Radioactive Decay and Half-Life Calculations

Understanding radioactive decay is essential not just for passing your IB Physics exams, but for grasping the fundamental behavior of matter at the nuclear level. This knowledge underpins technologies from medical imaging to archaeological dating and dictates safety protocols in industries from energy to aerospace. Mastering decay equations and half-life calculations allows you to predict how unstable nuclei transform over time, a cornerstone of modern physics.

The Nature of Radioactive Decay and Types of Radiation

At its core, radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This instability typically arises from an imbalance in the ratio of protons to neutrons. The decay process continues until a stable nucleus is formed. The three primary types of decay—alpha, beta, and gamma—differ dramatically in their composition, mechanism, and interaction with matter.

Alpha ($\alpha$) decay involves the emission of an alpha particle, which is identical to a helium-4 nucleus (two protons and two neutrons). Because it is relatively massive and carries a double positive charge, it has high ionizing power, meaning it can knock electrons out of atoms very effectively. However, this also means it has very low penetrating power. A sheet of paper or a few centimeters of air can stop it. An example is the decay of radium-226 into radon-222.

Beta decay comes in two forms. Beta-minus (${\beta }^{-}$) decay occurs when a neutron transforms into a proton, emitting an electron (the beta particle) and an antineutrino. Beta-plus (${\beta }^{+}$) decay, or positron emission, occurs when a proton transforms into a neutron, emitting a positron and a neutrino. Beta particles are much smaller and faster than alpha particles. They have moderate ionizing power and moderate penetrating power, being stopped by a few millimeters of aluminum.

Gamma ($\gamma$) decay is the emission of high-energy electromagnetic radiation (photons) from a nucleus that is in an excited state. Gamma rays carry no charge and have almost no mass. Consequently, they have very low ionizing power but extremely high penetrating power, requiring several centimeters of lead or meters of concrete to attenuate significantly. Gamma emission often accompanies alpha or beta decay as the newly formed nucleus sheds excess energy.

Writing Balanced Nuclear Decay Equations

A nuclear equation must balance for both mass number (A, the total number of nucleons) and atomic number (Z, the number of protons). The sum of these numbers on the left must equal the sum on the right.

For alpha decay, the parent nucleus loses 4 in mass number and 2 in atomic number. The general form is: $${}_Z^{A}X{\to }_{Z-2}^{A-4}Y{+}_2^4\alpha$$ Example: The decay of uranium-238: $${}_{92}^{238}U{\to }_{90}^{234}Th{+}_2^4\alpha$$

For beta-minus decay, a neutron converts to a proton, so the mass number stays the same, but the atomic number increases by 1. $${}_Z^{A}X{\to }_{Z+1}^{A}Y{+}_{-1}^0\beta +{\bar{\nu }}_e$$ Example: The decay of carbon-14 into nitrogen-14: $${}_6^{14}C{\to }_7^{14}N{+}_{-1}^0\beta +{\bar{\nu }}_e$$

For beta-plus decay, a proton converts to a neutron, so the mass number stays the same, but the atomic number decreases by 1. $${}_Z^{A}X{\to }_{Z-1}^{A}Y{+}_{+1}^0\beta +{\nu }_e$$ Example: The decay of fluorine-18 into oxygen-18, common in PET scans: $${}_9^{18}F{\to }_8^{18}O{+}_{+1}^0\beta +{\nu }_e$$

Gamma decay does not change the identity of the nucleus; it only changes its energy state. It is often written alongside another decay product, e.g., ${}_Z^A{X}^{\ast }{\to }_Z^{A}X+\gamma$, where the asterisk (*) denotes an excited state.

The Concept of Half-Life and Quantitative Calculations

The half-life ($t_{1/2}$) is defined as the time taken for half the number of radioactive nuclei in a sample to decay, or equivalently, for the activity of the sample to halve. It is a constant for a given isotope, unaffected by temperature, pressure, or chemical state.

The exponential nature of decay is described by the equation: $$N=N_0{\left(\frac{1}{2}\right)}^n$$ where $N$ is the final number of nuclei, $N_0$ is the initial number, and $n$ is the number of half-lives elapsed ($n=t/t_{1/2}$). The same form applies to mass or activity (decays per second, measured in Becquerels, Bq).

To calculate half-life from a decay curve, plot activity (or count rate) against time on a linear scale. The curve will be exponential. Find the time at which the initial activity has dropped to half. This time is the half-life. You can verify it by checking the time it takes for that new value to halve again.

Worked Example: A sample has an initial activity of 800 Bq. After 12 days, its activity is 100 Bq. What is the half-life?

1. Find the number of half-lives ($n$): $800\to 400\to 200\to 100$. That's 3 half-lives.
2. Time elapsed = 12 days = 3 half-lives.
3. Therefore, half-life $t_{1/2}=12\text{days}/3=4\text{days}$.

You can also use the decay constant $\lambda$, related to half-life by $t_{1/2}=\frac{\ln 2}{\lambda }$. The fundamental decay law is $N=N_0{e}^{-\lambda t}$.

Applications of Radioactive Isotopes

The properties of different decays make specific isotopes suitable for distinct applications.

In medicine, gamma emitters like technetium-99m (with a 6-hour half-life) are ideal for diagnostic imaging because gamma rays exit the body to be detected by a camera. Beta emitters like iodine-131 are used in radiotherapy to treat thyroid cancer, as the beta radiation destroys targeted tissue with minimal exit dose. Positron emitters are used in PET scans.

In dating, the predictable half-life of long-lived isotopes acts as a clock. Radiocarbon dating uses carbon-14 ($t_{1/2}\approx 5730$ years) to date organic materials up to about 50,000 years old. For geological timescales, isotopes like potassium-40 ($t_{1/2}\approx 1.25$ billion years) are used in potassium-argon dating.

In industry, radioactive tracers can monitor fluid flow in pipes or detect leaks. Beta sources are used in thickness gauges for paper or metal production, as the amount of radiation absorbed depends on the material's thickness. Gamma radiography can inspect welds in critical structures for defects.

Common Pitfalls

1. Misbalancing Decay Equations: A common error is to forget to balance the atomic number (charge). Always check that the sum of the bottom numbers (Z) on the left equals the sum on the right. For example, in beta-minus decay, writing the emitted electron as ${}_0^0\beta$ instead of ${}_{-1}^0\beta$ will cause an imbalance.
2. Confusing Decay Constant and Half-Life: Students often mistakenly think $\lambda$ and $t_{1/2}$ are directly proportional. They are inversely related: $t_{1/2}=\ln 2/\lambda$. A larger decay constant means a shorter half-life.
3. Misinterpreting Decay Curves on Linear Graphs: On a linear activity vs. time graph, the decay is exponential, not linear. You cannot simply take two points and use a linear interpolation to find the half-life. You must use the method of successive halvings or use the logarithmic form of the equation.
4. Overlooking the Random Nature of Decay: While half-life is a statistical constant for a large population of atoms, you cannot predict which specific nucleus will decay next. This intrinsic randomness is a key conceptual point in the IB syllabus.

Summary

• Radioactive decay occurs in three primary forms: alpha (heavy, +2 charge, low penetration), beta (light electrons/positrons, moderate penetration), and gamma (high-energy photons, very high penetration).
• Nuclear equations must balance for both mass number (A) and atomic number (Z). Alpha decay reduces A by 4 and Z by 2; beta-minus increases Z by 1; beta-plus decreases Z by 1.
• The half-life is the constant time for half the nuclei in a sample to decay. Calculations use the relationship $N=N_0(1/2{)}^{t/t_{1/2}}$, where $t/t_{1/2}$ is the number of elapsed half-lives.
• Isotopes are selected for applications based on their decay type and half-life: medicine (diagnosis/therapy), dating (archaeological/geological clocks), and industry (tracers, gauging, inspection).