Question

Gold-192 is a radioactive isotope with a half-life of 4.95 hours. How long would it take for a 25 gram sample of Gold-192 to decay until only 2 grams remain? a. about 9.1 hours b. about 31 hours c. about 18 hours d. about 61.9 hours

Answer 1

Answer:

Step-by-step explanation:

The formula you will need for this is:

$A=A_0(\frac{1}{2})^{\frac{t}{H}$ where A is the amount after some decay has happened, A₀ is the intial amount, t is the time in hours, and H is the half-life in hours. Those values for us are:

A = 2 g

A₀ = 25 g

H = 4.95 hrs and

t = ?

Filling in:

$2=25(\frac{1}{2})^{\frac{t}{4.95}$ Keep in mind that, because of the nature of the exponential form of this equation, you CANNOT simply multiply the 25 by the 1/2. Exponential equations don't work that way. Begin instead by dividing both sides by 25 to get

$.08=(\frac{1}{2})^{\frac{t}{4.95}$ The goal is to get that t out from its exponential position. Do that by taking the natural log of both sides:

$ln(.08)=ln(.5)^{\frac{t}{4.95}$

After you take the natural log of the right side, the property allows you to bring the exponent down out front:

$ln(.08)=\frac{t}{4.95}ln(.5)$

Now divide both sides by ln(.5) to get

$\frac{ln(.08)}{ln(.5)}=\frac{t}{4.95}$

Simplify the left side out on your calculator to get

$\frac{-2.525728644}{-.6931471806}=\frac{t}{4.95}$ and then divide:

$3.643856=\frac{t}{4.95}$

Finally, multiply both sides by 4.95 to get

3.643856(4.95) = t so

t = 18.0 hours which is choice C