Question

Tritium \(\left({ }^{3} \mathrm{H}\right)\) is radioactive and decays by electron emission. Its half-life is 12.5 years. In ordinary water the ratio of \({ }^{1} \mathrm{H}\) to \({ }^{3} \mathrm{H}\) atoms is \(1.0 \times 10^{17}\) to \(1 .\) (a) Write a balanced nuclear equation for tritium decay. (b) How many disintegrations will be observed per minute in a 1.00-kg sample of water?

Short answer

35 disintegrations per minute.

Step-by-step solution

Understanding Tritium Decay

Tritium, denoted as \( ^3\mathrm{H} \), is a radioactive isotope of hydrogen that decays through beta decay. In this process, tritium emits an electron to become helium-3 (\( ^3\mathrm{He} \)) and an electron (\( \beta^- \)).

Writing the Nuclear Equation

In beta decay, a neutron in the tritium nucleus is converted to a proton, resulting in the emission of an electron and an antineutrino. Therefore, the balanced nuclear equation for tritium decay is:\[ ^3_1\mathrm{H} \rightarrow ^3_2\mathrm{He} + \beta^- + \bar{u}_e \]

Calculating Number of Tritium Atoms

Calculate the number of tritium atoms in a 1.00 kg sample of water. Since 1 kg of water is 1000 g and water's molar mass is approximately 18 g/mol, the amount of water is \( \frac{1000}{18} = 55.56 \) moles. This equals \( 55.56 \times 6.022 \times 10^{23} = 3.34 \times 10^{25} \) water molecules. With a \( ^1\mathrm{H} \) to \( ^3\mathrm{H} \) ratio of \( 1.0 \times 10^{17}:1 \), the number of \( ^3\mathrm{H} \) atoms is \( \frac{3.34 \times 10^{25}}{1.0 \times 10^{17}} = 3.34 \times 10^8 \).

Calculating Decay Constant

The decay constant (\( \lambda \)) can be calculated using the half-life formula \( \lambda = \frac{\ln 2}{T_{1/2}} \), where \( T_{1/2} \) is the half-life (12.5 years). Convert the half-life to minutes: \( 12.5 \times 365.25 \times 24 \times 60 = 6.570 \times 10^6 \) minutes. Therefore, \( \lambda = \frac{\ln 2}{6.570 \times 10^6} = 1.055 \times 10^{-7} \text{ min}^{-1} \).

Calculating Disintegrations Per Minute

The total disintegrations per minute is given by \( N \lambda \), where \( N \) is the number of \( ^3\mathrm{H} \) atoms. Using \( N = 3.34 \times 10^8 \) and \( \lambda = 1.055 \times 10^{-7} \text{ min}^{-1} \), the number of disintegrations per minute is \( 3.34 \times 10^8 \times 1.055 \times 10^{-7} = 35.23 \).

Key concepts

Radioactivity

Radioactivity refers to the process by which unstable atomic nuclei lose energy by emitting radiation. This happens when certain isotopes, like tritium, undergo spontaneous transformations, resulting in the release of particles. The process is a natural phenomenon observed in various isotopes and leads to the emission of particles or gamma rays. In the context of tritium, a type of hydrogen isotope, this occurs through beta decay, where the unstable nucleus tries to reach a stable state by transforming a neutron into a proton. This transformation releases an electron and an antineutrino, a process integral to understanding radioactivity.

Some important points about radioactivity include:

• It is a random, spontaneous process.
• A radioactive material continuously emits radiation over time.
• The rate of emission decreases as the material becomes more stable.

Understanding radioactivity is key to various fields, including nuclear physics, medicine, and environmental science.

Half-Life Calculation

The half-life of an isotope is the time required for half the quantity of the isotope in a sample to decay. For tritium, this half-life is 12.5 years. This concept helps in determining how long a radioactive material remains active and can be used to calculate decay rates. The longer the half-life, the slower the decay process, indicating a stable, long-lasting radioisotope compared to isotopes with shorter half-lives.

To calculate the decay constant (\(\lambda\)), we use: \(\lambda = \frac{\ln 2}{T_{1/2}}\), where \(T_{1/2}\) is the given half-life. For tritium:

• Convert years to minutes for precision in scientific calculations.
• Use the decay constant in determining the rate of decay over time.

The half-life calculation is crucial in fields such as radiocarbon dating, nuclear medicine, and environmental monitoring.

Beta Decay

Beta decay is a type of radioactive decay where a beta particle, which is either an electron or a positron, is emitted from an atomic nucleus. In the case of tritium decay, it involves the emission of an electron, effectively changing the isotope of hydrogen into an isotope of helium.

During beta decay:

• A neutron is converted into a proton, leading to the emission of an electron and an antineutrino.
• This conversion increases the atomic number of the element by one, thus transforming tritium into helium.
• The mass number remains the same since the transformation involves particles with similar masses.

Understanding beta decay helps explain not only the process of nuclear transmutation but also has practical applications in medicine, like in radiotherapy and diagnostic imaging techniques.

Nuclear Equations

Nuclear equations are used to represent radioactive decay and other nuclear reactions. These equations show the transformation of the original nucleus into the new nucleus, along with the emitted particles. For tritium decay, a balanced nuclear equation is essential for understanding the process:\[^{3}_{1}\mathrm{H} \rightarrow ^{3}_{2}\mathrm{He} + \beta^- + \bar{u}_{e}\]In this equation:

• The superscripts denote the atomic mass number, indicating the number of protons and neutrons.
• The subscripts denote the atomic number, representing the number of protons in the nucleus.
• The equation maintains the balance of mass and charge, ensuring both are conserved in the reaction.

Nuclear equations allow individuals to accurately predict and understand the outcome of nuclear reactions, which is foundational in fields like astrophysics and energy production.