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?

Name: Chemistry: The Central Science
Author: Brown
ISBN: 9780137542970

Verified step by step guidance

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: \( \text{moles of water} = \frac{26.00 \text{ g}}{18.02 \text{ g/mol}} \).

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.

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.

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: \( N = \frac{R}{\lambda} \).

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

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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The nuclear masses of 7Be, 9Be, and 10Be are 7.0147, 9.0100, and 10.0113 amu, respectively. Which of these nuclei has the largest binding energy per nucleon?

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Textbook Question

Each of the following transmutations produces a radionuclide used in positron emission tomography (PET).

(a) In equations (i) and (ii), identify the species signified as 'X.'

(i) ¹⁴N(p,α)X

(ii) ¹⁸O(p,X)¹⁸F

(iii) ¹⁴N(d,n)¹⁵O

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Textbook Question

In 2010, a team of scientists from Russia and the United States reported creation of the first atom of element 117, which is named tennessine, and whose symbol is Ts. The synthesis involved the collision of a target of ²⁴⁹₉₇Bk with accelerated ions of an isotope which we will denote Q. The product atom, which we will call Z, immediately releases neutrons and forms ²⁹⁴₁₁₇Ts: ²⁴⁹₉₇Bk + Q → Z → ²⁹⁴₁₁₇Ts + 3 ¹₀n (a) What are the identities of isotopes Q and Z? (c) Collision of ions of isotope Q with a target was also used to produce the first atoms of livermorium, Lv. The initial product of this collision was ²⁹⁶₁₁₆Lv. What was the target isotope with which Q collided in this experiment?