CHEM 132 - ADVANCED GENERAL
CHEMISTRY II - Radioactive decay
calculations
Question Bank - Set 2
Liberty University
Question 1
Question
A certain radioactive substance decays exponentially with a half-life of 10 days.
If we start with an initial amount of 500 grams, determine the amount remaining
after 30 days.
Solution
Step 1: Determine the decay constant, λ, using the formula λ=ln(2)
t1/2where t1/2
is the half-life.
λ=ln(2)
10 ≈0.0693 per day
Step 2: Use the exponential decay formula N(t) = N0e−λt to find the amount
remaining after 30 days.
N(30) = 500e−0.0693×30
N(30) = 500e−2.079 ≈123.85 grams
Therefore, the amount remaining after 30 days is approximately 123.85
grams.
Question 2
Question
A sample of radon-222 has an initial activity of 5000 Bq. If the half-life of
radon-222 is 3.8 days, determine the activity of the sample after 10 days.
Solution
Step 1: Determine the decay constant (λ) using the formula λ=ln(2)
t1/2
, where
t1/2is the half-life of the element. Step 2: Calculate the activity of the sample
after 10 days using the formula A(t) = A0·e−λt, where A0is the initial activity
of the sample. Step 3: Substitute the values into the formula from step 2 to
find the activity after 10 days.
Step 1: Calculate the decay constant λGiven: t1/2= 3.8 days
λ=ln(2)
t1/2
=ln(2)
3.8≈0.182 day−1
Step 2: Calculate the activity after 10 days Given: A0= 5000 Bq, t=
10 days
A(10) = 5000 ·e−0.182·10
Step 3: Substitute the values to find the final activity
A(10) = 5000 ·e−1.82 ≈1403 Bq
Therefore, the activity of the sample after 10 days will be approximately
1403 Bq.
Question 3
Question
A certain radioactive substance decays according to the equation N(t) = N0e−kt,
where N(t) is the amount of substance remaining at time t,N0is the initial
amount of substance, kis the decay constant, and tis time in years. Given that
60
Solution
Step 1: We are given that 60
0.4N0=N0e−30k
Step 2: Divide both sides by N0to simplify the equation:
0.4 = e−30k
Step 3: Take the natural logarithm (ln) of both sides to solve for k:
ln(0.4) = lne−30k
ln(0.4) = −30k
Step 4: Solve for kby dividing both sides by −30:
k=−ln(0.4)
30 ≈0.0231 year−1
Therefore, the value of the decay constant kis approximately 0.0231 year−1.
2
Question 4
Question
A sample of a radioactive isotope decays according to the formula Q(t) =
Q0e−kt, where Q(t) is the quantity of the isotope remaining at time t(in years),
Q0is the initial quantity of the isotope, and kis the decay constant. Suppose
that a sample of a radioactive isotope has an initial quantity of 100 grams and
the decay constant is 0.02. How many years will it take for the quantity of the
isotope to reduce to 50 grams?
Solution
Step 1: Substitute the given values into the formula Q(t) = Q0e−kt to form an
equation we can solve for t.
We have Q(t) = 100e−0.02t= 50
Step 2: Solve for t by dividing both sides by 100 and applying natural
logarithm to both sides.
100e−0.02t
100 =50
100
e−0.02t= 0.5
lne−0.02t= ln(0.5)
−0.02t= ln(0.5)
Step 3: Solve for t by dividing by -0.02.
t=ln(0.5)
−0.02
Step 4: Calculate the final answer using a calculator.
t≈ln(0.5)
−0.02 ≈34.657 years
Therefore, it will take approximately 34.657 years for the quantity of the
isotope to reduce to 50 grams.
Question 5
Question
A certain radioactive substance has a half-life of 10 days. If you start with an
initial amount of 100 grams, find:
1. The amount of the substance remaining after 30 days.
2. How long it will take for the initial amount to decay to 25 grams.
3
Solution
Let’s denote: - Aas the amount of the substance remaining at any time t, - A0
as the initial amount of the substance, - tas the time that has passed, - T1
2as
the half-life of the substance.
1. We can use the formula for radioactive decay:
A=A0·1
2t
T1
2
After 30 days:
A= 100 ·1
230
10
A= 100 ·1
23
A= 100 ·1
8
A= 12.5 grams
Therefore, the amount of the substance remaining after 30 days is 12.5
grams.
2. To find the time it will take for the initial amount to decay to 25 grams,
we set A= 25 and solve for t:
25 = 100 ·1
2t
10
1
4=1
2t
10
2=2t
10
1 = t
10
t= 10 days
Therefore, it will take 10 days for the initial amount to decay to 25 grams.
Question 6
Question
A sample of a radioactive substance has an initial activity of 10,000 counts per
minute. After 3 hours, the activity has decreased to 5000 counts per minute.
Assuming the decay is exponential, find the half-life of the substance.
4
Solution
Step 1: The decay of a radioactive substance can be modeled by the equation:
N(t) = N0·e−λt
where: - N(t) is the activity at time t, - N0is the initial activity, - λis the decay
constant, - eis the base of the natural logarithm (approximately 2.71828).
Step 2: Substituting the given values into the equation:
N(0) = 10000 (initial activity)
N(3) = 5000 (activity after 3 hours)
Step 3: We can set up the following equations:
10000 = 10000 ·e−3λ
5000 = 10000 ·e−3λ
Step 4: Divide the second equation by the first to eliminate N0:
5000
10000 =e−3λ
1
2=e−3λ
Step 5: Take the natural logarithm of both sides to solve for λ:
ln 1
2=−3λ
−ln 2 = −3λ
λ=ln 2
3
Step 6: The half-life of the substance can be found using the relation between
half-life (T1/2) and the decay constant (λ):
T1/2=ln 2
λ=ln 2
ln 2
3
= 3 hours
Therefore, the half-life of the substance is 3 hours.
Question 7
Question
A sample of a radioactive substance decays according to the formula N(t) =
N0e−kt, where N(t) is the amount of the substance remaining after tdays, N0
is the initial amount of the substance, and kis the decay constant. If 30
5
Solution
Step 1: Using the provided information to set up equations.
Let N0be the initial amount of the substance. From the information given,
we have:
N(100) = 0.3N0and N(200) = 0.2N0
Substitute the formula N(t) = N0e−kt into the equations to get:
0.3N0=N0e−100kand 0.2N0=N0e−200k
Step 2: Solving the equations to find the decay constant k.
Divide the two equations to eliminate N0:
0.3
0.2=e−100k
e−200k
Simplify the right side: 3
2=e100k
Taking the natural logarithm of both sides:
ln 3
2= 100k
Solving for k:
k=1
100 ln 3
2≈ −0.0051
Therefore, the decay constant k≈ −0.0051.
Question 8
Question
A sample of a radioactive isotope decays according to the formula N(t) =
N0e−0.02t, where N(t) is the amount of the isotope remaining after tyears
and N0is the initial amount of the isotope. If the initial amount is 100 grams,
determine the amount of the isotope remaining after 30 years.
Solution
Step 1: Given that the initial amount N0= 100 grams, we can substitute this
value into the formula to get N(t) = 100e−0.02t.
Step 2: We are asked to find the amount of the isotope remaining after 30
years. Substituting t= 30 into the formula, we get N(30) = 100e−0.02(30).
Step 3: Simplifying the expression, we have N(30) = 100e−0.6.
Step 4: Using the value of e≈2.71828, we can further simplify the expression
to get N(30) ≈100 ×2.71828−0.6.
6
Step 5: Calculating the value, we find N(30) ≈100 ×0.54881 ≈54.881
grams.
Step 6: Therefore, the amount of the isotope remaining after 30 years is
approximately 54.881 grams.
Question 9
Question
A sample of an unknown radioactive material has an initial mass of 200 grams
and after 10 days, its mass is reduced to 100 grams. The half-life of the material
is 5 days. Determine the decay constant and the mass of the material after 30
days.
Solution
Step 1: Determine the decay constant using the half-life formula.
Half-life (T) = ln(2)
k
Given that the half-life (T) is 5 days, we can solve for the decay constant (k).
5
ln(2) =k
k≈0.693 days−1
Step 2: Using the decay constant, we can determine the mass of the material
after 30 days using the exponential decay formula.
m(t) = m0·e−kt
where: - m(t) is the mass of the material after time t, - m0is the initial mass,
-kis the decay constant, - tis the time elapsed.
Plug in the values m0= 200g, k= 0.693 days−1, and t= 30 days into the
formula.
m(30) = 200 ·e−0.693·30
m(30) = 200 ·e−20.79
m(30) ≈200 ·1.604 ×10−10
m(30) ≈3.208 ×10−8grams
Therefore, the mass of the material after 30 days is approximately 3.208 ×
10−8grams.
7
Question 10
Question
A sample of a radioactive substance has an initial mass of 400 grams. After 5
hours, only 100 grams of the substance remains. If the half-life of the substance
is 3 hours, determine the decay constant and the age of the sample.
Solution
Step 1: Determine the decay constant.
The decay constant, denoted by λ, can be found using the formula:
N(t) = N0·e−λt
where N(t) is the amount of substance remaining at time t,N0is the initial
amount of substance, and eis Euler’s number (≈2.71828).
Given that N(5) = 100 and N0= 400, we can plug these values into the
formula to solve for λ:
100 = 400 ·e−λ·5
100
400 =e−5λ
1
4=e−5λ
ln 1
4=−5λ
λ=ln(4)
5≈0.2773 hours−1
Step 2: Determine the age of the sample.
To find the age of the sample, we need to solve for tin the formula N(t) = 100
using the decay constant λwe found.
100 = 400 ·e−0.2773t
1
4=e−0.2773t
ln 1
4=−0.2773t
t=ln(4)
0.2773 ≈7.9 hours
Therefore, the decay constant is approximately 0.2773 hours−1and the age
of the sample is approximately 7.9 hours.
8
Question 11
Question
A sample of a radioactive substance has an initial mass of 200 grams. After 10
days, the mass of the substance has decreased to 100 grams. If the half-life of
the substance is 5 days, what is the decay constant of the substance?
Solution
Step 1: Calculate the fraction of the substance remaining after 10 days. Step
2: Use the fraction of the substance remaining to calculate the decay constant.
Step 1: The fraction of the substance remaining after 10 days can be cal-
culated using the formula:
Fraction remaining = 1
2time elapsed
half-life
Plugging in the values:
Fraction remaining = 1
210
5
=1
22
=1
4
Therefore, after 10 days, 1
4of the substance remains.
Step 2: The decay constant (λ) can be calculated using the formula:
Fraction remaining = e−λ×time elapsed
Plugging in the known values (1
4remaining after 10 days):
1
4=e−λ×10
Taking the natural logarithm of both sides:
ln 1
4=−λ×10
Solving for λ:
λ=ln(4)
10 ≈0.07918 days−1
Therefore, the decay constant of the substance is approximately 0.07918
days−1.
Question 12
Question
A sample of radioactive material has an initial activity of 800 Becquerels. After
10 minutes, the activity decreases to 200 Becquerels. If the decay is exponential,
what is the half-life of the material?
9
Solution
Step 1: Let’s denote the initial activity as A0= 800 Becquerels, the activity
after 10 minutes as A= 200 Becquerels, and the half-life as T1
2.
Step 2: We can use the exponential decay formula to relate the activities at
different times:
A=A0×1
2t
T1
2
Step 3: From the given data, we have:
200 = 800 ×1
210
T1
2
Step 4: Simplifying the equation, we get:
1
210
T1
2=1
4
Step 5: To solve for T1
2, we can write 1/4 as 2−2:
2
−10
T1
2= 2−2
Step 6: Equating the exponents, we get:
−10
T1
2
=−2
Step 7: Solving for T1
2, we find:
T1
2= 5 minutes
Step 8: Therefore, the half-life of the radioactive material is 5 minutes.
Question 13
Question
The half-life of a radioactive isotope is 24 hours. If you start with a sample
containing 2 ×1012 atoms, how many atoms are left after 3 days?
Solution
Step 1: Calculate the decay constant, λ. Given the half-life, T1/2= 24 hours,
we can use the formula λ=ln(2)
T1/2
to find the decay constant.
λ=ln(2)
24 ≈0.02892 hours−1
10
Step 2: Use the decay formula to find the number of atoms remaining after
3 days. The number of atoms remaining can be found using the formula N(t) =
N0·e−λt, where: - N(t) is the number of atoms remaining after time t-N0is
the initial number of atoms - λis the decay constant - tis the time elapsed
Substitute the values, N0= 2 ×1012,λ≈0.02892, and t= 3 ×24 = 72
hours, into the formula to find N(72).
N(72) = 2 ×1012 ·e−(0.02892·72) ≈5.55 ×1011 atoms
Therefore, after 3 days, approximately 5.55 ×1011 atoms are left in the
radioactive sample.
Question 14
Question
A sample of a radioactive material has an activity of 800 Bq at t= 0. The
activity decays to 400 Bq after 10 days. Calculate the half-life of the material.
Solution
Step 1: Use the formula for radioactive decay to find the decay constant.
λ=ln(2)
t1
2
where λis the decay constant, t1
2is the half-life of the material.
Step 2: Calculate the decay constant using the initial and final activity
values.
λ=ln(2)
10 , λ = 0.0693 days−1
Step 3: Use the decay constant to find the half-life of the material.
0.0693 = ln(2)
t1
2
, t1
2=ln(2)
0.0693
Step 4: Calculate the half-life of the material.
t1
2
≈0.693
0.0693 ≈10 days
Therefore, the half-life of the material is 10 days.
Question 15
Question
A sample of a radioactive substance decays according to the model A(t) =
A0·e−kt, where A(t) is the amount of the substance after time t,A0is the
11
initial amount of the substance, and k > 0 is a decay constant. Given that the
initial amount of the substance is 100 grams and that after 10 hours, the amount
remaining is 55 grams, calculate the decay constant kfor this substance.
Solution
Step 1: We are given that the initial amount of the substance is 100 grams, so
A0= 100.
Step 2: After 10 hours, the amount remaining is 55 grams, so we have
A(10) = 55. Substituting these values into the decay model, we get:
55 = 100 ·e−10k
Step 3: Divide both sides by 100 to isolate the exponential term:
0.55 = e−10k
Step 4: Take the natural logarithm of both sides to solve for k:
ln(0.55) = −10k
Step 5: Solve for kby dividing by −10:
k=ln(0.55)
−10
Step 6: Use a calculator to find the approximate value of k:
k≈ln(0.55)
−10 ≈
−0.5978
−10 ≈0.05978
Step 7: Therefore, the decay constant for this radioactive substance is ap-
proximately 0.05978.
Question 16
Question
A sample of a radioactive isotope has an initial mass of 200 grams. After 60
hours, only 50 grams of the isotope remains. If the half-life of this isotope is 30
hours, calculate the decay constant and the activity of the sample in becquerels
at t= 60 hours.
Solution
Step 1: Calculate the decay constant (λ). The decay constant can be calculated
using the formula:
λ=−ln(2)
t1/2
12
where ln denotes the natural logarithm and t1/2is the half-life of the isotope.
Plugging in the values for t1/2= 30 hours and ln(2) ≈0.693, we get:
λ=−0.693
30
λ=−0.0231 hours−1
Step 2: Calculate the activity of the sample at t= 60 hours. The activity of
a radioactive sample is given by:
A=λN
where Nis the number of radioactive atoms at any time t.
From the question, we are given the half-life of the isotope and the mass
of the remaining sample. We can calculate the number of radioactive atoms
remaining using the relationship between mass and number of atoms:
N=N0·1
2t
t1/2
where N0is the initial number of radioactive atoms, tis the time elapsed, and
t1/2is the half-life.
Given that N0is proportional to the mass of the sample, the initial number
of radioactive atoms (N0) can be calculated as:
N0=200 g
M
where Mis the molar mass of the isotope in grams per mole.
Substitute N0and t= 60 hours into the equation for N:
N=200
M·1
260
30
Finally, substitute λ=−0.0231 hours−1and Ninto the formula for activity:
A= (−0.0231) ·200
M·1
22
Therefore, the decay constant is approximately −0.0231 hours−1and the
activity of the sample at t= 60 hours is given by (−0.0231)·200
M·1
22becquerels.
Question 17
Question
A radioactive isotope has a half-life of 200 days. If the initial amount of the
isotope is 500 grams, how much of the isotope remains after 3 years?
13
Solution
Step 1: Calculate the number of half-lives that have passed in 3 years. Since
the half-life of the isotope is 200 days, we have:
Number of half-lives = 3 years ×365 days/year
200 days/half-life
Number of half-lives = 1095
200 ≈5.475
Step 2: Determine the amount of the isotope remaining after 5 half-lives.
The amount of the isotope remaining after each half-life is given by:
Amount remaining = 1
2×initial amount
After 5 half-lives, the amount remaining is:
Amount remaining = 1
25
×500 grams
Amount remaining = 1
32×500 = 15.625 grams
Therefore, after 3 years, there are 15.625 grams of the isotope remaining.
Question 18
Question
A sample of a radioactive isotope has an initial mass of 10 grams. After 8 hours,
only 2.5 grams of the isotope remain. If the half-life of the isotope is 5 hours,
determine the decay constant and the age of the sample.
Solution
Step 1: Determine the decay constant (λ) using the formula:
λ=−ln(0.5)
t1/2
where t1/2is the half-life of the isotope. The half-life is given as 5 hours, so
t1/2= 5 hours.
λ=−ln(0.5)
5
λ=−ln(0.5)
5
λ≈0.1386 hours−1
14
Step 2: Determine the age of the sample using the formula:
N(t) = N0e−λt
where N(t) is the remaining mass at time t,N0is the initial mass, λis the
decay constant, and tis the time elapsed. Rearrange the formula and plug in
the given values.
e−λt =N(t)
N0
lne−λt= ln N(t)
N0
−λt = ln N(t)
N0
t=−
ln N(t)
N0
λ
Substitute N(t)=2.5 grams, N0= 10 grams, and λ≈0.1386 hours−1.
t=−ln 2.5
10
0.1386
t=−ln(0.25)
0.1386
t≈1.386
0.1386 ≈10 hours
Therefore, the decay constant is approximately 0.1386 hours−1and the age
of the sample is 10 hours.
Question 19
Question
A radioactive substance decays according to the formula N(t) = N0e−kt, where
N(t) represents the amount of the substance remaining at time t,N0is the
initial amount of the substance, and kis the decay constant. If a sample of this
substance originally contained 100 grams of the substance and after 10 hours
only 40 grams remain, what is the half-life of the substance?
Solution
Step 1: Use the given information to set up the equation for the decay of the
substance. We are given that N0= 100 grams, N(t) = 40 grams, and t= 10
hours. We need to find the value of k. Using the formula N(t) = N0e−kt, we
have:
40 = 100e−10k
15
Step 2: Solve for k. Divide both sides by 100:
e−10k=40
100
e−10k= 0.4
Take the natural logarithm of both sides:
lne−10k= ln(0.4)
−10k= ln(0.4)
k=−ln(0.4)
10
Step 3: Find the half-life of the substance. The half-life of a radioactive
substance is the amount of time it takes for half of the substance to decay. The
half-life can be found using the formula:
T1/2=ln(2)
k
Substitute the value of kwe found in Step 2:
T1/2=ln(2)
−ln(0.4)
10
Simplify the expression:
T1/2=−10 ln(2)
ln(0.4)
T1/2≈6.91 hours
Therefore, the half-life of the substance is approximately 6.91 hours.
Question 20
Question
A sample of a radioactive element has an initial mass of 200 grams and decays
according to the formula m(t) = 200e−0.02t, where tis the time in years. Find
the mass of the element after 20 years.
Solution
Step 1: Determine the mass of the element after 20 years by substituting t= 20
into the formula m(t) = 200e−0.02t.
m(20) = 200e−0.02(20)
= 200e−0.4
16
Step 2: Evaluate the exponential term.
m(20) = 200e−0.4
= 200(0.6703)
= 134.06 grams
The mass of the element after 20 years is 134.06 grams.
Question 21
Question
A certain radioactive substance decays according to the formula A(t) = A0e−kt,
where A(t) is the amount of the substance remaining after tyears, A0is the
initial amount of the substance, and kis a positive constant. If 80% of a sample
of this substance decays in 10 years, determine the value of kand find the
half-life of the substance.
Solution
Step 1: Given the formula A(t) = A0e−kt, we know that after 10 years, 80%
of the sample has decayed, leaving 20% of the sample remaining. Therefore,
A(10) = 0.20A0.
0.20A0=A0e−10k
Step 2: Divide both sides by A0.
0.20 = e−10k
Step 3: Take the natural logarithm of both sides to solve for k.
ln(0.20) = −10k
k=−ln(0.20)
10
Step 4: Calculate the value of k.
k≈ln(5)
10 ≈0.0693
Step 5: The half-life of a radioactive substance is the time it takes for half
of the initial substance to decay. The half-life T1
2can be found by letting
A(t) = 1
2A0(as we want half of the substance remaining) and solving for t.
1
2A0=A0e−kt
1
2=e−kt
17
Step 6: Again, take the natural logarithm of both sides and solve for t.
ln 1
2=−kt
t=ln(2)
k
Step 7: Substituting the value of k, we find the half-life T1
2.
T1
2=ln(2)
0.0693 ≈10.0 years
Therefore, the value of k≈0.0693 and the half-life of the substance is
approximately 10.0 years.
Question 22
Question
A sample of a radioactive substance has an initial mass of 500 grams. After 10
hours, the mass of the sample is measured to be 200 grams. If the half-life of
the substance is 8 hours, what is the decay constant for this substance?
Solution
Step 1: Calculate the fraction of the sample remaining after 10 hours. Step 2:
Use the half-life formula to find the decay constant.
Step 1: Calculate the fraction of the sample remaining after 10 hours. Let
m0be the initial mass of the sample and mbe the mass of the sample after 10
hours. The fraction of the sample remaining is given by:
m
m0
=1
2t
T1
2
Where: m0= 500 grams (initial mass) m= 200 grams (mass after 10 hours)
t= 10 hours T1
2= 8 hours (half-life) Substitute the given values into the
formula:
200
500 =1
210
8
2
5=1
25
4
2
5=1
21.25
18
Step 2: Use the half-life formula to find the decay constant. The half-life
formula relates the decay constant λto the half-life T1
2:
λ=ln(2)
T1
2
Given that T1
2= 8 hours, we can find the decay constant:
λ=ln(2)
8≈0.0866 hours−1
Therefore, the decay constant for this radioactive substance is approximately
0.0866 hours−1.
Question 23
Question
A sample of a radioactive isotope has an initial mass of 50 grams. After 24
hours, only 10 grams of the isotope remains. If the half-life of the isotope is 12
hours, what is the decay constant for the isotope?
Solution
Step 1: Determine the fraction of the isotope remaining after 24 hours. Step
2: Use the formula N(t) = N0·e−kt to find the decay constant k. Step 3:
Substitute the known values into the equation and solve for k.
Step 1: The fraction of the isotope remaining after 24 hours can be calcu-
lated using the formula:
N(t)
N0
=e−kt
Where: - N(t) = final mass of the isotope after 24 hours = 10 grams - N0=
initial mass of the isotope = 50 grams We can now plug these values into the
equation: 10
50 =e−k·24
1
5=e−24k
Step 2: The formula N(t) = N0·e−kt can be rearranged to solve for the
decay constant k:
k=ln(N0/N(t))
t
Step 3: Substitute the known values N0= 50, N(t) = 10, and t= 24 into
the equation for k:
k=ln(50/10)
24
19
k=ln(5)
24
k≈1.609
24
k≈0.067% per hour
Therefore, the decay constant for the isotope is approximately 0.067% per
hour.
Question 24
Question
A sample of a radioactive material has an initial mass of 100 grams. After
10 hours, the mass of the sample decreases to 50 grams. The half-life of this
material is 8 hours. (a) What is the decay constant for this material? (b)
Determine the time it will take for the sample to decay to 10 grams.
Solution
(a) Let N(t) be the mass of the sample at time t, and N0be the initial mass.
Given that the half-life of the material is 8 hours, we can use the formula for
exponential decay:
N(t) = N0·2
−t
T1
2
where T1
2is the half-life of the material. Since N(10) = 50 and N0= 100, we
have:
50 = 100 ·2−10
8
0.5=2−5
4
1
2= 2−5
4
To find the decay constant (λ), we use the formula:
λ=ln(2)
T1
2
λ=ln(2)
8
Therefore, the decay constant for this material is λ=ln(2)
8.
(b) To find the time it will take for the sample to decay to 10 grams, we use
the formula:
N(t) = N0·e−λt
Since N(0) = 100 and N(t) = 10, we have:
10 = 100 ·e−ln(2)
8·t
20
0.1 = e−ln(2)
8·t
Taking the natural logarithm of both sides, we get:
ln(0.1) = −ln(2)
8·t
t=−8·ln(0.1)
ln(2)
t≈20.78 hours
Therefore, it will take approximately 20.78 hours for the sample to decay to 10
grams.
Question 25
Question
A certain radioactive substance has a half-life of 10 days. If you start with 100
grams of the substance, how many grams will remain after 30 days?
Solution
Step 1: Determine the fraction of substance remaining after each half-life period.
The formula to calculate the amount of substance remaining after a certain
time period is given by:
Amount remaining = Initial amount ×1
2time elapsed
half-life
For this problem, the initial amount is 100 grams, the half-life is 10 days,
and we want to find the amount remaining after 30 days.
Step 2: Calculate the amount remaining after the first half-life (10 days).
Amount remaining after 10 days = 100 ×1
210
10
= 100 ×1
2= 50 grams
Step 3: Calculate the amount remaining after the second half-life (20 days).
Amount remaining after 20 days = 50 ×1
210
10
= 50 ×1
2= 25 grams
Step 4: Calculate the amount remaining after the third half-life (30 days).
Amount remaining after 30 days = 25 ×1
210
10
= 25 ×1
2= 12.5 grams
Therefore, after 30 days, there will be 12.5 grams of the radioactive substance
remaining.
21
Question 26
Question
A radioactive isotope has a half-life of 5 days. If you start with a sample
containing 4 ×1012 atoms, how many atoms will remain after 20 days?
Solution
Step 1: Calculate the decay constant (λ) using the half-life formula T1/2=ln(2)
λ.
Step 1: λ=ln(2)
T1/2
=ln(2)
5≈0.1386 days−1
Step 2: Use the decay formula N(t) = N0e−λt to find the number of atoms
remaining after 20 days.
Step 2: N(20) = (4×1012)e−0.1386×20 ≈(4×1012)e−2.772 ≈(4×1012)×0.0637 ≈2.548×1012
After 20 days, approximately 2.548 ×1012 atoms will remain in the sample.
Question 27
Question
A sample of a radioactive isotope has an initial mass of 200 grams. After 9
hours, only 25 grams remain. If the half-life of the isotope is 6 hours, determine
the decay constant and the amount of the isotope that will remain after 20
hours.
Solution
Step 1: Determine the decay constant, λ, using the half-life formula:
λ=ln(2)
T1
2
=ln(2)
6≈0.1155 hours−1
Step 2: Use the exponential decay formula to find the amount of isotope
remaining after 20 hours, denoted as A:
A=A0·e−λt = 200 ·e−0.1155·20 ≈62.08 grams
Therefore, approximately 62.08 grams of the isotope will remain after 20
hours.
22
Question 28
Question
A sample of a radioactive isotope decays in such a way that at the end of 10
hours, only 39
Solution
Let N0be the original amount of the isotope and Ntbe the amount remaining
after time t. We are given that Nt= 0.39N0after 10 hours.
Step 1: Use the decay formula Nt=N01
2
t
T1
2where T1
2is the half-life of
the isotope.
0.39N0=N01
210
T1
2
Step 2: Divide both sides by N0and simplify.
0.39 = 1
210
T1
2
Step 3: Take the natural logarithm of both sides.
ln(0.39) = ln 1
210
T1
2!
ln(0.39) = 10
T1
2
ln 1
2
Step 4: Solve for T1
2.
T1
2=10 ln(2)
ln(0.39) ≈18.5 hours
Therefore, the half-life of the isotope is approximately 18.5 hours.
Question 29
Question
The half-life of a radioactive substance is 20 hours. If there are initially 100
grams of the substance, how much will remain after 60 hours?
23
Solution
Step 1: Determine the decay constant using the half-life formula:
Half-life (T) = ln 2
k
20 = ln 2
k
k=ln 2
20
Step 2: Write the decay model using the decay constant:
A(t) = A0e−kt
where: A(t) = amount remaining after time t,A0= initial amount, and k=
decay constant.
Step 3: Plug in the initial values:
A(t) = 100e−ln 2
20 ·60
Step 4: Simplify the expression:
A(t) = 100e−3 ln 2
Step 5: Recall the property of logarithms:
eln x=x
thus,
A(t) = 100eln 2−3
A(t) = 100 ·2−3
A(t) = 100 ·1
8
A(t) = 12.5 grams
Therefore, after 60 hours, there will be 12.5 grams of the substance remain-
ing.
Question 30
Question
A certain radioactive substance decays according to the equation N(t) = N0e−0.02t,
where N(t) represents the number of atoms remaining after tyears, N0is the
initial number of atoms, and tis the time in years. If the initial number of
atoms is 500, determine the half-life of the substance.
24
Solution
Step 1: Determine the half-life of the substance by finding the time at which
half of the initial number of atoms remain.
N(t)
N0
=1
2
N0e−0.02t
N0
=1
2
e−0.02t=1
2
lne−0.02t= ln 1
2
−0.02t= ln 1
2
t=ln 1
2
−0.02
t≈34.657 years
Therefore, the half-life of the substance is approximately 34.657 years.
Question 31
Question
An unknown radioactive substance has a half-life of 5 hours. If you start with
200 grams of the substance, how much will remain after 15 hours?
Solution
Step 1: Determine the decay constant λusing the formula T1/2=ln(2)
λ, where
T1/2is the half-life.
λ=ln(2)
5≈0.1386 hours−1
Step 2: Use the exponential decay formula N(t) = N0·e−λt to find the
amount of substance remaining after 15 hours.
N(15) = 200 ·e−0.1386·15
N(15) ≈200 ·e−2.079 ≈64.07 grams
Therefore, after 15 hours, approximately 64.07 grams of the substance will
remain.
25
Question 32
Question
A sample of a certain radioactive isotope decays according to the equation
N(t) = N0e−kt, where N(t) is the amount of the isotope remaining at time
t(in years), N0is the initial amount of the isotope, and kis the decay constant.
Given that the half-life of the isotope is 40 years, find the decay constant
k. Then, calculate the amount of the isotope remaining after 100 years if the
initial amount was 1000 grams.
Solution
Step 1: Find the decay constant k
The decay constant kcan be found using the formula k=ln(2)
T1
2
, where T1
2is the
half-life of the isotope. Given T1
2= 40 years, we have:
k=ln(2)
40
Step 2: Calculate the decay constant k
k=ln(2)
40 ≈0.0173 years−1
Step 3: Find the amount of the isotope remaining after 100 years
Substitute N0= 1000, k≈0.0173, and t= 100 into the decay equation N(t) =
N0e−kt:
N(100) = 1000e−0.0173·100
N(100) = 1000e−1.73
N(100) ≈1000 ·0.178
N(100) ≈178 grams
Therefore, after 100 years, the amount of the isotope remaining is approxi-
mately 178 grams.
Question 33
Question
A sample of a radioactive material decays according to the function N(t) =
N0e−kt, where N(t) is the amount of material at time t,N0is the initial amount
of material, and kis a decay constant. If it is known that 40% of the material
remains after 100 years, what is the half-life of this material?
26
Solution
Step 1: Determine the decay constant k. From the given information, we know
that N(100) = 0.4N0. Substituting this into the decay function, we get:
0.4N0=N0e−k·100
0.4 = e−100k
Taking the natural logarithm of both sides, we get:
ln(0.4) = −100k
k=−ln(0.4)
100
Step 2: Calculate the half-life of the material. The half-life (t1/2) is the
time it takes for half of the material to decay. We can find t1/2by setting
N(t1/2) = 0.5N0:
0.5N0=N0e−k·t1/2
0.5 = e−k·t1/2
Taking the natural logarithm of both sides and substituting the value of k, we
get:
ln(0.5) = −ln(0.4)
100 ·t1/2
t1/2=100 ln(0.5)
ln(0.4) ≈48.79 years
Therefore, the half-life of this material is approximately 48.79 years.
Question 34
Question
A sample of radioactive material has an initial mass of 500 grams. After 10
days, only 62.5 grams of the material remain. If the half-life of the material is
5 days, what is the decay constant of the material?
Solution
Step 1: Calculate the fraction of the original material remaining after 10 days.
Step 2: Use the half-life formula to find the decay constant.
Step 1: Let N0be the initial quantity of the material and Nbe the quantity
remaining after time t. The fraction of material remaining after 10 days is given
by: N
N0
=62.5
500 = 0.125
27
Step 2: The formula for radioactive decay is given by:
N
N0
=e−λt
where: - Nis the quantity remaining after time t, - N0is the initial quantity of
the material, - λis the decay constant, and - tis the time elapsed.
Using the information from Step 1, we have:
0.125 = e−λ·10
−λ·10 = ln(0.125)
λ=−ln(0.125)
10
Thus, the decay constant of the material is:
λ≈ln(8)
10 ≈2.079
10 ≈0.208 day−1
Therefore, the decay constant of the material is approximately 0.208 day−1.
Question 35
Question
An unknown radioactive substance has a half-life of 10 days. If you start with
100 grams of the substance, how much of it will remain after 30 days?
Solution
Step 1: Determine the decay constant by using the formula for radioactive decay:
N(t) = N0·1
2t
T1
2
where: - N(t) is the quantity of the substance after time t, - N0is the initial
quantity of the substance, - T1
2is the half-life of the substance.
Plugging in the values, we have:
N(30) = 100 ·1
230
10
Step 2: Calculate the remaining quantity of the substance after 30 days:
N(30) = 100 ·1
23
= 100 ·1
8= 12.5 grams
Therefore, after 30 days, there will be 12.5 grams of the radioactive substance
remaining.
28
Solution
Step 1: Determine the decay constant (λ) using the formula λ=ln(2)
t1/2
, where
t1/2is the half-life of the element. Step 2: Calculate the activity of the sample
after 10 days using the formula A(t) = A0·e−λt, where A0is the initial activity
of the sample. Step 3: Substitute the values into the formula from step 2 to
find the activity after 10 days.
Step 1: Calculate the decay constant λGiven: t1/2= 3.8 days
λ=ln(2)
t1/2
=ln(2)
3.8≈0.182 day−1
Step 2: Calculate the activity after 10 days Given: A0= 5000 Bq, t=
10 days
A(10) = 5000 ·e−0.182·10
Step 3: Substitute the values to find the final activity
A(10) = 5000 ·e−1.82 ≈1403 Bq
Therefore, the activity of the sample after 10 days will be approximately
1403 Bq.
Question 3
Question
A certain radioactive substance decays according to the equation N(t) = N0e−kt,
where N(t) is the amount of substance remaining at time t,N0is the initial
amount of substance, kis the decay constant, and tis time in years. Given that
60
Solution
Step 1: We are given that 60
0.4N0=N0e−30k
Step 2: Divide both sides by N0to simplify the equation:
0.4 = e−30k
Step 3: Take the natural logarithm (ln) of both sides to solve for k:
ln(0.4) = lne−30k
ln(0.4) = −30k
Step 4: Solve for kby dividing both sides by −30:
k=−ln(0.4)
30 ≈0.0231 year−1
Therefore, the value of the decay constant kis approximately 0.0231 year−1.
2
Question 4
Question
A sample of a radioactive isotope decays according to the formula Q(t) =
Q0e−kt, where Q(t) is the quantity of the isotope remaining at time t(in years),
Q0is the initial quantity of the isotope, and kis the decay constant. Suppose
that a sample of a radioactive isotope has an initial quantity of 100 grams and
the decay constant is 0.02. How many years will it take for the quantity of the
isotope to reduce to 50 grams?
Solution
Step 1: Substitute the given values into the formula Q(t) = Q0e−kt to form an
equation we can solve for t.
We have Q(t) = 100e−0.02t= 50
Step 2: Solve for t by dividing both sides by 100 and applying natural
logarithm to both sides.
100e−0.02t
100 =50
100
e−0.02t= 0.5
lne−0.02t= ln(0.5)
−0.02t= ln(0.5)
Step 3: Solve for t by dividing by -0.02.
t=ln(0.5)
−0.02
Step 4: Calculate the final answer using a calculator.
t≈ln(0.5)
−0.02 ≈34.657 years
Therefore, it will take approximately 34.657 years for the quantity of the
isotope to reduce to 50 grams.
Question 5
Question
A certain radioactive substance has a half-life of 10 days. If you start with an
initial amount of 100 grams, find:
1. The amount of the substance remaining after 30 days.
2. How long it will take for the initial amount to decay to 25 grams.
3
Solution
Let’s denote: - Aas the amount of the substance remaining at any time t, - A0
as the initial amount of the substance, - tas the time that has passed, - T1
2as
the half-life of the substance.
1. We can use the formula for radioactive decay:
A=A0·1
2t
T1
2
After 30 days:
A= 100 ·1
230
10
A= 100 ·1
23
A= 100 ·1
8
A= 12.5 grams
Therefore, the amount of the substance remaining after 30 days is 12.5
grams.
2. To find the time it will take for the initial amount to decay to 25 grams,
we set A= 25 and solve for t:
25 = 100 ·1
2t
10
1
4=1
2t
10
2=2t
10
1 = t
10
t= 10 days
Therefore, it will take 10 days for the initial amount to decay to 25 grams.
Question 6
Question
A sample of a radioactive substance has an initial activity of 10,000 counts per
minute. After 3 hours, the activity has decreased to 5000 counts per minute.
Assuming the decay is exponential, find the half-life of the substance.
4
Solution
Step 1: The decay of a radioactive substance can be modeled by the equation:
N(t) = N0·e−λt
where: - N(t) is the activity at time t, - N0is the initial activity, - λis the decay
constant, - eis the base of the natural logarithm (approximately 2.71828).
Step 2: Substituting the given values into the equation:
N(0) = 10000 (initial activity)
N(3) = 5000 (activity after 3 hours)
Step 3: We can set up the following equations:
10000 = 10000 ·e−3λ
5000 = 10000 ·e−3λ
Step 4: Divide the second equation by the first to eliminate N0:
5000
10000 =e−3λ
1
2=e−3λ
Step 5: Take the natural logarithm of both sides to solve for λ:
ln 1
2=−3λ
−ln 2 = −3λ
λ=ln 2
3
Step 6: The half-life of the substance can be found using the relation between
half-life (T1/2) and the decay constant (λ):
T1/2=ln 2
λ=ln 2
ln 2
3
= 3 hours
Therefore, the half-life of the substance is 3 hours.
Question 7
Question
A sample of a radioactive substance decays according to the formula N(t) =
N0e−kt, where N(t) is the amount of the substance remaining after tdays, N0
is the initial amount of the substance, and kis the decay constant. If 30
5
Solution
Step 1: Using the provided information to set up equations.
Let N0be the initial amount of the substance. From the information given,
we have:
N(100) = 0.3N0and N(200) = 0.2N0
Substitute the formula N(t) = N0e−kt into the equations to get:
0.3N0=N0e−100kand 0.2N0=N0e−200k
Step 2: Solving the equations to find the decay constant k.
Divide the two equations to eliminate N0:
0.3
0.2=e−100k
e−200k
Simplify the right side: 3
2=e100k
Taking the natural logarithm of both sides:
ln 3
2= 100k
Solving for k:
k=1
100 ln 3
2≈ −0.0051
Therefore, the decay constant k≈ −0.0051.
Question 8
Question
A sample of a radioactive isotope decays according to the formula N(t) =
N0e−0.02t, where N(t) is the amount of the isotope remaining after tyears
and N0is the initial amount of the isotope. If the initial amount is 100 grams,
determine the amount of the isotope remaining after 30 years.
Solution
Step 1: Given that the initial amount N0= 100 grams, we can substitute this
value into the formula to get N(t) = 100e−0.02t.
Step 2: We are asked to find the amount of the isotope remaining after 30
years. Substituting t= 30 into the formula, we get N(30) = 100e−0.02(30).
Step 3: Simplifying the expression, we have N(30) = 100e−0.6.
Step 4: Using the value of e≈2.71828, we can further simplify the expression
to get N(30) ≈100 ×2.71828−0.6.
6
Step 5: Calculating the value, we find N(30) ≈100 ×0.54881 ≈54.881
grams.
Step 6: Therefore, the amount of the isotope remaining after 30 years is
approximately 54.881 grams.
Question 9
Question
A sample of an unknown radioactive material has an initial mass of 200 grams
and after 10 days, its mass is reduced to 100 grams. The half-life of the material
is 5 days. Determine the decay constant and the mass of the material after 30
days.
Solution
Step 1: Determine the decay constant using the half-life formula.
Half-life (T) = ln(2)
k
Given that the half-life (T) is 5 days, we can solve for the decay constant (k).
5
ln(2) =k
k≈0.693 days−1
Step 2: Using the decay constant, we can determine the mass of the material
after 30 days using the exponential decay formula.
m(t) = m0·e−kt
where: - m(t) is the mass of the material after time t, - m0is the initial mass,
-kis the decay constant, - tis the time elapsed.
Plug in the values m0= 200g, k= 0.693 days−1, and t= 30 days into the
formula.
m(30) = 200 ·e−0.693·30
m(30) = 200 ·e−20.79
m(30) ≈200 ·1.604 ×10−10
m(30) ≈3.208 ×10−8grams
Therefore, the mass of the material after 30 days is approximately 3.208 ×
10−8grams.
7
Question 10
Question
A sample of a radioactive substance has an initial mass of 400 grams. After 5
hours, only 100 grams of the substance remains. If the half-life of the substance
is 3 hours, determine the decay constant and the age of the sample.
Solution
Step 1: Determine the decay constant.
The decay constant, denoted by λ, can be found using the formula:
N(t) = N0·e−λt
where N(t) is the amount of substance remaining at time t,N0is the initial
amount of substance, and eis Euler’s number (≈2.71828).
Given that N(5) = 100 and N0= 400, we can plug these values into the
formula to solve for λ:
100 = 400 ·e−λ·5
100
400 =e−5λ
1
4=e−5λ
ln 1
4=−5λ
λ=ln(4)
5≈0.2773 hours−1
Step 2: Determine the age of the sample.
To find the age of the sample, we need to solve for tin the formula N(t) = 100
using the decay constant λwe found.
100 = 400 ·e−0.2773t
1
4=e−0.2773t
ln 1
4=−0.2773t
t=ln(4)
0.2773 ≈7.9 hours
Therefore, the decay constant is approximately 0.2773 hours−1and the age
of the sample is approximately 7.9 hours.
8
Question 11
Question
A sample of a radioactive substance has an initial mass of 200 grams. After 10
days, the mass of the substance has decreased to 100 grams. If the half-life of
the substance is 5 days, what is the decay constant of the substance?
Solution
Step 1: Calculate the fraction of the substance remaining after 10 days. Step
2: Use the fraction of the substance remaining to calculate the decay constant.
Step 1: The fraction of the substance remaining after 10 days can be cal-
culated using the formula:
Fraction remaining = 1
2time elapsed
half-life
Plugging in the values:
Fraction remaining = 1
210
5
=1
22
=1
4
Therefore, after 10 days, 1
4of the substance remains.
Step 2: The decay constant (λ) can be calculated using the formula:
Fraction remaining = e−λ×time elapsed
Plugging in the known values (1
4remaining after 10 days):
1
4=e−λ×10
Taking the natural logarithm of both sides:
ln 1
4=−λ×10
Solving for λ:
λ=ln(4)
10 ≈0.07918 days−1
Therefore, the decay constant of the substance is approximately 0.07918
days−1.
Question 12
Question
A sample of radioactive material has an initial activity of 800 Becquerels. After
10 minutes, the activity decreases to 200 Becquerels. If the decay is exponential,
what is the half-life of the material?
9
Solution
Step 1: Let’s denote the initial activity as A0= 800 Becquerels, the activity
after 10 minutes as A= 200 Becquerels, and the half-life as T1
2.
Step 2: We can use the exponential decay formula to relate the activities at
different times:
A=A0×1
2t
T1
2
Step 3: From the given data, we have:
200 = 800 ×1
210
T1
2
Step 4: Simplifying the equation, we get:
1
210
T1
2=1
4
Step 5: To solve for T1
2, we can write 1/4 as 2−2:
2
−10
T1
2= 2−2
Step 6: Equating the exponents, we get:
−10
T1
2
=−2
Step 7: Solving for T1
2, we find:
T1
2= 5 minutes
Step 8: Therefore, the half-life of the radioactive material is 5 minutes.
Question 13
Question
The half-life of a radioactive isotope is 24 hours. If you start with a sample
containing 2 ×1012 atoms, how many atoms are left after 3 days?
Solution
Step 1: Calculate the decay constant, λ. Given the half-life, T1/2= 24 hours,
we can use the formula λ=ln(2)
T1/2
to find the decay constant.
λ=ln(2)
24 ≈0.02892 hours−1
10
Step 2: Use the decay formula to find the number of atoms remaining after
3 days. The number of atoms remaining can be found using the formula N(t) =
N0·e−λt, where: - N(t) is the number of atoms remaining after time t-N0is
the initial number of atoms - λis the decay constant - tis the time elapsed
Substitute the values, N0= 2 ×1012,λ≈0.02892, and t= 3 ×24 = 72
hours, into the formula to find N(72).
N(72) = 2 ×1012 ·e−(0.02892·72) ≈5.55 ×1011 atoms
Therefore, after 3 days, approximately 5.55 ×1011 atoms are left in the
radioactive sample.
Question 14
Question
A sample of a radioactive material has an activity of 800 Bq at t= 0. The
activity decays to 400 Bq after 10 days. Calculate the half-life of the material.
Solution
Step 1: Use the formula for radioactive decay to find the decay constant.
λ=ln(2)
t1
2
where λis the decay constant, t1
2is the half-life of the material.
Step 2: Calculate the decay constant using the initial and final activity
values.
λ=ln(2)
10 , λ = 0.0693 days−1
Step 3: Use the decay constant to find the half-life of the material.
0.0693 = ln(2)
t1
2
, t1
2=ln(2)
0.0693
Step 4: Calculate the half-life of the material.
t1
2
≈0.693
0.0693 ≈10 days
Therefore, the half-life of the material is 10 days.
Question 15
Question
A sample of a radioactive substance decays according to the model A(t) =
A0·e−kt, where A(t) is the amount of the substance after time t,A0is the
11
initial amount of the substance, and k > 0 is a decay constant. Given that the
initial amount of the substance is 100 grams and that after 10 hours, the amount
remaining is 55 grams, calculate the decay constant kfor this substance.
Solution
Step 1: We are given that the initial amount of the substance is 100 grams, so
A0= 100.
Step 2: After 10 hours, the amount remaining is 55 grams, so we have
A(10) = 55. Substituting these values into the decay model, we get:
55 = 100 ·e−10k
Step 3: Divide both sides by 100 to isolate the exponential term:
0.55 = e−10k
Step 4: Take the natural logarithm of both sides to solve for k:
ln(0.55) = −10k
Step 5: Solve for kby dividing by −10:
k=ln(0.55)
−10
Step 6: Use a calculator to find the approximate value of k:
k≈ln(0.55)
−10 ≈
−0.5978
−10 ≈0.05978
Step 7: Therefore, the decay constant for this radioactive substance is ap-
proximately 0.05978.
Question 16
Question
A sample of a radioactive isotope has an initial mass of 200 grams. After 60
hours, only 50 grams of the isotope remains. If the half-life of this isotope is 30
hours, calculate the decay constant and the activity of the sample in becquerels
at t= 60 hours.
Solution
Step 1: Calculate the decay constant (λ). The decay constant can be calculated
using the formula:
λ=−ln(2)
t1/2
12
where ln denotes the natural logarithm and t1/2is the half-life of the isotope.
Plugging in the values for t1/2= 30 hours and ln(2) ≈0.693, we get:
λ=−0.693
30
λ=−0.0231 hours−1
Step 2: Calculate the activity of the sample at t= 60 hours. The activity of
a radioactive sample is given by:
A=λN
where Nis the number of radioactive atoms at any time t.
From the question, we are given the half-life of the isotope and the mass
of the remaining sample. We can calculate the number of radioactive atoms
remaining using the relationship between mass and number of atoms:
N=N0·1
2t
t1/2
where N0is the initial number of radioactive atoms, tis the time elapsed, and
t1/2is the half-life.
Given that N0is proportional to the mass of the sample, the initial number
of radioactive atoms (N0) can be calculated as:
N0=200 g
M
where Mis the molar mass of the isotope in grams per mole.
Substitute N0and t= 60 hours into the equation for N:
N=200
M·1
260
30
Finally, substitute λ=−0.0231 hours−1and Ninto the formula for activity:
A= (−0.0231) ·200
M·1
22
Therefore, the decay constant is approximately −0.0231 hours−1and the
activity of the sample at t= 60 hours is given by (−0.0231)·200
M·1
22becquerels.
Question 17
Question
A radioactive isotope has a half-life of 200 days. If the initial amount of the
isotope is 500 grams, how much of the isotope remains after 3 years?
13
Solution
Step 1: Calculate the number of half-lives that have passed in 3 years. Since
the half-life of the isotope is 200 days, we have:
Number of half-lives = 3 years ×365 days/year
200 days/half-life
Number of half-lives = 1095
200 ≈5.475
Step 2: Determine the amount of the isotope remaining after 5 half-lives.
The amount of the isotope remaining after each half-life is given by:
Amount remaining = 1
2×initial amount
After 5 half-lives, the amount remaining is:
Amount remaining = 1
25
×500 grams
Amount remaining = 1
32×500 = 15.625 grams
Therefore, after 3 years, there are 15.625 grams of the isotope remaining.
Question 18
Question
A sample of a radioactive isotope has an initial mass of 10 grams. After 8 hours,
only 2.5 grams of the isotope remain. If the half-life of the isotope is 5 hours,
determine the decay constant and the age of the sample.
Solution
Step 1: Determine the decay constant (λ) using the formula:
λ=−ln(0.5)
t1/2
where t1/2is the half-life of the isotope. The half-life is given as 5 hours, so
t1/2= 5 hours.
λ=−ln(0.5)
5
λ=−ln(0.5)
5
λ≈0.1386 hours−1
14
Step 2: Determine the age of the sample using the formula:
N(t) = N0e−λt
where N(t) is the remaining mass at time t,N0is the initial mass, λis the
decay constant, and tis the time elapsed. Rearrange the formula and plug in
the given values.
e−λt =N(t)
N0
lne−λt= ln N(t)
N0
−λt = ln N(t)
N0
t=−
ln N(t)
N0
λ
Substitute N(t)=2.5 grams, N0= 10 grams, and λ≈0.1386 hours−1.
t=−ln 2.5
10
0.1386
t=−ln(0.25)
0.1386
t≈1.386
0.1386 ≈10 hours
Therefore, the decay constant is approximately 0.1386 hours−1and the age
of the sample is 10 hours.
Question 19
Question
A radioactive substance decays according to the formula N(t) = N0e−kt, where
N(t) represents the amount of the substance remaining at time t,N0is the
initial amount of the substance, and kis the decay constant. If a sample of this
substance originally contained 100 grams of the substance and after 10 hours
only 40 grams remain, what is the half-life of the substance?
Solution
Step 1: Use the given information to set up the equation for the decay of the
substance. We are given that N0= 100 grams, N(t) = 40 grams, and t= 10
hours. We need to find the value of k. Using the formula N(t) = N0e−kt, we
have:
40 = 100e−10k
15
Step 2: Solve for k. Divide both sides by 100:
e−10k=40
100
e−10k= 0.4
Take the natural logarithm of both sides:
lne−10k= ln(0.4)
−10k= ln(0.4)
k=−ln(0.4)
10
Step 3: Find the half-life of the substance. The half-life of a radioactive
substance is the amount of time it takes for half of the substance to decay. The
half-life can be found using the formula:
T1/2=ln(2)
k
Substitute the value of kwe found in Step 2:
T1/2=ln(2)
−ln(0.4)
10
Simplify the expression:
T1/2=−10 ln(2)
ln(0.4)
T1/2≈6.91 hours
Therefore, the half-life of the substance is approximately 6.91 hours.
Question 20
Question
A sample of a radioactive element has an initial mass of 200 grams and decays
according to the formula m(t) = 200e−0.02t, where tis the time in years. Find
the mass of the element after 20 years.
Solution
Step 1: Determine the mass of the element after 20 years by substituting t= 20
into the formula m(t) = 200e−0.02t.
m(20) = 200e−0.02(20)
= 200e−0.4
16
Step 2: Evaluate the exponential term.
m(20) = 200e−0.4
= 200(0.6703)
= 134.06 grams
The mass of the element after 20 years is 134.06 grams.
Question 21
Question
A certain radioactive substance decays according to the formula A(t) = A0e−kt,
where A(t) is the amount of the substance remaining after tyears, A0is the
initial amount of the substance, and kis a positive constant. If 80% of a sample
of this substance decays in 10 years, determine the value of kand find the
half-life of the substance.
Solution
Step 1: Given the formula A(t) = A0e−kt, we know that after 10 years, 80%
of the sample has decayed, leaving 20% of the sample remaining. Therefore,
A(10) = 0.20A0.
0.20A0=A0e−10k
Step 2: Divide both sides by A0.
0.20 = e−10k
Step 3: Take the natural logarithm of both sides to solve for k.
ln(0.20) = −10k
k=−ln(0.20)
10
Step 4: Calculate the value of k.
k≈ln(5)
10 ≈0.0693
Step 5: The half-life of a radioactive substance is the time it takes for half
of the initial substance to decay. The half-life T1
2can be found by letting
A(t) = 1
2A0(as we want half of the substance remaining) and solving for t.
1
2A0=A0e−kt
1
2=e−kt
17
Step 6: Again, take the natural logarithm of both sides and solve for t.
ln 1
2=−kt
t=ln(2)
k
Step 7: Substituting the value of k, we find the half-life T1
2.
T1
2=ln(2)
0.0693 ≈10.0 years
Therefore, the value of k≈0.0693 and the half-life of the substance is
approximately 10.0 years.
Question 22
Question
A sample of a radioactive substance has an initial mass of 500 grams. After 10
hours, the mass of the sample is measured to be 200 grams. If the half-life of
the substance is 8 hours, what is the decay constant for this substance?
Solution
Step 1: Calculate the fraction of the sample remaining after 10 hours. Step 2:
Use the half-life formula to find the decay constant.
Step 1: Calculate the fraction of the sample remaining after 10 hours. Let
m0be the initial mass of the sample and mbe the mass of the sample after 10
hours. The fraction of the sample remaining is given by:
m
m0
=1
2t
T1
2
Where: m0= 500 grams (initial mass) m= 200 grams (mass after 10 hours)
t= 10 hours T1
2= 8 hours (half-life) Substitute the given values into the
formula:
200
500 =1
210
8
2
5=1
25
4
2
5=1
21.25
18
Step 2: Use the half-life formula to find the decay constant. The half-life
formula relates the decay constant λto the half-life T1
2:
λ=ln(2)
T1
2
Given that T1
2= 8 hours, we can find the decay constant:
λ=ln(2)
8≈0.0866 hours−1
Therefore, the decay constant for this radioactive substance is approximately
0.0866 hours−1.
Question 23
Question
A sample of a radioactive isotope has an initial mass of 50 grams. After 24
hours, only 10 grams of the isotope remains. If the half-life of the isotope is 12
hours, what is the decay constant for the isotope?
Solution
Step 1: Determine the fraction of the isotope remaining after 24 hours. Step
2: Use the formula N(t) = N0·e−kt to find the decay constant k. Step 3:
Substitute the known values into the equation and solve for k.
Step 1: The fraction of the isotope remaining after 24 hours can be calcu-
lated using the formula:
N(t)
N0
=e−kt
Where: - N(t) = final mass of the isotope after 24 hours = 10 grams - N0=
initial mass of the isotope = 50 grams We can now plug these values into the
equation: 10
50 =e−k·24
1
5=e−24k
Step 2: The formula N(t) = N0·e−kt can be rearranged to solve for the
decay constant k:
k=ln(N0/N(t))
t
Step 3: Substitute the known values N0= 50, N(t) = 10, and t= 24 into
the equation for k:
k=ln(50/10)
24
19
k=ln(5)
24
k≈1.609
24
k≈0.067% per hour
Therefore, the decay constant for the isotope is approximately 0.067% per
hour.
Question 24
Question
A sample of a radioactive material has an initial mass of 100 grams. After
10 hours, the mass of the sample decreases to 50 grams. The half-life of this
material is 8 hours. (a) What is the decay constant for this material? (b)
Determine the time it will take for the sample to decay to 10 grams.
Solution
(a) Let N(t) be the mass of the sample at time t, and N0be the initial mass.
Given that the half-life of the material is 8 hours, we can use the formula for
exponential decay:
N(t) = N0·2
−t
T1
2
where T1
2is the half-life of the material. Since N(10) = 50 and N0= 100, we
have:
50 = 100 ·2−10
8
0.5=2−5
4
1
2= 2−5
4
To find the decay constant (λ), we use the formula:
λ=ln(2)
T1
2
λ=ln(2)
8
Therefore, the decay constant for this material is λ=ln(2)
8.
(b) To find the time it will take for the sample to decay to 10 grams, we use
the formula:
N(t) = N0·e−λt
Since N(0) = 100 and N(t) = 10, we have:
10 = 100 ·e−ln(2)
8·t
20
0.1 = e−ln(2)
8·t
Taking the natural logarithm of both sides, we get:
ln(0.1) = −ln(2)
8·t
t=−8·ln(0.1)
ln(2)
t≈20.78 hours
Therefore, it will take approximately 20.78 hours for the sample to decay to 10
grams.
Question 25
Question
A certain radioactive substance has a half-life of 10 days. If you start with 100
grams of the substance, how many grams will remain after 30 days?
Solution
Step 1: Determine the fraction of substance remaining after each half-life period.
The formula to calculate the amount of substance remaining after a certain
time period is given by:
Amount remaining = Initial amount ×1
2time elapsed
half-life
For this problem, the initial amount is 100 grams, the half-life is 10 days,
and we want to find the amount remaining after 30 days.
Step 2: Calculate the amount remaining after the first half-life (10 days).
Amount remaining after 10 days = 100 ×1
210
10
= 100 ×1
2= 50 grams
Step 3: Calculate the amount remaining after the second half-life (20 days).
Amount remaining after 20 days = 50 ×1
210
10
= 50 ×1
2= 25 grams
Step 4: Calculate the amount remaining after the third half-life (30 days).
Amount remaining after 30 days = 25 ×1
210
10
= 25 ×1
2= 12.5 grams
Therefore, after 30 days, there will be 12.5 grams of the radioactive substance
remaining.
21
Question 26
Question
A radioactive isotope has a half-life of 5 days. If you start with a sample
containing 4 ×1012 atoms, how many atoms will remain after 20 days?
Solution
Step 1: Calculate the decay constant (λ) using the half-life formula T1/2=ln(2)
λ.
Step 1: λ=ln(2)
T1/2
=ln(2)
5≈0.1386 days−1
Step 2: Use the decay formula N(t) = N0e−λt to find the number of atoms
remaining after 20 days.
Step 2: N(20) = (4×1012)e−0.1386×20 ≈(4×1012)e−2.772 ≈(4×1012)×0.0637 ≈2.548×1012
After 20 days, approximately 2.548 ×1012 atoms will remain in the sample.
Question 27
Question
A sample of a radioactive isotope has an initial mass of 200 grams. After 9
hours, only 25 grams remain. If the half-life of the isotope is 6 hours, determine
the decay constant and the amount of the isotope that will remain after 20
hours.
Solution
Step 1: Determine the decay constant, λ, using the half-life formula:
λ=ln(2)
T1
2
=ln(2)
6≈0.1155 hours−1
Step 2: Use the exponential decay formula to find the amount of isotope
remaining after 20 hours, denoted as A:
A=A0·e−λt = 200 ·e−0.1155·20 ≈62.08 grams
Therefore, approximately 62.08 grams of the isotope will remain after 20
hours.
22
Question 28
Question
A sample of a radioactive isotope decays in such a way that at the end of 10
hours, only 39
Solution
Let N0be the original amount of the isotope and Ntbe the amount remaining
after time t. We are given that Nt= 0.39N0after 10 hours.
Step 1: Use the decay formula Nt=N01
2
t
T1
2where T1
2is the half-life of
the isotope.
0.39N0=N01
210
T1
2
Step 2: Divide both sides by N0and simplify.
0.39 = 1
210
T1
2
Step 3: Take the natural logarithm of both sides.
ln(0.39) = ln 1
210
T1
2!
ln(0.39) = 10
T1
2
ln 1
2
Step 4: Solve for T1
2.
T1
2=10 ln(2)
ln(0.39) ≈18.5 hours
Therefore, the half-life of the isotope is approximately 18.5 hours.
Question 29
Question
The half-life of a radioactive substance is 20 hours. If there are initially 100
grams of the substance, how much will remain after 60 hours?
23
Solution
Step 1: Determine the decay constant using the half-life formula:
Half-life (T) = ln 2
k
20 = ln 2
k
k=ln 2
20
Step 2: Write the decay model using the decay constant:
A(t) = A0e−kt
where: A(t) = amount remaining after time t,A0= initial amount, and k=
decay constant.
Step 3: Plug in the initial values:
A(t) = 100e−ln 2
20 ·60
Step 4: Simplify the expression:
A(t) = 100e−3 ln 2
Step 5: Recall the property of logarithms:
eln x=x
thus,
A(t) = 100eln 2−3
A(t) = 100 ·2−3
A(t) = 100 ·1
8
A(t) = 12.5 grams
Therefore, after 60 hours, there will be 12.5 grams of the substance remain-
ing.
Question 30
Question
A certain radioactive substance decays according to the equation N(t) = N0e−0.02t,
where N(t) represents the number of atoms remaining after tyears, N0is the
initial number of atoms, and tis the time in years. If the initial number of
atoms is 500, determine the half-life of the substance.
24
Solution
Step 1: Determine the half-life of the substance by finding the time at which
half of the initial number of atoms remain.
N(t)
N0
=1
2
N0e−0.02t
N0
=1
2
e−0.02t=1
2
lne−0.02t= ln 1
2
−0.02t= ln 1
2
t=ln 1
2
−0.02
t≈34.657 years
Therefore, the half-life of the substance is approximately 34.657 years.
Question 31
Question
An unknown radioactive substance has a half-life of 5 hours. If you start with
200 grams of the substance, how much will remain after 15 hours?
Solution
Step 1: Determine the decay constant λusing the formula T1/2=ln(2)
λ, where
T1/2is the half-life.
λ=ln(2)
5≈0.1386 hours−1
Step 2: Use the exponential decay formula N(t) = N0·e−λt to find the
amount of substance remaining after 15 hours.
N(15) = 200 ·e−0.1386·15
N(15) ≈200 ·e−2.079 ≈64.07 grams
Therefore, after 15 hours, approximately 64.07 grams of the substance will
remain.
25
Question 32
Question
A sample of a certain radioactive isotope decays according to the equation
N(t) = N0e−kt, where N(t) is the amount of the isotope remaining at time
t(in years), N0is the initial amount of the isotope, and kis the decay constant.
Given that the half-life of the isotope is 40 years, find the decay constant
k. Then, calculate the amount of the isotope remaining after 100 years if the
initial amount was 1000 grams.
Solution
Step 1: Find the decay constant k
The decay constant kcan be found using the formula k=ln(2)
T1
2
, where T1
2is the
half-life of the isotope. Given T1
2= 40 years, we have:
k=ln(2)
40
Step 2: Calculate the decay constant k
k=ln(2)
40 ≈0.0173 years−1
Step 3: Find the amount of the isotope remaining after 100 years
Substitute N0= 1000, k≈0.0173, and t= 100 into the decay equation N(t) =
N0e−kt:
N(100) = 1000e−0.0173·100
N(100) = 1000e−1.73
N(100) ≈1000 ·0.178
N(100) ≈178 grams
Therefore, after 100 years, the amount of the isotope remaining is approxi-
mately 178 grams.
Question 33
Question
A sample of a radioactive material decays according to the function N(t) =
N0e−kt, where N(t) is the amount of material at time t,N0is the initial amount
of material, and kis a decay constant. If it is known that 40% of the material
remains after 100 years, what is the half-life of this material?
26
Solution
Step 1: Determine the decay constant k. From the given information, we know
that N(100) = 0.4N0. Substituting this into the decay function, we get:
0.4N0=N0e−k·100
0.4 = e−100k
Taking the natural logarithm of both sides, we get:
ln(0.4) = −100k
k=−ln(0.4)
100
Step 2: Calculate the half-life of the material. The half-life (t1/2) is the
time it takes for half of the material to decay. We can find t1/2by setting
N(t1/2) = 0.5N0:
0.5N0=N0e−k·t1/2
0.5 = e−k·t1/2
Taking the natural logarithm of both sides and substituting the value of k, we
get:
ln(0.5) = −ln(0.4)
100 ·t1/2
t1/2=100 ln(0.5)
ln(0.4) ≈48.79 years
Therefore, the half-life of this material is approximately 48.79 years.
Question 34
Question
A sample of radioactive material has an initial mass of 500 grams. After 10
days, only 62.5 grams of the material remain. If the half-life of the material is
5 days, what is the decay constant of the material?
Solution
Step 1: Calculate the fraction of the original material remaining after 10 days.
Step 2: Use the half-life formula to find the decay constant.
Step 1: Let N0be the initial quantity of the material and Nbe the quantity
remaining after time t. The fraction of material remaining after 10 days is given
by: N
N0
=62.5
500 = 0.125
27
Step 2: The formula for radioactive decay is given by:
N
N0
=e−λt
where: - Nis the quantity remaining after time t, - N0is the initial quantity of
the material, - λis the decay constant, and - tis the time elapsed.
Using the information from Step 1, we have:
0.125 = e−λ·10
−λ·10 = ln(0.125)
λ=−ln(0.125)
10
Thus, the decay constant of the material is:
λ≈ln(8)
10 ≈2.079
10 ≈0.208 day−1
Therefore, the decay constant of the material is approximately 0.208 day−1.
Question 35
Question
An unknown radioactive substance has a half-life of 10 days. If you start with
100 grams of the substance, how much of it will remain after 30 days?
Solution
Step 1: Determine the decay constant by using the formula for radioactive decay:
N(t) = N0·1
2t
T1
2
where: - N(t) is the quantity of the substance after time t, - N0is the initial
quantity of the substance, - T1
2is the half-life of the substance.
Plugging in the values, we have:
N(30) = 100 ·1
230
10
Step 2: Calculate the remaining quantity of the substance after 30 days:
N(30) = 100 ·1
23
= 100 ·1
8= 12.5 grams
Therefore, after 30 days, there will be 12.5 grams of the radioactive substance
remaining.
28
