Statements:
1. A radioactive substance has a half-life of eight months. In how much time will three-fourths of the substance decay?
2. In 420 days, the activity of a sample of polonium (Po) decreases to one-eighth of its initial value. What is the half-life of polonium?
3. The half-life of radium is about 1600 years. In how much time will 25 g remain un-decayed from an original 100 g?
4. One-eighth of the initial mass of a certain radioactive isotope remains undecayed after one hour. What is the half-life of the isotope in minutes?
5. What proportion of a radium sample would be decayed after 8000 years? (Half-life of radium is 1600 years.)
6. The half-life of a radioactive element is 15 days. If we initially have 50,000 atoms of this radioactive element, how many atoms will be left after 45 days?
7. What percentage of the original quantity of a radioactive material is left after five half-lives approximately?
1. A radioactive substance has a half-life of eight months. In how much time will three-fourths of the substance decay?
We know that when three-fourths decay, one-fourth remains.
The formula for remaining quantity is:
Where:
- N0 is the initial quantity
- N(t) is the remaining quantity after time
- is the half-life
We want to find when N(t)=1/4N0 and T=8 months. So,
1/4=(1/2)t/8
Taking the logarithm of both sides:
log(1/4)=t/8log(1/2)
Since log(1/4)=−2 and log(1/2)=−0.3010,
−2=t/8×(−0.3010)
Solving for :
t=2×8/0.3010≈53.32 months
So, it will take approximately 53.32 months for three-fourths of the substance to decay.
2. In 420 days, the activity of a sample of polonium (Po) decreases to one-eighth of its initial value. What is the half-life of polonium?
We use the formula:
1/8=(1/2)t/T
where t=420 days. Taking logarithms:
log(1/8)=420/T log(1/2)
Since log(1/8)=−0.9031 and log(1/2)=−0.3010,
−0.9031=420/T×(−0.3010)
Solving for :
T=420×0.3010/0.9031≈140 days
So, the half-life of polonium is approximately 140 days.
3. The half-life of radium is about 1600 years. In how much time will 25 g remain un-decayed from an original 100 g?
We want to find the time when 25 g remains, i.e., half of the substance has decayed. Using the formula:
1/2=(1/2)t/1600
This equation simplifies to:
t=1600 years
So, it will take 1600 years for 25 g to remain.
4. One-eighth of the initial mass of a certain radioactive isotope remains undecayed after one hour. What is the half-life of the isotope in minutes?
We are given that 1/8 of the mass remains after 1 hour, meaning the substance has decayed to 7/8. Using the formula:
N(t)=N0×(1/2)t/T
where t=60 minutes. Taking logarithms:
log(1/8)=60/T log(1/2)
Using log(1/8)=−0.9031 and log(1/2)=−0.3010,
−0.9031=60/T×(−0.3010)
Solving for :
T=60×0.3010/0.9031≈20 minutes
So, the half-life is approximately 20 minutes.
5. What proportion of a radium sample would be decayed after 8000 years? (Half-life of radium is 1600 years.)
We use the formula:
N(t)=N0×(1/2)t/T
where T=1600 years and t=8000 years. Substituting into the formula:
N(8000)=N0×(1/2)8000/1600=N0×(1/2)5
N(8000)=N0×1/32
So, the remaining proportion is 1/32, meaning the decayed proportion is:
1−1/32=31/32
Thus, 31/32 or about 96.88% of the radium sample would be decayed.
6. The half-life of a radioactive element is 15 days. If we initially have 50,000 atoms of this radioactive element, how many atoms will be left after 45 days?
The time t=45 days and the half-life T=15 days. We want to find how many atoms remain. Using the formula:
N(t)=N0×(1/2)t/T
Substituting:
N(45)=50000×(1/2)45/15
=50000×(1/2)3
=50000×1/8
N(45)
So, after 45 days, 6250 atoms will remain.
7. What percentage of the original quantity of a radioactive material is left after five half-lives approximately?
After half-lives, the remaining quantity is (1/2)n. For n=5:
Remaining quantity=(1/2)5=1/32≈0.03125
Thus, approximately 3.125% of the original quantity remains after five half-lives.