All calculatorsHalf-Life Calculator
Compute remaining amount N, initial amount N₀, decay constant k, elapsed time t, or half-life t₁/₂ using N = N₀·e−kt and t₁/₂ = ln(2)/k. Enter any three fields and we’ll solve the rest. See steps and a mini decay chart.
Background
First-order decay processes (radioactive decay or many chemical decompositions) follow N = N₀·e−kt. The half-life is the time for N to drop to N₀/2, so t₁/₂ = ln(2)/k. Fraction remaining is N/N₀ = e−kt.
Conceptual Connection to Thermodynamics
Although half-life is a kinetic concept that measures how fast a reaction or decay occurs, it’s often discussed alongside thermodynamics. Gibbs free energy (ΔG) determines whether a process is spontaneous, while half-life determines how quickly it happens. A reaction can be thermodynamically favorable (ΔG < 0) yet still have a very long half-life if its rate constant k is small — showing why kinetics and thermodynamics must always be considered together.
How this calculator works
- Model: N = N₀·e−kt (first-order decay)
- Half-life: t₁/₂ = ln(2)/k
- Fraction remaining: N/N₀ = e−kt
- Enter any three of N, N₀ (or N/N₀), k, t, t₁/₂; we solve the rest.
Formula & Equation Used
Exponential decay: N = N₀·e−kt
Half-life relation: t₁/₂ = ln(2)/k
Rearrangements: k = −(1/t)ln(N/N₀), t = −(1/k)ln(N/N₀), N/N₀ = e−kt
Example Problems & Step-by-Step Solutions
Example 1 — Find t from fraction
Given N/N₀ = 0.125 and k = 0.2310 (1/h).
t = −(1/k)·ln(N/N₀) = −(1/0.2310)·ln(0.125) ≈ 9.0 h.
Example 2 — Find k and t from N, N₀, t₁/₂
N₀ = 100 g, N = 25 g, t₁/₂ = 5.0 h.
k = ln(2)/t₁/₂ = 0.693/5.0 = 0.1386 h⁻¹.
N/N₀ = 0.25 → t = −(1/k)ln(0.25) ≈ 10.0 h.
Example 3 — Find t₁/₂ from k
k = 0.0693 (1/min).
t₁/₂ = ln(2)/k = 0.693/0.0693 ≈ 10.0 min.
Frequently Asked Questions
Q: Do units matter?
Yes. Use the same time unit for t and t₁/₂. Then k is 1/(that time).
Q: What if I only know N and N₀?
You can compute the fraction N/N₀. To get t or k, you need one more input (k, t, or t₁/₂).
Q: Does this assume first-order behavior?
Yes. Radioactive decay and many decompositions follow first-order kinetics.
