Problem 18 - University Physics with Modern Physics

Question

Tritium (\(^{3}_{1}\)H) is an unstable isotope of hydrogen; its mass, including one electron, is 3.016049 u. (a) Show that tritium must be unstable with respect to beta decay because the decay products ( \(^{3}_{2}\)He plus an emitted electron ) have less total mass than the tritium. \right. (b) Determine the total kinetic energy (in MeV) of the decay products, taking care to account for the electron masses correctly.

Step-by-Step Solution

Answer

(A) Tritium is unstable due to lower mass in decay products. (B) Total kinetic energy is 0.491 MeV.

Step 1: Compare Masses of Reactants and Products

Tritium ( ^{3}_{1} ext{H} ) has a mass of 3.016049 u. The decay product, ^{3}_{2} ext{He} , has an atomic mass of 3.016029 u. The mass of an electron is 0.000548 u. Adding the mass of an electron to ^{3}_{2} ext{He} gives 3.016029 u + 0.000548 u = 3.016577 u. Since ^{3}_{2} ext{He} plus an electron is lighter than tritium, the decay is energetically favorable.

Step 2: Calculate Mass Defect

The mass defect is the difference in mass between the reactants and products. Calculate it as follows: 3.016049 u (tritium mass) - 3.016577 u ( ^{3}_{2} ext{He} + electron mass) = -0.000528 u. This indicates a mass loss which is converted into energy in the form of kinetic energy.

Step 3: Convert Mass Defect to Energy

Use Einstein's relation E=mc² to convert the mass defect to energy. First, convert mass defect from atomic mass units to MeV: -0.000528 ext{ u} imes 931.5 ext{ MeV/u} = -0.491 ext{ MeV} . This is the energy released as kinetic energy of the decay products.

Key Concepts

Beta Decay

In nuclear physics, beta decay is a process where an unstable atomic nucleus transforms into a different nucleus. This happens through the emission of beta particles, which are either electrons or positrons.
Beta decay helps to stabilize the nucleus by changing a neutron into a proton (beta-minus decay) or a proton into a neutron (beta-plus decay).
For tritium, which is an unstable hydrogen isotope, beta minus decay occurs.

• This results in the transformation of tritium ( ^{3}_{1} H) into helium-3 ( ^{3}_{2}He) along with the emission of an electron.
• The process is driven by the quest for a more stable energy configuration, as seen in the differing masses.

The mass of helium-3 plus the emitted electron is less than the original tritium, suggesting that excess energy during the decay is liberated as kinetic energy.
This forms a fundamental understanding of why beta decay occurs and how it results in more stable, lower mass products.

Mass Defect

Mass defect is a fascinating concept in nuclear physics that explains how the mass of a nucleus differs from the sum of the individual masses of its protons and neutrons.
In the context of tritium decay, the mass defect is the difference between the mass of tritium and the total mass of its decay products, which include helium-3 and an electron.

• The calculation in the exercise shows that the tritium mass, at 3.016049 u, is greater than the combined mass of helium-3 and the electron, at 3.016577 u.
• This discrepancy is termed the "mass defect," and it is equal to -0.000528 u in this case.

The negative value signifies a mass decrease, meaning that this amount of mass has been converted to energy according to Einstein's famous equation, \( E=mc^2 \).
This energy safety measure accounts for the kinetic energy observed in the decay products.

Kinetic Energy

Kinetic energy emerges as a significant part of understanding nuclear decay processes like that of tritium.
When tritium undergoes beta decay, the resulting mass defect is converted into kinetic energy.
This transformation follows Einstein's relation \( E=mc^2 \), where the mass defect is turned into energy.

• The exercise converts the mass defect from atomic mass units to energy measured in MeV, the standard energy unit in particle physics.
• Specifically, the mass defect of -0.000528 u translates to -0.491 MeV after considering the conversion factor \( 931.5 \text{ MeV/u} \).

This released energy appears as the kinetic energy of the decay products, such as the electron and helium-3 nucleus.
Understanding kinetic energy in nuclear decay helps explain the behavior of particles and how they achieve a state of stability post-decay.

Atomic Mass Unit

The atomic mass unit (amu) is a standard unit used to express atomic and molecular masses.
It serves as a convenient way to measure tiny amounts of mass, especially useful when dealing with particles at the atomic scale.
The amu is defined as one-twelfth the mass of a carbon-12 atom, effectively serving as a "mass baseline" for atoms and molecules.

• In the context of the exercise, tritium's mass is measured in atomic mass units, showing how manageable these units are for nuclear calculations.
• Converting between amu and energy reveals the profound connections between mass and energy, guided by Einstein's principle of relativity (\( E=mc^2 \)).

Measuring and manipulating masses in amu helps physicists predict how atoms will behave, how they interact, and the energy exchanges during reactions like beta decay.
It's a small but powerful unit that makes significant contributions to the study of atomic physics.