Problem 38 - Chemistry and Chemical Reactivity

Question

Scandium occurs in nature as a single isotope, scandium-45. Neutron irradiation produces scandium-46, a \(\beta\) emitter with a half-life of 83.8 days. If the initial activity is \(7.0 \times 10^{4} \mathrm{dpm}\), draw a graph showing disintegrations per minute as a function of time during a period of 1 year.

Step-by-Step Solution

Answer

Plot activity decline using the radioactive decay formula for each month over a year.

Step 1: Understand the Problem

We need to find how the disintegrations per minute (dpm) of scandium-46 decay over time, specifically over one year. Given the half-life and initial activity, we can calculate the activity at different times using the radioactive decay law.

Step 2: Identify the Decay Formula

The radioactive decay of a substance can be described by the formula: \[ A(t) = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \] where \(A(t)\) is the activity at time \(t\), \(A_0\) is the initial activity, \(t_{1/2}\) is the half-life, and \(t\) is the time elapsed.

Step 3: Calculate Activities at Different Time Points

Using the decay formula, calculate the activity at several points in time (e.g., every month) for a year. For each month \(t = 30, 60, 90, \ldots, 360\) days, compute \(A(t)\). Example: - At \(t = 0\): \(A(0) = 7.0 \times 10^4\) dpm.- At \(t = 30\) days: \(A(30) = 7.0 \times 10^4 \times \left( \frac{1}{2} \right)^{\frac{30}{83.8}}\).- Repeat this calculation for all desired time points.

Step 4: Plot the Graph

Plot the calculated disintegration rates on a graph with time (in days) as the x-axis and activity (in dpm) as the y-axis. Make sure to label your axes and plot points at regular intervals to see the exponential decay over time.

Step 5: Interpret the Graph

The graph should show an exponential decay curve, starting from the initial activity of \(7.0 \times 10^4\) dpm, and decreasing over time as scandium-46 decays into another element. The curve will get closer to zero but will not actually reach zero within a year due to the nature of exponential decay.

Key Concepts

Half-life

Half-life is a crucial concept in the study of radioactive decay, useful in understanding how long it takes for a substance to reduce to half its initial quantity. Specifically, the half-life represents the time required for half of the radioactive atoms in a sample to disintegrate. This means if you start with a certain amount of a radioactive substance, after one half-life period, only half of it remains active while the other half has decayed into different elements or isotopes.
For Scandium-46, the half-life is noted as 83.8 days. This time frame is critical when calculating how the activity of a sample changes over months, directly impacting how calculations are performed to estimate activity levels at designated time points. This finite time frame reflects the constant potential for change in a radioactive substance's activity, which decreases predictably and measurably as time progresses.

Scandium-46

Scandium-46 is an artificially produced isotope created from its natural counterpart, scandium-45, through neutron bombardment. This isotope is of significant interest due to its radioactive properties and applications. As a beta emitter, Scandium-46 helps researchers and technologists use its decay in various applications, such as tracing or studying processes in metal and building material analysis.
Its decay emits beta particles, which are basically high-energy electrons. This emission type is useful in numerous industrial and scientific fields. The known half-life of Scandium-46 facilitates its use in controlled experimental settings where researchers need precise decay measurements over specific periods.

Exponential Decay

Exponential decay characterizes the pattern observed in the reduction of a radioactive substance's activity over time. Instead of proceeding linearly, radioactive decay accelerates such that substances lose a fixed percentage of their remaining radioactivity over each time interval equal to their half-life. This decreases activity in a curve-like fashion, sharply dropping initially and then flattening out over time.
The decay formula used to describe this process is: \[ A(t) = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]where

• \( A(t) \) is the activity at time \( t \)
• \( A_0 \) is the initial activity
• \( t_{1/2} \) is the half-life
• \( t \) is the time elapsed

This mathematical representation helps accurately predict activity levels at any given time interval and justifies why the graph of activity versus time is never truly linear.

Activity Calculation

Calculating the activity of a radioactive isotope involves understanding and using the decay formula effectively. This calculation predicts how a substance like Scandium-46 will behave over time, in terms of its ability to emit radiation. Starting with a known initial activity, such as \(7.0 \times 10^4\) disintegrations per minute (dpm), the activity at any future time can be precisely estimated to reflect how much of the isotope remains.
By substituting different values of \( t \) into the formula, it’s possible to calculate the remaining activity at monthly intervals or any desired points throughout the year. This method yields a clear, predictable decline in activity, illustrating the exponential decay nature of radioactive substances. Through these calculations, it's evident how the remaining radioactive material diminishes steadily, providing practical insights for both scientific and practical applications where precise timing and dosing matter.