Earth, planets and orbits

1956

The age of the Earth

Clair Patterson

How do you date the Earth when every rock gets recycled?

Patterson measured uranium and lead in meteorites, leftovers from the Solar System’s birth. Uranium turns into lead at a steady, known rate, so the ratio is a clock. He found about 4.55 billion years. While cleaning his lab of lead contamination, he also discovered how much lead pollution was everywhere, and campaigned until leaded petrol was banned.

First, make a guess

After three half-lives, what fraction of the uranium is left?

How they figured it out

Clair Patterson needed to measure tiny amounts of lead in rock, but lead from the air, his clothes and his equipment swamped every reading. He built one of the first ultra-clean labs. Rather than Earth rocks, he used a meteorite, Canyon Diablo, leftover material from the Solar System’s birth, and in 1953 measured its lead: 4.55 billion years.

His clean-room obsession revealed something else. Lead was everywhere, far above natural levels, and it came from leaded petrol. Patterson spent decades fighting the industry. Leaded petrol was phased out, and the lead in people’s blood fell dramatically.

The equation

N(t)=N0 e−λt,t1/2=ln⁡2λN(t) = N_0\, e^{-\lambda t},\qquad t_{1/2} = \frac{\ln 2}{\lambda}

In every half-life, half of what is left decays; so the amount left tells you the time passed.

Everyday echoCarbon dating of old bones and wood uses the same idea with a faster clock.

See it in real life

An hourglass you cannot watch

If you could not see an hourglass running but could weigh both halves, you could still tell how long it had been running. Uranium-to-lead ratios work the same way. Carbon dating of wood and bone uses a faster clock, carbon-14.

Where engineers use it

  • Dating

    Carbon-14 dates archaeology; uranium–lead dates rocks.

  • Medicine

    Doses of medical isotopes are timed by their half-lives.

  • Capacitors

    A capacitor discharging through a resistor follows the same curve, with time constant RCRC.

Worked example: Age from a ratio

  1. t=t1/2 log⁡2(N0/N)t = t_{1/2}\,\log_2(N_0/N)
  2. If an eighth remains: t=3 t1/2t = 3\,t_{1/2}.

Where it came from

  • Newton’s law of cooling 1701

    Same technique

    Both follow exponential decay: the rate of change is proportional to how much is left.

    Where the comparison stops: Cooling is smooth because trillions of molecules average out. Decay is random atom by atom, and only exponential on average.

  • Logarithms 1614

    Built on

    Reading an age from a ratio means undoing an exponential: a logarithm.