A sample of 201Tl, a radioisotope used to determine the function of the heart, decays initially at a rate of 25,700 disintegrations/min, but the decay rate falls to 15,990 disintegrations/min after 50.0 hours. What is the half-life of 201Tl, in hours? (a) 73.0 hours (b) 105 hours (c) 1.56 x 10^-2 hours (d) 3.84 x 10^2 hours

Verified step by step guidance

Step 1: Understand that the decay of a radioactive isotope follows first-order kinetics, which can be described by the equation: \( N_t = N_0 e^{-kt} \), where \( N_t \) is the remaining quantity at time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( t \) is the time elapsed.

Step 2: Use the given decay rates to set up the equation: \( 15,990 = 25,700 e^{-k \times 50} \). This equation will allow you to solve for the decay constant \( k \).

Step 3: Rearrange the equation to solve for \( k \): \( \frac{15,990}{25,700} = e^{-k \times 50} \). Take the natural logarithm of both sides to isolate \( k \): \( \ln\left(\frac{15,990}{25,700}\right) = -k \times 50 \).

Step 4: Solve for \( k \) by dividing both sides by -50: \( k = -\frac{1}{50} \ln\left(\frac{15,990}{25,700}\right) \).

Step 5: Use the relationship between the decay constant \( k \) and the half-life \( t_{1/2} \), which is \( t_{1/2} = \frac{\ln(2)}{k} \), to calculate the half-life of \( ^{201}\text{Tl} \).

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 an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a characteristic rate for each isotope, often described by its decay constant. The rate of decay can be quantified in terms of disintegrations per minute, which indicates how many nuclei decay in a given time period.

Half-Life

The half-life of a radioactive isotope is the time required for half of the radioactive nuclei in a sample to decay. It is a constant value unique to each isotope and is crucial for understanding the stability and longevity of the material. The half-life can be calculated from the decay rate and is essential for predicting how long a sample will remain radioactive.

Exponential Decay

Exponential decay describes the process where the quantity of a substance decreases at a rate proportional to its current value. In the context of radioactive decay, this means that the number of disintegrations per minute decreases exponentially over time. The relationship can be expressed mathematically, allowing for the calculation of remaining quantities after a certain period, which is fundamental in determining half-lives.

Recommended video:

Guided course