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Q 3.2-23E

Expert-verified
Found in: Page 101

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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In Problems 23–27, assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioactive substance is the time it takes for one-half of the substance to disintegrate. If initially there are 50 g of a radioactive substance and after 3 days there are only 10 g remaining, what percentage of the original amount remains after 4 days?

After 4 days, the remaining radioactive substance will be 11.7% of the original amount.

See the step by step solution

Step 1: Analyzing the given statement

Given that the rate of decay of a radioactive substance is directly proportional to the amount of the substance present. Let the present amount of the radioactive substance be N.

Therefore, $\frac{dN}{dt}\propto N$

Given that there are 50g of a radioactive substance and after 3 days there are only 10g remain. We have to find the mass of the substance, remaining after 4 days, and its percentage of the original amount.

Step 2: Determining the formula with the help of the given proportionality relation, to solve the question

Given,

$\frac{dN}{dt}\propto N\phantom{\rule{0ex}{0ex}}\frac{dN}{dt}=-\lambda N\phantom{\rule{0ex}{0ex}}$

where, $\lambda$is the constant of proportionality.

$\frac{dN}{N}=-\lambda \phantom{\rule{0ex}{0ex}}\int \frac{dN}{N}=-\lambda \int dt\phantom{\rule{0ex}{0ex}}lnN=-\lambda t+ln{N}_{0}\phantom{\rule{0ex}{0ex}}$

where, In N0 is an arbitrary constant.

$lnN-ln{N}_{0}=-\lambda t\phantom{\rule{0ex}{0ex}}ln\left(\frac{N}{{N}_{0}}\right)=-\lambda t\phantom{\rule{0ex}{0ex}}\frac{N}{{N}_{0}}={e}^{-\lambda t}\phantom{\rule{0ex}{0ex}}N={N}_{0}{e}^{-\lambda t}······\left(1\right)\phantom{\rule{0ex}{0ex}}$

One will use this formula to solve the question.

Step 3: Using the formula obtained in the step 2, we will find the value of λ

Let the initial amount of the radioactive substance be i.e., N0= 50g and given that the remaining amount of radioactive substance after 3 days is 10g i.e.,

t = 3 days and N = 10g

Now, from the equation (1),

$10=50{e}^{-3\lambda }\phantom{\rule{0ex}{0ex}}\frac{1}{5}={e}^{-3\lambda }\phantom{\rule{0ex}{0ex}}{e}^{3\lambda }=5\phantom{\rule{0ex}{0ex}}3\lambda =ln5\phantom{\rule{0ex}{0ex}}\lambda =\frac{ln5}{3}\phantom{\rule{0ex}{0ex}}\lambda =0.5365\phantom{\rule{0ex}{0ex}}$

One will use this value of $\lambda$ in the next step to find the value of the mass of the remaining radioactive substance after 4days.

Step 4: Finding the mass of the remaining radioactive substance after 4 days

Now we will find the mass of the remaining radioactive substance after 4 days.

For this, let N be the mass to be found,

N0=50 g

Time, t = 4 days

(From Step 3)

Using the equation (1),

$N={N}_{0}{e}^{-\lambda t}\phantom{\rule{0ex}{0ex}}N=\left(50\right)·{e}^{-\left(0.5365\right)4}\phantom{\rule{0ex}{0ex}}N=5.847g\phantom{\rule{0ex}{0ex}}$

Hence, the mass of the remaining radioactive substance after 4 days is 5.847 g.

Step 5: Determining what percentage of the original amount remains after 4 days

The mass of the remaining radioactive substance after 4 days is 5.847 g

Therefore,

Percentage of remaining mass

$=\frac{5.847}{{N}_{0}}×100\phantom{\rule{0ex}{0ex}}=\frac{5.847}{50}×100\phantom{\rule{0ex}{0ex}}=11.7%\phantom{\rule{0ex}{0ex}}$

Thus, the percentage of the original amount that remains after 4 days is 11.7%.

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