Question

In the following nuclear fusion reaction \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{3} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+{ }_{0} \mathrm{n}^{1}\) the repulsive potential energy between the two fusing nuclei is \(7.7 \times 10^{-14} \mathrm{~J}\). The Temperature to which the gas must be heated is nearly (Boltzman constant \(\mathrm{K}=1.38 \times 10^{-23} \mathrm{JK}^{-1}\) ) (A) \(10^{3} \mathrm{~K}\) (B) \(10^{5} \mathrm{~K}\) (C) \(10^{7} \mathrm{~K}\) (D) \(10^{9} \mathrm{~K}\)

Short answer

The temperature required to overcome the repulsive potential energy for this nuclear fusion reaction is around \(5.58 \times 10^{9} K\). Therefore, the closest answer is option (D) - \(10^{9} K\).

Step-by-step solution

Understand the problem

The problem is asking us to determine the minimal temperature required to overcome the repulsive potential energy for a nuclear fusion reaction. This is done by rearranging the formula of the most probable energy in a Maxwell-Boltzmann distribution, \(E = kT\) for some constant \(k\), to solve for \(T\) (Temperature).

Rearrange the formula to solve for T

We rearrange the formula \(E = kT\) to solve for \(T\). This gives us \(T = \frac{E}{k}\).

Substitute the values into the formula

We substitute the repulsive potential energy \(E = 7.7 \times 10^{-14} J\) and the Boltzmann constant \(k = 1.38 \times 10^{-23} \: JK^{-1}\) into the formula from step 2. When we do this, we get \(T = \frac{7.7 \times 10^{-14} \: J}{1.38 \times 10^{-23} \: JK^{-1}}\).

Calculate T (Temperature)

Evaluate \(T = \frac{7.7 \times 10^{-14} \: J}{1.38 \times 10^{-23} \: JK^{-1}}\) to find the temperature. A careful calculation yields \(T = 5.58 \times 10^{9} K\).

Make a conclusion

From our calculations, the temperature required to overcome the repulsive potential energy for this nuclear fusion reaction is around \(5.58 \times 10^{9} K\). This temperature is greater than all the options given except (D) - \(10^{9} K\). Hence, option D is the closest and is accepted as the answer.

Key concepts

Repulsive Potential Energy

In nuclear fusion, two or more atomic nuclei collide at very high speeds and fuse to form a single heavier nucleus. However, for fusion to occur, the reacting nuclei must overcome their repulsive potential energy. This repulsion occurs due to the positive charges of the protons in the nuclei. They naturally repel each other due to electrostatic forces, which is similar to how like charges repel.

• Potential energy represents the energy stored due to the position of an object.
• In this context, it's the energy needed to bring the nuclei close enough to fuse.
• The repulsive potential energy can be determined experimentally or through calculations based on the charge and distance between the nuclei.

Overcoming this repulsive potential energy is crucial for fusion, as it allows the nuclei to approach each other closely enough to allow the strong nuclear force to bind them together.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann Distribution is a statistical means of describing the velocities of particles in a gas. At any given temperature, particles in a gas are in continuous random motion, and their speeds vary. The distribution provides a function that predicts the likelihood of a particle having a certain speed.

• This concept is key to understanding how gases behave at different temperatures.
• The distribution is characterized by a peak, which signifies the most probable speed of particles at a given temperature.

In the context of fusion, the distribution helps us understand kinetic energies of particles. When we talk about the most probable energy, we're referring to the peak of this distribution, which is related to temperature by the expression \[E = kT\]. This helps us calculate the conditions necessary for the particles to overcome repulsive forces and react.

Temperature Calculation

Determining the temperature required for overcoming nuclear repulsive forces is a critical application of physics. The temperature needs to be high enough to provide particles with enough kinetic energy. This way, they can overcome repulsive potential energy and get close enough for nuclear forces to take over.

• The Maxwell-Boltzmann relation used for calculating temperature involves rearranging the expression \(E = kT\) to \(T = \frac{E}{k}\).
• This formula shows how temperature correlates with energy. As energy increases, the requisite temperature rises correspondingly.
• This formula is utilized to compute the precise temperature needed for fusion processes to occur, giving particles the energy needed to meet the conditions specified in a reaction problem.

Using this calculation, physicists can simulate the conditions needed for fusion in various environments, whether in the stars or in controlled experiments on Earth.

Boltzmann Constant

The Boltzmann constant, denoted by \( k \), is a fundamental constant in physics. It relates the average kinetic energy of particles in a gas with the temperature of the gas. This constant plays a pivotal role in statistical mechanics and thermodynamics.

• The Boltzmann constant has a value of \(1.38 \times 10^{-23} \text{JK}^{-1}\), which acts as a conversion factor between the energy scales and temperature scales.
• In calculations involving atomic or subatomic particles, it serves to link temperature with energy in a meaningful way.
• In our specific problem, \(k\) helps relate radiant energy to thermal energy, allowing us to calculate the temperature needed for nuclear reactions to proceed.

Understanding this constant allows scientists to predict and manipulate the behaviors of gases at different temperatures and energies, crucial for fields like nuclear physics, where precise energy measurements are necessary for controlled reactions.