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Chapter 20: Problem 26

A radioactive substance undergoes decay as follows: $$ \begin{array}{cc} \text { Time (days) } & \text { Mass (g) } \\ \hline 0 & 500 \\ 1 & 389 \\ 2 & 303 \\ 3 & 236 \\ 4 & 184 \\ 5 & 143 \\ 6 & 112 \end{array} $$ Calculate the first-order decay constant and the half-life of the reaction.

Short Answer

Expert verified
Decay constant \( k \approx 0.252 \), half-life \( \approx 2.75 \) days.

Step by step solution

01

Understand the decay model

Radioactive decay follows the first-order kinetics, described by the equation \( m(t) = m_0 e^{-kt} \), where \( m(t) \) is the mass at time \( t \), \( m_0 \) is the initial mass, and \( k \) is the decay constant. We are given the masses over time, and need to find \( k \).
02

Use the decay formula

Using the masses at each time, set up equations based on \( 389 = 500e^{-k} \), \( 303 = 500e^{-2k} \), etc. Solving these will help estimate \( k \). A common way is using the first time point to simplify: \( 389/500 = e^{-k} \).
03

Calculate decay constant \( k \)

Divide the mass at day 1 by the initial mass: \( \frac{389}{500} \approx 0.778 \). Then, solve for \( k \) using \( e^{-k} = 0.778 \). Taking the natural logarithm on both sides, \( k = -\ln(0.778) \approx 0.252 \).
04

Calculate the half-life

The half-life \( t_{1/2} \) is calculated using the relation \( t_{1/2} = \frac{0.693}{k} \). Plugging in the value of \( k \), \( t_{1/2} = \frac{0.693}{0.252} \approx 2.75 \) days.

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Key Concepts

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

First-Order Kinetics
When we talk about radioactive decay, it's essential to understand first-order kinetics. This means the rate at which the substance decays is proportional to the amount of substance present. In simpler terms, as the mass decreases over time, the speed at which it breaks down also declines. This characteristic is captured by the exponential decay equation \( m(t) = m_0 e^{-kt} \). Here, \( m(t) \) is the mass at any given time \( t \), \( m_0 \) is your starting mass, and \( k \) is known as the decay constant.
  • Proportional decay rate
  • Exponential decrease over time
  • Key contributors: initial mass and decay constant
Grasping first-order kinetics is crucial, as it simplifies predicting how long it will take for a radioactive sample to decrease to a certain mass.
Decay Constant
The decay constant \( k \) is a fundamental factor in first-order kinetics. It represents how quickly a radioactive isotope disintegrates. A larger \( k \) indicates a faster decay, whereas a smaller \( k \) suggests a slower process. You can identify \( k \) by investigating the change in mass over time. By re-arranging the exponential decay formula \( \frac{m}{m_0} = e^{-kt} \), we can take the natural logarithm of both sides to solve for \( k \). This allows you to determine the unique rate of decay for different isotopes, which is vital for applications like radiometric dating or understanding nuclear waste behavior.
  • Indicates speed of decay
  • Can be calculated using observed data points
  • Helpful in isotope identification
Half-Life
The half-life \( t_{1/2} \) of a radioactive isotope is the time required for half of the isotope to decay. It's a crucial parameter as it tells us how long a substance remains active or hazardous. The relationship between the half-life and decay constant is given by \( t_{1/2} = \frac{0.693}{k} \). Understanding this concept helps us predict how quickly a substance loses its radioactivity.
  • Time for half the sample to decay
  • Gives insight into stability of isotopes
  • Calculated directly from decay constant
This measure is essential in fields like medicine, archaeology, and geology, where isotopes must be monitored over specific timelines.
Natural Logarithm
Natural logarithms are used frequently in the context of radioactive decay to transform exponential decay equations into linear ones, making it easier to solve for unknown variables. Specifically, the natural logarithm helps in finding the decay constant \( k \) by rearranging \( m(t) = m_0 e^{-kt} \) into \( \ln \left( \frac{m}{m_0} \right) = -kt \). By plotting \( \ln(m) \) over time, radioactive decay processes can be represented as a straight line, simplifying analysis.
  • Translates exponential to linear relationships
  • Simplifies solving decay equations
  • Essential for k determination and half-life calculations
Being comfortable with natural logarithms is critical for interpreting radioactive decay data accurately.
Radioactive Isotopes
Radioactive isotopes, often called radioisotopes, are variants of chemical elements that are unstable due to an imbalance in their nucleus. This instability makes them prone to spontaneous decay, emitting radiation as they transform into more stable atoms over time. Here are key aspects:
  • Serve as tracers in the medical field
  • Used for dating ancient artifacts
  • Power sources for nuclear reactors and weapons
Understanding these isotopes and how they decay is essential in a range of scientific applications, providing tools to predict changes in their mass and energy output, and leveraging their unique properties for innovation and safety.

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Most popular questions from this chapter

Strontium-90 is one of the products of the fission of uranium-235. This strontium isotope is radioactive, with a half-life of 28.1 years. Calculate how long (in years) it will take for \(1.00 \mathrm{~g}\) of the isotope to be reduced to \(0.200 \mathrm{~g}\) by decay.

What is the difference between an electron and a positron?

Modern designs of atomic bombs contain, in addition to uranium or plutonium, small amounts of tritium and deuterium to boost the power of explosion. What is the role of tritium and deuterium in these bombs?

A 0.0100 -g sample of a radioactive isotope with a halflife of \(1.3 \times 10^{9}\) years decays at the rate of \(2.9 \times 10^{4}\) disintegrations per minute. Calculate the molar mass of the isotope.

The quantity of a radioactive material is often measured by its activity (measured in curies or millicuries) rather than by its mass. In a brain scan procedure, a \(70-\mathrm{kg}\) patient is injected with \(20.0 \mathrm{mCi}\) of \({ }^{99 \mathrm{~m}} \mathrm{Tc},\) which decays by emitting \(\gamma\) -ray photons with a half-life of \(6.0 \mathrm{~h}\). Given that the \(\mathrm{RBE}\) of these photons is 0.98 and only two-thirds of the photons are absorbed by the body, calculate the rem dose received by the patient. Assume all the \({ }^{99 \mathrm{~m}}\) Tc nuclei decay while in the body. The energy of a \(\gamma\) -ray photon is \(2.29 \times 10^{-14} \mathrm{~J}\).
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