You have just isolated a new radioactive element. If you can determine its half-life, you will win the Nobel Prize in physics. You purify a sample of 2 grams. One of your colleagues steals half of it, and eight days later you find that 0.1 gram of the radioactive material is still left. What is the half-life? (Round your answer to three significant digits.)
after the theft, there was 1 g left.
If the half-life is n days, then the fraction left after t days is
(1/2)^(t/n)
so, solve this for n:
0.1 = (1/2)^(8/n)
ln 0.1 = (8/n) ln 0.5
ln .1/ln .5 = 8/n
n = 8 * ln .5/ln .1
n = 2.4 days
To determine the half-life of the radioactive element, we can use the equation for exponential decay:
N = N0 * (1/2)^(t / T)
Where:
N is the final amount of radioactive material
N0 is the initial amount of radioactive material
t is the time that has passed
T is the half-life of the radioactive element
In this case, we have an initial amount of 2 grams and a final amount of 0.1 grams. Let's plug in these values into the equation:
0.1 = 2 * (1/2)^(t / T)
To simplify, divide both sides of the equation by 2:
0.05 = (1/2)^(t / T)
Now, take the logarithm of both sides to solve for t / T:
log(0.05) = log((1/2)^(t / T))
Using logarithmic properties, we can bring down the exponent:
log(0.05) = (t / T) * log(1/2)
Now, we need to solve for t / T, so divide both sides of the equation by log(1/2):
(t / T) = log(0.05) / log(1/2)
Evaluate the right side of the equation using a calculator:
(t / T) ≈ -2.995
Now, we can solve for T by multiplying both sides of the equation by T:
t ≈ -2.995 * T
Since the half-life cannot be negative, we take the absolute value:
t ≈ 2.995 * T
Now, we know that after eight days, 0.1 grams of the radioactive material is left. We need to convert this time into the units of the half-life (T) to find the value of T:
8 days ≈ 2.995 * T
Divide both sides of the equation by 2.995 to solve for T:
T ≈ 8 / 2.995
Evaluate the right side of the equation to get the value of T:
T ≈ 2.672 days
Rounding to three significant digits, the half-life of the radioactive element is approximately 2.67 days.