CHEM 132 - ADVANCED GENERAL
CHEMISTRY II - Radioactive decay
calculations
Question Bank - Set 3
Liberty University
Question 1
Question
A certain radioactive substance decays exponentially at a rate of 0.03 per year.
If there are 500 grams of the substance initially, determine the mass of the
substance remaining after 10 years. Round your answer to two decimal places.
Solution
Step 1: Determine the decay constant, λ, using the decay rate formula:
λ=−ln(1 −0.03)
1≈0.0305
Step 2: Use the radioactive decay formula to find the mass of the substance
remaining after 10 years:
Mass remaining = 500 ×e−λ·10
Step 3: Substitute the values of λ, initial mass, and time into the formula:
Mass remaining = 500 ×e−0.0305·10 ≈379.66 grams
Therefore, the mass of the substance remaining after 10 years is approxi-
mately 379.66 grams.
Question 2
Question
A certain radioactive substance decays at a rate proportional to the amount
present. If the initial amount of the substance is 100 grams and after 3 hours,
the amount remaining is 65 grams, what is the half-life of this substance?
Solution
Let A(t) be the amount of the substance remaining after thours. We can write
the differential equation for the decay as:
dA
dt =−kA
where kis the decay constant.
Step 1: Solve the differential equation to find the general formula for A(t).
dA
dt =−kA
Separate variables: dA
A=−kdt
Integrate both sides:
Z1
AdA =Z−kdt
ln |A|=−kt +C
A=Ce−kt
where Cis the constant of integration.
Step 2: Use the initial condition A(0) = 100 to find the value of C.
A(0) = Ce0=C= 100
So, the equation for A(t) is A(t) = 100e−kt.
Step 3: Use the information given that A(3) = 65 to find the decay constant
k.
A(3) = 100e−3k= 65
e−3k= 0.65
−3k= ln(0.65)
k=−ln(0.65)
3
Step 4: The half-life T1
2of a substance is the time it takes for half of the
substance to decay. We can find T1
2using the formula T1
2=ln(2)
k.
T1
2=ln(2)
−ln(0.65)
3
= 3 ln(2) ≈2.079hours
Therefore, the half-life of this substance is approximately 2.079 hours.
2
Question 3
Question
A certain radioactive substance decays according to the model A(t) = A0e−kt,
where A(t) is the amount of substance remaining at time t,A0is the initial
amount of substance, and kis the decay constant. If 25
Solution
Step 1: Given the model A(t) = A0e−kt, we know that after 100 years, the
amount of substance remaining is 75
0.75A0=A0e−k·100
Step 2: Divide both sides by A0to simplify the equation:
0.75 = e−100k
Step 3: Take the natural logarithm of both sides to eliminate the exponential
term:
ln(0.75) = lne−100k
Step 4: Recall that ln(ex) = xfor any real number x:
ln(0.75) = −100k
Step 5: Solve for k:
k=−ln(0.75)
100 ≈0.00547
Therefore, the value of kfor the radioactive substance is approximately
0.00547 per year.
Question 4
Question
A sample of radium-226 decays according to the function Q(t) = Q0e−0.000187t,
where Q(t) is the quantity of radium-226 remaining after tyears and Q0is the
initial quantity. If the initial quantity of radium-226 is 100 grams, determine
how long it will take for 25 grams of radium-226 to remain.
3
Solution
Step 1: Find the decay constant k
From the given function, we can see that the decay constant kis equal to
0.000187.
Step 2: Set up the equation to find the time when 25 grams remains
We are given that Q(t) = Q0e−kt and we want to find twhen Q(t) = 25.
Plugging in Q(t) = 25 and Q0= 100 into the equation gives:
25 = 100 ·e−0.000187t
Step 3: Solve for t
Divide both sides by 100 to isolate the exponential term:
0.25 = e−0.000187t
Step 4: Take the natural logarithm of both sides
We take the natural logarithm of both sides to solve for t:
ln(0.25) = lne−0.000187t
ln(0.25) = −0.000187t
Step 5: Solve for t
Divide by −0.000187 to solve for t:
t=ln(0.25)
−0.000187
t≈9189.26 years
Therefore, it will take approximately 9189 years for 25 grams of radium-226
to remain.
Question 5
Question
An unknown radioactive substance has an initial activity of 2000 decays per
second. After 15 minutes, the activity has decreased to 500 decays per second.
What is the half-life of this substance?
Solution
Step 1: We can use the formula for radioactive decay:
N(t) = N01
2t
T1
2
4
where: - N(t) is the final activity after time t, - N0is the initial activity, - T1
2
is the half-life of the substance.
Step 2: Substitute the given values into the formula. After 15 minutes,
the final activity N(t) = 500 decays per second, the initial activity N0= 2000
decays per second, and the time t= 15 minutes. We want to find T1
2.
Step 3: Convert the time from minutes to seconds since the unit of time in
the formula is seconds.
t= 15 minutes ×60 seconds/minute = 900 seconds
Step 4: Substitute the values into the formula and solve for T1
2:
500 = 2000 1
2900
T1
2
Step 5: Divide both sides by 2000 to isolate the exponential term:
500
2000 =1
2900
T1
2
Step 6: Simplify the left side to get 1
4:
1
4=1
2900
T1
2
Step 7: Rewrite the right side using base 2:
1
4= 2−2= 2
−2×900
T1
2
Step 8: Equate the exponents:
−2 = −1800
T1
2
Step 9: Solve for T1
2:
T1
2=1800
2= 900 seconds
Step 10: Therefore, the half-life of the radioactive substance is 900 seconds.
Question 6
Question
A sample of a radioactive isotope has an initial mass of 10 grams. After 20
hours, only 2.5 grams of the isotope remain. If the half-life of the isotope is 24
hours, what is the decay constant for this isotope?
5
Solution
Step 1: Determine the fraction of the isotope that remains after 20 hours. To
find the fraction that remains after a certain time, we use the formula:
N(t) = N01
2t
T1/2
where: - N(t) is the remaining mass after time thours, - N0is the initial mass,
-T1/2is the half-life of the isotope.
Plugging in the values:
N(20) = 10 1
220
24
N(20) = 10 1
25
6
N(20) = 10 (0.5)5
6
N(20) = 10 0.51
65
N(20) ≈10 ×0.57435
N(20) ≈5.7435 grams
Therefore, 5.7435 grams of the isotope remain after 20 hours.
Step 2: Calculate the decay constant. The decay constant (λ) can be found
from the formula:
λ=ln(2)
T1/2
where ln(2) is the natural logarithm of 2.
Plugging in the given half-life:
λ=ln(2)
24
λ≈0.6931
24
λ≈0.02888 hours−1
Therefore, the decay constant for this isotope is approximately 0.02888
hours−1.
Question 7
Question
A sample of a radioactive isotope has an initial mass of 100 grams. After 5
hours, only 25 grams of the isotope remain. If the half-life of the isotope is 3
hours, determine the decay constant and the age of the sample.
6
Solution
Step 1: Determine the decay constant.
Let Nbe the remaining mass of the isotope at time t.
The decay of the isotope follows the formula N(t) = N0·e−kt, where N0
is the initial mass, kis the decay constant, and tis the time elapsed.
Given that N0= 100 grams and N(5) = 25 grams, we have 25 = 100·e−5k.
Dividing both sides by 100 gives us 0.25 = e−5k.
Taking the natural logarithm of both sides, we get ln(0.25) = lne−5k.
Simplifying further, we have ln(0.25) = −5k.
Solving for kgives us k≈0.5108 per hour.
Step 2: Determine the age of the sample.
Since the half-life of the isotope is 3 hours, we know that N0/2 = 100/2 =
50 grams of the isotope remains after 3 hours.
Substituting N0= 100 grams, N(3) = 50 grams, and k≈0.5108 into the
decay formula N(t) = N0·e−kt, we get 50 = 100 ·e−3·0.5108.
Solving for tgives t≈3.09 hours.
Therefore, the decay constant of the isotope is approximately 0.5108 per
hour, and the age of the sample is approximately 3.09 hours.
Question 8
Question
A sample of a radioactive isotope decays such that the number of atoms remain-
ing after t days is given by the function N(t) = 100e−0.05t, where N(t) is the
number of atoms and tis the time in days. Determine the rate of decay of the
sample after 20 days.
Solution
Step 1: Find the derivative of the function N(t) with respect to t.
dN
dt =d
dt(100e−0.05t)
Step 2: Apply the chain rule to differentiate 100e−0.05t.
dN
dt =−100(0.05)e−0.05t
7
Step 3: Simplify the result.
dN
dt =−5e−0.05t
Step 4: Evaluate the rate of decay after 20 days by substituting t= 20 into
the derivative. dN
dt
t=20
=−5e−0.05×20
Step 5: Calculate the rate of decay after 20 days.
dN
dt
t=20
=−5e−1≈ −5(0.3679) ≈ −1.8395
Therefore, the rate of decay of the sample after 20 days is approximately
1.8395 atoms per day.
Question 9
Question
A certain radioactive substance has a half-life of 10 days. If a sample initially
contains 100 grams of the substance, determine the amount of the substance
remaining after 30 days.
Solution
Let’s denote the amount of the radioactive substance remaining after tdays as
A(t).
Step 1: Determine the decay constant λusing the half-life formula.
λ=ln(2)
T1
2
=ln(2)
10
Step 2: Write the differential equation for radioactive decay:
dA
dt =−λA(t)
Step 3: Solve the initial value problem:
A(t) = Ce−λt
Using the initial condition A(0) = 100, we find C:
100 = Ce−λ·0=C
So, the equation becomes:
A(t) = 100e−ln(2)
10 ·t
8
Step 4: Calculate the amount of substance remaining after 30 days:
A(30) = 100e−ln(2)
10 ·30
A(30) = 100e−3 ln(2)
A(30) = 100(2−3)
A(30) = 100 ·1
8
A(30) = 12.5 grams
Question 10
Question
A radioactive substance decays according to the equation N(t) = N0e−0.03t,
where N(t) is the amount of substance remaining after tyears, N0is the initial
amount of substance, and 0.03 is the decay constant. If the initial amount of
the substance is 500 grams, find the amount of substance remaining after 20
years.
Solution
Step 1: Find the amount of substance remaining after 20 years using the formula
N(t) = N0e−0.03t.
N(20) = 500e−0.03×20
= 500e−0.6
= 500 ×0.5488
= 274.4 grams
Therefore, after 20 years, there will be 274.4 grams of the substance remain-
ing.
Question 11
Question
A sample of radioactive material has an initial mass of 100 grams. After 10
days, only 12.5 grams of the material remain. If the half-life of the material is
3 days, determine the decay constant λassociated with this material.
9
Solution
Step 1: Calculate the decay constant using the formula: λ=ln(2)
t1/2
where
ln(2) ≈0.6931.
λ=ln(2)
t1/2
=0.6931
3≈0.2310 per day
Therefore, the decay constant λassociated with this material is approxi-
mately 0.2310 per day.
Question 12
Question
A certain radioactive substance has a half-life of 30 days. If you start with a
sample containing 2 ×106atoms, how many atoms will remain after 90 days?
Solution
Step 1: Calculate the decay constant λusing the formula T1/2=ln(2)
λ, where
T1/2is the half-life.
Step 1: λ=ln(2)
30 ≈0.0231 per day
Step 2: Use the formula for radioactive decay to find the number of atoms
remaining after 90 days.
N(t) = N0·e−λt
N(90) = 2 ×106·e−0.0231×90
N(90) = 2 ×106·e−2.079
Step 3: Solve for N(90).
N(90) ≈2×106·0.125 ≈2.5×105
Therefore, after 90 days, there will be approximately 2.5×105atoms re-
maining.
Question 13
Question
A radioactive substance decays at a rate of 3
10
Solution
Step 1: Let A(t) represent the amount of the radioactive substance remaining
after tdays. The rate of decay can be expressed as:
A′(t) = kA(t)
where kis the decay constant. Since the substance decays at a rate of 3
Step 2: The differential equation is now:
dA
dt =−0.03A(t)
Step 3: Solve the differential equation using separation of variables:
dA
A=−0.03dt
ZdA
A=Z−0.03dt
ln |A|=−0.03t+C
A=Ce−0.03t
Step 4: Use the initial condition A(0) = 500 to find the value of C:
500 = Ce−0.03×0
C= 500
Step 5: The equation for the amount of substance remaining is now:
A(t) = 500e−0.03t
Step 6: Calculate the amount of the substance remaining after 10 days:
A(10) = 500e−0.03×10
A(10) = 500e−0.3
A(10) ≈500 ×0.74082
A(10) ≈370.41 grams
Therefore, after 10 days, there are approximately 370.41 grams of the sub-
stance remaining.
Question 14
Question
A certain radioactive substance has a half-life of 25 years. If there are initially
200 grams of the substance, how much will remain after 75 years? Round your
answer to the nearest gram.
11
Solution
Step 1: Determine the decay constant using the half-life formula N(t) = N0·
(1/2)(t/T1/2), where: - N(t) is the amount remaining after time t-N0is the
initial amount - T1/2is the half-life of the substance
Plugging in the values, we have:
200 = 200 ·(1/2)(75/25) = 200 ·0.53= 200 ·0.125
200 = 25
Only 25 grams will remain after 75 years.
Question 15
Question
A sample of a radioactive material has an initial mass of 500 grams. After 10
minutes, only 150 grams of the material remains. If the half-life of the material
is 5 minutes, what is the decay constant of the material?
Solution
Step 1: Calculate the decay constant using the formula for radioactive decay.
Decay constant = −ln(1/2)
T1/2
where T1/2is the half-life of the material.
Step 2: Given T1/2= 5 minutes and the initial mass M0= 500 grams, and
the final mass M= 150 grams after 10 minutes, we can find the decay constant.
λ=−ln(1/2)
5
Step 3: Calculate the fraction of material remaining after 10 minutes.
M
M0
=e−λt
150
500 =e−λ×10
Step 4: Solve for the decay constant by plugging in the values and solving
the equation. 3
10 =e−10λ
Step 5: Take the natural logarithm of both sides to solve for the decay
constant.
ln 3
10= lne−10λ
12
ln 3
10=−10λ
Step 6: Finally, solve for the decay constant.
λ=−ln 3
10
10
λ≈0.0790 minutes−1
Therefore, the decay constant of the material is approximately 0.0790 minutes−1.
Question 16
Question
A radioactive substance has a half-life of 30 days. If the initial mass of the
substance is 100 grams, determine the mass of the substance remaining after 90
days.
Solution
Step 1: Determine the decay constant λ. The decay constant λis related to the
half-life T1
2by the formula:
λ=ln(2)
T1
2
Substitute T1
2= 30 days into the formula:
λ=ln(2)
30 ≈0.0231 days−1
Step 2: Use the exponential decay formula N(t) = N0·e−λt. Given that
N0= 100 grams and t= 90 days, we can find N(90):
N(90) = 100 ·e−0.0231·90 ≈31.71 grams
So, the mass of the substance remaining after 90 days is approximately 31.71
grams.
Question 17
Question
A sample of a radioactive isotope decays according to the equation N(t) =
N0e−kt, where N(t) is the amount of the isotope present at time t,N0is the
initial amount of the isotope, kis the decay constant, and tis the time elapsed.
Suppose a sample of a radioactive isotope has an initial amount of 200 grams
and a decay constant of 0.05 per year.
Calculate: (a) The amount of the isotope that remains after 10 years. (b)
The time it takes for 95
13
Solution
(a) To find the amount of the isotope that remains after 10 years, we can use the
given equation N(t) = N0e−kt and substitute N0= 200, k= 0.05, and t= 10:
N(10) = 200 ·e−0.05·10
Step 1: Calculate the exponent.
N(10) = 200 ·e−0.5
Step 2: Evaluate the exponential term.
N(10) = 200 ·e−0.5≈200 ·0.6065 ≈121.3
So, the amount of the isotope that remains after 10 years is approximately
121.3 grams.
(b) To find the time it takes for 95
0.05N0=N0·e−kt
Step 1: Simplify the equation.
0.05 = e−kt
Step 2: Take the natural logarithm of both sides.
ln(0.05) = lne−kt
Step 3: Use the property of logarithms to bring down the exponent.
ln(0.05) = −kt ln(e)
Step 4: Simplify the right side.
ln(0.05) = −kt
Step 5: Solve for t.
t=ln(0.05)
−k≈ln(0.05)
−0.05 ≈13.5
Therefore, it takes approximately 13.5 years for 95
Question 18
Question
A sample of a radioactive isotope has an initial mass of 100 grams. After 30
days, only 25 grams of the isotope remain. If the half-life of the isotope is 15
days, determine the decay constant and the age of the sample.
14
Solution
Step 1: Calculate the decay constant using the half-life formula N=N0·1
2t
T1/2,
where: - Nis the final mass (25 grams), - N0is the initial mass (100 grams), -
tis the elapsed time (30 days), - T1/2is the half-life (15 days).
Step 2: Substitute the given values into the formula and solve for the decay
constant.
Step 3: Once you have the decay constant, use the decay equation N=
N0·e−λt to find the age of the sample.
Step 4: Substitute the calculated decay constant and the final mass into the
decay equation. Solve for the age of the sample.
Step 1: Calculate the decay constant using the half-life formula:
25 = 100 ·1
230
15
Step 2: Solve for the decay constant:
1
4=1
22
=e−2λ
Step 3: The decay equation is:
25 = 100 ·e−2λ·30
Step 4: Solve for the age of the sample:
e−60λ=1
4⇒ −60λ= ln 1
4⇒λ=ln(4)
60
Therefore, the decay constant is λ=ln(4)
60 and the age of the sample is
t=30
λ=30
ln(4)/60 ≈87.31 days.
Question 19
Question
A sample of a radioactive isotope has an activity of 600 counts per minute
at 12:00 PM. The activity decreases to 300 counts per minute at 12:10 PM.
Assuming the decay follows an exponential model, calculate the half-life of the
isotope.
Solution
Step 1: Find the decay constant (λ) using the formula:
N(t) = N0·e−λt
15
where N(t) is the number of radioactive atoms at time t,N0is the initial number
of atoms, and λis the decay constant.
Given that N(t) = A
Activity per minute , we can rewrite the formula as:
A(t) = A0·e−λt
where A(t) is the activity at time t, and A0is the initial activity.
At t= 0, A0= 600 counts per minute. At t= 10 minutes, A(10) = 300
counts per minute.
Substitute into the formula:
300 = 600 ·e−λ·10
Step 2: Solve for λ:
e−10λ=300
600 =1
2
−10λ= ln 1
2
λ=ln(2)
10
Step 3: Calculate the half-life (T1/2) using the formula:
T1/2=ln(2)
λ=ln(2)
ln(2)
10
= 10 minutes
Therefore, the half-life of the isotope is 10 minutes.
Question 20
Question
A sample of a radioactive substance decays exponentially. After 10 minutes,
only 60% of the original substance remains. If the half-life of the substance is 5
minutes, what percentage of the substance will remain after 30 minutes?
Solution
Step 1: Find the decay constant, λ, using the half-life formula:
1
2=e−5λ
e−5λ=1
2
−5λ= ln 1
2
16
λ=−1
5ln 1
2
λ≈0.1386
Step 2: Use the exponential decay formula to find the percentage of substance
remaining after 30 minutes:
N(t) = N0e−λt
N(30) = 0.60 ×e−0.1386×30
N(30) ≈0.60 ×e−4.158
N(30) ≈0.60 ×0.0157
N(30) ≈0.0094
Step 3: Convert the result to a percentage to find the percentage of substance
remaining after 30 minutes:
0.0094 ×100% = 0.94%
Therefore, approximately 0.94% of the substance will remain after 30 min-
utes.
Question 21
Question
A sample of a radioactive substance initially contains 5,000 atoms. After 10
hours, only 1,250 atoms remain. Determine the half-life of the substance.
Solution
Step 1: Determine the decay constant using the formula:
N(t) = N0·e−kt
where: - N(t) is the number of atoms remaining after time t, - N0is the initial
number of atoms, - kis the decay constant, - tis the time elapsed.
We are given:
N(0) = 5,000
N(10) = 1,250
Substitute these values into the formula to get two equations:
5,000 = 5,000 ·e−k·0
1,250 = 5,000 ·e−k·10
17
Step 2: Solve the equations to find the decay constant k. Using the first
equation, we get:
1 = e0
1=1
Using the second equation, we have:
e−10k=1,250
5,000
e−10k= 0.25
−10k= ln(0.25)
k=−ln(0.25)
10
Step 3: Calculate the half-life t1/2using the formula:
t1/2=ln(2)
k
Substitute the value of kinto the formula:
t1/2=ln(2)
−ln(0.25)
10
t1/2=ln(2) ·10
ln(0.25)
t1/2≈0.693 ·10
−1.386
t1/2≈6.93
−1.386
t1/2≈ −5 hours
Therefore, the half-life of the substance is 5 hours.
Question 22
Question
A sample of a radioactive isotope has an initial mass of 100 g. After 50 hours,
only 25 g remain. If the half-life of the isotope is 20 hours, what is the decay
constant λand the decay rate of the isotope?
18
Solution
Step 1: Calculate the decay constant λusing the half-life formula:
λ=−ln(2)
t1/2
where t1/2is the half-life of the isotope.
λ=−ln(2)
20 ≈ −0.03465 hours−1
Step 2: Calculate the decay rate using the formula:
R=−λN
where Nis the remaining mass of the isotope.
R=−(−0.03465)(25) ≈0.8663 g/hour
Therefore, the decay constant λ≈0.03465 hours−1and the decay rate of
the isotope is approximately 0.8663 g/hour.
Question 23
Question
A sample of a radioactive element has an initial mass of 100 grams. After 5
days, the mass of the sample has decreased to 84 grams. The element has a
half-life of 3 days. Determine the decay constant and the mass of the sample
after 10 days.
Solution
Step 1: Calculate the decay constant using the half-life formula N(t) = N01
2t
T1
2,
where N(t) is the final mass, N0is the initial mass, tis the time, and T1
2is the
half-life.
84
100 =1
25
3
21
25 =1
25
3
1
25
3
=1
25
3
Since the bases are equal, we can equate the exponents:
5
3=k
19
k=5
3= 1.67 per day
Step 2: Use the decay constant to find the mass of the sample after 10 days.
N(t) = N0e−kt
N(10) = 100e−1.67(10)
N(10) = 100e−16.7
N(10) ≈100 ×3.43 ×10−8
N(10) ≈3.43 ×10−6grams
Therefore, the decay constant is 1.67 per day, and the mass of the sample
after 10 days is approximately 3.43 ×10−6grams.
Question 24
Question
A sample of a radioactive material has an initial mass of 200 grams. After 24
hours, only 50 grams of the material remain. If the half-life of the material is
12 hours, calculate the decay constant and the age of the material.
Solution
Step 1: Calculate the decay constant.
Given that the half-life of the material is 12 hours, we can use the formula for
radioactive decay:
mass remaining = initial mass ×e−kt
where: - mass remaining = 50 grams - initial mass = 200 grams - t = 24
hours
Substitute these values into the formula and solve for k:
50 = 200 ×e−12k
0.25 = e−12k
Take the natural logarithm of both sides to solve for k:
ln(0.25) = lne−12k
ln(0.25) = −12k
k=ln(0.25)
−12
k≈0.05774
20
So, the decay constant is approximately 0.05774.
Step 2: Calculate the age of the material.
To find the age of the material, we can use the formula:
t=
ln initial mass
mass remaining
k
Substitute the values of the initial mass, remaining mass, and decay constant
into the formula:
t=ln 200
50
0.05774
t=ln(4)
0.05774
t≈1.3863
0.05774
t≈23.97 hours
Therefore, the age of the material is approximately 23.97 hours.
Question 25
Question
A sample of a radioactive material decays according to the function N(t) =
N0e−kt, where N(t) is the amount of material remaining after tyears, N0is
the initial amount of material, and kis a decay constant. Given that the initial
amount of material is 100 grams and the half-life of the material is 10 years,
what is the amount of material remaining after 20 years?
Solution
Step 1: Determine the decay constant k
The half-life of the material is 10 years. We can use the half-life to find the
decay constant k. The half-life is the time it takes for half of the material to
decay. Thus, we have: N0
2=N0e−10k
Solving for k:
e−10k=1
2
−10k= ln 1
2
k=ln(2)
10
21
Step 2: Determine the amount of material remaining after 20 years
We are asked to find the amount of material remaining after 20 years, so t= 20.
Substituting N0= 100, k=ln(2)
10 , and t= 20 into the formula N(t) = N0e−kt:
N(20) = 100e−ln(2)
10 ×20
N(20) = 100e−ln(2)×2
N(20) = 100eln(2)−2
N(20) = 100 ×1
22
N(20) = 25
Therefore, the amount of material remaining after 20 years is 25 grams.
Question 26
Question
A certain radioactive element has a half-life of 25 minutes. If a sample initially
contains 200 grams of the element, how much of the element will remain after
2 hours?
Solution
Step 1: Determine the number of half-lives that have elapsed in 2 hours.
Number of half-lives = Time elapsed
Half-life =120 minutes
25 minutes = 4.8
Step 2: Since only whole numbers of half-lives can occur, we know that 4
complete half-lives have elapsed in 2 hours.
Step 3: Calculate the remaining amount of the radioactive element after 4
half-lives.
Amount remaining = Initial amount×1
2Number of half-lives
= 200×1
24
= 200×1
16= 12.5 grams
Therefore, after 2 hours, there will be 12.5 grams of the radioactive element
remaining.
Question 27
Question
A sample of a radioactive substance has an activity of 600 counts per minute at
3:00 PM. The activity is measured again at 6:00 PM and found to be 75 counts
per minute. Assuming that the decay is exponential, determine the half-life of
the substance.
22
Solution
Step 1: Find the decay constant λusing the formula:
λ=ln(N0/Nt)
t
where: - N0is the initial activity at 3:00 PM (600 counts per minute), - Nt
is the activity at 6:00 PM (75 counts per minute), and - tis the time interval
between the two measurements (3 hours).
λ=ln(600/75)
3
λ=ln(8)
3≈0.693
Step 2: Calculate the half-life T1/2using the formula:
T1/2=ln(2)
λ
T1/2=ln(2)
0.693 ≈1.001 hours
Therefore, the half-life of the radioactive substance is approximately 1.001
hours.
Question 28
Question
A certain radioactive substance decays according to the formula N(t) = N0e−0.015t,
where N(t) is the amount of substance present at time tin years, N0is the ini-
tial amount of the substance, and tis the time in years. If the initial amount of
the substance is 100 grams, find the amount of the substance remaining after
30 years.
Solution
Step 1: Substitute the given values into the formula to find the amount of the
substance remaining after 30 years.
N(30) = 100e−0.015×30
= 100e−0.45
≈100 ×0.6366
≈63.66 grams
Therefore, the amount of the substance remaining after 30 years is approxi-
mately 63.66 grams.
23
Question 29
Question
A sample of a radioactive isotope has an initial mass of 100 grams. After 10
days, only 25 grams of the isotope remains. If the half-life of the isotope is 5
days, what is the decay constant (λ) for this isotope?
Solution
Step 1: We can use the radioactive decay formula to find the decay constant λ:
N(t) = N0·e−λt
where: - N(t) is the amount remaining after time t, - N0is the initial amount,
-λis the decay constant, - tis the time elapsed.
Step 2: Given that the initial amount N0= 100 grams and the amount
remaining after 10 days is N(10) = 25 grams, we can substitute these values
into the formula:
25 = 100 ·e−λ·10
Step 3: To find the decay constant, we need to solve the equation for λ:
e−10λ=25
100 =1
4
Step 4: Taking the natural logarithm of both sides to solve for λ:
−10λ= ln 1
4
Step 5: Solving further:
λ=−ln(1/4)
10 =ln(4)
10
Step 6: Therefore, the decay constant (λ) for this isotope is ln(4)
10 .
Question 30
Question
A certain radioactive isotope has a half-life of 1000 years. If a sample initially
contains 1 gram of the isotope, how much of the isotope will be left after 3000
years?
24
Solution
Step 1: Determine the fraction of the isotope remaining after 3000 years. Step
2: Calculate the amount of the isotope remaining after 3000 years.
Step 1: The fraction of the isotope remaining after 3000 years can be
calculated using the formula:
Fraction remaining = 1
23000
1000
Step 2: Now, let’s calculate the amount of the isotope remaining after 3000
years. Using the fraction of the isotope remaining calculated in Step 1:
Amount remaining = Initial amount×Fraction remaining = 1×1
23
= 1×1
8=1
8grams
Therefore, after 3000 years, there will be 1
8grams of the radioactive isotope
remaining.
Question 31
Question
A radioactive substance has a half-life of 15 hours. If initially there are 200
grams of the substance, how much will be left after 45 hours?
Solution
Step 1: Determine the decay constant λusing the half-life formula: T1
2=ln 2
λ.
λ=ln 2
T1
2
λ=ln 2
15
λ≈0.0462 hours−1
Step 2: Use the formula for radioactive decay to find the amount of substance
remaining after 45 hours: N(t) = N0·e−λt.
N(45) = 200 ·e−0.0462·45
N(45) ≈104.26 grams
So, after 45 hours, there will be approximately 104.26 grams of the substance
remaining.
25
Question 32
Question
A sample of a radioactive isotope has an initial mass of 4 grams and a half-life
of 10 days. After 30 days, what is the mass of the sample remaining?
Solution
Step 1: Determine the decay constant. The decay constant (λ) can be calculated
using the formula:
λ=ln(2)
T1/2
where T1/2is the half-life of the isotope. Substitute T1/2= 10 days into the
formula to find λ.
Step 2: Calculate the remaining mass. The amount of radioactive material
remaining after time tcan be found using the formula:
m(t) = m0·e−λt
where: - m(t) is the mass remaining after time t, - m0is the initial mass, - tis
the time elapsed, - λis the decay constant.
Substitute m0= 4 grams, t= 30 days, and the previously calculated λinto
the formula to find the mass of the sample remaining after 30 days.
Question 33
Question
A sample of radioactive material has an initial mass of 100 grams and a half-life
of 10 days. After 30 days, what is the mass of the sample remaining?
Solution
Step 1: Calculate the decay constant, λ, using the formula:
λ=ln(2)
t1
2
where t1
2is the half-life of the material (10 days).
λ=ln(2)
10
λ≈0.0693 days−1
26
Step 2: Use the formula for radioactive decay to find the mass remaining
after 30 days:
m(t) = m0·e−λt
where m(t) is the mass at time t,m0is the initial mass, and tis the time elapsed.
m(30) = 100 ·e−0.0693·30
m(30) = 100 ·e−2.079 ≈100 ·0.125
m(30) ≈12.5 grams
Therefore, after 30 days, the mass of the sample remaining is 12.5 grams.
Question 34
Question
A certain radioactive substance decays according to the equation N(t) = N0·
e−kt, where N(t) is the amount of substance remaining after tyears, N0is the
initial amount of substance, and kis the decay constant. If 40% of the substance
decays in the first 500 years, find the value of kfor this substance.
Solution
Step 1: Let’s first express the given information in terms of the decay equation.
Since 40% of the substance decays in the first 500 years, this means that after 500
years, 60% of the substance remains. We can write this as: N(500) = 0.6·N0.
Step 2: Substitute the given information into the decay equation to get:
0.6·N0=N0·e−k·500.
Step 3: Simplify the equation by dividing both sides by N0: 0.6 = e−500k.
Step 4: Take the natural logarithm of both sides to solve for k: ln(0.6) =
−500k.
Step 5: Finally, solve for kby dividing both sides by −500: k=−ln(0.6)
500 .
Therefore, the value of kfor this substance is k=−ln(0.6)
500 .
Question 35
Question
A certain radioactive substance decays at a rate proportional to the amount
present. The initial amount of the substance is 100 grams, and after 5 hours,
only 30 grams are left. Find the half-life of the substance.
27
Solution
Let A(t) be the amount of the radioactive substance left after thours. Since
the substance decays at a rate proportional to the amount present, we have the
differential equation dA
dt =−kA
where k > 0 is the decay constant.
Step 1: Solve the differential equation to find an expression for A(t).
We separate variables and integrate:
ZdA
A=Z−kdt
ln |A|=−kt +C
where Cis the constant of integration.
Step 2: Use the initial condition A(0) = 100 to find the value of the constant
C.
ln |100|=C
C= ln(100) = ln102= 2 ln(10) = 2
So, the equation for A(t) is
A(t) = Ce−kt = 100e−kt
Step 3: Use the information that A(5) = 30 to find the value of the decay
constant k.
A(5) = 100e−5k= 30
e−5k=30
100 = 0.3
−5k= ln(0.3)
k=−1
5ln(0.3) ≈0.4621
Step 4: Find the half-life T1
2, which is the time taken for half of the sub-
stance to decay.
The half-life T1
2is related to the decay constant kby the formula T1
2=ln(2)
k.
T1
2=ln(2)
0.4621 ≈1.50 hours
Therefore, the half-life of the radioactive substance is approximately 1.50
hours.
28
Question 2
Question
A certain radioactive substance decays at a rate proportional to the amount
present. If the initial amount of the substance is 100 grams and after 3 hours,
the amount remaining is 65 grams, what is the half-life of this substance?
Solution
Let A(t) be the amount of the substance remaining after thours. We can write
the differential equation for the decay as:
dA
dt =−kA
where kis the decay constant.
Step 1: Solve the differential equation to find the general formula for A(t).
dA
dt =−kA
Separate variables: dA
A=−kdt
Integrate both sides:
Z1
AdA =Z−kdt
ln |A|=−kt +C
A=Ce−kt
where Cis the constant of integration.
Step 2: Use the initial condition A(0) = 100 to find the value of C.
A(0) = Ce0=C= 100
So, the equation for A(t) is A(t) = 100e−kt.
Step 3: Use the information given that A(3) = 65 to find the decay constant
k.
A(3) = 100e−3k= 65
e−3k= 0.65
−3k= ln(0.65)
k=−ln(0.65)
3
Step 4: The half-life T1
2of a substance is the time it takes for half of the
substance to decay. We can find T1
2using the formula T1
2=ln(2)
k.
T1
2=ln(2)
−ln(0.65)
3
= 3 ln(2) ≈2.079hours
Therefore, the half-life of this substance is approximately 2.079 hours.
2
Question 3
Question
A certain radioactive substance decays according to the model A(t) = A0e−kt,
where A(t) is the amount of substance remaining at time t,A0is the initial
amount of substance, and kis the decay constant. If 25
Solution
Step 1: Given the model A(t) = A0e−kt, we know that after 100 years, the
amount of substance remaining is 75
0.75A0=A0e−k·100
Step 2: Divide both sides by A0to simplify the equation:
0.75 = e−100k
Step 3: Take the natural logarithm of both sides to eliminate the exponential
term:
ln(0.75) = lne−100k
Step 4: Recall that ln(ex) = xfor any real number x:
ln(0.75) = −100k
Step 5: Solve for k:
k=−ln(0.75)
100 ≈0.00547
Therefore, the value of kfor the radioactive substance is approximately
0.00547 per year.
Question 4
Question
A sample of radium-226 decays according to the function Q(t) = Q0e−0.000187t,
where Q(t) is the quantity of radium-226 remaining after tyears and Q0is the
initial quantity. If the initial quantity of radium-226 is 100 grams, determine
how long it will take for 25 grams of radium-226 to remain.
3
Solution
Step 1: Find the decay constant k
From the given function, we can see that the decay constant kis equal to
0.000187.
Step 2: Set up the equation to find the time when 25 grams remains
We are given that Q(t) = Q0e−kt and we want to find twhen Q(t) = 25.
Plugging in Q(t) = 25 and Q0= 100 into the equation gives:
25 = 100 ·e−0.000187t
Step 3: Solve for t
Divide both sides by 100 to isolate the exponential term:
0.25 = e−0.000187t
Step 4: Take the natural logarithm of both sides
We take the natural logarithm of both sides to solve for t:
ln(0.25) = lne−0.000187t
ln(0.25) = −0.000187t
Step 5: Solve for t
Divide by −0.000187 to solve for t:
t=ln(0.25)
−0.000187
t≈9189.26 years
Therefore, it will take approximately 9189 years for 25 grams of radium-226
to remain.
Question 5
Question
An unknown radioactive substance has an initial activity of 2000 decays per
second. After 15 minutes, the activity has decreased to 500 decays per second.
What is the half-life of this substance?
Solution
Step 1: We can use the formula for radioactive decay:
N(t) = N01
2t
T1
2
4
where: - N(t) is the final activity after time t, - N0is the initial activity, - T1
2
is the half-life of the substance.
Step 2: Substitute the given values into the formula. After 15 minutes,
the final activity N(t) = 500 decays per second, the initial activity N0= 2000
decays per second, and the time t= 15 minutes. We want to find T1
2.
Step 3: Convert the time from minutes to seconds since the unit of time in
the formula is seconds.
t= 15 minutes ×60 seconds/minute = 900 seconds
Step 4: Substitute the values into the formula and solve for T1
2:
500 = 2000 1
2900
T1
2
Step 5: Divide both sides by 2000 to isolate the exponential term:
500
2000 =1
2900
T1
2
Step 6: Simplify the left side to get 1
4:
1
4=1
2900
T1
2
Step 7: Rewrite the right side using base 2:
1
4= 2−2= 2
−2×900
T1
2
Step 8: Equate the exponents:
−2 = −1800
T1
2
Step 9: Solve for T1
2:
T1
2=1800
2= 900 seconds
Step 10: Therefore, the half-life of the radioactive substance is 900 seconds.
Question 6
Question
A sample of a radioactive isotope has an initial mass of 10 grams. After 20
hours, only 2.5 grams of the isotope remain. If the half-life of the isotope is 24
hours, what is the decay constant for this isotope?
5
Solution
Step 1: Determine the fraction of the isotope that remains after 20 hours. To
find the fraction that remains after a certain time, we use the formula:
N(t) = N01
2t
T1/2
where: - N(t) is the remaining mass after time thours, - N0is the initial mass,
-T1/2is the half-life of the isotope.
Plugging in the values:
N(20) = 10 1
220
24
N(20) = 10 1
25
6
N(20) = 10 (0.5)5
6
N(20) = 10 0.51
65
N(20) ≈10 ×0.57435
N(20) ≈5.7435 grams
Therefore, 5.7435 grams of the isotope remain after 20 hours.
Step 2: Calculate the decay constant. The decay constant (λ) can be found
from the formula:
λ=ln(2)
T1/2
where ln(2) is the natural logarithm of 2.
Plugging in the given half-life:
λ=ln(2)
24
λ≈0.6931
24
λ≈0.02888 hours−1
Therefore, the decay constant for this isotope is approximately 0.02888
hours−1.
Question 7
Question
A sample of a radioactive isotope has an initial mass of 100 grams. After 5
hours, only 25 grams of the isotope remain. If the half-life of the isotope is 3
hours, determine the decay constant and the age of the sample.
6
Solution
Step 1: Determine the decay constant.
Let Nbe the remaining mass of the isotope at time t.
The decay of the isotope follows the formula N(t) = N0·e−kt, where N0
is the initial mass, kis the decay constant, and tis the time elapsed.
Given that N0= 100 grams and N(5) = 25 grams, we have 25 = 100·e−5k.
Dividing both sides by 100 gives us 0.25 = e−5k.
Taking the natural logarithm of both sides, we get ln(0.25) = lne−5k.
Simplifying further, we have ln(0.25) = −5k.
Solving for kgives us k≈0.5108 per hour.
Step 2: Determine the age of the sample.
Since the half-life of the isotope is 3 hours, we know that N0/2 = 100/2 =
50 grams of the isotope remains after 3 hours.
Substituting N0= 100 grams, N(3) = 50 grams, and k≈0.5108 into the
decay formula N(t) = N0·e−kt, we get 50 = 100 ·e−3·0.5108.
Solving for tgives t≈3.09 hours.
Therefore, the decay constant of the isotope is approximately 0.5108 per
hour, and the age of the sample is approximately 3.09 hours.
Question 8
Question
A sample of a radioactive isotope decays such that the number of atoms remain-
ing after t days is given by the function N(t) = 100e−0.05t, where N(t) is the
number of atoms and tis the time in days. Determine the rate of decay of the
sample after 20 days.
Solution
Step 1: Find the derivative of the function N(t) with respect to t.
dN
dt =d
dt(100e−0.05t)
Step 2: Apply the chain rule to differentiate 100e−0.05t.
dN
dt =−100(0.05)e−0.05t
7
Step 3: Simplify the result.
dN
dt =−5e−0.05t
Step 4: Evaluate the rate of decay after 20 days by substituting t= 20 into
the derivative. dN
dt
t=20
=−5e−0.05×20
Step 5: Calculate the rate of decay after 20 days.
dN
dt
t=20
=−5e−1≈ −5(0.3679) ≈ −1.8395
Therefore, the rate of decay of the sample after 20 days is approximately
1.8395 atoms per day.
Question 9
Question
A certain radioactive substance has a half-life of 10 days. If a sample initially
contains 100 grams of the substance, determine the amount of the substance
remaining after 30 days.
Solution
Let’s denote the amount of the radioactive substance remaining after tdays as
A(t).
Step 1: Determine the decay constant λusing the half-life formula.
λ=ln(2)
T1
2
=ln(2)
10
Step 2: Write the differential equation for radioactive decay:
dA
dt =−λA(t)
Step 3: Solve the initial value problem:
A(t) = Ce−λt
Using the initial condition A(0) = 100, we find C:
100 = Ce−λ·0=C
So, the equation becomes:
A(t) = 100e−ln(2)
10 ·t
8
Step 4: Calculate the amount of substance remaining after 30 days:
A(30) = 100e−ln(2)
10 ·30
A(30) = 100e−3 ln(2)
A(30) = 100(2−3)
A(30) = 100 ·1
8
A(30) = 12.5 grams
Question 10
Question
A radioactive substance decays according to the equation N(t) = N0e−0.03t,
where N(t) is the amount of substance remaining after tyears, N0is the initial
amount of substance, and 0.03 is the decay constant. If the initial amount of
the substance is 500 grams, find the amount of substance remaining after 20
years.
Solution
Step 1: Find the amount of substance remaining after 20 years using the formula
N(t) = N0e−0.03t.
N(20) = 500e−0.03×20
= 500e−0.6
= 500 ×0.5488
= 274.4 grams
Therefore, after 20 years, there will be 274.4 grams of the substance remain-
ing.
Question 11
Question
A sample of radioactive material has an initial mass of 100 grams. After 10
days, only 12.5 grams of the material remain. If the half-life of the material is
3 days, determine the decay constant λassociated with this material.
9
Solution
Step 1: Calculate the decay constant using the formula: λ=ln(2)
t1/2
where
ln(2) ≈0.6931.
λ=ln(2)
t1/2
=0.6931
3≈0.2310 per day
Therefore, the decay constant λassociated with this material is approxi-
mately 0.2310 per day.
Question 12
Question
A certain radioactive substance has a half-life of 30 days. If you start with a
sample containing 2 ×106atoms, how many atoms will remain after 90 days?
Solution
Step 1: Calculate the decay constant λusing the formula T1/2=ln(2)
λ, where
T1/2is the half-life.
Step 1: λ=ln(2)
30 ≈0.0231 per day
Step 2: Use the formula for radioactive decay to find the number of atoms
remaining after 90 days.
N(t) = N0·e−λt
N(90) = 2 ×106·e−0.0231×90
N(90) = 2 ×106·e−2.079
Step 3: Solve for N(90).
N(90) ≈2×106·0.125 ≈2.5×105
Therefore, after 90 days, there will be approximately 2.5×105atoms re-
maining.
Question 13
Question
A radioactive substance decays at a rate of 3
10
Solution
Step 1: Let A(t) represent the amount of the radioactive substance remaining
after tdays. The rate of decay can be expressed as:
A′(t) = kA(t)
where kis the decay constant. Since the substance decays at a rate of 3
Step 2: The differential equation is now:
dA
dt =−0.03A(t)
Step 3: Solve the differential equation using separation of variables:
dA
A=−0.03dt
ZdA
A=Z−0.03dt
ln |A|=−0.03t+C
A=Ce−0.03t
Step 4: Use the initial condition A(0) = 500 to find the value of C:
500 = Ce−0.03×0
C= 500
Step 5: The equation for the amount of substance remaining is now:
A(t) = 500e−0.03t
Step 6: Calculate the amount of the substance remaining after 10 days:
A(10) = 500e−0.03×10
A(10) = 500e−0.3
A(10) ≈500 ×0.74082
A(10) ≈370.41 grams
Therefore, after 10 days, there are approximately 370.41 grams of the sub-
stance remaining.
Question 14
Question
A certain radioactive substance has a half-life of 25 years. If there are initially
200 grams of the substance, how much will remain after 75 years? Round your
answer to the nearest gram.
11
Solution
Step 1: Determine the decay constant using the half-life formula N(t) = N0·
(1/2)(t/T1/2), where: - N(t) is the amount remaining after time t-N0is the
initial amount - T1/2is the half-life of the substance
Plugging in the values, we have:
200 = 200 ·(1/2)(75/25) = 200 ·0.53= 200 ·0.125
200 = 25
Only 25 grams will remain after 75 years.
Question 15
Question
A sample of a radioactive material has an initial mass of 500 grams. After 10
minutes, only 150 grams of the material remains. If the half-life of the material
is 5 minutes, what is the decay constant of the material?
Solution
Step 1: Calculate the decay constant using the formula for radioactive decay.
Decay constant = −ln(1/2)
T1/2
where T1/2is the half-life of the material.
Step 2: Given T1/2= 5 minutes and the initial mass M0= 500 grams, and
the final mass M= 150 grams after 10 minutes, we can find the decay constant.
λ=−ln(1/2)
5
Step 3: Calculate the fraction of material remaining after 10 minutes.
M
M0
=e−λt
150
500 =e−λ×10
Step 4: Solve for the decay constant by plugging in the values and solving
the equation. 3
10 =e−10λ
Step 5: Take the natural logarithm of both sides to solve for the decay
constant.
ln 3
10= lne−10λ
12
ln 3
10=−10λ
Step 6: Finally, solve for the decay constant.
λ=−ln 3
10
10
λ≈0.0790 minutes−1
Therefore, the decay constant of the material is approximately 0.0790 minutes−1.
Question 16
Question
A radioactive substance has a half-life of 30 days. If the initial mass of the
substance is 100 grams, determine the mass of the substance remaining after 90
days.
Solution
Step 1: Determine the decay constant λ. The decay constant λis related to the
half-life T1
2by the formula:
λ=ln(2)
T1
2
Substitute T1
2= 30 days into the formula:
λ=ln(2)
30 ≈0.0231 days−1
Step 2: Use the exponential decay formula N(t) = N0·e−λt. Given that
N0= 100 grams and t= 90 days, we can find N(90):
N(90) = 100 ·e−0.0231·90 ≈31.71 grams
So, the mass of the substance remaining after 90 days is approximately 31.71
grams.
Question 17
Question
A sample of a radioactive isotope decays according to the equation N(t) =
N0e−kt, where N(t) is the amount of the isotope present at time t,N0is the
initial amount of the isotope, kis the decay constant, and tis the time elapsed.
Suppose a sample of a radioactive isotope has an initial amount of 200 grams
and a decay constant of 0.05 per year.
Calculate: (a) The amount of the isotope that remains after 10 years. (b)
The time it takes for 95
13
Solution
(a) To find the amount of the isotope that remains after 10 years, we can use the
given equation N(t) = N0e−kt and substitute N0= 200, k= 0.05, and t= 10:
N(10) = 200 ·e−0.05·10
Step 1: Calculate the exponent.
N(10) = 200 ·e−0.5
Step 2: Evaluate the exponential term.
N(10) = 200 ·e−0.5≈200 ·0.6065 ≈121.3
So, the amount of the isotope that remains after 10 years is approximately
121.3 grams.
(b) To find the time it takes for 95
0.05N0=N0·e−kt
Step 1: Simplify the equation.
0.05 = e−kt
Step 2: Take the natural logarithm of both sides.
ln(0.05) = lne−kt
Step 3: Use the property of logarithms to bring down the exponent.
ln(0.05) = −kt ln(e)
Step 4: Simplify the right side.
ln(0.05) = −kt
Step 5: Solve for t.
t=ln(0.05)
−k≈ln(0.05)
−0.05 ≈13.5
Therefore, it takes approximately 13.5 years for 95
Question 18
Question
A sample of a radioactive isotope has an initial mass of 100 grams. After 30
days, only 25 grams of the isotope remain. If the half-life of the isotope is 15
days, determine the decay constant and the age of the sample.
14
Solution
Step 1: Calculate the decay constant using the half-life formula N=N0·1
2t
T1/2,
where: - Nis the final mass (25 grams), - N0is the initial mass (100 grams), -
tis the elapsed time (30 days), - T1/2is the half-life (15 days).
Step 2: Substitute the given values into the formula and solve for the decay
constant.
Step 3: Once you have the decay constant, use the decay equation N=
N0·e−λt to find the age of the sample.
Step 4: Substitute the calculated decay constant and the final mass into the
decay equation. Solve for the age of the sample.
Step 1: Calculate the decay constant using the half-life formula:
25 = 100 ·1
230
15
Step 2: Solve for the decay constant:
1
4=1
22
=e−2λ
Step 3: The decay equation is:
25 = 100 ·e−2λ·30
Step 4: Solve for the age of the sample:
e−60λ=1
4⇒ −60λ= ln 1
4⇒λ=ln(4)
60
Therefore, the decay constant is λ=ln(4)
60 and the age of the sample is
t=30
λ=30
ln(4)/60 ≈87.31 days.
Question 19
Question
A sample of a radioactive isotope has an activity of 600 counts per minute
at 12:00 PM. The activity decreases to 300 counts per minute at 12:10 PM.
Assuming the decay follows an exponential model, calculate the half-life of the
isotope.
Solution
Step 1: Find the decay constant (λ) using the formula:
N(t) = N0·e−λt
15
where N(t) is the number of radioactive atoms at time t,N0is the initial number
of atoms, and λis the decay constant.
Given that N(t) = A
Activity per minute , we can rewrite the formula as:
A(t) = A0·e−λt
where A(t) is the activity at time t, and A0is the initial activity.
At t= 0, A0= 600 counts per minute. At t= 10 minutes, A(10) = 300
counts per minute.
Substitute into the formula:
300 = 600 ·e−λ·10
Step 2: Solve for λ:
e−10λ=300
600 =1
2
−10λ= ln 1
2
λ=ln(2)
10
Step 3: Calculate the half-life (T1/2) using the formula:
T1/2=ln(2)
λ=ln(2)
ln(2)
10
= 10 minutes
Therefore, the half-life of the isotope is 10 minutes.
Question 20
Question
A sample of a radioactive substance decays exponentially. After 10 minutes,
only 60% of the original substance remains. If the half-life of the substance is 5
minutes, what percentage of the substance will remain after 30 minutes?
Solution
Step 1: Find the decay constant, λ, using the half-life formula:
1
2=e−5λ
e−5λ=1
2
−5λ= ln 1
2
16
λ=−1
5ln 1
2
λ≈0.1386
Step 2: Use the exponential decay formula to find the percentage of substance
remaining after 30 minutes:
N(t) = N0e−λt
N(30) = 0.60 ×e−0.1386×30
N(30) ≈0.60 ×e−4.158
N(30) ≈0.60 ×0.0157
N(30) ≈0.0094
Step 3: Convert the result to a percentage to find the percentage of substance
remaining after 30 minutes:
0.0094 ×100% = 0.94%
Therefore, approximately 0.94% of the substance will remain after 30 min-
utes.
Question 21
Question
A sample of a radioactive substance initially contains 5,000 atoms. After 10
hours, only 1,250 atoms remain. Determine the half-life of the substance.
Solution
Step 1: Determine the decay constant using the formula:
N(t) = N0·e−kt
where: - N(t) is the number of atoms remaining after time t, - N0is the initial
number of atoms, - kis the decay constant, - tis the time elapsed.
We are given:
N(0) = 5,000
N(10) = 1,250
Substitute these values into the formula to get two equations:
5,000 = 5,000 ·e−k·0
1,250 = 5,000 ·e−k·10
17
Step 2: Solve the equations to find the decay constant k. Using the first
equation, we get:
1 = e0
1=1
Using the second equation, we have:
e−10k=1,250
5,000
e−10k= 0.25
−10k= ln(0.25)
k=−ln(0.25)
10
Step 3: Calculate the half-life t1/2using the formula:
t1/2=ln(2)
k
Substitute the value of kinto the formula:
t1/2=ln(2)
−ln(0.25)
10
t1/2=ln(2) ·10
ln(0.25)
t1/2≈0.693 ·10
−1.386
t1/2≈6.93
−1.386
t1/2≈ −5 hours
Therefore, the half-life of the substance is 5 hours.
Question 22
Question
A sample of a radioactive isotope has an initial mass of 100 g. After 50 hours,
only 25 g remain. If the half-life of the isotope is 20 hours, what is the decay
constant λand the decay rate of the isotope?
18
Solution
Step 1: Calculate the decay constant λusing the half-life formula:
λ=−ln(2)
t1/2
where t1/2is the half-life of the isotope.
λ=−ln(2)
20 ≈ −0.03465 hours−1
Step 2: Calculate the decay rate using the formula:
R=−λN
where Nis the remaining mass of the isotope.
R=−(−0.03465)(25) ≈0.8663 g/hour
Therefore, the decay constant λ≈0.03465 hours−1and the decay rate of
the isotope is approximately 0.8663 g/hour.
Question 23
Question
A sample of a radioactive element has an initial mass of 100 grams. After 5
days, the mass of the sample has decreased to 84 grams. The element has a
half-life of 3 days. Determine the decay constant and the mass of the sample
after 10 days.
Solution
Step 1: Calculate the decay constant using the half-life formula N(t) = N01
2t
T1
2,
where N(t) is the final mass, N0is the initial mass, tis the time, and T1
2is the
half-life.
84
100 =1
25
3
21
25 =1
25
3
1
25
3
=1
25
3
Since the bases are equal, we can equate the exponents:
5
3=k
19
k=5
3= 1.67 per day
Step 2: Use the decay constant to find the mass of the sample after 10 days.
N(t) = N0e−kt
N(10) = 100e−1.67(10)
N(10) = 100e−16.7
N(10) ≈100 ×3.43 ×10−8
N(10) ≈3.43 ×10−6grams
Therefore, the decay constant is 1.67 per day, and the mass of the sample
after 10 days is approximately 3.43 ×10−6grams.
Question 24
Question
A sample of a radioactive material has an initial mass of 200 grams. After 24
hours, only 50 grams of the material remain. If the half-life of the material is
12 hours, calculate the decay constant and the age of the material.
Solution
Step 1: Calculate the decay constant.
Given that the half-life of the material is 12 hours, we can use the formula for
radioactive decay:
mass remaining = initial mass ×e−kt
where: - mass remaining = 50 grams - initial mass = 200 grams - t = 24
hours
Substitute these values into the formula and solve for k:
50 = 200 ×e−12k
0.25 = e−12k
Take the natural logarithm of both sides to solve for k:
ln(0.25) = lne−12k
ln(0.25) = −12k
k=ln(0.25)
−12
k≈0.05774
20
So, the decay constant is approximately 0.05774.
Step 2: Calculate the age of the material.
To find the age of the material, we can use the formula:
t=
ln initial mass
mass remaining
k
Substitute the values of the initial mass, remaining mass, and decay constant
into the formula:
t=ln 200
50
0.05774
t=ln(4)
0.05774
t≈1.3863
0.05774
t≈23.97 hours
Therefore, the age of the material is approximately 23.97 hours.
Question 25
Question
A sample of a radioactive material decays according to the function N(t) =
N0e−kt, where N(t) is the amount of material remaining after tyears, N0is
the initial amount of material, and kis a decay constant. Given that the initial
amount of material is 100 grams and the half-life of the material is 10 years,
what is the amount of material remaining after 20 years?
Solution
Step 1: Determine the decay constant k
The half-life of the material is 10 years. We can use the half-life to find the
decay constant k. The half-life is the time it takes for half of the material to
decay. Thus, we have: N0
2=N0e−10k
Solving for k:
e−10k=1
2
−10k= ln 1
2
k=ln(2)
10
21
Step 2: Determine the amount of material remaining after 20 years
We are asked to find the amount of material remaining after 20 years, so t= 20.
Substituting N0= 100, k=ln(2)
10 , and t= 20 into the formula N(t) = N0e−kt:
N(20) = 100e−ln(2)
10 ×20
N(20) = 100e−ln(2)×2
N(20) = 100eln(2)−2
N(20) = 100 ×1
22
N(20) = 25
Therefore, the amount of material remaining after 20 years is 25 grams.
Question 26
Question
A certain radioactive element has a half-life of 25 minutes. If a sample initially
contains 200 grams of the element, how much of the element will remain after
2 hours?
Solution
Step 1: Determine the number of half-lives that have elapsed in 2 hours.
Number of half-lives = Time elapsed
Half-life =120 minutes
25 minutes = 4.8
Step 2: Since only whole numbers of half-lives can occur, we know that 4
complete half-lives have elapsed in 2 hours.
Step 3: Calculate the remaining amount of the radioactive element after 4
half-lives.
Amount remaining = Initial amount×1
2Number of half-lives
= 200×1
24
= 200×1
16= 12.5 grams
Therefore, after 2 hours, there will be 12.5 grams of the radioactive element
remaining.
Question 27
Question
A sample of a radioactive substance has an activity of 600 counts per minute at
3:00 PM. The activity is measured again at 6:00 PM and found to be 75 counts
per minute. Assuming that the decay is exponential, determine the half-life of
the substance.
22
Solution
Step 1: Find the decay constant λusing the formula:
λ=ln(N0/Nt)
t
where: - N0is the initial activity at 3:00 PM (600 counts per minute), - Nt
is the activity at 6:00 PM (75 counts per minute), and - tis the time interval
between the two measurements (3 hours).
λ=ln(600/75)
3
λ=ln(8)
3≈0.693
Step 2: Calculate the half-life T1/2using the formula:
T1/2=ln(2)
λ
T1/2=ln(2)
0.693 ≈1.001 hours
Therefore, the half-life of the radioactive substance is approximately 1.001
hours.
Question 28
Question
A certain radioactive substance decays according to the formula N(t) = N0e−0.015t,
where N(t) is the amount of substance present at time tin years, N0is the ini-
tial amount of the substance, and tis the time in years. If the initial amount of
the substance is 100 grams, find the amount of the substance remaining after
30 years.
Solution
Step 1: Substitute the given values into the formula to find the amount of the
substance remaining after 30 years.
N(30) = 100e−0.015×30
= 100e−0.45
≈100 ×0.6366
≈63.66 grams
Therefore, the amount of the substance remaining after 30 years is approxi-
mately 63.66 grams.
23
Question 29
Question
A sample of a radioactive isotope has an initial mass of 100 grams. After 10
days, only 25 grams of the isotope remains. If the half-life of the isotope is 5
days, what is the decay constant (λ) for this isotope?
Solution
Step 1: We can use the radioactive decay formula to find the decay constant λ:
N(t) = N0·e−λt
where: - N(t) is the amount remaining after time t, - N0is the initial amount,
-λis the decay constant, - tis the time elapsed.
Step 2: Given that the initial amount N0= 100 grams and the amount
remaining after 10 days is N(10) = 25 grams, we can substitute these values
into the formula:
25 = 100 ·e−λ·10
Step 3: To find the decay constant, we need to solve the equation for λ:
e−10λ=25
100 =1
4
Step 4: Taking the natural logarithm of both sides to solve for λ:
−10λ= ln 1
4
Step 5: Solving further:
λ=−ln(1/4)
10 =ln(4)
10
Step 6: Therefore, the decay constant (λ) for this isotope is ln(4)
10 .
Question 30
Question
A certain radioactive isotope has a half-life of 1000 years. If a sample initially
contains 1 gram of the isotope, how much of the isotope will be left after 3000
years?
24
Solution
Step 1: Determine the fraction of the isotope remaining after 3000 years. Step
2: Calculate the amount of the isotope remaining after 3000 years.
Step 1: The fraction of the isotope remaining after 3000 years can be
calculated using the formula:
Fraction remaining = 1
23000
1000
Step 2: Now, let’s calculate the amount of the isotope remaining after 3000
years. Using the fraction of the isotope remaining calculated in Step 1:
Amount remaining = Initial amount×Fraction remaining = 1×1
23
= 1×1
8=1
8grams
Therefore, after 3000 years, there will be 1
8grams of the radioactive isotope
remaining.
Question 31
Question
A radioactive substance has a half-life of 15 hours. If initially there are 200
grams of the substance, how much will be left after 45 hours?
Solution
Step 1: Determine the decay constant λusing the half-life formula: T1
2=ln 2
λ.
λ=ln 2
T1
2
λ=ln 2
15
λ≈0.0462 hours−1
Step 2: Use the formula for radioactive decay to find the amount of substance
remaining after 45 hours: N(t) = N0·e−λt.
N(45) = 200 ·e−0.0462·45
N(45) ≈104.26 grams
So, after 45 hours, there will be approximately 104.26 grams of the substance
remaining.
25
Question 32
Question
A sample of a radioactive isotope has an initial mass of 4 grams and a half-life
of 10 days. After 30 days, what is the mass of the sample remaining?
Solution
Step 1: Determine the decay constant. The decay constant (λ) can be calculated
using the formula:
λ=ln(2)
T1/2
where T1/2is the half-life of the isotope. Substitute T1/2= 10 days into the
formula to find λ.
Step 2: Calculate the remaining mass. The amount of radioactive material
remaining after time tcan be found using the formula:
m(t) = m0·e−λt
where: - m(t) is the mass remaining after time t, - m0is the initial mass, - tis
the time elapsed, - λis the decay constant.
Substitute m0= 4 grams, t= 30 days, and the previously calculated λinto
the formula to find the mass of the sample remaining after 30 days.
Question 33
Question
A sample of radioactive material has an initial mass of 100 grams and a half-life
of 10 days. After 30 days, what is the mass of the sample remaining?
Solution
Step 1: Calculate the decay constant, λ, using the formula:
λ=ln(2)
t1
2
where t1
2is the half-life of the material (10 days).
λ=ln(2)
10
λ≈0.0693 days−1
26
Step 2: Use the formula for radioactive decay to find the mass remaining
after 30 days:
m(t) = m0·e−λt
where m(t) is the mass at time t,m0is the initial mass, and tis the time elapsed.
m(30) = 100 ·e−0.0693·30
m(30) = 100 ·e−2.079 ≈100 ·0.125
m(30) ≈12.5 grams
Therefore, after 30 days, the mass of the sample remaining is 12.5 grams.
Question 34
Question
A certain radioactive substance decays according to the equation N(t) = N0·
e−kt, where N(t) is the amount of substance remaining after tyears, N0is the
initial amount of substance, and kis the decay constant. If 40% of the substance
decays in the first 500 years, find the value of kfor this substance.
Solution
Step 1: Let’s first express the given information in terms of the decay equation.
Since 40% of the substance decays in the first 500 years, this means that after 500
years, 60% of the substance remains. We can write this as: N(500) = 0.6·N0.
Step 2: Substitute the given information into the decay equation to get:
0.6·N0=N0·e−k·500.
Step 3: Simplify the equation by dividing both sides by N0: 0.6 = e−500k.
Step 4: Take the natural logarithm of both sides to solve for k: ln(0.6) =
−500k.
Step 5: Finally, solve for kby dividing both sides by −500: k=−ln(0.6)
500 .
Therefore, the value of kfor this substance is k=−ln(0.6)
500 .
Question 35
Question
A certain radioactive substance decays at a rate proportional to the amount
present. The initial amount of the substance is 100 grams, and after 5 hours,
only 30 grams are left. Find the half-life of the substance.
27
Solution
Let A(t) be the amount of the radioactive substance left after thours. Since
the substance decays at a rate proportional to the amount present, we have the
differential equation dA
dt =−kA
where k > 0 is the decay constant.
Step 1: Solve the differential equation to find an expression for A(t).
We separate variables and integrate:
ZdA
A=Z−kdt
ln |A|=−kt +C
where Cis the constant of integration.
Step 2: Use the initial condition A(0) = 100 to find the value of the constant
C.
ln |100|=C
C= ln(100) = ln102= 2 ln(10) = 2
So, the equation for A(t) is
A(t) = Ce−kt = 100e−kt
Step 3: Use the information that A(5) = 30 to find the value of the decay
constant k.
A(5) = 100e−5k= 30
e−5k=30
100 = 0.3
−5k= ln(0.3)
k=−1
5ln(0.3) ≈0.4621
Step 4: Find the half-life T1
2, which is the time taken for half of the sub-
stance to decay.
The half-life T1
2is related to the decay constant kby the formula T1
2=ln(2)
k.
T1
2=ln(2)
0.4621 ≈1.50 hours
Therefore, the half-life of the radioactive substance is approximately 1.50
hours.
28
