Question

A fossil you are radiometrically dating contains 4 micrograms of Uranium 235 and 4 micrograms of Lead 207. Uranium 235 decays to Lead 207; the half-life of Uranium 235 is 710 million years. How old is the fossil

Answer 1

The rate at which radioactive substances disintegrate is a constant for a

given radioactive material.

• The fossil is 710 million years old.

Reasons:

The half-life of a radioactive material is the time it takes half of the nucleus

of a radioactive material to disintegrate into other forms of materials

through the given off of energy and particles.

The half life of Uranium 235 = 710 million years

The product of the decay of Uranium 235 = Lead 207

The mass of Uranium 235 in the fossil = 4 micrograms

The mass of Lead 207 in the sample = 4 micrograms

Therefore, the mass of Lead 207 in the fossil is equal to the mass of

Uranium 235, therefore, a minimum of half of the Uranium 235 has

decomposed, which gives;

The time of decomposition of the Uranium 235 = 1 Half life = 710 million years

The age of the fossil = The time in which the Uranium has been

decomposing = The time of decomposition = 1 half life of Uranium 235 = 710

million years

• The age of the fossil = 710 million years

Using the formula for half-life, we get;

$\displaystyle N(t) = N_0 \left (\dfrac{1}{2} \right )^{\dfrac{t}{t_{1/2}}$

The fossil contains initially only Uranium 235 with a minimum mass of 4 mg

+ 4 mg = 8 mg, which gives;

N₀ = 8 mg

N(t) = The current mass of Uranium 235 = 4 mg

$\displaystyle t_{1/2}$ = 710 million years

$\displaystyle 4 = 8 \cdot \left (\dfrac{1}{2} \right )^{\dfrac{t}{710}$

$\displaystyle \frac{4}{8} = \frac{1}{2} = \left (\dfrac{1}{2} \right )^{\dfrac{t}{710}$

Therefore;

$\displaystyle \left( \frac{1}{2}\right)^1 = \left (\dfrac{1}{2} \right )^{\dfrac{t}{710}$

$\displaystyle 1 = {\dfrac{t}{710}$

t = 710

• The age of the fossil, t = 710 million years

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