The Decay Constant and the Exponential Decay Law
Every unstable nucleus of one kind has the same chance of decaying each second, so the number left falls exponentially. The decay constant is that chance per nucleus per second.
What a learner can do afterwards
- Uses the exponential decay law to find either the number left or the time elapsed
- Explains why decay is random for a single nucleus yet predictable for a large sample
- Plots the natural log of the count against time and takes the decay constant from the gradient
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Every unstable nucleus of one kind shares the same chance of decaying each second. That chance per nucleus per second is the decay constant, written as lambda. Big lambda means fast decay. The survivors follow N equals N0 times e to the minus lambda t, linking start count, constant, and time.
Doctors choosing a medical tracer need to know how fast it fades, and half-lives give the answer. After one half-life half the nuclei remain, after two a quarter, after three one eighth. Start with 4000 nuclei and three half-lives later only 500 are left, since one eighth of 4000 is 500.
One nucleus is pure chance, like a single dice throw you cannot call. Trillions of nuclei together average out rock steady, so the sample shrinks on a smooth predictable curve. Plot the natural log of the count against time and the points form a straight line whose slope is minus lambda.
Trade half-life and lambda with care. Lambda equals 0.693 divided by the half-life, so a short half-life means a large lambda. With a 3 s half-life, lambda is 0.693 over 3, which is 0.231 per second. Never treat the half-life as the average lifetime.
One random chance per nucleus builds a smooth exponential fall whose slope and half-life both reveal lambda.
The rewritten readings were written for this dataset.
More to read
Random decays, predictable rates
An unstable nucleus can emit particles or radiation, but its exact decay time is unpredictable. Across a large sample, roughly the same fraction of the remaining nuclei decays during each equal time interval. The decay constant, lambda, is the decay probability rate per surviving nucleus, characteristic of the isotope. Larger lambda means faster decay; smaller lambda means slower decay. Alpha emission releases two protons and two neutrons together. Beta-minus decay converts a neutron to a proton. Gamma emission removes energy without changing the nucleon count or element.
Connecting half-life and activity
A half-life is the time over which half the original radioactive nuclei decay on average. The remaining fractions after one, two and three half-lives are ½, ¼ and ⅛. Activity, measured in decays per second, also decreases exponentially. These relationships help with rock dating and medical dosing. Half-life = ln 2 / lambda, while activity = lambda × N. Either half-life or lambda determines the other; activity also requires the number of radioactive nuclei, N.
Using the exponential decay equation
Exponential models describe compound money growth, radioactive decay and, under suitable conditions, a cooling object's temperature difference from its surroundings. The rate of change is proportional to the quantity being modelled. Radioactive survivors follow N = N0 e^(−lambda t), where N0 is the initial count and N is the count remaining at time t. Plotting ln N against time gives a line with slope −lambda, a common laboratory way to find the decay constant. With N0 and lambda known, calculate survivors at a chosen time or use logarithms to find elapsed time from a measured count.
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Where this leads
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.