Question

A \(26.00-\mathrm{g}\) sample of water containing tritium, \({ }^{3} \mathrm{H}\), 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 hydrogen atoms in the water sample that are tritium is approximately 4.85 x 10^(-13).

Step-by-step solution

Calculate the number of hydrogen atoms in the water sample

First, we need to determine the total number of hydrogen atoms in the water sample. To do this, we will use the mass of the sample, the mass of a water molecule, and Avogadro's number. Water has a molecular formula of H2O, which means it consists of 2 hydrogen atoms and 1 oxygen atom. The atomic mass of hydrogen is approximately 1 g/mol, and the atomic mass of oxygen is approximately 16 g/mol. Therefore, the molar mass of water is approximately 18 g/mol. Given that the water sample has a mass of 26.00 g, we can find the number of moles of water in the sample: Number of moles of water = \( \frac{Mass_{water}}{Molar \, mass_{water}} \) Number of moles of water = \( \frac{26.00 \, g}{18 \, g/mol} \) Number of moles of water = 1.44 moles Next, we can find the number of water molecules in the sample using Avogadro's number, which is approximately 6.022 x 10^23 molecules/mol: Number of water molecules = Number of moles × Avogadro's number Number of water molecules = 1.44 moles × 6.022 x 10^23 molecules/mol Number of water molecules = 8.67 x 10^23 molecules Since each water molecule contains 2 hydrogen atoms, we can find the total number of hydrogen atoms in the sample: Number of hydrogen atoms = 2 × Number of water molecules Number of hydrogen atoms = 2 × 8.67 x 10^23 Number of hydrogen atoms = 1.73 x 10^24

Calculate the number of tritium atoms in the sample

To find the number of tritium atoms in the sample, we will use the given information about the amount of beta particles emitted per second and the half-life of tritium. We can convert the half-life of tritium from years to seconds: 1 year = 3.154 x 10^7 seconds Half-life of tritium in seconds = 12.3 years × 3.154 x 10^7 seconds/year Half-life of tritium in seconds = 3.88 x 10^8 seconds From the half-life, we can find the decay constant: Decay constant = \( \frac{ln(2)}{Half-life} \) Decay constant = \( \frac{ln(2)}{3.88 \times 10^8 \, s} \) Decay constant = 1.79 x 10^(-9) s^(-1) Next, we can use the decay constant and the number of beta particles emitted per second to find the number of tritium atoms in the sample: Number of tritium atoms = \( \frac{Number \, of \, beta \, particles}{Decay \, constant} \) Number of tritium atoms = \( \frac{1.50 \times 10^3 \, particles/s}{1.79 \times 10^{-9} \, s^{-1}} \) Number of tritium atoms = 8.38 x 10^11 atoms

Calculate the fraction of hydrogen atoms that are tritium

Now that we have the number of hydrogen atoms in the sample and the number of tritium atoms in the sample, we can find the fraction of hydrogen atoms that are tritium: Fraction of tritium = \( \frac{Number \, of \, tritium \, atoms}{Number \, of \, hydrogen \, atoms} \) Fraction of tritium = \( \frac{8.38 \times 10^{11}}{1.73 \times 10^{24}} \) Fraction of tritium = 4.85 x 10^(-13) So, the fraction of hydrogen atoms in the water sample that are tritium is approximately 4.85 x 10^(-13).

Key concepts

Beta Particle Emission

Beta particle emission occurs when a neutron in an unstable nucleus transforms into a proton while releasing a beta particle, which is essentially an electron, and a neutrino. This process is a type of beta decay, and it is represented by the emission of a beta particle from the nucleus of an atom. In the context of tritium, which is a radioisotope of hydrogen with a nucleus containing one proton and two neutrons, beta decay transforms a neutron into a proton, and in the process, a beta particle is emitted. Tritium's beta decay is important as it enables us to detect and measure the amount of tritium through the particles it emits. Beta particles can be detected with a Geiger counter or similar device, allowing scientists to track the rate of decay and calculate the activity of the sample.

Half-life Calculation

The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample to decay. Half-life calculation is critical in understanding how long a radioactive substance remains active. It is independent of the sample size and is a characteristic property of each radioactive element.For instance, tritium has a half-life of about 12.3 years. This means every 12.3 years, half of the tritium atoms in a given sample will have transformed into helium-3 through beta decay. To find the fraction of tritium in the original exercise, we take into account the half-life to indirectly infer the number of tritium atoms present in our water sample.

Avogadro's Number

Avogadro's number is a fundamental constant of nature that is crucial in chemistry and physics. It represents the number of units in one mole of any substance and is approximately equal to 6.022 x 10^23. This large number allows scientists to work with the convenient unit of moles rather than counting an incomprehensible number of atoms or molecules.In our exercise, Avogadro's number helps convert the mass of the water sample into the number of hydrogen atoms it contains. Since each water molecule, H2O, is formed by two hydrogen atoms, knowing the total number of water molecules through Avogadro's constant enables us to determine the total number of hydrogen atoms present in the sample.

Decay Constant

The decay constant is a probability factor that describes the likelihood of a single atom decaying per unit time. It is inversely proportional to the half-life, meaning a shorter half-life corresponds to a higher decay constant. The decay constant is a key factor in the calculations of radioactive decay as it allows us to determine the activity, or rate of decay, of a radioactive substance.In the case of our exercise, by calculating the decay constant for tritium, we are able to find the number of tritium atoms in the sample based on the measured emission rate of beta particles. This calculation is essential because it bridges the half-life of tritium and the observable quantities, like beta particle emission rates, enabling us to figure out how much of the hydrogen in our water sample is actually tritium.