Question

Element X decays radioactively with a half life of 13 minutes. If there are 110 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 2 grams?

Answer 1

Answer : The time taken by the element to decay to 2 grams is, 75.2 minutes

Step-by-step explanation:

Half-life = 13 min

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

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

$k=\frac{0.693}{13\text{ min}}$

$k=0.0533\text{ min}^{-1}$

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  = $0.0533\text{ min}^{-1}$

t = time passed by the sample  = ?

a = initial amount of the reactant  = 110 g

a - x = amount left after decay process =  2 g

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

$t=\frac{2.303}{0.0533}\log\frac{110}{2}$

$t=75.2\text{ min}$

Therefore, the time taken by the element to decay to 2 grams is, 75.2 minutes