A radioactive nucleus formed with an increase in (N-2) and in consequence heavier
than the known stable one tends to exhibit:
a) alpha-Activity
b) complex behavior
c) electron activity
d)none
Answers
EXAMPLE 1. HOW OLD IS THE SHROUD OF TURIN?
Calculate the age of the Shroud of Turin given that the amount of 14C found in it is 92% of that in living tissue.
Strategy
Knowing that 92% of the 14C remains means that
N
N
0
=
0.92
. Therefore, the equation
N
=
N
0
e
−
λ
t
can be used to find λt. We also know that the half-life of 14C is 5730 y, and so once λt is known, we can use the equation
λ
=
0.693
t
1
/
2
to find λ and then find t as requested. Here, we postulate that the decrease in 14C is solely due to nuclear decay.
Solution
Solving the equation N = N0e−λt for
N
N
0
gives
N
N
0
=
e
−
λ
t
.
Thus, 0.92 = e−λt.
Taking the natural logarithm of both sides of the equation yields
ln 0.92 = −λt
so that
−0.0834 = −λt.
Rearranging to isolate t gives
t
=
0.0834
λ
Now, the equation
λ
=
0.693
t
1
/
2
can be used to find λ for 14C. Solving for λ and substituting the known half-life gives
λ
=
0.693
t
1
/
2
=
0.693
5730
y
.
We enter this value into the previous equation to find t:
t
=
0.0834
0.693
5730
y
=
690
y
Discussion
This dates the material in the shroud to 1988 – 690 = AD 1300. Our calculation is only accurate to two digits, so that the year is rounded to 1300. The values obtained at the three independent laboratories gave a weighted average date of AD 1320 ± 60. The uncertainty is typical of carbon-14 dating and is due to the small amount of 14C in living tissues, the amount of material available, and experimental uncertainties (reduced by having three independent measurements). It is meaningful that the date of the shroud is consistent with the first record of its existence and inconsistent with the period in which Jesus lived.