An unknown radioactive element decays into non-radioactive substances. In 100 days the radioactivity of a sample decreases by 64 percent.
a)(a) What is the half-life of the element? half-life:
b)How long will it take for a sample of 100 mg to decay to 51 mg?
time needed:
no
.36 = (1/2)^(100/k)
k = 67.84
so, the half-life is 67.84 days
Makes sense, since the sample has decreased by a little more than 1/2 in 100 days.
51/100 = (1/2)^(t/67.84)
t = 65.9 days
GREETINGS
a) To determine the half-life of the element, we can use the fact that the radioactivity decreases by 64 percent in 100 days. Half-life is the time it takes for half of the radioactive substance to decay.
To find the half-life, we can set up the equation:
(1 - 0.64) = 0.5^(100/half-life)
Simplifying the equation:
0.36 = 0.5^(100/half-life)
We can solve this equation by taking the logarithm of both sides. Using the natural logarithm, the equation becomes:
ln(0.36) = (100/half-life) * ln(0.5)
Now we can solve for the half-life by rearranging the equation:
half-life = 100 / (ln(0.36) / ln(0.5))
Evaluating this expression, we find that the half-life of the unknown radioactive element is approximately 35.09 days.
b) To determine how long it will take for a sample of 100 mg to decay to 51 mg, we can use the half-life we found in part a.
We know that the amount of radioactive substance remaining after a certain period of time is given by the equation:
amount remaining = initial amount * (0.5)^(time elapsed / half-life)
We can set up the equation and solve for time elapsed:
51 mg = 100 mg * (0.5)^(time elapsed / 35.09)
Simplifying the equation:
0.51 = (0.5)^(time elapsed / 35.09)
Taking the logarithm of both sides and rearranging the equation, we have:
time elapsed = 35.09 * log(0.51) / log(0.5)
Evaluating this expression, we find that it will take approximately 22.51 days for a sample of 100 mg to decay to 51 mg.