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Deffense [45]
3 years ago
13

The half-life of cesium-137 is 30 years. Suppose we have a 40 mg sample. Exercise (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 = · ekt Exercise (b) How much of the sample remains after 20 years? Step 1 After 20 years we have the following. y(20) = 40 · 2 = mg (Round your answer to two decimal places.)
Chemistry
1 answer:
Norma-Jean [14]3 years ago
5 0

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>

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