Four tools on this site are running the same equation. Its parameter has four names.
A capacitor reaches 63.21% of its supply in one time constant. A radioactive sample has 36.79% left after one of its own. That is not a coincidence, it is the same subtraction.
Open the RC circuit tool. With a 10 kΩ resistor and a 100 µF capacitor it reports a time constant τ = RC = 1.00 s, and then three readings: after one τ the capacitor holds 3.161 V of its 5 V supply, after three τ it holds 4.751 V, and after five it holds 4.966 V, which the tool labels ≈ 100%.
Now open radioactive decay with carbon-14. It reports a decay constant λ = ln 2 / t½ = 0.693 / 5730 = 1.21 × 10⁻⁴ per year, and after five half-lives, 3.125% of the sample remains.
These two pages are solving the same equation. Not an analogous equation — the same one, with the letters renamed.
The equation, and its one parameter
Both are instances of
dQ/dt = −kQ
"the rate at which the quantity changes is proportional to how much of it there is." A capacitor charging is the same statement about the gap remaining to the supply voltage. Fill in what Q is and you have picked a subject:
- Electronics. Q is the voltage still missing. k = 1/RC.
- Radioactivity. Q is the number of undecayed nuclei. k = λ.
- Chemistry. Q is the concentration of a reactant in a first-order reaction. k is the rate constant.
- Physiology. Q is the voltage across a nerve membrane. k = 1/RmCm, which is what Hodgkin and Huxley measured in 1952 when they described the axon as an RC circuit and got a Nobel Prize partly for taking the analogy literally.
One parameter with four names, and its reciprocal 1/k is the time constant τ. For the capacitor above, τ = 1.00 s. For carbon-14, τ = t½ / ln 2 = 8,267 years. That number does not appear on the decay page, but its reciprocal does: the printed λ = 1.21 × 10⁻⁴ inverts to 8,264, the same value to within the rounding the page shows.
Where the 63% comes from, and why it is on both pages
Put t = τ into the solution and the exponent becomes exactly −1, whatever the domain:
Q(τ) = Q₀ e⁻¹ = 0.3679 Q₀
So after one time constant, 36.79% of the original is left. The RC page prints the complement, because a charging capacitor is filling rather than emptying: 1 − e⁻¹ = 0.6321, which is where 3.161 V of 5 V comes from. The 63.2% an electronics textbook drills into you and the 36.8% a physics textbook drills into you are the same subtraction from the same number.
The 5τ convention has the same origin. e⁻⁵ = 0.0067, so after five time constants 0.67% remains, near enough to nothing that engineers call the transient over and the RC tool prints ≈ 100%. It is an arbitrary threshold, honestly chosen, and it is arbitrary in every one of the four domains at once.
The conversion everyone gets wrong
Half-life is not the time constant. It is shorter, by a factor of ln 2:
t½ = τ · ln 2 = 0.693 τ
Carbon-14's half-life of 5,730 years and its time constant of 8,267 years are the same physics described with two different thresholds: one asks when half is gone, the other when 1/e is left. Nothing about the atom changes.
The practical consequence is a translation table between two professions that never talk to each other. An engineer's "wait five time constants and it has settled" is 7.2 half-lives. A physicist's "ten half-lives and it is gone" is 6.9 time constants. Both mean the same wait; each sounds like a different quantity of patience.
And the error runs the other way too. Reading a decay curve off a chart and calling the visible knee the half-life gives you τ, which is 44% too long. The exponential tool keeps separate rows for the doubling time and the half-life, beside the rate k you set, for exactly this reason. Since t½ = ln 2 / k, one decay reports as two different numbers, and which one a dataset gives is a convention rather than a fact.
Why bother noticing
Because the transfer is free, and it runs in both directions.
If you know that a capacitor's charging time depends on RC and not on the supply voltage — a bigger battery reaches the same fraction at the same moment — then you already know why a radioactive sample's half-life does not depend on how much you started with, and why a first-order reaction's half-time does not depend on concentration. That is one fact, not three, and it is the fact that makes carbon dating possible at all: the clock rate cannot be affected by how much carbon the sample happens to contain.
It runs the other way too. Anyone comfortable with half-lives already has the intuition for why an oversized capacitor makes a circuit slow to respond, and why a neuron with a leaky membrane integrates its inputs over a longer window and therefore fires differently. The 1952 paper in the references is what happens when someone takes that transfer seriously enough to measure it.
Where it stops being true
The isomorphism holds only while the rate is proportional to the amount, and that assumption is a claim about mechanism, not a mathematical convenience.
Reaction kinetics is the place to break it, because it lets you set the order directly. A zero-order reaction removes a fixed amount per unit time no matter how much is present, so its graph is a straight line and it has no time constant to speak of. The concept does not survive the change. Alcohol clearance is the standard example, and it is why the "one drink per hour" rule of thumb can exist at all: a fixed amount leaves per hour, so the last drink takes as long to clear as the first. Make it first-order and the rule is useless, because the final drink would then be the slowest of the lot to go.
Second-order is a third shape again. Two of the four costumes above are first-order by construction — a capacitor and a nucleus have no choice — and two are first-order only when the chemistry or the biology cooperates. The equation is worth recognising precisely because the recognition is falsifiable: if the log plot is not straight, you are not in this family, and the entire table of translations above does not apply.
References (1)
- the membrane as an RC circuit, with the time constant measured Hodgkin & Huxley (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology 117(4).