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Chapter 6: Problem 15

The half-life of radium- 226 is 1590 years. Suppose we start with a sample of radium- 226 whose mass is \(50 \mathrm{mg}\) (a) Find a formula for the mass \(m(t)\) remaining after years and use this formula to predict the mass remaining after 1000 years. (b) When will only \(10 \mathrm{mg}\) remain?

Short Answer

Expert verified
(a) After 1000 years, about 34.9 mg remains. (b) It will take approximately 3712 years for the mass to reduce to 10 mg.

Step by step solution

01

Understand the half-life concept

The half-life of a substance is the time required for it to reduce to half its initial amount. For radium-226, this time is 1590 years. We will use this information to model the exponential decay of the substance.
02

Set up the decay formula

The general formula for exponential decay is \( m(t) = m_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \), where \( m_0 \) is the initial mass, \( t \) is time, and \( T_{1/2} \) is the half-life. Here, \( m_0 = 50 \) mg and \( T_{1/2} = 1590 \) years.
03

Write the specific formula for radium-226

Plug the known values into the decay formula to get the specific equation for radium-226: \( m(t) = 50 \times \left(\frac{1}{2}\right)^{\frac{t}{1590}} \).
04

Calculate mass after 1000 years

Use the formula \( m(t) = 50 \times \left(\frac{1}{2}\right)^{\frac{1000}{1590}} \) to find the mass remaining after 1000 years. This simplifies to approximately \( m(1000) = 50 \times 0.698 \approx 34.9 \) mg.
05

Solve for time when 10 mg remains

Set \( m(t) = 10 \) and solve for \( t \) using the equation: \( 10 = 50 \times \left(\frac{1}{2}\right)^{\frac{t}{1590}} \). Divide both sides by 50 to get \( 0.2 = \left(\frac{1}{2}\right)^{\frac{t}{1590}} \). Take the natural logarithm of both sides to solve for \( t \): \( \ln(0.2) = \frac{t}{1590} \times \ln(0.5) \). Finally, solve for \( t \): \( t = 1590 \times \frac{\ln(0.2)}{\ln(0.5)} \approx 3712 \) years.

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Key Concepts

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

Half-life
Half-life is an important concept in understanding the process of decay in substances. It's the time required for a quantity to reduce to half its initial value. For radium-226, the half-life is 1590 years.
This means that after every 1590 years, the mass of radium-226 will be half of what it initially was. The idea of half-life is crucial, especially in fields like archaeology and geology, as it helps in dating fossils and rocks.
  • In practical terms, half-life helps us predict how quickly a substance will lose its potency over time.
  • The concept is fundamental in a variety of scientific applications including medicine, where half-life tells us how long a drug stays active in the body.
Radium-226
Radium-226 is a radioactive isotope of radium and has a significant role in the history of radiation studies. It has a very long half-life of 1590 years, which means it decays slowly, making it useful for certain long-term applications.
Historically, radium was used in luminescent paint for clocks and watch dials due to its glowing properties under certain conditions.
  • Due to its radioactivity, radium-226 has to be handled carefully to avoid the harmful effects of radiation exposure.
  • Understanding radium-226's half-life helps scientists know how long it remains hazardous.
Despite its dangerous nature, radium-226 has provided vital insights into the nature of radioactive decay and has been instrumental in developing safety standards for handling radioactive materials.
Exponential Functions
Exponential functions are mathematical expressions that describe processes that change rapidly over time, like the decay of radioactive substances such as radium-226.
In these functions, the rate of change is proportional to the current amount, making them ideal for modeling scenarios of growth and decay.
The general formula for exponential decay is: \[ m(t) = m_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]
  • Where \( m(t) \) is the mass after time \( t \), \( m_0 \) is the initial mass, and \( T_{1/2} \) is the half-life.
  • This equation naturally incorporates the concept of half-life, showing how a substance decreases exponentially through successive half-lives.
Exponential functions are widely used not just in radioactive decay but also in finance for compound interest, population growth models, and many other areas where change accelerates over time.

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Most popular questions from this chapter

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The set of all linear transformations from a vector space \(V\) to a vector space \(W\) is denoted by \(\mathscr{L}(V, W)\). If \(S\) and \(T\) are in \(\mathscr{L}(V, W),\) we can define the sum \(S+T\) of \(\operatorname{Sand} T b y\) \\[(S+T)(\mathbf{v})=S(\mathbf{v})+T(\mathbf{v})\\] for all v in \(V\). If \(c\) is a scalar, we define the scalar multiple \(c T\) of T by c to be \\[(c T)(\mathbf{v})=c T(\mathbf{v})\\] for all \(\mathbf{v}\) in \(V\). Then \(S+T\) and \(c\) T are both transformations from \(V\) to \(W\). Let \(R, S\), and \(T\) be linear transformations such that the following operations make sense. Prove that: (a) \(R \circ(S+T)=R \circ S+R \circ T\) (b) \(c(R \circ S)=(c R) \circ S=R \circ(c S)\) for any scalar \(c\)

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