Decay Curve
Side-by-side linear N(t) + semilog lnN(t) — slope = −λ on the log plot.
Key Notes
The decay curve N(t) = N₀ e^(−λt) is an exponential drop.
It looks STRAIGHT on a semi-log plot (ln N vs t): slope = −λ, intercept = ln N₀.
Half-lives evenly space along t-axis: at T_½, 2T_½, 3T_½, … the count halves each time.
Tail of the curve never reaches zero (exponentials are asymptotic) — but after many T_½ the count is negligibly small.
Daughter accumulation in a chain (Parent → Daughter): daughter builds up to a maximum, then decays if it too is unstable.
Secular equilibrium: when parent is much longer-lived than daughter, daughter activity equals parent activity at equilibrium.
Transient equilibrium: parent slightly longer than daughter; daughter exceeds parent for a while before equilibrium.
Formulas
Exponential decay
Smooth curve; never reaches zero.
Logarithmic form (linearised)
Slope of ln N vs t = −λ.
After n half-lives
Useful discrete form.
Bateman (parent → daughter)
Daughter builds up then decays in a chain.
Important Points
Plot N vs t: smooth exponential curve. Plot ln N vs t: straight line. Use semi-log to identify single exponentials.
Half-life can be read off the linear (semi-log) plot — successive halvings spaced by ΔT = T_½.
In sequential decays (A → B → C), the intermediate B builds up, peaks, then decays.
Activity curve has the SAME shape as N(t) — both decay exponentially with the same λ.
Tail: even after 10 T_½ (1/1024), activity remains — important for nuclear waste storage.
Real plots include statistical noise — short timescales need many counts to see the exponential cleanly.
Decay Curve notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.