16. A technician places 50 mg of a radioactive substance into a laboratory chamber.
After 10 minutes, 41 mg remain.
a) Determine the half-life of the substance.
b) How long will it take the substance to decay to 1% of its initial amount?
a) To determine the half-life of the substance, we first need to calculate the decay constant (k) using the formula:
Remaining mass = Initial mass * e^(-k*t)
where:
Remaining mass = 41 mg
Initial mass = 50 mg
t = 10 minutes
Rearranging the formula, we can solve for k:
k = ln(Remaining mass/Initial mass) / (-t)
Plugging in the given values, we get:
k = ln(41/50) / (-10)
k ≈ -0.035
The half-life (t1/2) of a radioactive substance can be determined using the equation:
t1/2 = ln(2) / k
Plugging in the calculated value of k, we get:
t1/2 ≈ ln(2) / (-0.035)
t1/2 ≈ 19.8 minutes
Therefore, the half-life of the substance is approximately 19.8 minutes.
b) To determine how long it will take the substance to decay to 1% of its initial amount, we can use the decay equation again:
Remaining mass = Initial mass * e^(-k*t)
where:
Remaining mass = 1% of the initial mass = 0.01 * 50 mg = 0.5 mg
Initial mass = 50 mg
t = time in minutes
Rearranging the formula, we can solve for t:
t = (-1/k) * ln(Remaining mass/Initial mass)
Plugging in the given values, we get:
t = (-1/-0.035) * ln(0.5/50)
t ≈ 68.84 minutes
Therefore, it will take approximately 68.84 minutes for the substance to decay to 1% of its initial amount.