Problem 40 - Calculus

Question

(a) The half-life of radium-226 is 1620 years. If the initial quantity of radium is \(Q_{0},\) explain why the quantity, \(Q,\) of radium left after \(t\) years, is given by $$ Q=Q_{0}\left(\frac{1}{2}\right)^{t / 1620} $$ (b) What percentage of the original amount of radium is left after 500 years?

Step-by-Step Solution

Answer

After 500 years, approximately 80.4% of the original radium remains.

Step 1: Understanding Half-life Concept

The half-life of a substance is the time it takes for half of the amount to decay. For radium-226, the half-life is 1620 years. This means that after 1620 years, only half of the original quantity of radium will remain.

Step 2: Deriving the Exponential Decay Formula

The given formula for exponential decay of radium is derived from the concept of half-life. Since each half-life results in the substance being reduced by half, the decay formula is given by:\[ Q = Q_0 \left(\frac{1}{2}\right)^{t / 1620} \]This formula calculates the remaining quantity \(Q\) by considering \(t\) in terms of how many half-lives (1620 years each) have passed.

Step 3: Calculating the Amount Left After 500 Years

To find what percentage of the original radium remains after 500 years, substitute \(t = 500\) into the decay formula:\[ Q = Q_0 \left(\frac{1}{2}\right)^{500 / 1620} \]

Step 4: Evaluating the Exponential Expression

Calculate the exponent:\[ \frac{500}{1620} \approx 0.3086 \]Now compute \(\left(\frac{1}{2}\right)^{0.3086}\), which approximately equals 0.804. Thus:\[ Q \approx Q_0 \times 0.804 \]

Step 5: Determining the Percentage Remaining

The result 0.804 indicates that approximately 80.4% of the original radium amount remains after 500 years, since 0.804 as a percentage is 80.4%.

Key Concepts

Exponential Decay

Exponential decay is a process that describes how the amount of a substance decreases over time. Unlike linear decay, where the amount decreases by a set amount each time, exponential decay reduces the quantity by a consistent percentage or fraction. In other words, the change in amount is proportional to the current amount. This is why it is called "exponential"; the decay process is rapid initially and gradually becomes slower.

In mathematical terms, if you start with an initial quantity called \( Q_0 \), after a specific time \( t \), the remaining quantity \( Q \) can be expressed with the formula:

• \( Q = Q_0 \, e^{-kt} \)

where \( e \) is the base of the natural logarithms, and \( k \) is a constant that depends on the material's decay rate. However, in the case of substances like radium-226, the process is often modeled using half-life because it simplifies the problem by using the time it takes for half the material to decay instead of a continuous rate.

Radium-226

Radium-226 is a naturally occurring radioactive element that was famously discovered by Marie and Pierre Curie. It has a long half-life of 1620 years, making it particularly interesting for studying long-term radioactive decay processes. Radium-226 gradually transforms into radon-222, emitting radiation in the process, which is a characteristic of radioactive elements.

This radium isotope is used in various applications including medical treatment and radiography, albeit less so now due to better alternatives. Its long half-life makes it essential to understand how it decays, particularly in environmental studies where it might be present.

The lengthy half-life means that radium-226 retains a significant amount of radioactivity over centuries. This extended decay time is why its decay is described using exponential decay formulas to accurately predict its remaining quantity over time.

Decay Formula

The decay formula for substances with a known half-life helps to calculate how much of the substance remains after a certain period. Specifically, for radium-226, if you know the initial quantity \( Q_0 \), the formula to find the quantity \( Q \) remaining after \( t \) years is:

• \( Q = Q_0 \left(\frac{1}{2}\right)^{t/1620} \)

This formula is derived from the concept of a half-life, where each cycle of 1620 years reduces the amount of radium to half its prior quantity.

Put easily, for every 1620 years that pass, multiply the remaining radium amount by 0.5. After any period \( t \), you can calculate how many half-lives have transpired with \( \frac{t}{1620} \). This exponent tells you how much the initial quantity has been halved. For example, after 500 years, the fraction of the radium remaining can be calculated to give approximately 80.4% of the initial amount, using this decay formula. This is crucial in fields like archaeology or environmental science to predict the presence of radioactive materials over time.