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3 years ago

7

(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.

1 answer:

Cloud [144]3 years ago

6 0

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

<u>Explanation:</u>

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

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