Half-Life Calculator - मुफ़्त ऑनलाइन कैलकुलेटर

Half-life is the foundational concept for understanding any exponential-decay process - and there are many. In nuclear physics, half-life determines how long radioactive isotopes remain dangerous (Plutonium-239's 24,000-year half-life is why nuclear waste storage is a centuries-long problem). In medicine, drug half-life determines dosing intervals (Paracetamol's 2-hour half-life means it's nearly gone after 8-10 hours, so 4-6 hourly dosing makes sense). In archaeology, carbon-14 dating relies on the 5,730-year half-life to date organic remains up to ~50,000 years old. In capacitor discharge circuits, RC time constants describe the same exponential pattern.

For Class 11-12 physics in India (radioactivity chapter), half-life calculations appear in board exams every year. NEET aspirants encounter it in modern physics; JEE Main and Advanced test pharmacokinetics-style problems. Pharmacy and medicine students study drug half-life in pharmacology. This calculator works for any exponential decay process - just specify the initial amount, half-life, and elapsed time in consistent units. It also shows the decay constant λ = ln(2)/T and mean lifetime τ = T/ln(2) ≈ 1.443×T, useful in radiochemistry and continuous decay equations.

Exponential decay law: N(t) = N₀ × (1/2)^(t/T), where N(t) is the amount remaining at time t, N₀ is initial amount, and T is the half-life. After one half-life (t = T), the result is N₀/2. After two half-lives (t = 2T), N₀/4. After n half-lives, N₀ × (1/2)ⁿ. The formula works for any t including non-integer multiples of T - the continuous form makes calculations possible at any time point.

Alternative formulations: N(t) = N₀ × e^(−λt) where λ = ln(2)/T ≈ 0.693/T is the decay constant. Both formulations give identical results; the e-form is preferred in continuous mathematics and differential equations. Mean lifetime τ = 1/λ = T/ln(2) ≈ 1.443T - the average time an individual particle 'lives' before decaying, longer than half-life because the distribution is skewed.

Practical interpretation: after 5 half-lives, only 1/32 ≈ 3.1% remains. After 7 half-lives, ~0.8%. After 10 half-lives, ~0.1%. Clinical pharmacology uses '5 half-lives' as the rule of thumb for 'drug effectively eliminated' - by then, residual is negligible for most practical purposes. For radioactive waste storage, '10 half-lives' is often cited as 'safe' (though some short-lived isotopes produce hazardous intermediate products).