Calculate the kinetic energy of the αparticle in Problem 25 . This time, do not assume that the

Calculate the kinetic energy of the αparticle in Problem 25...

Key Concepts

Conservation of Momentum

In any isolated system, including nuclear reactions, the total momentum remains constant. During a decay process, the momentum carried by one fragment must be balanced by the momentum of the remaining fragment(s), leading to specific relationships between their velocities and, as a consequence, their kinetic energies. This concept is crucial in analyzing the recoil of the nucleus when an alpha particle is emitted.

Conservation of Energy

In nuclear reactions, the total energy is conserved, meaning that the sum of the kinetic energy of the products and any other forms of energy (such as binding energy) remains constant. When calculating the kinetic energy of reaction products, it is important to consider that energy is partitioned between them in accordance with this conservation principle.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object and is given by the expression 1/2 mv², where m is the mass and v is the velocity. In nuclear decay, understanding kinetic energy is essential because it helps explain how the energy released in the reaction is distributed among the emitted particles.

Recoil Effect in Nuclear Decay

The recoil effect occurs in decay processes like alpha decay when the emitted particle carries away momentum, causing the remaining nucleus to recoil. This phenomenon must be considered to correctly calculate the kinetic energies of both the alpha particle and the recoiling nucleus, rather than assuming one of the products is stationary.

Mass Ratio and Energy Partitioning

The partitioning of kinetic energy between the decaying products is influenced by their mass ratio. Typically, the lighter particle ends up with a larger fraction of the kinetic energy because, under momentum conservation, a smaller mass requires a greater velocity to compensate. This inverse relationship is critical when determining the relative kinetic energies of decay products.

Transcript

00:01 In this question, we're looking at the decay of radium 226 into radon 222 via alpha decay.

00:10 So this is the decay reaction.

00:13 And we want to find the kinetic energy of the alpha particle.

00:18 But this time we are not assuming that the radon is stationary after the decay.

00:24 So we can take some of the kinetic energy away.

00:27 But first we have to find what is the total energy that is.

00:31 Is converted into kinetic energy or the total kinetic energy and that we can find by taking the mass difference between the final product and the initial parent nucleate.

00:47 So the mass we can get online.

00:52 So the difference, right, difference in the mass between the products and the initial parent is the mass of your alpha particle plus the mass of your radon subtracted by the mass of the radium.

01:14 Mass of radium can be found to be 226 .025 -4026 u.

01:23 Mass of radon is 222 .101757550u.

01:33 Mass of the alpha particle is 4 .002.

01:40 603 to you.

01:44 So plugging these numbers in, we get negative 0 .005 -228 -9u.

01:54 And it is negative which tells us that mass is being lost and that energy is being released.

02:01 So now we can find the total kinetic energy total.

02:07 To be delta m, c square, where c squared is 9.

02:15 4941, sorry 931 .931 .494 m ev per u, right, and then we can get 4 .8707 mega electron volts.

02:41 Now next to get the final kinetic energy of the alpha particle, we need to make use of the conservation of momentum...