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Answer :
Final answer:
After 6 hours, the mass of the radioactive metal X would be 5 mg. It would take approximately 796 minutes to reduce the mass of the radioactive metal to 1.25 mg.
Explanation:
In radioactive decay, the concept of half-life is used to determine the amount of radioactive material that remains after a certain period of time. The half-life is the time it takes for half of the radioactive material to decay. In this case, the half-life of metal X is 80 minutes.
A. What is the mass of this radioactive metal remaining after 6 hours?
To determine the mass remaining after 6 hours, we need to convert the given time into minutes. Since there are 60 minutes in an hour, 6 hours is equal to 6 x 60 = 360 minutes.
Next, we can use the half-life information to calculate the number of half-life periods that have passed in 360 minutes. Dividing 360 by 80, we find that approximately 4.5 half-lives have passed.
To calculate the mass remaining after 4.5 half-lives, we can use the formula: remaining mass = initial mass * (1/2)^(number of half-lives).
Given that the initial mass is 160 mg, we can calculate the remaining mass as follows:
Remaining mass = 160 * (1/2)^(4.5) = 160 * 0.03125 = 5 mg.
Therefore, the mass of the radioactive metal remaining after 6 hours is 5 mg.
B. How long does it take to reduce the mass to 1.25 mg?
To determine the time it takes to reduce the mass to 1.25 mg, we can use a similar approach. We'll use the formula: number of half-lives = log0.5(final mass/initial mass).
Substituting the given values, we have:
number of half-lives = log0.5(1.25/160) = log0.5(0.0078125).
Using a scientific calculator, we can find that the number of half-lives is approximately 9.95.
To convert the number of half-lives into minutes, we multiply it by the half-life time of 80 minutes:
time = 9.95 * 80 = 796 minutes.
Therefore, it takes approximately 796 minutes to reduce the mass of the radioactive metal to 1.25 mg.
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