumental challenge. It's the high temperatures calculated via the Boltzmann constant that characterize the energy-rich, albeit difficult to achieve, conditions needed for fusion energy production.