Question

In a comparison of two radioisotopes, isotope \(A\) requires 18.0 hours for its decay rate to fall to \(\frac{1}{16}\) its initial value, while isotope B has a half-life that is 2.5 times that of A. How long does it take for the decay rate of isotope \(B\) to decrease to \(\frac{1}{32}\) of its initial value?

Short answer

The decay rate of isotope B decreases to 1/32 of its initial value in 56.25 hours.

Step-by-step solution

Determine the half-life of isotope A

Let's assume the half-life of isotope A as \(T_A\). As mentioned, it takes four \(T_A\) for isotope A to decay to 1/16 of its initial value. So we can set up the equation: \(4T_A = 18\), from which we get \(T_A = \frac{18}{4} = 4.5\) hours.

Determine the half-life of isotope B

It's stated that the half-life of isotope B is 2.5 times the half-life of isotope A. Hence, the half-life of isotope B, which we'll denote as \(T_B\), is \(T_B = 2.5 * T_A = 2.5 * 4.5 = 11.25\) hours.

Find the time for isotope B to decay to 1/32 of initial value

To reduce to 1/32 of its initial value, isotope B would require five half-lives. Five half-lives of isotope B would take \(Time = 5 * T_B = 5 * 11.25 = 56.25\) hours.

Key concepts

Radioisotopes Half-Life

Understanding the half-life of radioisotopes is crucial in fields ranging from archaeology to medicine. The half-life of a radioisotope is the time required for half of the atoms in a given sample to undergo radioactive decay. This property is intrinsic to the radioisotope and is unaffected by external factors such as temperature, pressure, or chemical state.

Let's consider the exercise where we identified the half-life of isotope A as 4.5 hours. The term 'half-life' is frequently used because radioactive decay is a random, yet statistically predictable process. After one half-life, the quantity of isotope A would decrease by half, after two half-lives to one quarter, and so on.

This geometric progression continues and can be described by the equation \( N = N_0(\frac{1}{2})^{\frac{t}{T}} \) where \( N \) is the remaining quantity of the isotope after time \( t \) has elapsed, \( N_0 \) is the initial quantity, and \( T \) is the half-life of the isotope. In our example, the decay to 1/16 signifies that four half-lives have passed since the initial measurement.

Exponential Decay

Exponential decay is a mathematical model used to describe the process of decreasing an amount by a consistent percentage over a period of time. It's widely applied in nuclear chemistry to express the rate at which radioisotopes decay. This model is characterized by its swift decline in the beginning, which slows down as time passes.

In the case of isotope A, the decay rate drops to 1/16 after four half-lives. This quantifiable pattern of decay is the essence of the 'exponential' characteristic, emphasizing that the quantity decreases by the same proportion in each equal time period. The formula representing exponential decay is \( N = N_0e^{-kt} \) where \( N_0 \) is the initial amount, \( k \) is the decay constant specific to the radioisotope, and \( t \) is the time elapsed. In simpler terms for our specific exercise scenario, we observed that the radioisotope decay progresses through powers of 1/2 for each half-life elapsed. This pattern shows the clear exponential nature of the decay.

Nuclear Chemistry

Nuclear chemistry is a fascinating and complex field that deals with the reactions and properties of atomic nuclei. Within this field, concepts like radioisotope decay rates and half-lives are paramount. These concepts help in understanding the behavior of materials at an atomic level, which has implications across science and engineering disciplines.

For isotope B in our exercise, we applied the understanding of half-lives to calculate its decay rate. Nuclear chemistry encompasses more than just the decay of radioisotopes; it also includes topics such as nuclear synthesis, fission, and fusion processes. The principles and calculations are not just academic exercises but have real-world applications, such as radioactive dating, medical diagnostics and treatment, nuclear power generation, and the safe handling and disposal of nuclear materials.