A sample of 37Ar undergoes 8540 disintegrations/min initially but undergoes 6990 disintegrations/min after 10.0 days. What is the half-life of 37Ar in days?

Verified step by step guidance

1. Identify the initial and final disintegration rates: \( R_0 = 8540 \) disintegrations/min and \( R = 6990 \) disintegrations/min after 10 days.
2. Use the radioactive decay formula: \( R = R_0 \times e^{-\lambda t} \), where \( \lambda \) is the decay constant and \( t \) is the time in days.
3. Rearrange the formula to solve for \( \lambda \): \( \lambda = \frac{1}{t} \ln\left(\frac{R_0}{R}\right) \).
4. Substitute the known values into the equation to calculate \( \lambda \): \( \lambda = \frac{1}{10} \ln\left(\frac{8540}{6990}\right) \).
5. Use the relationship between the decay constant and half-life: \( t_{1/2} = \frac{\ln(2)}{\lambda} \) to find the half-life in days.

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate, characterized by the half-life, which is the time required for half of the radioactive nuclei in a sample to decay. Understanding this concept is crucial for analyzing the changes in disintegration rates over time.

Half-Life

Half-life is a specific measure of the time it takes for half of a given quantity of a radioactive substance to decay. It is a constant for each radioactive isotope and is essential for calculating the remaining quantity of the substance after a certain period. In this question, determining the half-life of 37Ar involves using the initial and final disintegration rates over the specified time.

Exponential Decay Formula

The exponential decay formula describes how the quantity of a radioactive substance decreases over time. It is expressed as N(t) = N0 * (1/2)^(t/T), where N(t) is the quantity at time t, N0 is the initial quantity, and T is the half-life. This formula is fundamental for solving problems related to radioactive decay, as it allows for the calculation of remaining quantities based on time elapsed.