Calculus

posted by .

The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 4000 years?

  • Calculus -

    Equation for exponential growth/decay:
    y = Ce^(kt)
    Where y is the amount leftover and t is the time elapsed.

    Notice if you plug in time (4000 years) right away, you get
    y = Ce^(4000k), and you can't solve for y (the amount)--there's too many unknowns (C and k).

    So first you have to find C and k in order to find y.
    Use what you know.
    At time t=0 years, you still have the whole sample of 100 mg. Therefore (0,100) is a solution to the equation. Plug these values in.

    100 = Ce^(0*k) so 100 = Ce^(0)
    Anything to the zero power is just one,
    so 100 = C*1 or C = 100

    Now you have to find k. Think of what else we know about half life. At time 1590 years, we'll have half the sample, or 50 mg. Therefore (1590, 50) is a solution to the equation. Plug it in.
    y = Ce^(kt)
    we know C=100, t=1590, and y=50, so we can solve for k.
    50 = 100*e^(1590k)
    1/2 = e^(1590k)
    take the natural log of both sides to bring down the exponent
    ln (1/2) = ln [ e^(1590k)]
    ln (1/2) = 1590k
    therefore k = ln(1/2) / 1590

    You now have your final equation
    y = Ce^(kt) or y = 100e^[(ln(1/2) / 1590)t]
    Now you can plug in any value for time and find the amount left over:

    y = 100e^[(ln(1/2) / 1590)*4000]

    Plugging this into a calculator returns the result 17.5 grams.

  • Calculus -

    Hmmm. If you start with 100mg, I don't see how it will grow to 17.5g after 4000 years.

    Since we are dealing with half-life, it's really not necessary to go through the gyrations of a general exponential function.

    The amount remaining after t years is
    a(t) = 100 (1/2)^(t/1590)
    a(4000) = 17.5 mg

    I guess you are correct numerically. Just the units are off.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Chemistry

    Which radioactive sample would contain the greatest remaining mass of the radioactive isotope after 10 years?
  2. nuclear chem

    i need help solving this. Can you guys show me how it's done. Radium-226 decays by alpha emission to radon-222, a noble gas.What volume of pure Radon- 222 at 23oC and 785mmHg could be obtained from 543.0 mg of radium bromide,RaBr2 …
  3. Chem 3A

    please help me with this chemistry problem. Half-Life A radium-226 sample initially contains 0.112 mol. How much radium-226 is left in the sample after 6400 years?
  4. Calculus

    Radius decays at a rate of that is proportional to its mass, and has a half-life of 1590 years. If 20 g of radium is present initially, how long will it take for 90% of this mass to decay?
  5. chemistry

    A radium-226 sample initially contains 0.112mol. How much radium-226 is left in the sample after 6400 years. The half-life of radium-226 is 1600 years.
  6. Chemistry

    Uranium-238 has half life of 4.51 * 10^9 years Uranium-234 has half life of 2.48 * 10^5 years Thorium-230 has half life of 8.0 * 10^4 years Radium-226 has half life of 1.62 * 10^3 years Lead-210 has half life of 20.4 years. The rate …
  7. Calc

    Radioactive radium () has a half-life of 1599 years. What percent of a given amount will remain after years?
  8. MATHS

    experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment . if half life is 1590 years what percentage will disappear in one year?
  9. MATHS

    please help me ...Experiments show that radium at rate proportional to the amount of radium present at the moment . if half life is 1590 years what percentage will disappear in one year ?
  10. physics

    the half life of radium is 1600 years of 100 grams of radium existing now 25 grams will remain after how many years

More Similar Questions