Question

The isotope radium-226 decays into radon-222, with a half-life of around 1,600 years. If a rock contained 6 grams of radium-226 when it reached its closure temperature but only 0.375 grams when it was discovered, which two statements about the rock are true?
A] The rock reached its closure temperature 6,400 years ago.
B] The rock reached its closure temperature 4,800 years ago.
C]The rock had 2.625 grams of radon-222 1,600 years ago.
D]When the rock was discovered, it had 5.625 grams of radon-222.
E]When the rock was discovered, it had 3.375 grams of radon-222.

Answer 1

A and D is the answer

Answer 2

Answer : The correct option is, (A) The rock reached its closure temperature 6,400 years ago.

Explanation :

Half-life = 1600 years

First we have to calculate the rate constant, we use the formula :

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

$k=\frac{0.693}{1600\text{ years}}$

$k=4.33\times 10^{-4}\text{ years}^{-1}$

Now we have to calculate the time passed.

Expression for rate law for first order kinetics is given by:

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

where,

k = rate constant  = $4.33\times 10^{-4}\text{ years}^{-1}$

t = time passed by the sample  = ?

a = initial amount of the reactant  = 6 g

a - x = amount left after decay process = 0.375 g

Now put all the given values in above equation, we get

$t=\frac{2.303}{4.33\times 10^{-4}}\log\frac{6}{0.375}$

$t=6404.36\text{ years}\approx 6400\text{ years}$

Therefore, the correct option is, (A) The rock reached its closure temperature 6,400 years ago.