Question

Using only symbolic variables and constants, write an expression that defines that time necessary for \(95 \%\) of a radioactive isotope to decay. Hint: interpret this to mean that we seek an expression for t when \(\mathrm{N} / \mathrm{N}_{0}=0.05\)

Short answer

The time necessary for 95% of a radioactive isotope to decay is given by the expression \(t = \ln(0.05)/-k\)

Step-by-step solution

Write the equation for nuclear decay

Write the equation for nuclear decay, which is \(N = N_0 e^{-kt}\). This models the decay of radioactive isotopes.

Solve for decay

Plug in the given condition that \(N/N_0 = 0.05\). This gives the equation \(0.05 = e^{-kt}\).

Solve for t

To solve for \(t\), first take the natural log on both sides to remove the exponential on the right-hand side. This yields \(\ln(0.05) = -kt\). Then, solve for \(t\) by dividing both sides by \(-k\). This gives us \(t = \ln(0.05)/-k\).

Key concepts

Nuclear Decay Equation

In the study of radioactive decay, the nuclear decay equation is a crucial formula used to describe how a radioactive substance decreases over time. This equation is typically represented as \[ N = N_0 e^{-kt} \] where:

• \( N \) is the amount of the substance remaining after time \( t \).
• \( N_0 \) is the initial amount at time \( t = 0 \).
• \( k \) is the decay constant, a positive number that is unique to each substance.
• \( t \) is the time that has elapsed.

The equation is an example of an exponential decay model, which means that the rate of decrease of the substance is proportional to its current value.
The higher the value of \( k \), the faster the substance decays. Understanding this equation is foundational to studies in nuclear physics and helps us predict how long it will take for a certain fraction of the isotopes to decay.

Symbolic Variables

Symbolic variables are used in equations where specific numerical values are not initially needed. Instead, they represent general quantities and are manipulated algebraically. In the context of nuclear decay, symbolic variables are helpful to derive generalized formulas that can apply to different radioactive isotopes, regardless of their initial quantity or the exact decay constant.
For instance:

• \( N \) and \( N_0 \) are used symbolically to denote the amount of substance at any given time and its initial amount, respectively.
• \( k \) serves symbolically to represent the decay constant and varies based on the particular isotope concerned.
• \( t \) symbolically reflects the time variable, allowing for exploration into how time affects decay.

Using symbolic variables makes it easier to solve problems universally and to later substitute specific values as needed for particular cases.

Exponential Functions

Exponential functions feature prominently in modelling radioactive decay because they describe processes that change at rates proportional to their current values.
This is seen in the equation:\[ N = N_0 e^{-kt} \] Key features of exponential functions include:

• The rate of change is not constant; rather, it accelerates or decelerates depending on the current amount.
• The function involves the base \( e \), a mathematical constant approximately equal to 2.71828.
• Exponential decay implies the quantity decreases rapidly at first, then more slowly over time, never actually reaching zero.

Exponential functions come into play when simplifying decay problems, such as determining when a certain percentage of an isotope has decayed. They make using mathematical operations more straightforward, especially logarithms, when solving for unknown variables like time.

Decay Constant

The decay constant, denoted as \( k \), is integral to understanding a substance's decay rate. It is determined by the properties of the isotope itself and represents how quickly the substance undergoes decay.
The decay constant has these characteristics:

• It is a positive value that determines the speed of decay; larger values signify faster decay.
• In the nuclear decay equation, \( k \) functions as a proportionality factor.
• The units of the decay constant are typically inverse time (e.g., \( s^{-1} \), \( min^{-1} \)), which correlate with the time unit used for \( t \).

To solve decay problems, determining the decay constant is crucial because it directly influences the time it takes for a substance to decay to a specific amount. When using the nuclear decay equation and rearranging to find \( t \), a clear understanding of \( k \) is essential to accurate calculations.