SME
Grade 12 Science · Problem type
Dating the oldest rocks with uranium-238
Use this for “A meteorite has as much lead-206 as uranium-238. How old is it?”
Diagram
After 4.47 × 10⁹ years, 1 half-life has passed: 100 × (1/2)¹ = 50% left.
In the grid 50 of 100 U-238 atoms are left; each half-life, half of those still there decay.
This is an example. Type your own values to start a new problem.
p
%
=
100%
÷
(1
+
R
)
=
100%
×
(0.5)
÷
(4.47
×
10⁹)
Each lead-206 atom was once a U-238 atom, so 1 + R atoms started for each one left: p = 100% ÷ (1 + R). The half-lives in p then give the age.
n
Half-lives passed
worked out (not typed)
Formulas
p
=
100
1 +
R
50
=
100
1 + 1
p
=
100
×
0.5
n
50
=
100
×
0.5
1
t
=
n
×
4.47 ×
10
9
4.47 ×
10
9
=
1
×
4.47 ×
10
9
Assumptions
•
U-238 decays to lead-206 with a half-life of 4.47 × 10⁹ years.
•
The rock started with no lead-206 and lost none.
•
Most surface rocks were melted or weathered since Earth formed. Meteorites were not, so they date the solar system.
•
Nothing in the solar system is older than about 4.57 × 10⁹ years, so R is at most about 1.
Step-by-step
WE KNOW
R = 1
FIND
U-238 left (p), half-lives passed (n), age (t)
Step 1 · Find U-238 left (p)
Use p = 100 ÷ (1 + R)
Each
lead
atom
was
once
a
uranium
atom:
the
uranium’s
share
of
1
+
R
atoms.
p
=
100
1 + 1
p
=
100
2
p
=
50%
Step 2 · Find half-lives passed (n)
Use p = 100 × 0.5ⁿ
Count
the
halvings:
n
is
the
power
of
2
that
equals
100
p
.
n
=
ln(
100
p
)/ln(2)
n
=
ln(
100
50
)/ln(2)
n
=
0.6931/0.6931
n
=
1
Step 3 · Find age (t)
Use t = n × 4.47 × 10⁹
The age is the half-lives passed times the half-life.
t
=
1
×
4.47 ×
10
9
t
=
4.47 ×
10
9
years
Check
Put the numbers back in. Both sides should match.
50
=
100
1 + 1
✓
50
=
100
×
0.5
1
✓
4.47 ×
10
9
=
1
×
4.47 ×
10
9
✓
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