The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 4000 years?
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.
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.
To calculate the amount of Radium-226 remaining after 4000 years, we need to determine the number of half-lives that have elapsed.
The formula to calculate the remaining amount of a substance is:
Remaining amount = Initial amount × (1/2)^(number of half-lives)
The half-life of Radium-226 is 1590 years. To find the number of half-lives that have elapsed after 4000 years, we divide the total time by the half-life:
Number of half-lives = Total time / Half-life
Number of half-lives = 4000 years / 1590 years
Now let's calculate it step by step:
Number of half-lives = 4000 / 1590
Number of half-lives ≈ 2.52
Since we cannot have fractions of a half-life, we can round this to 2 half-lives.
Now, we can use the formula to calculate the remaining amount of Radium-226:
Remaining amount = 100 mg × (1/2)^(number of half-lives)
Remaining amount = 100 mg × (1/2)^2
Remaining amount = 100 mg × (1/4)
Remaining amount = 25 mg
After 4000 years, approximately 25 mg of Radium-226 will remain in the sample.
To find out how many milligrams (mg) of Radium-226 will remain after 4000 years, we can use the following formula:
Amount remaining = Initial amount × (1/2)^(time elapsed / half-life)
Let's break down the formula and substitute the given values:
Initial amount = 100 mg (given)
Time elapsed = 4000 years (given)
Half-life of Radium-226 = 1590 years (given)
Amount remaining = 100 mg × (1/2)^(4000 / 1590)
To calculate (1/2)^(4000 / 1590), we can plug it into a calculator or use logarithm properties.
Using a calculator, we get:
Amount remaining ≈ 100 mg × 0.313 ≈ 31.3 mg
Therefore, after 4000 years, approximately 31.3 mg of Radium-226 will remain.