The half-life of ²³²Th is 1.4⨉10¹⁰ yr. Find the number of disintegrations per hour emitted by 1.0 mol of ²³²Th.

Verified step by step guidance

1. Step 1: Understand the concept of half-life and decay constant.
   The half-life (t_1/2) is the time required for half of the radioactive nuclei in a sample to decay. The decay constant (\( \lambda \)) is related to the half-life by the formula: \( \lambda = \frac{\ln(2)}{t_{1/2}} \).
2. Step 2: Calculate the decay constant (\( \lambda \)).
   Use the given half-life of ²³²Th, which is 1.4⨉10¹⁰ years, to find \( \lambda \) using the formula: \( \lambda = \frac{\ln(2)}{1.4 \times 10^{10} \text{ yr}} \).
3. Step 3: Convert the decay constant to per hour.
   Since the decay constant is initially in terms of years, convert it to hours by using the conversion factor: 1 year = 365.25 days ⨉ 24 hours/day.
4. Step 4: Calculate the number of disintegrations per hour.
   Use the formula for activity (A), which is the number of disintegrations per unit time: \( A = \lambda N \), where \( N \) is the number of atoms. For 1.0 mol of ²³²Th, \( N = 6.022 \times 10^{23} \) atoms (Avogadro's number).
5. Step 5: Substitute the values into the activity formula.
   Substitute the decay constant (in per hour) and the number of atoms into the formula \( A = \lambda N \) to find the number of disintegrations per hour.

Key Concepts

Half-life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a crucial concept in nuclear chemistry, as it helps determine the stability and longevity of radioactive isotopes. For example, if the half-life of an isotope is known, one can calculate how much of the substance remains after a certain period, which is essential for understanding radioactive decay processes.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This can occur in various forms, including alpha, beta, and gamma decay. The rate of decay is characterized by the half-life, and understanding this process is vital for calculating the number of disintegrations over a given time period, such as per hour.

Avogadro's Number

Avogadro's number, approximately 6.022 × 10²³, is the number of atoms, ions, or molecules in one mole of a substance. This concept is essential for converting between moles and the actual number of particles, allowing chemists to relate macroscopic quantities to atomic-scale phenomena. In the context of the question, it helps determine the total number of thorium atoms in 1.0 mol, which is necessary for calculating the total disintegrations per hour.