Question

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

Short answer

Decay constant $k\approx 0.252$, half-life $\approx 2.75$ days.

Step-by-step solution

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 $k$.

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: $389/500=e^{-k}$.

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, $k=-\ln (0.778)\approx 0.252$.

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.

Key concepts

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.