Physics 9630: Comprehensive Study Notes (A2 Level)
3.9.2 Exponential Changes in Radioactivity
Welcome to the fascinating world of nuclear decay! This chapter introduces the mathematical framework needed to describe how radioactive substances decrease over time. Don't worry if the exponential mathematics look intimidating at first—they are just tools to model a natural, random process.
The key idea here is that radioactive decay isn't a linear process (it doesn't decrease at a constant rate). Instead, it follows an exponential decay pattern, meaning the rate of decay is proportional to the amount of substance currently present. Fast decay when there's lots of material, slow decay when there's little left.
Section 1: The Random Nature of Decay
The Fundamentals of Radioactive Decay
All radioactive processes are inherently random and spontaneous.
- Random: We cannot predict when a specific nucleus will decay. It's totally unpredictable on an individual level.
- Spontaneous: The decay is not affected by external factors like temperature, pressure, or chemical state.
Analogy: Imagine a huge bowl of popcorn kernels. We can't predict exactly which kernel will pop next, but we can predict that half of them will have popped after a certain time (the half-life).
The Decay Constant (\(\lambda\))
While individual decays are random, the overall probability for a large number of identical nuclei is constant. This is described by the decay constant, \(\lambda\).
- Definition: The decay constant \(\lambda\) is the probability that an individual nucleus will decay per unit time.
- Units: \(\lambda\) is measured in units of inverse time, typically \(\text{s}^{-1}\) or \(\text{year}^{-1}\).
- Meaning: A large \(\lambda\) means the substance decays quickly (high probability of decay). A small \(\lambda\) means it decays slowly.
Quick Review: The constant \(\lambda\) links the probability of decay to the overall rate of decay.
Section 2: Activity and the Decay Law
Activity (\(A\))
The rate at which nuclei decay is called the Activity, \(A\).
- Definition: Activity is the number of decays per unit time.
- SI Unit: The Becquerel (\(\text{Bq}\)), where \(1 \text{ Bq} = 1\) decay per second.
The Fundamental Decay Equation
The rate of change of the number of nuclei (\(\frac{\Delta N}{\Delta t}\)) is directly proportional to the number of undecayed nuclei remaining (\(N\)).
$$\frac{\Delta N}{\Delta t} = -\lambda N$$
- \(\frac{\Delta N}{\Delta t}\): This is the rate of change of the number of nuclei \(N\) with respect to time \(t\).
- \(N\): The number of undecayed nuclei present at time \(t\).
- \(\lambda\): The decay constant.
- Negative Sign: This is crucial! It shows that the number of nuclei \(N\) is decreasing over time.
Linking Activity and Nuclei (\(A = \lambda N\))
Since activity \(A\) is the magnitude of the rate of decay (\(A = |\frac{\Delta N}{\Delta t}|\)), we can write a simple relationship:
$$A = \lambda N$$
Key Takeaway: The activity is directly proportional to the number of nuclei present. If you double the nuclei, you double the activity.
The Exponential Decay Formulas (A2 Requirement)
When the fundamental decay equation is solved using calculus, we get the two essential exponential formulas that describe decay over time:
1. Nuclei Remaining (\(N\))
$$N = N_0 e^{-\lambda t}$$
2. Activity Remaining (\(A\))
$$A = A_0 e^{-\lambda t}$$
- \(N\): Number of undecayed nuclei remaining at time \(t\).
- \(N_0\): Initial number of undecayed nuclei (at \(t=0\)).
- \(A\): Activity at time \(t\).
- \(A_0\): Initial activity (at \(t=0\)).
- \(e\): Euler's number (approximately 2.718, a mathematical constant).
Did You Know? Because \(A = \lambda N\), and \(\lambda\) is constant, the activity and the number of nuclei follow exactly the same exponential decay curve! If the number of nuclei halves, the activity halves.
Connecting to Mass and Moles
Sometimes, questions provide the mass of a radioactive sample instead of the number of nuclei \(N\).
- Since the mass of the radioactive substance is directly proportional to the number of radioactive nuclei \(N\), the mass (\(M\)) also decays exponentially: \(M = M_0 e^{-\lambda t}\).
- To convert mass (or moles) into \(N\), you must use the Avogadro constant (\(N_A\)) and the molar mass (or atomic mass unit, \(\text{u}\)). You should be ready to apply these constants if required by a question.
Section 3: Half-Life (\(T_{1/2}\))
Definition and Importance
The half-life (\(T_{1/2}\)) is the time taken for half of the number of radioactive nuclei to decay, or equivalently, the time taken for the activity to halve.
This is a characteristic property for a specific isotope. Carbon-14 has a half-life of 5,730 years, while Technetium-99m (used in medicine) has a half-life of 6 hours.
The Half-Life Equation
The half-life \(T_{1/2}\) is inversely related to the decay constant \(\lambda\). A short half-life means a high \(\lambda\).
$$T_{1/2} = \frac{\ln 2}{\lambda}$$
- \(T_{1/2}\): Half-life (measured in seconds, minutes, years, etc.).
- \(\ln 2\): The natural logarithm of 2 (approximately 0.693).
Step-by-Step Derivation Tip:
- Start with the decay formula: \(N = N_0 e^{-\lambda t}\)
- At \(t = T_{1/2}\), we know \(N = N_0 / 2\).
- Substitute: \(N_0 / 2 = N_0 e^{-\lambda T_{1/2}}\)
- Simplify: \(1 / 2 = e^{-\lambda T_{1/2}}\)
- Take the natural logarithm of both sides: \(\ln(1/2) = -\lambda T_{1/2}\)
- Remember \(\ln(1/2) = -\ln 2\).
- Result: \(-\ln 2 = -\lambda T_{1/2} \implies T_{1/2} = \frac{\ln 2}{\lambda}\).
Simple Half-Life Calculations (Whole Numbers)
For times that are a whole number multiple of the half-life, you can use simple fractions:
After 1 half-life: $N = N_0 \times \frac{1}{2}$
After 2 half-lives: $N = N_0 \times \frac{1}{2} \times \frac{1}{2} = N_0 \times \frac{1}{4}$
After \(n\) half-lives: $N = N_0 \times (\frac{1}{2})^n$
Example: If a sample has an initial activity of 800 Bq and a half-life of 5 hours, after 10 hours (2 half-lives), the activity will be \(800 \times \frac{1}{4} = 200 \text{ Bq}\).
Section 4: Graphical Determination of Half-Life
In the lab, we often measure the activity \(A\) over time \(t\). Since the decay is exponential, plotting \(A\) vs \(t\) gives a curve, which is hard to analyze accurately. To find \(\lambda\) or \(T_{1/2}\) accurately, we must linearize the data.
1. The Exponential Decay Curve
If you plot Activity (\(A\)) against Time (\(t\)), you get a curve that starts high and flattens out. We can find \(T_{1/2}\) by picking any activity value, halving it, and finding the time interval between them. However, this method is prone to measurement error.
2. Linearizing the Data using Logarithms
We start with our activity equation:
$$A = A_0 e^{-\lambda t}$$
To turn this into the linear equation \(Y = mX + C\), we take the natural logarithm (\(\ln\)) of both sides:
$$\ln(A) = \ln(A_0) + \ln(e^{-\lambda t})$$
Since \(\ln(e^x) = x\), this simplifies to:
$$\ln(A) = -\lambda t + \ln(A_0)$$
This is a linear equation:
- Y-axis: \(\ln(A)\) (The dependent variable)
- X-axis: \(t\) (The independent variable)
- Gradient (\(m\)): \(-\lambda\) (The negative decay constant)
- Y-intercept (\(C\)): \(\ln(A_0)\) (The natural log of the initial activity)
Quick Tip for Struggling Students: To find \(\lambda\), calculate the gradient of the straight line you plot (\(\text{Gradient} = \frac{\Delta Y}{\Delta X}\)). The value of the decay constant \(\lambda\) is simply the magnitude of this gradient.
$$ \lambda = -(\text{Gradient}) $$
Once you have \(\lambda\), you can easily calculate the half-life:
$$T_{1/2} = \frac{\ln 2}{\lambda}$$
Section 5: Common Mistakes and Applications
Mistakes to Avoid
- Units of \(\lambda\): Ensure the units of \(\lambda\) match the units of time \(t\). If \(\lambda\) is in \(\text{s}^{-1}\), \(t\) must be in seconds.
- Background Radiation: Always subtract the background count rate from your measured count rate before calculating Activity or performing graphical analysis. Radioactivity measured in a lab always includes counts from the environment.
- Negative Sign: Remember the gradient of the \(\ln(A)\) vs \(t\) graph is negative (\(-\lambda\)). The decay constant \(\lambda\) itself is always a positive value.
Real-World Application: Medical Diagnosis
Radioactive isotopes used in medical diagnosis (like Technetium-99m) are chosen specifically because they have a short half-life (around 6 hours).
Why is a short half-life important?
This ensures the radiation exposure to the patient is minimized. The isotope performs its diagnostic role quickly, and then its activity rapidly drops, limiting the hazard while still having a high initial activity (\(A_0\)) to allow imaging.
Key Takeaway for Calculations: Whether dealing with nuclei \(N\), activity \(A\), or mass \(M\), they all follow the same exponential decay pattern governed by the decay constant \(\lambda\).