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Chapter 4: Problem 13

Decay of Radium Find the half-life of radium- 226 , which decays according to the function \(A(t)=A_{0} e^{-0.00043 t},\) where \(t\) is time in years.

Short Answer

Expert verified
The half-life of radium-226 is approximately 1611.63 years.

Step by step solution

01

Understand the Given Function

The decay of radium-226 is described by the function \(A(t) = A_0 e^{-0.00043 t}\), where \(A(t)\) represents the amount of radium-226 left after time \(t\) years, and \(A_0\) is the initial amount.
02

Concept of Half-Life

The half-life of a substance is the time it takes for half of the initial amount to decay. This means at half-life, \(A(t) = \frac{A_0}{2}\).
03

Set Up the Half-Life Equation

Using the definition of half-life, substitute \(A(t) = \frac{A_0}{2}\) into the decay function: \(\frac{A_0}{2} = A_0 e^{-0.00043 t_{1/2}}\). Here, \(t_{1/2}\) represents the half-life.
04

Simplify the Equation

Cancel \(A_0\) on both sides of the equation: \(\frac{1}{2} = e^{-0.00043 t_{1/2}}\).
05

Solve for \(t_{1/2}\) (Half-Life)

Take the natural logarithm (ln) on both sides to solve for \(t_{1/2}\): \[ \ln \left( \frac{1}{2} \right) = \ln \left( e^{-0.00043 t_{1/2}} \right) \] Simplifying this yields: \[ \ln \left( \frac{1}{2} \right) = -0.00043 t_{1/2} \] Next, calculate \( \ln \left( \frac{1}{2} \right)\), which is \(\ln \left( \frac{1}{2} \right) = -\ln(2)\). Use the value \( \ln(2) = 0.693\).
06

Calculate the Half-Life

Substitute the value into the equation: \[ -0.693 = -0.00043 t_{1/2} \] Solve for \(t_{1/2}\): \[ t_{1/2} = \frac{0.693}{0.00043} \approx 1611.63 \] Thus, the half-life of radium-226 is approximately 1611.63 years.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Decay
Exponential decay refers to the process by which a quantity decreases at a rate proportional to its current value. It's commonly used in contexts such as radioactive decay, population decline, and cooling processes. The general form of the exponential decay function is given by \(A(t) = A_0 e^{-kt}\), where:
  • \(A(t)\) is the amount of substance at time \(t\)
  • \(A_0\) is the initial amount of the substance
  • \(k\) is the decay constant
  • \
Natural Logarithm
The natural logarithm, denoted \( \text{ln} \), is the logarithm to the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. The natural logarithm has several important properties in mathematics, particularly in dealing with exponential functions. For example:
  • \
Radium-226 Decay
Radium-226 is a naturally occurring radioactive isotope of radium with a half-life of approximately 1600 years. It decays by alpha decay to form radon-222, which is also radioactive. The decay of radium-226 is described by the function \(A(t) = A_0 e^{-0.00043 t}\), making it an example of exponential decay. Understanding radium-226 decay is crucial in fields such as geology, archeology, and medicine.

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