(III) The αparticle emitted when (238)/(92) U decays has 4.20 MeV of kinetic energy. Calculate t

(III) The αparticle emitted when (238)/(92) U decays has...

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, which consists of two protons and two neutrons. This process reduces the original nucleus’s atomic number by 2 and mass number by 4, resulting in a daughter nucleus with different chemical properties. It is a common decay mode for heavy elements and is crucial for understanding nuclear transmutations.

Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant before and after a decay event. In the context of alpha decay, the emitted alpha particle and the recoiling daughter nucleus must have equal and opposite momenta, allowing us to relate the kinetic energies of the two particles based on their masses.

Conservation of Energy

Conservation of energy is a fundamental principle stating that energy cannot be created or destroyed in an isolated system; it only transforms from one form to another. In nuclear decay, the total kinetic energy of the emitted particles (alpha particle and daughter nucleus) and any other forms of energy (such as recoil energy) must equal the Q-value, or the net energy released in the process.

Q-value

The Q-value of a nuclear decay process is the total amount of energy released during the decay, calculated as the difference in mass energy between the initial nucleus and the final products. It provides insight into the energetics of the reaction and is partitioned among the decay products as kinetic energy according to conservation of energy.

Kinetic Energy Distribution in Nuclear Decay

In nuclear decay, the energy released (Q-value) is shared between the decay products—in this case, the alpha particle and the recoiling daughter nucleus—based on their masses. Due to conservation of momentum, lighter particles (such as the alpha particle) generally receive a larger fraction of the kinetic energy, while heavier particles (like the daughter nucleus) acquire a smaller fraction, leading to the phenomenon known as recoil kinetic energy.

Transcript

00:01 In this question, we have uranium 238 decaying, and its emitted alpha particle has a kinetic energy of 4 .2 m .ev.

00:10 And we want to find the recoil kinetic energy and the q value of the decay.

00:18 So let's begin by writing our decay equation.

00:22 So we have uranium 238, it's going to some daughter nucleus, and an end.

00:33 Alpha particle or a helium nucleus, however you want to write it.

00:39 So if we write down all our numbers, we're taking away four from the top, so we have two, three, four, and two from the bottom, so we have 90.

00:50 The atom with 90 protons is thorium, so that is our daughter nucleus.

00:58 So we've been asked to find the recoil kinetic energy of this daughter nucleus.

01:05 And to start this question, i am going to assume that the uranium nucleus begins at rest.

01:14 So it decays at rest.

01:18 So because we've said this, we know that the momentum of the thorium and the alpha have to be equal to each other because we need to have momentum conservation before and after the decay.

01:35 So the momentums need to be equal to each other.

01:39 So we have that the momentum of the helium or the alpha particle is equal to the momentum of the thorium.

01:52 So that's the first step we have.

01:55 We then know that kinetic energy can be written as kinetic energy is equal to momentum squared over 2m.

02:06 So we can say that the kinetic energy of the thorium e thorium is equal to the momentum of the thorium squared over 2m.

02:22 But then we've just said that the momentum of the thorium is equal to the momentum of the helium.

02:26 So we can then write that the momentum of the helium squared divided by 2m is equal to the energy of the thorium.

02:37 We can then rearrange this equation to make p squared the subject.

02:43 So we get that kinetic energy times by 2m is equal to p squared.

02:50 So the two times the kinetic energy of the helium is equal to the helium momentum squared.

03:02 So i'm going to sub that into this equation here.

03:06 So we get 2 e -helium, mass of helium, over two times the mass of, sorry, these should be labeled as thorium, two times the mass of thorium...

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