Question

Geologist know that potassium 40 decays to argon 40, with a half life of 1.3 billion years. analysis of a hypothetical sample of granite reveals that 75 percent of the potassium 40 atoms have decyaed to form argon 40. what is the age of the sample of granite

Answer 1

First let us calculate for the rate constant k from the formula:

k = ln(2) / t0.5

where t0.5 is the half life

k = ln(2) / 1.3x10^9 years

k = 5.33x10^-10 years-1

 

Then we use the formula:

A/Ao = e^-kt

where A/Ao is the amount remaining = 25% = 0.25, t is time

 

Rearranging to get t:

t = ln(A/Ao) / -k

t = ln(0.25) / (-5.33x10^-10 years-1)

t = 2.6x10^9 years

Answer 2

Answer : The age of the sample of granite is, 2.6 billion years

Solution : Given,

As we know that the radioactive decays follow the first order kinetics.

First we have to calculate the rate constant.

Formula used : $t_{1/2}=\frac{0.693}{k}$

$1.3\text{ billion years}=\frac{0.693}{k}$

$k=0.533(\text{billion years})^{-1}$

Now we have to calculate the age of the sample of granite.

The expression for rate law for first order kinetics is given by :

$k=\frac{2.303}{t}\log\frac{a}{a-x}$

where,

k = rate constant  = $0.533$

t = time taken for decay process  = ?

a = initial amount of the reactant  = 100 g

a - x = amount left after decay process  = 100 - 75 = 25 g

Putting values in above equation, we get the age of the sample of granite.

$0.533=\frac{2.303}{t}\log\frac{100}{25}$

$t=2.6\text{ billion years}$

Therefore, the age of the sample of granite is, 2.6 billion years