Radioactive Half-Life Practice Problem: Decay Constant & Fraction Remaining

The activity (A) of a radioactive sample is given by A = λN, where λ is the decay constant and N is the number of radioactive nuclei. Activity represents the number of decays per unit time and is measured in becquerels (Bq).

The activity (A) of a radioactive sample is calculated using A = λN, where λ is the decay constant and N is the number of radioactive nuclei. For example, if λ = 1.57 × 10^-5 s⁻¹ and N = 3.62 × 10^19 nuclei, then A = 1.57 × 10^-5 × 3.62 × 10^19 = 5.68 × 10^14 Bq. This calculation shows how the decay constant and number of nuclei determine the rate of radioactive decay, with larger values indicating higher activity.

The activity (A) of a radioactive sample is given by A = λN, where λ is the decay constant and N is the number of nuclei. The SI unit of activity is the becquerel (Bq), where 1 Bq = 1 decay per second. The older unit is the curie (Ci), where 1 Ci = 3.7 × 10¹⁰ Bq.

The activity of a radioactive sample is given by A = λN, where λ is the decay constant and N is the number of radioactive nuclei. The number of nuclei is N = (mass/molar mass) × N_A, where N_A is Avogadro's number. The activity can be expressed in becquerels (Bq) or curies (Ci), where 1 Ci = 3.7 × 10¹⁰ Bq.

Radioactivity (activity) is the rate of nuclear decay, measured in becquerels (Bq), where 1 Bq = 1 decay per second. The activity formula A = λN shows activity is directly proportional to both the decay constant and number of nuclei. The becquerel is the SI unit, while the curie (Ci) is another common unit (1 Ci = 3.7 × 10¹⁰ Bq). Time must be in seconds for results in becquerels.

Integrating the differential equation -dN/dt = λN gives ln(N₀/N) = λt, or equivalently N = N₀ × e^(-λt). This is the integrated rate law for radioactive decay, which is mathematically identical to the first-order kinetics integrated rate law, with λ replacing the rate constant k.

Radioactive decay follows first-order reaction kinetics where the rate of decay is proportional to the number of radioactive nuclei present. The differential rate equation is: dN/dt = -λN. The integrated rate law is: ln([A]₀/[A]) = kt, which can be rearranged to [A] = [A]₀ × e^(-kt). This exponential decay formula shows how concentration decreases over time. The decay constant λ represents the probability of decay per unit time, and is related to half-life by λ = ln(2)/T₁/₂.

For first order reaction: Rate = k[A], so -d[A]/dt = k[A]. Integrating gives ln[A] = -kt + ln[A]₀, or 2.303 log([A]₀/[A]) = kt. Plotting ln[A] vs t gives a straight line with slope = -k. Half-life t₁/₂ = 0.693/k, independent of initial concentration. This is characteristic of radioactive decay.

Radioactive decay follows first order kinetics. The rate law is: Rate = k[A], where [A] is the amount of radioactive material. The integrated rate law is: ln(A/A₀) = -kt, or A = A₀e^(-kt). The half-life is constant: t₁/₂ = 0.693/k. The instructor explains that radioactive decay is a classic example of a first order process.

Radioactivity is the phenomenon where unstable atomic nuclei break down and release large amounts of energy. Unlike chemical reactions, radioactive decay follows first-order kinetics, meaning the rate of decay is directly proportional to the number of unstable nuclei present. The differential rate law is -dn/dt = λn, where λ is the decay constant. The integrated rate law is ln(n₀/n) = λt. The decay constant is analogous to the rate constant in chemical kinetics but is specifically called the decay constant because it describes radioactive decay. This fundamental principle allows scientists to calculate how much radioactive material remains after any given time period.

In radioactive decay series, the relative amounts of daughter isotopes and all decay series members are determined by the decay constants of each isotope. When a parent isotope has a much longer half-life than its daughter, the system approaches secular equilibrium over time. This means the daughter isotope's activity becomes proportional to the parent's activity, and the ratio of daughter to parent atoms becomes constant.

Radioactive series describe chains of nuclear decay where unstable parent nuclei transform into daughter nuclei through successive decays. The process involves parent nuclei (like Uranium-238) decaying into daughter nuclei (like Thorium-234), which themselves are unstable and continue decaying. Two types of equilibrium exist: transient equilibrium occurs when parent half-life is much longer than daughter half-life (e.g., 78 hours vs 2-3 hours), allowing daughter activity to reach a steady state. Secular equilibrium occurs when parent half-life is extremely long (e.g., 1620 years for Radium-226), making parent activity change negligible over practical time periods.

Some unstable nuclei undergo multiple decay steps before reaching a stable nucleus. For example, Uranium-238 decays through a series of transformations (alpha and beta decays) before finally becoming stable Lead-206. This is called a radioactive decay series. The term 'stable' means the nucleus maintains its composition indefinitely without further decay.

Nuclear stability depends on the neutron-to-proton ratio relative to the stability line. Nuclei above the line (neutron-rich) undergo beta minus decay to convert neutrons to protons. Nuclei below the line (proton-rich) undergo beta plus decay to convert protons to neutrons. Heavy nuclei (Z > 82) undergo alpha decay to reduce mass. Radioactive decay series are sequences where unstable parent nuclei decay into daughter nuclei, which may themselves be unstable and undergo further decay. These series continue through multiple generations (up to 15 decays) until stable nuclei are formed. For example, Uranium-238 decays through a series of alpha and beta decays to form stable Lead-206.

A radioactive decay series is a sequence of decays where an unstable parent nucleus transforms through multiple steps into a stable daughter nucleus. For example, Uranium-238 decays through a series: U-238 → Th-234 (alpha) → Pa-234 (beta) → U-234 (alpha) → Th-230 (alpha) → Ra-226 (alpha) → Rn-222 (alpha) → Po-218 (alpha) → Pb-214 (beta) → Bi-214 (beta) → Po-214 (alpha) → Pb-210 (beta) → Bi-210 (beta) → Po-210 (alpha) → Pb-206 (stable). Each step involves alpha or beta decay until a stable isotope is reached.