Problem 58 - Chemistry Matter and Change

Question

An unknown radioisotope exhibits 8540 decays per second. After 350.0 min, the number of decays has decreased to 1250 per second. What is the half-life?

Step-by-Step Solution

Answer

The half-life of the radioisotope is found using the decay formula and natural logarithm.

Step 1: Understand the Problem

We are given a radioisotope with an initial decay rate of 8540 decays per second and a decay rate that decreases to 1250 decays per second after 350.0 minutes. We need to find the half-life of the radioisotope.

Step 2: Convert Time to Seconds

The time given is 350.0 minutes. Convert this time into seconds because decay rates are given per second: \(350.0 \text{ minutes} \times 60 \text{ seconds/minute} = 21000 \text{ seconds}\).

Step 3: Apply the Decay Formula

The decay of radioisotopes can be described by the formula \(N(t) = N_0 \times e^{-\lambda t}\), where \(N(t)\) is the remaining activity at time \(t\), \(N_0\) is the initial activity, and \(\lambda\) is the decay constant. Given \(N_0 = 8540\) and \(N(t) = 1250\), substitute into the formula: \[1250 = 8540 \times e^{-\lambda \times 21000}\]

Step 4: Solve for λ (Decay Constant)

Rearrange the equation to find \(\lambda\):\[e^{-\lambda \times 21000} = \frac{1250}{8540}\]Take the natural logarithm of both sides:\[-\lambda \times 21000 = \,\ln \left(\frac{1250}{8540}\right)\]Solving for \(\lambda\):\[\lambda = -\frac{\ln \left(\frac{1250}{8540}\right)}{21000}\]

Step 5: Convert Decay Constant to Half-Life

The half-life \(T_{1/2}\) is related to the decay constant by \[T_{1/2} = \frac{\ln(2)}{\lambda}\].Calculate \(T_{1/2}\) using the \(\lambda\) obtained in the previous step.

Key Concepts

Half-life Calculation

The concept of half-life is central to understanding radioactive decay. Half-life represents the time needed for half of the radioactive nuclei in a sample to decay. It is a fixed property of each radioactive isotope and does not change over time. To determine the half-life, you first need the decay constant, \(\lambda\). Once \(\lambda\) is calculated, apply the formula:

• \(T_{1/2} = \frac{\ln(2)}{\lambda}\)

In this equation, \(\ln(2)\) is the natural logarithm of 2, a constant approximately equal to 0.693. This formula allows us to compute the half-life once the decay constant is known, giving insight into the stability of the isotope and the rate at which it transforms.

Decay Constant

The decay constant \(\lambda\) provides an understanding of how quickly a radioactive isotope decays. It is a crucial parameter in the exponential decay model, signifying the probability per unit time that a given nucleus will decay.To find \(\lambda\), we manipulate the exponential decay equation:

• \(N(t) = N_0 \times e^{-\lambda t}\)

In solving real-world problems, like in our exercise, substitute known values \(N_0\) (initial decay rate) and \(N(t)\) (decay rate after a time \(t\)) into this formula. Then rearrange and solve for \(\lambda\) using logarithmic operations:

• \(\lambda = -\frac{\ln\left(\frac{N(t)}{N_0}\right)}{t}\)

This expression uses the natural logarithm because radioactive decay processes are inherently exponential, with the rate of decay proportional to the current amount of substance.

Exponential Decay Formula

The exponential decay formula describes how the quantity of radioactive material decreases over time. In its general form:

• \(N(t) = N_0 \times e^{-\lambda t}\)

Here, \(N(t)\) is the remaining quantity at time \(t\), \(N_0\) is the initial quantity, \(\lambda\) is the decay constant, and \(t\) is the elapsed time. The formula reveals that the decay process is characterized by a constant relative rate of change per unit time, resulting in an exponential decrease.The exponential nature means that the process speeds up or slows down consistently, relative to the amount present. This formula is pivotal for calculating how much of a radioactive isotope remains after a certain duration, helping us predict future decay rates and understand the material's longevity.

Conversion of Time Units

Converting time units is essential in solving problems involving radioactive decay because decay rates are commonly measured per second. In our given problem, time was initially in minutes, so conversion to seconds was necessary:

• 1 minute = 60 seconds
• 350 minutes = 350 \(\times\) 60 = 21000 seconds

Ensuring consistency in your units when applying the decay formula is crucial. When all the time measurements are in seconds, the decay rates, typically per second, align with the temporal unit in the equations. This alignment ensures that calculations of quantities like the decay constant and half-life are accurate and reliable.