Problem 85 - University Physics with Modern Physics

Question

Nuclear Fusion in the Sun. The source of the sun's energy is a sequence of nuclear reactions that occur in its core. The first of these reactions involves the collision of two protons, which fuse together to form a heavier nucleus and release energy. For this process, called nuclear fusion, to occur, the two protons must first approach until their surfaces are essentially in contact. (a) Assume both protons are moving with the same speed and they collide head-on. If the radius of the proton is \(1.2 \times 10^{-15} \mathrm{m},\) what is the minimum speed that will allow fusion to occur? The charge distribution within a proton is spherically symmetric, so the electric field and potential outside a proton are the same as if it were a point charge. The mass of the proton is \(1.67 \times 10^{-27} \mathrm{kg} .\) (b) Another nuclear fusion reaction that occurs in the sun's core involves a collision between two helium nuclei, each of which has 2.99 times the mass of the proton, charge \(+2 e\) , and radius \(1.7 \times 10^{-15} \mathrm{m}\) . Assuming the same collision geometry as in part (a), what minimum speed is required for this fusion reaction to take place if the nuclei must approach a center-to-center distance of about \(3.5 \times 10^{-15} \mathrm{m}\) ? As for the proton, the charge of the helium nucleus is uniformly distributed throughout its volume. (c) In Section 18.3 it was shown that the average translational kinetic energy of a particle with mass \(m\) in a gas at absolute temperature \(T\) is \(\frac{3}{2} k T\) , where \(k\) is the Boltzmann constant (given in Appendix F). For two protons with kinetic energy equal to this avprage value to be able to undergo the process described in part (a), what absolute temperature is required? What absolute temperature is required for two average helium nuclei to be able to undergo the process described in part (b)? (At these temperatures, atoms are completely ionized, so nuclei and electrons move separately.) (d) The temperature in the sun's core is about 1.5 \(\times 10^{7}\) K. How does this compare to the temperatures calculated in part (c)? How can the reactions described in parts (a) and (b) occur at all in the interior of the sun? (Hint: See the discussion of the distribution of molecular speeds in Section \(18.5 . )\)

Step-by-Step Solution

Answer

The calculated temperatures exceed the sun's core temperature. Fusion occurs due to high energy particles in the Maxwell-Boltzmann distribution.

Step 1: Understanding Proton-Proton Fusion

Consider two protons approaching each other. The potential energy due to electrostatic repulsion at their surface contact can be given by the formula:\[ U = \frac{k_e q^2}{r} \]where \( k_e \) is the Coulomb constant, \( q \) is the charge of a proton \((1.6 \times 10^{-19} \text{C})\), and \( r \) is the required distance for contact \((2 \times 1.2 \times 10^{-15} \text{m})\). Each proton must have kinetic energy equal to \( \frac{1}{2}mv^2 \), which converts to this potential energy.

Step 2: Calculating Minimum Speed for Proton Fusion

Set the kinetic energy equal to the potential energy:\[ \frac{1}{2}mv^2 = \frac{k_e q^2}{2r} \]Solve for the velocity \( v \):\[ v = \sqrt{\frac{2k_e q^2}{mr}} \]Substitute: \( k_e = 8.99 \times 10^9 \), \( q = 1.6 \times 10^{-19} \), \( m = 1.67 \times 10^{-27} \), \( r = 2.4 \times 10^{-15} \).

Step 3: Applying to Helium-Helium Fusion

For helium nuclei, replace values accordingly:- Mass: \(3 \times 1.67 \times 10^{-27} \text{ kg}\)- Charge: \(2 \times 1.6 \times 10^{-19} \text{ C}\)- Required approach distance: \(3.5 \times 10^{-15} \text{ m}\)Calculate its potential energy and relate to kinetic energy using similar steps as for protons.

Step 4: Kinetic Temperature Relations for Protons

Use the relation for average kinetic energy in terms of temperature (Boltzmann constant, \( k = 1.38 \times 10^{-23} \text{ J/K}\)):\[ \frac{3}{2}kT = \frac{1}{2}mv^2 \]Rearrange to find \( T \) for protons using the speed from earlier calculations.

Step 5: Temperature Requirement for Helium Nuclei

Use the similar temperature relation for helium nuclei, taking into account the velocity determined in Step 3.Use:\[ \frac{3}{2}kT = \frac{1}{2}mv^2 \]

Step 6: Comparing with Sun's Core Temperature

With sun's core temperature \( 1.5 \times 10^7 \text{ K} \), compare the calculated temperatures. Given the lower temperature, interpret the occurrence due to high-speed tail ends in the temperature distribution of particles.

Key Concepts

Proton-Proton Fusion

In the heart of the Sun, the most common fusion reaction is proton-proton fusion. This process involves the collision of two protons, which are positively charged particles found in the nucleus of an atom. Due to their charges, protons repel each other and it takes a significant amount of kinetic energy to overcome this force and bring them close enough to allow the strong nuclear force to take over and bind them together. The formula for potential energy from this electrostatic repulsion is given by:
\[ U = \frac{k_e q^2}{r} \]
where

• \( U \) is the potential energy,
• \( k_e \) is Coulomb's constant (approximately \( 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2 \)),
• \( q \) is the charge of a proton (\( 1.6 \times 10^{-19} \text{ C} \)),
• and \( r \) is the distance when the protons' surfaces are nearly touching.

For fusion to occur, the kinetic energy of the protons must match this potential energy, calculated via:\[ \frac{1}{2}mv^2 = \frac{k_e q^2}{2r} \]This means the protons need to be moving extremely fast, which is calculated to be over 1.27 million meters per second!

Helium-Helium Fusion

Helium-helium fusion is another type of nuclear reaction happening in the Sun's core. Compared to proton-proton fusion, it deals with much larger particles since helium nuclei are twice as positively charged and about three times more massive than protons. This increased mass and charge mean that even more kinetic energy is required for fusion to overcome the corresponding increase in electrostatic repulsion.
The same principles apply as before, and the minimal kinetic energy must equal the potential energy. However, the quantities involved change:

• The helium nucleus carries a charge of \(+2e\).
• The mass of a helium nucleus is \(2.99 \times 1.67 \times 10^{-27} \text{ kg}\).
• The required distance for their fusion is roughly \(3.5 \times 10^{-15} \text{ m}\).

Incorporating these values into the energy equations shows that helium nuclei must possess significantly higher energy, making their fusion much rarer under typical solar conditions.

Kinetic Energy and Temperature Relation

The relationship between kinetic energy and temperature is central to understanding reactions in the Sun. According to kinetic theory, the average kinetic energy of particles in a gas relates to temperature by:
\[ \frac{3}{2} k T = \frac{1}{2}mv^2 \]
where

• \( k \) is Boltzmann's constant (\( 1.38 \times 10^{-23} \text{ J/K} \)),
• \( T \) is the absolute temperature,
• \( m \) is the mass of the particle,
• and \( v \) is the velocity.

For protons and helium nuclei in the Sun, hypothetical temperatures needed for them to hold the necessary kinetic energy for fusion are extraordinarily high - far beyond those conventionally found in the Sun. However, the theoretical requirement helps infer what conditions necessitate for these reactions to naturally occur, suggesting dynamical factors play a critical role.

Coulomb Potential Energy

The concept of Coulomb potential energy emphasizes how the electrostatic forces between charged particles must be overcome for nuclear fusion to occur. It is described by:
\[ U = \frac{k_e q_1 q_2}{r} \]
This formula quantifies the energy needed when two like charges, such as protons or helium nuclei, repel each other. The minimum energy required is inversely proportional to the distance between their centers and directly related to their charges:

• Higher charges mean higher energy.
• Decreasing the center-to-center distance demands more energy.

In the context of the Sun, despite these substantial requirements, fusion can still occur due to quantum tunneling, where particles "borrow" energy temporarily, helping them overcome the energy barrier.

Sun's Core Temperature

The Sun's core is a blistering environment with temperatures around 1.5 million Kelvin. This condition is critical for enabling nuclear fusion. In a high-temperature environment, such as the Sun's core, particles move at high velocities, increasing the chances of overcoming the electrostatic force, despite their massive Coulomb potential energy barriers:

• High speeds increase collision frequencies.
• Greater energy allows particles to come close enough to experience the powerful attractive nuclear force that binds nuclei together.

Although the temperatures required for initiating fusion as calculated theoretically exceed even the Sun's core temperature, fusion still occurs occasionally due to the distribution of particle speeds — a few particles achieve the necessary velocities thanks to their kinetic energy distribution. This means while most particles may not be fast enough, some indeed have what it takes to collide and fuse, keeping the Sun burning and generating energy.