(III) What is the energy of the αparticle emitted in the decay 84^210 Po → 82^206 Pb+α? Take i

(III) What is the energy of the αparticle emitted in the...

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, which consists of 2 protons and 2 neutrons. This process changes the parent nucleus into a daughter nucleus with a reduced mass and atomic number, and it is a key mechanism by which heavy nuclei achieve greater stability.

Conservation of Energy

Conservation of energy in nuclear reactions dictates that the total energy before and after the decay remains constant. In the context of alpha decay, the total energy released (the Q-value) is divided between the kinetic energies of the emitted alpha particle and the recoiling daughter nucleus.

Conservation of Momentum

The conservation of momentum is fundamental in nuclear decay processes. When an alpha particle is emitted, the daughter nucleus recoils in the opposite direction to conserve momentum. This sharing of momentum ensures that the energy is partitioned accurately between the two products based on their masses.

Q-Value

The Q-value of a nuclear reaction represents the net energy released by the decay, which is determined by the mass difference between the initial and final products. In alpha decay, this energy is allocated between the recoiling daughter nucleus and the emitted alpha particle, with the precise distribution depending on the conservation laws.

Recoil Kinetic Energy

Recoil kinetic energy refers to the energy carried by the daughter nucleus as a result of conserving momentum during the decay. Because the daughter nucleus has a much larger mass compared to the alpha particle, it receives a smaller portion of the total kinetic energy, affecting the calculation of the alpha particle's energy.

Transcript

00:01 So in this task we should calculate the energy of alpha particle emitted in this reaction.

00:08 Reaction of alpha decay here.

00:13 And we can use these two laws.

00:16 In fact, it is recommended to use both laws, law of conservation of energy, and the law of conservation of momentum.

00:26 Okay, now, first step.

00:30 Is to express vd, in other words, the velocity of daughter, vd means daughter, index d means daughter.

00:43 Vd, it is the velocity of the daughter nuclei equals m alpha times v alpha divided by md.

00:56 Now, if we recall the expression for kinetic energy in general is of any particle, is, but we will, of course, use that expression for d 'oter nuclei, daughter nucleus here.

01:21 M d half, vd squared.

01:26 Now we will include this term vd into equation for the kinetic energy and we will obtain the following expression md half.

01:40 Vd squared is m alpha times valpha divided by md squared.

01:46 After rearranging and canceling some terms, md and md squared, of course, we will obtain the following equation.

02:00 M -alpha divided by md times m -alpha v.

02:08 .

02:10 .

02:10 This is, in other words, expression for kinetic energy of alpha particle therefore we will write in the next step of our derivation kinetic energy of alpha particle.

02:26 Okay now we will write down this derived expression for kinetic energy of alpha particle here with red color, k -e -h.

02:45 D equals m alpha divided by md times k e alpha.

02:55 Okay, which had derived next expression using the law conservation of energy.

03:07 And we will replace this term here, ked, by this expression here.

03:14 We have just derived in the previous step and it follows q equals k -e -alpha plus.

03:24 Instead of k -e -d, we will write here this expression.

03:29 It is m -alpha divided by md times k -e -alpha.

03:36 Okay, if we put k -e -alpha in front of the brackets, inside of the brackets we will have, 1 plus m alpha divided by md.

03:49 Ok.

03:51 Now, if we order this equation, we will have md in brackets.

03:59 K -a alpha will write here.

04:01 And here we will add these two terms, md and m -alpha...

Please give Ace some feedback