Question

Fission of a uranium nucleus (containing 92 protons) produces a barium nucleus ( 56 protons) and a krypton nucleus ( 36 protons). The fragments fly apart as a result of electrostatic repulsion; they ultimately emerge with a total of \(200 .\) MeV of kinetic energy. Use this information to estimate the size of the uranium nucleus; that is, treat the barium and krypton nuclei as point charges and calculate the separation between them at the start of the process.

Short answer

Question: Estimate the size of the uranium nucleus by calculating the separation between the barium and krypton nuclei at the start of the fission process, given that the fragments ultimately emerge with a total of 200 MeV of kinetic energy. Answer: The estimated size of the uranium nucleus is approximately \(1.24\times 10^{-14}\) meters.

Step-by-step solution

Identify the initial and final energies

Initially, the barium and krypton nuclei have electrostatic potential energy due to their charges. This energy gets converted into their kinetic energy after fission.

Calculate the electrostatic potential energy between the nuclei

We will use the formula for electrostatic potential energy between two point charges: \[U = k\frac{q_1q_2}{r}\] where k is the electrostatic constant (\(8.9875\times 10^9 Nm^2/C^2\)), \(q_1\) and \(q_2\) are the charges of the barium and krypton nuclei, and r is the separation between them. The charges of the barium and krypton nuclei are given by the number of protons in each nucleus multiplied by the elementary charge (e = \(1.602\times 10^{-19} C\)): \(q_1 = 56e\) \(q_2 = 36e\)

Equate the initial potential energy to the final kinetic energy

As the energy is conserved, we equate the initial electrostatic potential energy between the nuclei to their final kinetic energy: \[k\frac{q_1q_2}{r} = K\] where K is the total kinetic energy of the fragments, which is given as 200 MeV or \(3.204\times 10^{-11} J\).

Solve for the separation r

Now, we have all the values to solve for the separation r: \[r = \frac{kq_1q_2}{K}\] \[r = \frac{(8.9875\times 10^9 Nm^2/C^2)(56\times 1.602\times 10^{-19} C)(36\times 1.602\times 10^{-19} C)}{3.204\times 10^{-11} J}\] After calculating the values, we get: \[r \approx 1.24\times 10^{-14} m\] The estimated size of the uranium nucleus is approximately \(1.24\times 10^{-14}\) meters, which is the separation between the barium and krypton nuclei at the start of the process.

Key concepts

Electrostatic Potential Energy

When dealing with particles that have an electric charge, a form of potential energy called electrostatic potential energy comes into play. This energy is stored when charged particles are positioned within an electric field created by other charges. Imagine placing two magnets near each other; they either pull closer or push apart depending on their poles. Similarly, charged particles exert forces over a distance, attracting or repelling each other without touching.

Mathematically, the electrostatic potential energy, denoted as U, between two point charges is expressed using the equation:
\[U = k\frac{q_1q_2}{r}\]
where k is the Coulomb's constant, representing the strength of the electrostatic force, q_1 and q_2 are the electric charges of the particles, and r is the distance separating them. If the charges have the same sign (like two protons), they repel, and it takes work to push them closer, which increases their potential energy.

Nuclear Physics

Nuclear physics is the branch of physics that studies the constituents and interactions of atomic nuclei. The most famous nuclear processes are fission and fusion. Nuclear fission, in particular, occurs when a heavy nucleus, such as uranium-235, splits into smaller fragments. This process releases a significant amount of energy and is used in nuclear reactors and atomic bombs.

The monumental energy release during fission originates from the strong nuclear forces that hold protons and neutrons together in the nucleus. Overcoming this force requires a great amount of energy, but once overcome, as in fission, the energy released often exceeds the energy input, leading to a net release of energy.

Kinetic Energy

As the term suggests, kinetic energy is the energy of motion. Any object that is moving—whether it's a car cruising down the highway, a ball thrown through the air, or particles shooting out from a nuclear reaction—is carrying kinetic energy. The faster the object moves, the more kinetic energy it has.

The formula for kinetic energy (K) for an object with mass m moving at velocity v is:
\[K = \frac{1}{2}mv^2\]
In the context of nuclear fission, the kinetic energy of the fragments (like barium and krypton nuclei) comes from the transformation of potential energy they had when they were part of the larger uranium nucleus. As they split and repel each other due to their like charges, the potential energy is converted into kinetic energy, propelling them away at high speeds.

Conservation of Energy

The conservation of energy is a fundamental concept in physics, stating that energy cannot be created or destroyed, only transformed from one form to another. When observing any process, including nuclear fission, the total energy before the process must equal the total energy after the process.

In the case of a uranium nucleus undergoing fission, the electrostatic potential energy that was present due to the repulsive forces between the protons is converted into kinetic energy. That's how we determine the energy of the fission fragments in our calculation above. By equating the electrostatic potential energy and the kinetic energy, we employ this principle of conservation to estimate the initial conditions of the fission event.

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative. Like charges repel each other, and unlike charges attract. Protons carry a positive charge, while electrons carry a negative charge.

When we talk about nuclear reactions like fission, we mainly consider the protons, which are positively charged. The number of protons determines the overall charge of the nucleus and subsequently the strength of the electrostatic forces at play. These charges are crucial in calculations involving electrostatic potential energy because they determine the magnitude of the repulsive forces that cause the fission fragments to fly apart.