Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the table for the radioactive isotope. $$\begin{array}{lccc} & \text {Half-Life} & \text {Initial} & \text {Amount After} \\ \text {Isotope} & \text {(years)} & \text {Quantity} & \text {1000 Years} \\ ^{226} \mathrm{Ra} & 1600 & 10 \mathrm{g} & \end{array}$$

Step-by-Step Solution

Answer

The amount of ^{226} Ra remaining after 1000 years is approximately 7.98 g.

Step 1: Identify Variables and Plug into Formula

We know the initial amount \(A_{0}\) of the isotope, which is 10 g, the half-life \(T\) of the Isotope, which is 1600 years, and the time elapsed \(t\), which is 1000 years. Now, we substitute these values into the half-life decay formula \(A=A_{0} * (0.5)^{\frac{t}{T}}\).

Step 2: Calculating

Now, we calculate the final amount of isotopes: \(A = 10 * (0.5)^\frac{1000}{1600}\).

Step 3: Evaluate

After calculating, we find \(A \approx 7.98 g\). So after 1000 years, there will be approximately 7.98 grams of the isotope left.

Key Concepts

Understanding the Half-Life of Isotopes

The concept of half-life is critical in the realm of nuclear physics and chemistry, particularly when dealing with isotopes, which are forms of an element with different numbers of neutrons. The half-life of an isotope is the time it takes for half of the atoms in a sample to decay. This period does not vary; it remains the same for a given isotope regardless of the initial amount and is a characteristic property of the radioactive element.

For example, consider a hypothetical isotope X with a half-life of 4 years. If we start with 10 grams of isotope X, after 4 years, only 5 grams would remain. After another 4 years (a total of 8 years), the mass would again be reduced by half, leaving us with 2.5 grams, and so on. This process of radioactive decay is a probabilistic event and is unaffected by external factors such as temperature, pressure, or chemical forms.

To master half-life calculations, one must become comfortable with the concept that the decay reduces the amount of the substance exponentially, rather than linearly, highlighting the importance of understanding the exponential decay formula which is discussed next.

Grasping the Exponential Decay Formula

Radioactive decay follows an exponential decay pattern, which is expressed mathematically by the formula:
\( A = A_0 * (0.5)^{\frac{t}{T}} \)
where \( A \) is the amount of the isotope remaining after time \( t \), \( A_0 \) is the initial quantity of the isotope, and \( T \) is the half-life of the isotope.

The key to understanding this formula is recognizing that the amount of isotope decreases by half every time a period equivalent to the half-life passes. This 'halving' effect is due to the nature of exponential decay, where the rate of decay is proportional to the amount of substance present. In simple terms, the more of the isotope you have, the more of it decays in each half-life. As time goes on, even though the same fraction (half) decays, the actual amount getting decayed becomes smaller because the quantity of the isotope itself is decreasing.

Exponential decay can be tricky because our everyday experiences usually involve linear changes. However, by consistently applying this formula and practicing with various values, the concept of exponential decay becomes clearer and calculations become second nature.

Radioactive Decay Calculations

Solving radioactive decay problems involves a few steps that are straightforward once you comprehend the exponential decay formula. Here, we'll walk through an example to solidify our understanding.

Let's return to the original exercise with isotope Ra-226. To find how much of this isotope remains after 1000 years, we first identify the known variables as done in Step 1 of the solution: the initial amount \( A_0 = 10 \) grams, the half-life \( T = 1600 \) years, and the time passed \( t = 1000 \) years. This leads us to the formula in Step 2, \( A = 10 * (0.5)^{\frac{1000}{1600}} \), where we calculate the amount remaining, found in Step 3 to be approximately 7.98 grams.

When dealing with these calculations, always ensure your units match, particularly time units, to avoid errors. Also, using a calculator with an exponent function greatly simplifies the process, making it accessible to perform even complex decay calculations reliably. Practice by changing the values of \( t \) and \( T \) to see how they impact the amount remaining, and soon you'll be able to tackle any radioactive decay problem with confidence.