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Chapter 19: Problem 53

A \(100.0\) -g sample of water containing tritium, \({ }_{1}^{3} \mathrm{H}\), emits \(2.89 \times 10^{3}\) beta particles per second. Tritium has a half-life of \(12.3\) years. What percentage of all the hydrogen atoms in the water sample is tritium?

Short Answer

Expert verified
The percentage of hydrogen atoms in the water sample that are tritium is approximately 2.41 x 10^-11%.

Step by step solution

01

Calculate the number of moles of tritium atoms in the sample

To determine the number of tritium atoms in the sample, we'll use the half-life and the decay constant: - Decay constant lambda = ln(2) / half-life For tritium, lambda = ln(2) / (12.3 years x 3.154 x 10^7 seconds/year) ≈ 1.784 x 10^-9 s^-1 Now using this decay constant, we can calculate the number of tritium atoms by dividing the emission rate of beta particles by lambda: - N_t = (emission rate) / lambda N_t ≈ (2.89 x 10^3 particles/s) / (1.784 x 10^-9 s^-1) ≈ 1.616 x 10^12 tritium atoms
02

Calculate the moles of tritium atoms

To determine the moles of tritium atoms, we can use the relation: - moles = (number of atoms) / (Avogadro's number) The number of moles of tritium atoms can be calculated as follows: moles_t ≈ 1.616 x 10^12 atoms / 6.022 x 10^23 atoms/mol ≈ 2.68 x 10^-12 moles
03

Calculate the total moles of hydrogen atoms

To calculate the total moles of hydrogen atoms, we can use the mass of the water sample and water's molecular formula (H₂O): - Moles_h = (mass of water sample) / (molar mass of water) * 2 since water contains two hydrogen atoms in its molecular formula H₂O. Given that the mass of the water sample is 100 g, and the molar mass of water is 18.015 g/mol, we can calculate the moles of hydrogen atoms: Moles_h ≈ (100 g) / (18.015 g/mol) * 2 ≈ 11.10 moles
04

Calculate the percentage of hydrogen atoms that are tritium

To calculate the percentage of hydrogen atoms that are tritium atoms, we can use the relation: - Percentage = (moles of tritium atoms / total moles of hydrogen atoms) * 100 Percentage ≈ (2.68 x 10^-12 moles / 11.10 moles) * 100 ≈ 2.41 x 10^-11 % Thus, the percentage of hydrogen atoms in the water sample that are tritium is approximately 2.41 x 10^-11%.

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