Answer:
1) y(t) = (40.0 mg)(e∧(-(0.0231 year⁻¹)t).
2) 25.2 mg.
Explanation:
(a) Find the mass that remains after t years. Step 1 Let y(t) be the mass (in mg) remaining after t years. Then we know the following. y(t) = y(0)ekt
For first order reactions: y(t) = y(0)(e∧-kt)
where, y(t) is the mass of the substance at any time (t).
y(0) is the initial concentration of the substance at (t = 0) (y(0) = 40.0 mg).
k is the rate constant of the reaction.
t is the time of the reaction.
For first order reactions:<em> k = ln2/(t1/2)</em> = 0.693/(30 years) = 0.0231 year⁻¹.
<em>∴ y(t) = y(0)(e∧-kt)</em>
<em> y(t) = (40.0 mg)(e∧(-(0.0231 year⁻¹)t).</em>
<em>Exercise (b) How much of the sample remains after 20 years?</em>
<em>∵ y(t) = y(0)(e∧kt)</em>
k = 0.03465 year⁻¹, t = 20.0 years, y(0) = 40.0 mg.
<em>∴ y(t) = y(0)(e∧-kt) </em>= (40.0 mg)e∧-(0.0231 year⁻¹)(20.0 years) = <em>25.2 mg.</em>
