im stumped on this one

Question

A  lab is currently in possession of 980 grams Curium-243, which has a half-life of 29.1 years. Assuming they leave the material alone, how many grams of Cm-243 will they have in the year 2170 to the nearest gram?

Answer 1

One way is ln[A₀]/[A_t] = kt, and k = 0.693/t_1/2, where t_1/2 = half-life.

Another is to calculate the number of half-lives, and multiply by 1/2 or 0.5 to that number as a power.

Try the 2nd way:

from 2020 to 2170 is 150 years, /29.1 = 5.1546 half-lives

0.5^5.1546 = 0.028074

The amount remaining is this X 980 = 27.5129. Rounded to the nearest gram, 28 g.

Answer 2

There's a formula for this sort of thing, but let's start from how you might tackle this if you didn't know the formula. Let's let the variable t mean how many years after Jan 29, 2020.

We might already know that radioactive decay is an exponential decay, meaning it follows a curve that looks something like

y = Ae^ct

Where y is the amount of Curium left after t years. We already have one bit of info: when t=0 (i.e. today), there are 980 grams left. Plugging that in, we have

980 = Ae⁰ = A

So we've already solved for A. Next, we know that when t=29.1, y will be half of 980 or 490. Let's plug that in:

490 = 980e^c(29.1)

Now solve for c. first divide both sides by 980:

0.5 = e^c(29.1)

To get rid of the e on the left, do the opposite operation to exponentiation, which is take the natural logarithm of both sides:

ln(0.5) = 29.1*c

c = -ln(2)/29.1 ≈ -0.02382

So we've solved for c, and we have the final equation. Depending on your teacher's preferences you might either want to leave c in exact form, or in its decimal approximation. I'm just going to use a decimal here since it's neater to format.

y = 980e^-0.02382t

To determine how many grams there will be in 2170, figure out what value of t that represents, then plug that value into your equation and simplify to get y. I'll leave the last step to you!

Answer 3

The initial amount is 980 grams

The rate = .5

Since the half life is 29.1 years, and the total time is 150 years (2020 to 2170), the exponent needs to represent the number of half-lives. We can get that by dividing the total time by the half-life: 150/29.1

P = A(1 - r)^t

P = 980(.5)^150/29.1

Enter that into the calculator to get the answer.