Simple Nature - Light and Matter

hours. Since 8 hours is the amount of time required for half of the
atoms to decay, it is known as the half-life, writtent 1 / 2. The general
rule is as follows:

Exponential Decay Equation

Psurv(t) = 0.5t/t^1 /^2

Using the rule for calculating averages, we can also find the num-
ber of atoms,N(t), remaining in a sample at timet:

N(t) =N(0)×0.5t/t^1 /^2

Both of these equations have graphs that look like dying-out expo-
nentials, as in the example below.

Radioactive contamination at Chernobyl example 2

.One of the most dangerous radioactive isotopes released by the

Chernobyl disaster in 1986 was^90 Sr, whose half-life is 28 years.

(a) How long will it be before the contamination is reduced to one

tenth of its original level? (b) If a total of 10^27 atoms was released,

about how long would it be before not a single atom was left?

.(a) We want to know the amount of time that a^90 Sr nucleus

has a probability of 0.1 of surviving. Starting with the exponential

decay formula,

Psur v= 0.5t/t^1 /^2 ,

we want to solve fort. Taking natural logarithms of both sides,

lnP=

t

t 1 / 2

ln 0.5,

so

t=

t 1 / 2

ln 0.5

lnP

Plugging inP= 0.1 andt 1 / 2 = 28 years, we gett= 93 years.

(b) This is just like the first part, butP = 10−^27. The result is

about 2500 years.

(^14) CDating example 3
Almost all the carbon on Earth is^12 C, but not quite. The isotope
(^14) C, with a half-life of 5600 years, is produced by cosmic rays in
the atmosphere. It decays naturally, but is replenished at such a
rate that the fraction of^14 C in the atmosphere remains constant,
at 1.3× 10 −^12. Living plants and animals take in both^12 C and
(^14) C from the atmosphere and incorporate both into their bodies.
Once the living organism dies, it no longer takes in C atoms from
the atmosphere, and the proportion of^14 C gradually falls off as it
undergoes radioactive decay. This effect can be used to find the
Section 13.1 Rules of randomness 865