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.