Question

A \(26.00-g\) sample of water containing tritium, \({ }_{1}^{3} \mathrm{H},\) emits \(1.50 \times 10^{3}\) beta particles per second. Tritium is a weak beta emitter with a half-life of \(12.3 \mathrm{yr}\). What fraction of all the hydrogen in the water sample is tritium?

Short answer

The fraction of tritium in the water sample is approximately \(4.85 \times 10^{-13}\).

Step-by-step solution

Recall radioactive decay laws

We know that the decay rate of a radioactive isotope is proportional to the number of atoms of the isotope. Therefore, we can use the following formula to find out the decay constant, λ, for tritium: \[\lambda = \frac{ln(2)}{t_{\frac{1}{2}}}\] Where \(t_{\frac{1}{2}}\) is the half-life of the isotope.

Calculate the decay constant, λ, for tritium

We are given the half-life of tritium, \({t_{\frac{1}{2}} = 12.3}\) years. Now, use the above formula to find the decay constant: \[\lambda = \frac{ln(2)}{12.3}\] \[\lambda \approx 0.0563 \mathrm{yr}^{-1}\]

Find the tritium activity, A

We are given that the tritium in the sample emits 1.50 x \(10^3\) beta particles per second. This is its activity (number of decays per unit time). To account for the difference in units (seconds and years), we'll convert the activity to decays per year using the following conversion: \[A = 1.50 \times 10^3 \frac{\mathrm{decays}}{\mathrm{s}} \times \frac{3.1536 \times 10^7 \mathrm{s}}{\mathrm{yr}}\] \[A \approx 4.73 \times 10^{10} \mathrm{decays/yr}\]

Calculate the number of tritium atoms, N

We can calculate the number of tritium atoms, N, using the following formula, which relates activity, decay constant, and number of atoms: \[A = \lambda N\] We have already calculated A and λ, so we can find the number of tritium atoms: \[N = \frac{A}{\lambda}\] \[N \approx 8.4 \times 10^{11} \mathrm{atoms}\]

Calculate the total number of hydrogen atoms in the sample

We are given that the mass of the water sample is 26.00 g. Water has a molecular formula \(H_2O\), which means there are two hydrogen atoms for each water molecule. We can find the total number of hydrogen atoms in the sample using the following steps: 1. Calculate the number of moles of water: \(\frac{26.00 g}{18.015 g/mol} = 1.44 mol\), where 18.015 g/mol is the molar mass of water. 2. Multiply the number of moles of water by Avogadro's number to find the number of water molecules: \(1.44 mol \times 6.022 \times 10^{23} mol^{-1} = 8.67 \times 10^{23}\) water molecules. 3. Since there are two hydrogen atoms in each water molecule, multiply the number of water molecules by two to get the total number of hydrogen atoms: \(8.67 \times 10^{23} \times 2 = 1.73 \times 10^{24}\) hydrogen atoms.

Calculate the fraction of tritium in the sample

Now we can calculate the fraction of the tritium in the sample by dividing the number of tritium atoms by the total number of hydrogen atoms: \[\mathrm{Fraction\: of\: tritium} = \frac{N}{\mathrm{Total\: hydrogen\: atoms}}\] \[\mathrm{Fraction\: of\: tritium} = \frac{8.4 \times 10^{11}}{1.73 \times 10^{24}}\] \[\mathrm{Fraction\: of\: tritium} \approx 4.85 \times 10^{-13}\] Thus, the fraction of tritium in the water sample is approximately \(4.85 \times 10^{-13}\).

Key concepts

Tritium

Tritium, denoted as \(^3_1H\), is a rare and radioactive isotope of hydrogen. Unlike the more common isotope of hydrogen, called protium, tritium contains one proton and two neutrons. While normal hydrogen is stable, tritium is known for its radioactive properties. - Tritium naturally occurs in very small amounts and is often produced in nuclear reactors.- It is a key subject of study in both nuclear physics and environmental science.Since tritium emits radiation in the form of beta particles, it is often used in research and in various industries, such as self-powered lighting. It's fascinating due to its potential for nuclear fusion applications.

Decay Constant

The decay constant, denoted by \(\lambda\), is a crucial factor in understanding the rate at which a radioactive isotope decays. It relates to how quickly or slowly an isotope undergoes radioactive decay, which is the process by which an unstable atomic nucleus loses energy by radiation.To calculate the decay constant, the half-life of the substance is used in the formula:\[\lambda = \frac{ln(2)}{t_{\frac{1}{2}}}\]Here, \(t_{\frac{1}{2}}\) represents the half-life, or the time it takes for half of the radioactive atoms in a sample to decay. For tritium, which has a half-life of 12.3 years, the calculated decay constant is approximately 0.0563 yr\(^{-1}\). - The decay constant helps scientists predict the behavior of radioactive materials over time.- It plays a pivotal role in calculating the number of remaining unstable atoms at any given time point.

Beta Particles

Beta particles are high-energy, high-speed electrons or positrons that are emitted from the nucleus of an atom during radioactive decay. Tritium is known to emit beta particles during its decay process. - These particles are a form of ionizing radiation, meaning they have the potential to ionize atoms or molecules they encounter. - They are typically stopped by relatively thin materials, such as a sheet of paper or human skin, which makes them less penetrating and less hazardous compared to other types of radiation. Beta particles are used in medical treatments, radiation therapies, and industrial applications. In the context of environmental science and health, understanding beta radiation is important for assessing exposure risks and implementing safety measures, particularly in water or soil samples containing small amounts of tritium.

Half-Life

The concept of half-life is fundamental to the study of radioactive materials. It is defined as the amount of time it takes for half of the radioactive atoms in a sample to decay. - For tritium, this half-life is 12.3 years. - It is a critical measure that indicates the rate of decay and informs predictions about the future presence of the isotope. Half-life does not vary with the amount of substance, which means that it is a constant characteristic of a given isotope under particular conditions. In practical applications:

• The half-life helps in determining the age of archaeological findings and is a key measure in carbon dating techniques.
• It is used in medicine to assess the efficacy and duration of radioactive drugs.

Understanding half-life aids in managing long-term exposure to radioactive materials, ensuring safety protocols are followed in environments dealing with radioactive isotopes.