Physics · nuclei · NEET
Mean life is always greater than half-life. Since tau = 1.44 x T-half, the mean life is about 44 percent longer than the half-life. Reason: after one half-life, half the nuclei are still alive and keep living, pulling the average survival time up. So mean life > half-life, always. Memory line: mean life is longer.
Both come from the decay constant lambda. Mean life tau = 1/lambda. Half-life T-half = 0.693 / lambda (the 0.693 is ln 2). Divide them: tau / T-half = (1/lambda) / (0.693/lambda) = 1/0.693 = 1.44. So tau = 1.44 x T-half, and T-half = 0.693 x tau. Learn all three: tau = 1/lambda, T-half = 0.693/lambda, tau = 1.44 T-half.
The number 1.44 is just 1 divided by 0.693 (which is 1/ln 2). Half-life uses ln 2 = 0.693 because we ask when N drops to N0/2. Mean life is the mathematical average lifetime, which works out to 1/lambda. When you take the ratio, lambda cancels and you get 1/0.693 = 1.44. It is a fixed number for every radioactive substance.
Yes. Mean life is the average time a single nucleus exists before decaying, averaged over all nuclei in the sample. Half-life instead describes the whole sample: the time for the count to fall to half. So mean life is a per-nucleus average; half-life is a sample-level time. Both give the same information because both are fixed by lambda.
Just multiply. Mean life = 1.44 x half-life. Example: if T-half = 5 hours, then tau = 1.44 x 5 = 7.2 hours. To go the other way, T-half = 0.693 x tau. Also remember tau = 1/lambda, so if lambda is given, mean life is simply its reciprocal.
It lets you switch between mean life and half-life in one step. If a question gives half-life and asks mean life (or the reverse), multiply by 1.44 or 0.693. It saves time in decay numericals.
No. Because tau = 1.44 x T-half, they can never be equal for a real radioactive substance. Mean life is always about 1.44 times the half-life.
Both are times, so their unit is seconds (or minutes, hours, years). The decay constant lambda has the unit per second, and both mean life and half-life are found from lambda.
Both become half. Since tau = 1/lambda and T-half = 0.693/lambda, doubling lambda halves both. A larger decay constant means faster decay, so shorter mean life and shorter half-life.
No. A sample never fully decays in a fixed finite time in this model; N falls exponentially. Mean life is the average survival time of one nucleus, about 1.44 half-lives, not a total-decay time.
