Radioactive Decay and Potassium-Argon Dating · CAP 2010 #P3

Physics · modern-physics

Problem

Sir Ernest Rutherford, a very famous New Zealander who worked at McGill University in Montreal and the Cavendish Laboratory in Cambridge, won the Nobel Prize for his studies of the disintegration of nuclei. He established an important relationship for a nuclear reaction, in which the rate of decay

\frac{dN}{dt}

is proportional to the number N of radioactive nuclei present:

\frac{dN}{dt} = -\lambda N

where λ is called the decay constant. We can integrate this expression to get the relationship

N(t)=N_0 e^{-λt}

where

N_0

represents the number of radioactive nuclei at time t = 0. a) Show that the time taken for N(t) =

N_0

/2 (known as the half-life), is given by

T_{1/2} = (\ln 2)/ λ

. Potassium-argon (K-Ar) dating is used in earth science to determine the age of a rock sample. The potassium isotope

^{40}K

is radioactive and decays over time to the argon isotope

^{40}Ar

. When a rock is molten the argon gas is released into the atmosphere but as the rock cools and crystallizes, the daughter argon atoms are trapped within the rock matrix. Time since crystallization is calculated by measuring the ratio of the amount of

^{40}Ar

accumulated to the amount of

^{40}K

remaining. The two possible nuclear reaction for

^{40}K

to change to

^{40}Ar

are:

^{40}K + ^0_{-1}e → ^{40}Ar +

a neutrino

^{40}K → ^{40}Ar + ^0_{+1}e +

a neutrino b) Describe the physical process that is occurring in these reactions. c) Which of the two is the more likely to occur? d) Most of the potassium

^{40}K

(89.1%) decays to

^{40}Ca

. What is the equation for this reaction? The approximate ratio of the amount of

^{40}Ar

to that of

^{40}K

is directly related to the time elapsed since the rock was cool enough to trap the argon by the following equation:

t= \frac{T_{1/2}}{\log_e (2)} \log_e \left(\frac{Ar_f}{K_f + f \cdot 0.109}\right)

Where

T_{1/2}

is the half-life of

^{40}K

K_f

is the amount of

^{40}K

remaining in the sample, and

Ar_f

is the amount of

^{40}Ar

found in the sample. e) If a rock sample has remained undisturbed for 50 [million years, calculate the ratio of

^{40}Ar

^{40}K

that would be expected. The half-life of

^{40}K

is 1.25 × 10⁹ years.]

Intuitive explanation

The key to part (a) is recognizing that the half-life is simply the time when $N(t)$ drops to exactly half its initial value—substitute $N(t) = N_0/2$ into the exponential decay formula and solve for $t$ . You'll need to use logarithms to isolate the time variable from the exponential.

Intuitive explanation

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