AkCalculators

Half-Life and Exponential Decay — concepts, equations and applications

Half-life is a central concept in processes governed by exponential change. Originally developed in the context of radioactive decay, the concept applies broadly to chemical reactions, pharmacokinetics (drug elimination), and population decline/growth when changes follow a constant fractional rate. This article explains the discrete half-life formula and its continuous exponential equivalent, how to switch between them, how to solve for any variable (remaining amount, initial amount, elapsed time, or half-life), units and measurement considerations, and worked examples to make the math concrete.

Discrete half-life form

The familiar half-life expression is

N = N₀ × (1/2)^{t / T½}

Here N₀ is the initial quantity at time t = 0, N is the remaining quantity after elapsed time t, and T½ is the half-life. Each interval of duration T½ reduces the remaining quantity by half. This discrete form is convenient when thinking in integral half-life steps, but it is mathematically equivalent to a continuous exponential form when t is allowed to vary continuously.

Continuous exponential form

The continuous model uses an exponential with the decay constant λ:

N = N₀ × e^{−λ t}

The decay constant λ is related to T½ by λ = ln(2) / T½. Conversely T½ = ln(2) / λ. The continuous form is often used in mathematical modeling, differential equations and when processes are described by a constant proportional rate of change.

Switching between forms

Given T½ you can compute λ and vice versa. For example, T½ = 5 hours implies λ = ln(2) / 5 ≈ 0.138629 hours⁻¹. Using either form will produce the same numeric N for continuous t when units are consistent.

Solving for different variables

Rearrange algebraically to solve for the unknown you need:

• Find N: N = N₀ × (1/2)^{t/T½} (or N = N₀ e^{−λ t}).
• Find N₀: N₀ = N / (1/2)^{t/T½} (or N₀ = N e^{λ t}).
• Find t (discrete half form): t = T½ × log₂(N₀ / N) = (T½ / ln 2) × ln(N₀ / N). For continuous form t = (1/λ) × ln(N₀ / N).
• Find T½: T½ = t / log₂(N₀ / N) = (t × ln 2) / ln(N₀ / N). For continuous form T½ = ln 2 / λ.

Growth vs decay

Growth processes can be handled by toggling sign in the exponent or choosing 'Growth' mode in this tool. For growth, the continuous form is often written N = N₀ × e^{+k t} with growth constant k (k = ln 2 / doubling time). The discrete half-life equivalent for doubling is replace 1/2 with 2 and T½ with doubling time.

Units and practical tips

Time unit consistency is critical. If T½ is in years, convert t to years. This calculator provides unit selectors for time to prevent mismatches. For radioactivity, N may represent activity (Bq, Ci), number of atoms, or mass — ensure N and N₀ use consistent units. Use natural logarithms (ln) for continuous forms and base-2 logs (log₂) or ln with conversion factors for discrete forms.

Measurement considerations

Experimental determination of half-life requires careful measurement and statistical treatment. Count rates in radioactivity are subject to Poisson statistics; use proper background subtraction and uncertainty propagation. Many substances exhibit multi-component decay (more than one exponential), which requires curve-fitting to separate components. Always consult domain-specific literature for experimental protocols.

Worked examples

Example 1 — Remaining amount (discrete): N₀ = 1000 units, T½ = 3 hours, t = 7 hours. N = 1000 × (1/2)^{7/3} ≈ 1000 × 0.31498 ≈ 315 units.

Example 2 — Find time: N₀ = 5000, N = 1250, T½ = 2 hours. t = 2 × log₂(5000/1250) = 2 × log₂(4) = 2 × 2 = 4 hours.

Example 3 — Continuous form: T½ = 10 days → λ = ln 2 / 10 ≈ 0.0693147 day⁻¹. After 25 days N/N₀ = e^{−0.0693147×25} ≈ 0.185 (18.5%).

Limitations and domain notes

This calculator is an educational tool for single-exponential decay/growth. Multi-exponential or non-exponential kinetics, feedback-driven growth, or processes with thresholds require more advanced models. For medical, environmental or legal decisions, use validated instruments and expert consultation.

Using this tool effectively

Select decay/growth and the formula form. Enter any two independent values among N, N₀, t, T½ (or λ for continuous) and leave the one you want to compute blank. Use the step-by-step output for reports and copy/CSV export for saving results.