Chemistry

Half-life measures how long it takes for half a sample to decay, while the decay constant describes the rate at which decay happens—they're two ways of expressing the same underlying exponential process. The key insight is that when exactly half the nuclei remain, the exponential decay equation $N(t) = N_0 e^{-\lambda t}$ gives you $0.5 = e^{-\lambda T_{1/2}}$. Solve for $T_{1/2}$ by taking the natural logarithm, and you'll find a simple inverse relationship between these two quantities.

Half-life measures how long it takes for half a sample to decay, while the decay constant describes the rate at which decay happens—they're two ways of expressing the same underlying exponential process. The key insight is that when exactly half the nuclei remain, the exponential decay equation $N(t) = N_0 e^{-\lambda t}$ gives you $0.5 = e^{-\lambda T_{1/2}}$. Solve for $T_{1/2}$ by taking the natural logarithm, and you'll find a simple inverse relationship between these two quantities.