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Ch 25: The Electric Potential
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 25: The Electric PotentialProblema 57
Capitolo 25, Problema 57

In the form of radioactive decay known as alpha decay, an unstable nucleus emits a helium-atom nucleus, which is called an alpha particle. An alpha particle contains two protons and two neutrons, thus having mass m=4 u and charge q=2e. Suppose a uranium nucleus with 92 protons decays into thorium, with 90 protons, and an alpha particle. The alpha particle is initially at rest at the surface of the thorium nucleus, which is 15 fm in diameter. What is the speed of the alpha particle when it is detected in the laboratory? Assume the thorium nucleus remains at rest.

Guida verificata passo dopo passo
1
Step 1: Understand the problem. The uranium nucleus undergoes alpha decay, emitting an alpha particle (mass m = 4 u, charge q = 2e) and leaving behind a thorium nucleus. The alpha particle starts at rest at the surface of the thorium nucleus and gains kinetic energy due to the electrostatic repulsion between the thorium nucleus and the alpha particle. The goal is to find the speed of the alpha particle when it is detected in the laboratory.
Step 2: Apply the principle of energy conservation. The initial potential energy of the system (due to the electrostatic interaction between the thorium nucleus and the alpha particle) is converted into the kinetic energy of the alpha particle. The thorium nucleus remains at rest, so all the kinetic energy is attributed to the alpha particle. The potential energy is given by the formula: Ui = k q1q2/r, where k is Coulomb's constant, q1 and q2 are the charges of the thorium nucleus and alpha particle, and r is the distance between their centers (equal to the radius of the thorium nucleus).
Step 3: Write the expression for the kinetic energy of the alpha particle. The kinetic energy is given by: K = 12mv2, where m is the mass of the alpha particle and v is its speed. Using energy conservation, set the initial potential energy equal to the final kinetic energy: Ui = K.
Step 4: Substitute the known values into the energy conservation equation. The charge of the thorium nucleus is q1 = 90e, the charge of the alpha particle is q2 = 2e, and the distance between their centers is r = 15 fm = 15 × 10-15 m. The mass of the alpha particle is m = 4 u = 4 × 1.66 × 10-27 kg. Coulomb's constant is k = 8.99 × 109 N·m2/C2. Substitute these values into the equation: 12mv2 = k q1q2/r.
Step 5: Solve for the speed of the alpha particle. Rearrange the equation to isolate v: v = 2m(k q1q2/r). Substitute the known values into this equation to calculate the speed of the alpha particle.

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Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable atomic nucleus emits an alpha particle, consisting of two protons and two neutrons. This process reduces the atomic number of the original nucleus by two, resulting in the formation of a new element. The emitted alpha particle carries a positive charge and has a relatively low penetration power compared to other forms of radiation.
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Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event. In the context of alpha decay, when the uranium nucleus emits an alpha particle, the momentum of the alpha particle and the remaining thorium nucleus must balance out, allowing us to calculate the speed of the alpha particle based on the initial conditions.
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Kinetic Energy and Speed

Kinetic energy is the energy possessed by an object due to its motion, defined mathematically as KE = 1/2 mv², where m is mass and v is velocity. In the case of the alpha particle, its kinetic energy can be derived from the energy released during the decay process. By applying the conservation of energy, we can relate the kinetic energy of the alpha particle to its speed after it is emitted from the thorium nucleus.
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