Problem 62 - College Physics

Question

\({ }_{1}^{3} \mathrm{H}\) (tritium) can be produced in water surrounding a strong source of neutrons, such as that occuring in nuclear reactors. One of the ways tritium can form is via neutron capture by deuterium. (a) Write down the equation for this capture reaction. (b) Tritium has a half-life of 12.33 years. What percentage of a sample containing \({ }_{1}^{3} \mathrm{H}\) will remain after exactly 6 years? (c) Determine the gamma-ray energy emitted during the capture (assuming the tritium ends up in its ground state and the incoming neutron kinetic energy is negligible). (d) Write down the reaction for the subsequent beta decay of the tritium and determine the stable daughter identity. (e) If all the energy released in the beta decay went into the beta particle, determine its energy.

Step-by-Step Solution

Answer

(a) \( \text{n} + {}_{1}^{2} \mathrm{D} \rightarrow {}_{1}^{3} \mathrm{H} + \gamma \); (b) 73.9%; (c) Depends on mass values; (d) \( {}_{1}^{3} \mathrm{H} \rightarrow {}_{2}^{3} \mathrm{He} + \beta^- + \bar{\nu}_e \); (e) Approx. 18.6 keV for \( \beta^- \).

Step 1: Neutron Capture Reaction Equation

When a neutron is captured by deuterium ({}_{1}^{2} \mathrm{D}), tritium ({}_{1}^{3} \mathrm{H}) is formed. The reaction is written as:\[ \text{n} + {}_{1}^{2} \mathrm{D} \rightarrow {}_{1}^{3} \mathrm{H} + \gamma \] where \( \text{n} \) is a neutron and \( \gamma \) represents the gamma photon released.

Step 2: Calculate Remaining Tritium after 6 Years

The half-life of tritium is 12.33 years. To find the percentage remaining after 6 years, use the equation \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \] where \( N(t) \) is the remaining quantity after time \( t \), \( N_0 \) is the initial quantity, and \( T_{1/2} \) is the half-life. Plugging in the values:\[ N(6) = N_0 \left(\frac{1}{2}\right)^{\frac{6}{12.33}} \approx N_0 \times 0.739 \] Thus, approximately 73.9% of the tritium remains after 6 years.

Step 3: Gamma-ray Energy Emission

The energy emitted as gamma radiation during the neutron capture can be found by calculating the mass difference before and after the reaction and converting it into energy using \( E = \Delta m \cdot c^2 \). However, calculating the exact gamma-ray energy requires the specific mass data of isotopes, typically found in nuclear tables.For the purpose of this exercise without exact values, acknowledge that since the ground state energy differences define it, more detailed resources are needed to calculate the precise value.

Step 4: Beta Decay Equation and Daughter Isotope

Tritium undergoes beta decay, transforming into helium-3:\[ {}_{1}^{3} \mathrm{H} \rightarrow {}_{2}^{3} \mathrm{He} + \beta^- + \bar{u}_e \] where \( \beta^- \) denotes a beta particle (electron) and \( \bar{u}_e \) is an antineutrino. The stable daughter product is helium-3 ({}_{2}^{3} \mathrm{He}).

Step 5: Beta Particle Energy

The energy released during the beta decay can be calculated considering the mass difference converted into energy, although exact numerical calculation requires precise isotope mass differences. Assume it is shared between the beta particle and antineutrino, typically measured as approximately 18.6 keV for tritium.

Key Concepts

Tritium

Tritium, often represented as \({ }_{1}^{3} \mathrm{H}\), is a radioactive isotope of hydrogen. It contains one proton and two neutrons, making it heavier than the most common hydrogen isotope with just one proton. Tritium is found in trace amounts in nature but can be produced artificially in nuclear reactors.

A common method of production is through neutron capture reactions involving deuterium, which is another, more stable, isotope of hydrogen. In this reaction, a deuterium nucleus, \({ }_{1}^{2} \mathrm{D}\), captures a neutron, becoming tritium and releasing a gamma photon in the process:

• \(\text{n} + {}_{1}^{2} \mathrm{D} \rightarrow {}_{1}^{3} \mathrm{H} + \gamma\)

This reaction typically occurs in environments where free neutrons are abundant, such as in the vicinity of nuclear reactors.

Tritium is radioactive, meaning it decays over time, which we will explore further with its beta decay process.

Beta Decay

Beta decay is a type of radioactive decay where an unstable atomic nucleus transforms into a different element by emitting a beta particle, which is essentially an electron or a positron. Tritium undergoes this process to become a stable form of helium:

• \({ }_{1}^{3} \mathrm{H} \rightarrow {}_{2}^{3} \mathrm{He} + \beta^- + \bar{u}_e\)

In this reaction, a neutron in the tritium nucleus is converted into a proton, which results in the emission of a beta particle (an electron, \(\beta^-\)) and an antineutrino (\(\bar{u}_e\)). This decay process changes the element from hydrogen to helium while keeping the mass number constant.

The beta decay of tritium is particularly important in various applications including nuclear fusion research and as a beta radiation source in scientific instruments.

Gamma-ray Energy

When tritium forms through neutron capture, a gamma-ray photon is emitted. Gamma rays are high-energy electromagnetic waves. Unlike alpha and beta particles, gamma rays carry no charge and have a very small wavelength. They can penetrate materials and are often detected using special equipment.

The energy of these emitted gamma rays can be determined by the energy difference involved in the nuclear reaction that produces tritium from deuterium. This energy difference is converted into gamma radiation. Calculating the gamma-ray energy of such a reaction involves precise mass data from nuclear tables to find the mass defect (i.e., the difference in mass between the reactants and the products) and use the equation:

• \( E = \Delta m \cdot c^2 \)

Where \(E\) is energy, \(\Delta m\) is the mass difference, and \(c\) is the speed of light. This highlights the conservation of energy principle in nuclear reactions.

Half-life Calculations

The concept of half-life is crucial in understanding radioactive decay. It describes the time required for half of a sample of a radioactive substance to decay. For tritium, half-life is noted as 12.33 years.

To find out how much of a tritium sample remains after a certain period, you can use the decay equation:

• \( N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \)

Where:

• \(N(t)\) is the remaining quantity after time \(t\)
• \(N_0\) is the initial quantity
• \(T_{1/2}\) is the half-life

This formula helps calculate the percentage of tritium that would remain after any given time. For example, after 6 years, approximately 73.9% of a tritium sample would remain, as derived from substituting the values into the formula. Such calculations are essential to applications involving radioactive materials, particularly in fields like medical imaging and nuclear power.