Radioactive decay

1. Overview

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting ionizing radiation to reach a more stable state. This process is a nuclear property, meaning it originates within the nucleus itself and is independent of the atom's chemical or physical environment. On a microscopic level, decay is governed by quantum mechanical chance; however, when dealing with the vast number of atoms in a macroscopic sample, this randomness averages out into a highly predictable exponential decay law. The rate of decay is determined solely by the internal stability of the specific nuclide, characterized by its decay constant.

Key Definitions

• Radioactive Decay: The spontaneous and random emission of ionizing radiation (alpha, beta, or gamma) from unstable nuclei.
• Spontaneous: A process that is unaffected by external factors. The rate of decay cannot be increased or decreased by changes in temperature, pressure, or the chemical compound in which the isotope is found.
• Random: A process in which it is impossible to predict which nucleus will decay next or exactly when a specific nucleus will decay. There is a constant probability of decay per unit time for every nucleus of a particular isotope.
• Activity ($A$): The number of nuclear decays occurring in a radioactive source per unit time. It is the rate of disintegration.
  • Unit: Becquerel (Bq), where $1 \text{ Bq} = 1 \text{ decay per second} \text{ (s}^{-1})$.
• Decay Constant ($\lambda$): The probability of decay of a nucleus per unit time. It represents the fraction of the total number of nuclei expected to decay in a very short time interval.
  • Unit: $\text{s}^{-1}$, $\text{h}^{-1}$, $\text{day}^{-1}$, or $\text{year}^{-1}$.
• Half-life ($t_{1/2}$): The mean time taken for the number of undecayed nuclei in a sample to reduce to half of its original value. It is also the time taken for the activity (or count rate) to halve.

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Content

Evidence for the Random Nature of Decay

The random nature of radioactive decay is not just a theoretical concept; it is directly observable through experimental measurement.

1. Fluctuations in Count Rate: If a Geiger-Müller (GM) tube is placed near a radioactive source with a long half-life, the number of counts recorded in successive, equal time intervals (e.g., every 10 seconds) will not be constant.
2. Observation: You might record counts such as 45, 38, 52, 41, and 47. These fluctuations in count rate provide experimental evidence that the timing of individual decay events is unpredictable.
3. Statistical Significance: As the total number of counts increases, the relative size of these fluctuations decreases, but they never disappear. This confirms that while we can predict the average behavior of the population, we cannot predict individual events.

Spontaneity and the Nucleus

The fact that decay is spontaneous implies that the trigger for decay is entirely internal to the nucleus. In chemical reactions, increasing the temperature increases the rate of reaction by providing activation energy. In radioactive decay, heating a sample of Radium-226 does absolutely nothing to its activity. This distinguishes nuclear processes from atomic/chemical processes.

The Fundamental Law of Radioactive Decay

The activity $A$ of a sample is directly proportional to the number of undecayed nuclei $N$ currently present. This relationship is defined by the equation:

$$\mathbf{A = \lambda N}$$

Where:

• $A$ is the activity (Bq)
• $\lambda$ is the decay constant ($\text{s}^{-1}$)
• $N$ is the number of undecayed nuclei remaining in the sample.

Because activity is the rate at which the number of nuclei $N$ decreases over time, we can write: $$A = -\frac{dN}{dt} = \lambda N$$ The negative sign indicates that $N$ is decreasing as time $t$ increases.

The Exponential Decay Law

The differential equation $\frac{dN}{dt} = -\lambda N$ leads to an exponential solution. This describes how the quantity of a radioactive substance diminishes over time. The law applies to the number of nuclei ($N$), the activity ($A$), and the received count rate ($C$):

$$\mathbf{x = x_0 e^{-\lambda t}}$$

Where:

• $x$ is the value at time $t$ (can be $N, A,$ or $C$).
• $x_0$ is the initial value at $t = 0$.
• $e$ is the base of the natural logarithm ($\approx 2.718$).
• $\lambda$ is the decay constant.
• $t$ is the time elapsed.

Graphical Representation:

• The Decay Curve: A plot of $N$ against $t$ shows a smooth curve that starts at $N_0$ and approaches the $x$-axis asymptotically (it never actually reaches zero).
• Half-life on the Graph: To find $t_{1/2}$, locate $N_0/2$ on the $y$-axis, move horizontally to the curve, and then vertically down to the $x$-axis. This interval is the half-life. Repeating this from $N_0/2$ to $N_0/4$ will yield the same time interval.

The Relationship between $\lambda$ and $t_{1/2}$

The decay constant and half-life are inversely related: a high probability of decay ($\lambda$) means a short half-life ($t_{1/2}$).

1. Start with: $N = N_0 e^{-\lambda t}$
2. At $t = t_{1/2}$, the number of nuclei is $N = \frac{N_0}{2}$.
3. Substitute: $\frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}}$
4. Simplify: $0.5 = e^{-\lambda t_{1/2}}$
5. Take natural logs: $\ln(0.5) = -\lambda t_{1/2}$
6. Since $\ln(0.5) = -0.693$: $-0.693 = -\lambda t_{1/2}$
7. Rearrange: $$\mathbf{\lambda = \frac{\ln 2}{t_{1/2}} \approx \frac{0.693}{t_{1/2}}}$$

Linearizing Exponential Data

In many practical applications, scientists plot the natural logarithm of activity against time. Taking the natural log of $A = A_0 e^{-\lambda t}$: $$\ln A = \ln(A_0 e^{-\lambda t})$$ $$\ln A = \ln A_0 + \ln(e^{-\lambda t})$$ $$\mathbf{\ln A = -\lambda t + \ln A_0}$$ This is in the form $y = mx + c$:

• $y$-axis: $\ln A$
• $x$-axis: $t$
• Gradient: $-\lambda$
• $y$-intercept: $\ln A_0$

Worked Example 1 — Calculating Activity from Mass

A sample contains $5.0 \text{ \mu g}$ of pure Sodium-24. The decay constant of Sodium-24 is $1.28 \times 10^{-5} \text{ s}^{-1}$ and its molar mass is $24 \text{ g mol}^{-1}$. Calculate the initial activity of the sample.

Step 1: Calculate the number of nuclei ($N$) First, find the number of moles ($n$): $$n = \frac{\text{mass}}{\text{molar mass}} = \frac{5.0 \times 10^{-6} \text{ g}}{24 \text{ g mol}^{-1}} = 2.083 \times 10^{-7} \text{ mol}$$ Now, find $N$ using Avogadro's constant ($N_A = 6.02 \times 10^{23} \text{ mol}^{-1}$): $$N = n \times N_A = (2.083 \times 10^{-7}) \times (6.02 \times 10^{23}) = 1.254 \times 10^{17} \text{ nuclei}$$

Step 2: Calculate Activity ($A$) $$A = \lambda N$$ $$A = (1.28 \times 10^{-5} \text{ s}^{-1}) \times (1.254 \times 10^{17})$$ $$A = 1.605 \times 10^{12} \text{ Bq}$$

Answer: $$A = 1.6 \times 10^{12} \text{ Bq (to 2 s.f.)}$$

Worked Example 2 — Determining Time Elapsed

The initial activity of a radioactive isotope is $400 \text{ Bq}$. The half-life of the isotope is $8.0 \text{ days}$. Calculate the time taken for the activity to fall to $50 \text{ Bq}$.

Method A: Using the half-life ratio (simplest for integer half-lives)

1. $400 \rightarrow 200$ (1 half-life)
2. $200 \rightarrow 100$ (2 half-lives)
3. $100 \rightarrow 50$ (3 half-lives) Total time $= 3 \times t_{1/2} = 3 \times 8.0 = 24 \text{ days}$.

Method B: Using the exponential equation (required for non-integer half-lives) Step 1: Find $\lambda$ $$\lambda = \frac{0.693}{8.0 \text{ days}} = 0.0866 \text{ days}^{-1}$$ Step 2: Use $A = A_0 e^{-\lambda t}$ $$50 = 400 e^{-0.0866t}$$ $$0.125 = e^{-0.0866t}$$ Step 3: Solve for $t$ $$\ln(0.125) = -0.0866t$$ $$-2.079 = -0.0866t$$ $$t = \frac{-2.079}{-0.0866} = 24.0 \text{ days}$$

Answer: $$t = 24 \text{ days}$$

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Key Equations

Equation	Description	Status
$A = \lambda N$	Relationship between activity, decay constant, and number of nuclei.	Data Sheet
$x = x_0 e^{-\lambda t}$	Exponential decay law (where $x$ is $N, A,$ or $C$).	Data Sheet
$\lambda = \frac{0.693}{t_{1/2}}$	Relationship between decay constant and half-life.	Memorize
$N = \frac{m}{M} \times N_A$	Calculating number of nuclei from mass $m$ and molar mass $M$.	Memorize
$\ln A = -\lambda t + \ln A_0$	Linearized form of the decay equation for graphing.	Memorize

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Common Mistakes to Avoid

• ❌ Unit Mismatch in Exponents: Using $\lambda$ in $\text{s}^{-1}$ but $t$ in hours.
  • ✓ Right: The product $\lambda t$ must be dimensionless. If $\lambda$ is in $\text{s}^{-1}$, $t$ must be in seconds. If $t$ is in years, $\lambda$ must be in $\text{year}^{-1}$.
• ❌ Confusing $N$ with Mass: Substituting the mass of the sample (e.g., 5g) directly for $N$ in $A = \lambda N$.
  • ✓ Right: $N$ is the number of nuclei. You must use the molar mass and Avogadro's constant to convert mass to $N$.
• ❌ Ignoring Background Radiation: Using the raw count rate from a GM tube directly in the decay equation.
  • ✓ Right: Background radiation is constant and does not decay with the source. You must subtract the background count rate ($C_b$) from the measured count rate ($C_m$) to get the corrected count rate ($C_c$) before doing any calculations: $C_c = C_m - C_b$.
• ❌ Sign Errors in Logarithms: Forgetting that $\ln(x)$ is negative if $x < 1$.
  • ✓ Right: When calculating $\ln(N/N_0)$, the result will be negative, which cancels the $-\lambda t$ term.
• ❌ Rounding Too Early: Rounding $\lambda$ to 2 s.f. and then using it to calculate a large $N$.
  • ✓ Right: Keep at least 4 significant figures in your calculator for $\lambda$ to avoid large rounding errors in the final answer.

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Exam Tips

1. Defining Randomness: If asked to define "random," ensure you mention both the unpredictability of a single nucleus and the constant probability of decay for the population.
2. Defining Spontaneity: Always mention that the rate is unaffected by external factors such as temperature or pressure.
3. Graph Skills: If you need to find the half-life from a graph, draw construction lines. Show that you have checked at least two different half-life intervals to ensure the decay is truly exponential.
4. The "ln" Graph: If a question provides a graph of $\ln(\text{Count Rate})$ against time, immediately identify the gradient as $-\lambda$. This is a very common Paper 4 and Paper 5 technique.
5. Activity vs. Count Rate: Be aware that "Activity" is the total rate of decay of the source (emitting in all directions), while "Count Rate" is what a detector measures (only a small fraction of the emissions). However, both follow the same $x = x_0 e^{-\lambda t}$ law.
6. Calculating $N$: Remember that $N$ is the number of undecayed nuclei. If a question asks for the number of nuclei that have decayed, calculate $N$ (remaining) and subtract it from $N_0$ (initial).
7. Significant Figures: Usually, 2 or 3 s.f. are appropriate. Look at the data provided in the question; if the mass is given as $5.0 \text{ g}$, provide your answer to 2 s.f.