Question

(6 points) Alice owns 20 grams of a radioactive isotope that has a half-life of ln(4) years. (a) Find an equation for the mass m(t) of the remaining isotope Alice owns after t years.

Answer 1

Answer: The equation to calculate the mass of remaining isotope is $[A]=\frac{20}{10^{-0.217t}}$

Explanation:

The equation used to calculate rate constant from given half life for first order kinetics:

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

where,

$t_{1/2}$ = half life of the reaction = $\ln 4=1.386yrs$

Putting values in above equation, we get:

$k=\frac{0.693}{1.386yrs}=0.5yrs^{-1}$

Rate law expression for first order kinetics is given by the equation:

$k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}$

where,

k = rate constant = $0.5yr^{-1}$

t = time taken for decay process

$[A_o]$ = initial amount of the sample = 20 grams

[A] = amount left after decay process =  ? grams

Putting values in above equation, we get:

$0.5=\frac{2.303}{t}\log\frac{20}{[A]}$

$[A]=\frac{20}{10^{-0.217t}}$

Hence, the equation to calculate the mass of remaining isotope is $[A]=\frac{20}{10^{-0.217t}}$