A 26.00-g sample of water containing tritium, ³¹H, emits 1.50 * 10³ beta particles per second. Tritium is a weak beta emitter with a half-life of 12.3 years. What fraction of all the hydrogen in the water sample is tritium?

Verified step by step guidance

1. Step 1: Calculate the number of moles of water in the 26.00-g sample. Use the molar mass of water (H₂O), which is approximately 18.02 g/mol, to find the moles: \( \text{moles of water} = \frac{26.00 \text{ g}}{18.02 \text{ g/mol}} \).
2. Step 2: Determine the total number of hydrogen atoms in the water sample. Since each water molecule (H₂O) contains 2 hydrogen atoms, multiply the moles of water by Avogadro's number (6.022 \times 10^{23} \text{ molecules/mol}) and then by 2.
3. Step 3: Calculate the decay constant (\( \lambda \)) for tritium using its half-life. The decay constant is given by \( \lambda = \frac{0.693}{\text{half-life}} \), where the half-life is 12.3 years. Convert the half-life to seconds for consistency with the decay rate.
4. Step 4: Use the decay rate formula \( R = N \lambda \) to find the number of tritium atoms (N) in the sample. Here, R is the decay rate (1.50 \times 10^3 \text{ beta particles/second}). Rearrange the formula to solve for N: \( N = \frac{R}{\lambda} \).
5. Step 5: Calculate the fraction of hydrogen that is tritium by dividing the number of tritium atoms (N) by the total number of hydrogen atoms calculated in Step 2.

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This can occur in various forms, including alpha, beta, and gamma decay. In the case of tritium, it undergoes beta decay, where a neutron is converted into a proton, resulting in the emission of a beta particle. Understanding this process is crucial for calculating the amount of tritium remaining in a sample over time.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For tritium, the half-life is 12.3 years, meaning that after this period, only half of the original amount of tritium will remain. This concept is essential for determining how much tritium is present in the water sample after a certain period, allowing for calculations related to its decay and current quantity.

Molar Mass and Composition

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole. Water (H₂O) has a molar mass of approximately 18.02 g/mol, with hydrogen contributing about 2.02 g/mol. To find the fraction of tritium in the water sample, one must calculate the total amount of hydrogen present and compare it to the amount of tritium, which requires understanding the composition of the sample and the molar mass of tritium.