Math / Calculus

QuestionThe radioactive isotope ${ }^{226} \mathrm{Ra}$ has a half-life of approximately 1599 years. Consider a lab that currently has 45 g of ${ }^{226} \mathrm{Ra}$ . (a.) How much of the isotope remains after 1200 years? Round your answer to three decimal places.

Studdy Solution

STEP 1

1. The decay of the isotope follows an exponential decay model.
2. The half-life of ${}^{226} \mathrm{Ra}$ is 1599 years.
3. The initial amount of ${}^{226} \mathrm{Ra}$ is 45 grams.
4. We are calculating the remaining amount after 1200 years.

STEP 2

1. Recall the formula for exponential decay.
2. Identify and substitute the known values.
3. Calculate the remaining amount after 1200 years.

STEP 3

Recall the formula for exponential decay:
$N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$
where: - $N(t)$ is the remaining quantity of the substance after time $t$ , - $N_0$ is the initial quantity of the substance, - $T_{1/2}$ is the half-life of the substance.

STEP 4

Identify and substitute the known values into the formula:
- $N_0 = 45$ grams, - $T_{1/2} = 1599$ years, - $t = 1200$ years.
$N(1200) = 45 \left(\frac{1}{2}\right)^{\frac{1200}{1599}}$

STEP 5

Calculate the remaining amount after 1200 years:
$N(1200) = 45 \left(\frac{1}{2}\right)^{\frac{1200}{1599}}$
First, calculate the exponent:
$\frac{1200}{1599} \approx 0.750469$
Now calculate the power of $\frac{1}{2}$ :
$\left(\frac{1}{2}\right)^{0.750469} \approx 0.594$
Finally, calculate the remaining amount:
$N(1200) = 45 \times 0.594 \approx 26.730$
The remaining amount of ${}^{226} \mathrm{Ra}$ after 1200 years is:
$\boxed{26.730 \text{ grams}}$

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