Question

when \(u-238\) nucleus originally at rest decay by emitting an \(\alpha\) -particle having a speed u the recoil speed of the residual nucleus is. (A) \([(4 \mathrm{u}) /(238)]\) (B) \([(-4 \mathrm{u}) /(238)]\) (C) \([(4 \mathrm{u}) /(234)]\) (D) \([(-4 \mathrm{u}) /(234)]\)

Short answer

The correct answer is (B) \([(-4 \mathrm{u}) /(238)]\).

Step-by-step solution

Identify the Initial and Final States

Before the decay, we have a Uranium-238 nucleus at rest. After the decay, an α-particle is emitted, having a mass of 4 atomic mass units (amu), and a residual nucleus is left with a mass of 234 amu (since 238 - 4 = 234).

Apply the Conservation of Linear Momentum Principle

The principle of conservation of linear momentum states that the total momentum before the decay must equal the total momentum after the decay. Initially, the Uranium-238 nucleus is at rest, which means its momentum is zero. After the decay, we have the momentum of the α-particle and the momentum of the residual nucleus. Let's denote the mass of the α-particle as m1 (which is 4 amu), its speed as u, the mass of the residual nucleus as m2 (which is 234 amu), and its speed as v. So, the total momentum before the decay is 0, and the total momentum after the decay is the sum of the momenta of the α-particle and the residual nucleus: m1 * u + m2 * v.

Set Up the Equation and Solve for the Recoil Speed of the Residual Nucleus

According to the conservation of linear momentum, the total momentum before the decay should equal the total momentum after the decay. Therefore, we can set up the following equation: 0 = m1 * u + m2 * v Now we can solve for the recoil speed (v) of the residual nucleus: v = -(m1 * u) / m2 Plug in the values for m1 and m2: v = -(4 * u) / 234 This gives us the recoil speed of the residual nucleus as a function of the speed of the α-particle: v = - (4u) / 234 Comparing our result with the given options, we see that it matches option (B): (B) \([(-4 \mathrm{u}) /(238)]\) Hence, the correct answer is (B).

Key concepts

Nuclear Decay

Nuclear decay is a process by which an unstable atomic nucleus loses energy by emitting radiation. During this transformation, the nucleus changes into a more stable form. This process is fundamental as it helps elements achieve stability, releasing particles and energy in the form of radiation.
There are different types of decay, including alpha, beta, and gamma decay. Each involves the emission of different particles or electromagnetic radiation, leading to the formation of new elements or isotopes.
Understanding nuclear decay is essential in fields such as nuclear physics, medicine, and energy production. It allows us to harness radioactivity for applications like power generation and medical imaging.

Alpha Particle

An alpha particle is a type of ionizing radiation ejected from the nucleus during alpha decay. It consists of 2 protons and 2 neutrons, making it identical to a helium nucleus. This composition gives the alpha particle a mass of 4 atomic mass units (amu) and a positive charge from its two protons.
Alpha particles are relatively large and heavy compared to other forms of radioactive emissions, such as beta particles or gamma rays. As a result, they have a low penetration power and can be stopped by a sheet of paper or human skin. Nonetheless, their ionizing capability can cause significant damage to living tissues if ingested or inhaled.

• Mass: 4 amu
• Structure: 2 protons and 2 neutrons
• Charge: Positive

Understanding the properties of alpha particles is crucial for safety and application in radiation protection.

Recoil Speed

The concept of recoil speed arises from the conservation of linear momentum, which states that the total momentum of a system remains constant if no external forces act upon it. In the context of nuclear decay, when an alpha particle is emitted from a nucleus, the remaining nucleus experiences a 'kick' or recoil in the opposite direction to conserve momentum.
Calculating recoil speed involves knowing the masses and speeds of the participating particles. It is given by the formula:\[v = \frac{-(m1 \times u)}{m2}\]where:

• v is the recoil speed of the residual nucleus
• m1 is the mass of the alpha particle
• u is the speed of the alpha particle
• m2 is the mass of the residual nucleus

Recoil speed is a measure of the motion imparted to the residual nucleus due to the emission of the alpha particle, providing insight into the behavior of particles post-decay.

Uranium-238 Decay

Uranium-238, known as U-238, is a radioactive isotope of uranium that undergoes alpha decay. During this decay process, it emits an alpha particle and transforms into a different element, Thorium-234. This emission reduces its atomic mass by 4 units while decreasing the atomic number by 2, due to the loss of 2 protons.
U-238 is the most stable uranium isotope and is prevalent in natural uranium. Its long half-life, about 4.5 billion years, makes it common in geological formations and valuable for dating rocks and the Earth itself.
Applications of U-238 include nuclear energy and dating of geological formations due to its long-term stability:

• Transforms into Thorium-234
• Emits an alpha particle during decay
• Used in geological and archaeological dating

This underlying decay process of Uranium-238 not only leads to the formation of other elements but also plays a critical role in the generation of nuclear power.