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

Explanation:

The formula you will need for this is:

`A=A_0((1)/(2))^{(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((1)/(2))^{(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=((1)/(2))^{(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)^{(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)=(t)/(4.95)ln(.5)`

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

`(ln(.08))/(ln(.5))=(t)/(4.95)`

Simplify the left side out on your calculator to get

`(-2.525728644)/(-.6931471806)=(t)/(4.95)` and then divide:

`3.643856=(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