CHEM 132 - ADVANCED GENERAL
CHEMISTRY II - Radioactive decay
calculations
Question Bank - Set 1
Liberty University
Question 1
Question
A certain radioactive sample decays according to the equation N(t) = N0e−0.01t,
where N(t) represents the amount of the radioactive substance at time t(in
years), N0is the initial amount of the substance, and tis the time elapsed since
the initial observation. If the initial amount of the substance is 100 grams, find:
(a) The amount of the substance remaining after 10 years. (b) The time it takes
for 80
Solution
(a) To find the amount of the substance remaining after 10 years, we substitute
t= 10 years and N0= 100 grams into the equation N(t) = N0e−0.01t.
N(10) = 100e−0.01(10)
= 100e−0.1
≈90.48 grams
Therefore, the amount of the substance remaining after 10 years is approxi-
mately 90.48 grams.
(b) To find the time it takes for 80
0.8N0=N0e−0.01t
0.8 = e−0.01t
ln(0.8) = lne−0.01t
−0.2231 = −0.01t
t≈22.31 years
Therefore, it takes approximately 22.31 years for 80
Question 2
Question
A sample of a radioactive isotope decays according to the function N(t) =
N0e−kt, where N0is the initial quantity of the isotope, N(t) is the quantity
remaining after time t, and kis the decay constant. Suppose a sample of a
radioactive isotope has an initial quantity of 100 grams and decays to 25 grams
after 10 days. Find the decay constant kfor this isotope.
Solution
Step 1: Use the given information to set up an equation. Since we are given
that the initial quantity N0= 100 grams decays to a quantity of 25 grams after
10 days, we have:
25 = 100 ·e−10k
Step 2: Solve for the decay constant k. Dividing both sides by 100, we get:
0.25 = e−10k
Taking the natural logarithm of both sides:
ln(0.25) = lne−10k
ln(0.25) = −10k
Solving for k:
k=−ln(0.25)
10
k=ln(4)
10
Therefore, the decay constant kfor this isotope is ln(4)
10 .
Question 3
Question
A certain radioactive isotope has a half-life of 10 days. If you start with a sample
of 100 grams, how many grams will remain after 30 days?
Solution
Step 1: Determine the decay constant λusing the formula:
λ=ln(2)
T1
2
2
where T1
2is the half-life of the isotope. Substitute T1
2= 10 days into the
formula:
λ=ln(2)
10
Step 2: Calculate the remaining amount Ausing the formula:
A=A0·e−λt
where: - Ais the remaining amount after time t, - A0is the initial amount, - λ
is the decay constant, and - tis the time elapsed. Substitute A0= 100 grams,
λ=ln(2)
10 , and t= 30 days into the formula:
A= 100 ·e−ln(2)
10 ·30
Question 4
Question
A sample of a radioactive element has an activity of 800 decays per second. The
half-life of the element is 5 hours. Calculate the initial number of radioactive
nuclei in the sample.
Solution
Step 1: Determine the decay constant λusing the formula T1/2=ln(2)
λ.
Step 1: λ=ln(2)
T1/2
=ln(2)
5 hours
Step 2: Convert the half-life to seconds for consistency.
Step 2: T1/2= 5 hours ×3600 s/hour = 18000 s
Step 3: Substitute the values into the formula to calculate the decay con-
stant:
Step 3: λ=ln(2)
18000 s
Step 4: Calculate the decay constant λ.
Step 4: λ≈0.6931
18000 ≈3.846 ×10−5s−1
Step 5: Use the formula for the decay of radioactive nuclei N(t) = N0e−λt,
where N(t) is the number of nuclei at time t,N0is the initial number of nuclei,
tis the time elapsed, and λis the decay constant.
Step 6: We know that the activity Ais related to the number of decays
per second and is given by A=λN. Given that the activity is 800 decays per
second, we have:
800 = (3.846 ×10−5)N0
3
Step 7: Solve for N0, the initial number of radioactive nuclei:
N0=800
3.846 ×10−5
Step 8: Calculate N0.
N0≈800
3.846 ×10−5≈2.08 ×107nuclei
Therefore, the initial number of radioactive nuclei in the sample is approxi-
mately 2.08 ×107.
Question 5
Question
A sample of a radioactive isotope decays such that its mass decreases by 25
Solution
Step 1: Determine the fraction of the sample remaining after 4 years.
The fraction of the sample remaining after 4 years is given by 1−0.25 = 0.75
(since the mass decreases by 25
Step 2: Calculate the mass of the sample after 4 years.
The mass of the sample after 4 years is 0.75 ×50 g = 37.5 grams.
Step 3: Determine the fraction of the sample remaining after 8 years.
After 8 years, the mass of the sample would decrease by another 25
Step 4: Calculate the mass of the sample after 8 years.
The mass of the sample after 8 years is 0.5625 ×50 g = 28.125 grams.
Step 5: Determine the fraction of the sample remaining after 12 years.
After 12 years, the mass of the sample would decrease by another 25
Step 6: Calculate the mass of the sample after 12 years.
The mass of the sample after 12 years is 0.421875 ×50 g = 21.09375 grams.
Therefore, the mass of the sample after 12 years is 21.09375 grams.
Question 6
Question
A sample of a radioactive element has an activity of 600 decays per second.
After 5 hours, the activity of the sample has decreased to 200 decays per second.
Determine the half-life of the element.
4
Solution
Step 1: Calculate the decay constant λfrom the given information. Step 2: Use
the decay constant to find the half-life of the element.
Step 1: Let A0be the initial activity and Atbe the activity after time t.
The relationship between the two is given by:
At=A0·e−λt
Given that A0= 600 decays per second and At= 200 decays per second after
t= 5 hours, we have:
200 = 600 ·e−5λ
Solving for λ:
e−5λ=1
3
−5λ= ln 1
3
λ=ln(3)
5
Step 2: The half-life T1/2is related to the decay constant λby:
T1/2=ln(2)
λ
Substitute the value of λ:
T1/2=ln(2)
ln(3)
5
=5 ln(2)
ln(3) ≈2.73 hours
Therefore, the half-life of the radioactive element is approximately 2.73
hours.
Question 7
Question
A sample of a radioactive isotope has an initial mass of 20 grams and decays
exponentially. After 3 days, the mass of the sample has decreased to 15 grams.
If the half-life of the isotope is 2 days, what is the decay constant λof the
isotope?
5
Solution
Step 1: Recall the formula for radioactive decay:
N(t) = N0e−λt
where: - N(t) is the remaining mass at time t, - N0is the initial mass, - λis
the decay constant, - tis the time elapsed.
Step 2: We are given: - N0= 20 grams, - N(3) = 15 grams, and - the
half-life T1/2= 2 days.
Step 3: Since the half-life is 2 days, we know that after 2 days, the remaining
mass will be half of the initial mass. Therefore, after 2 days:
N(2) = N0e−λ(2) =N0
2
Step 4: Substitute the given values:
20e−2λ= 10
Step 5: Solve for λ:
e−2λ=10
20
e−2λ=1
2
−2λ= ln 1
2
λ=−1
2ln 2
Step 6: Approximate λ:
λ≈ −1
2×0.6931
λ≈ −0.3465
Step 7: Therefore, the decay constant λof the isotope is approximately
−0.3465.
Question 8
Question
A sample of a radioactive isotope has an initial mass of 500 grams. The isotope
decays at a rate of 10
1. Determine the mass of the isotope after 5 hours.
2. How long will it take for the mass to reduce to 100 grams?
6
Solution
Let m(t) be the mass of the isotope at time thours, where m(0) = 500 grams.
The rate of decay is 10
1. Step 1: Calculate the mass of the isotope after 5 hours.
After 1 hour, the mass decreases by 10After 2 hours, the mass decreases
by another 10Similarly, after 3 hours: 0.9×405 = 364.5 grams, 4 hours:
0.9×364.5 = 328.05 grams, 5 hours: 0.9×328.05 = 295.245 grams.
Therefore, the mass of the isotope after 5 hours is 295.245 grams.
Step 2: Determine the time when the mass reduces to 100 grams.
Let tbe the time in hours when the mass reduces to 100 grams.
We can set up the equation:
500 ×(0.9)t= 100
Solving for t:
(0.9)t=100
500
(0.9)t= 0.2
t=ln(0.2)
ln(0.9)
t≈−1.609
−0.105 ≈15.323
So, it will take approximately 15.323 hours for the mass to reduce to 100
grams.
Question 9
Question
A sample of a radioactive substance decays at a rate of 6
Solution
Step 1: Determine the decay factor per hour.
Let rbe the decay rate as a decimal. We know that the decay rate is 6
Step 2: Use the decay formula to find the amount of substance remaining
after 24 hours.
The amount of substance remaining after thours is given by the formula:
A(t) = A0(1 −r)t
7
where: - A(t) is the amount of substance remaining after thours, - A0is the
initial amount of the substance (100 grams), - ris the decay rate (0.06), and -
tis the time period in hours (24 hours).
Substitute these values into the formula:
A(24) = 100(1 −0.06)24
Step 3: Calculate the amount of substance remaining after 24 hours.
A(24) = 100(0.94)24
A(24) ≈32.53 grams
Therefore, after 24 hours, there will be approximately 32.53 grams of the
substance remaining.
Question 10
Question
A sample of a radioactive isotope decays at a rate of 2.5% per hour. If the
initial amount of the sample was 100 grams, determine the amount of the sample
remaining after 12 hours, to the nearest gram.
Solution
Step 1: Find the decay constant. Step 2: Use the decay constant to determine
the amount of the sample remaining after 12 hours.
Step 1: Let N(t) be the amount of the sample remaining after time t,N0
be the initial amount of the sample, and kbe the decay constant. The rate of
decay is given as 2.5% per hour, which means k= 0.025.
Step 2: The formula for radioactive decay is given by N(t) = N0·e−kt.
Substitute N0= 100, k= 0.025, and t= 12 into the formula:
N(12) = 100 ·e−(0.025·12)
N(12) = 100 ·e−0.3
N(12) ≈100 ·0.740818
N(12) ≈74.08 grams
Therefore, the amount of the sample remaining after 12 hours is approxi-
mately 74 grams.
8
Question 11
Question
A sample of a radioactive substance has an initial mass of 100 grams. If the
half-life of the substance is 20 days, determine the mass of the substance after
60 days. Assume that the decay follows an exponential model.
Solution
Step 1: Determine the decay constant. Step 2: Use the decay constant to
calculate the mass after 60 days.
Step 1: The decay constant, denoted by λ, is related to the half-life, T1/2,
by the formula:
λ=ln(2)
T1/2
Given that the half-life T1/2is 20 days:
λ=ln(2)
20 ≈0.0346 days−1
Step 2: The amount of substance present at any time t,m(t), is given by
the formula:
m(t) = m0e−λt
where - m0is the initial mass of the substance (100 grams in this case), - λis
the decay constant, - tis the time elapsed.
Substitute λ= 0.0346 and t= 60 into the formula:
m(60) = 100e−0.0346∗60 = 100e−2.076 ≈13.55 grams
Therefore, the mass of the substance after 60 days is approximately 13.55
grams.
Question 12
Question
A certain radioactive substance decays according to the equation N(t) = N0e−kt,
where N(t) is the quantity of the substance remaining at time t,N0is the initial
quantity of the substance, kis the decay constant, and tis the time elapsed.
Given that the half-life of this substance is 10 days, find the decay constant
kfor this substance.
9
Solution
Step 1: Recall that the decay constant kis related to the half-life T1/2of the
substance by the formula:
k=ln 2
T1/2
Step 2: We are given that the half-life T1/2of the substance is 10 days.
Substituting this into the formula, we get:
k=ln 2
10
Step 3: Calculating the value of k, we have:
k≈0.6931
10 ≈0.06931 days−1
Therefore, the decay constant kfor this substance is approximately 0.06931
days−1.
Question 13
Question
A radioactive substance decays according to the equation N(t) = N0·e−kt,
where N(t) is the amount remaining after tyears, N0is the initial amount, kis
the decay constant, and tis the time in years.
Suppose an initial amount of a radioactive substance decays to 75
Solution
Step 1: The half-life of a radioactive substance is the time it takes for half of
the substance to decay. Given that the half-life is 10 years, we can determine
the decay constant kusing this information. We know that after one half-life,
the amount remaining is 1
2N0, so we can set up the equation:
1
2N0=N0·e−k·10
Step 2: We can simplify the equation by cancelling out N0on both sides and
taking the natural logarithm of both sides to solve for k:
ln 1
2= ln e−10k
ln 1
2=−10k
Step 3: Since we know that after 8 years the amount remaining is 75
10
0.75N0=N0·e−8k
Step 4: Simplifying the equation using the same steps as before, we get:
ln (0.75) = ln e−8k
ln (0.75) = −8k
Step 5: Now, we have a system of two equations with two unknowns. Solving
for kin both equations, we get:
−10k= ln 1
2
−8k= ln (0.75)
Step 6: Solving these equations gives us:
k=ln 1
2
10
k=ln (0.75)
8
Step 7: Using a calculator to evaluate the natural logarithms, we find:
k≈0.0693
Therefore, the decay constant k≈0.0693.
Question 14
Question
A sample of a radioactive isotope decays at a rate proportional to the amount
of the isotope present at any time. Suppose we have a sample of a radioactive
isotope with a half-life of 5 days that initially weighs 100 grams. How much of
the isotope remains after 15 days?
Solution
Step 1: Determine the decay constant, k, using the formula:
half-life = ln(2)
k
Given that the half-life is 5 days, we have:
5 = ln(2)
k
11
Solving for kgives:
k=ln(2)
5
Step 2: Use the exponential decay model to find the amount of isotope
remaining after 15 days, A(15), where A(t) = A0e−kt and A0= 100 grams:
A(15) = 100e−ln(2)
5×15
Step 3: Calculate the final amount A(15).
A(15) = 100e−3 ln(2)
A(15) = 100eln(2−3)
A(15) = 100 ×2−3
A(15) = 12.5 grams
Therefore, after 15 days, there will be 12.5 grams of the radioactive isotope
remaining.
Question 15
Question
A sample of radioactive material has a half-life of 20 days. If the initial mass
of the sample is 100 grams, find the mass of the sample after 60 days. Round
your answer to the nearest whole number.
Solution
Step 1: We can use the radioactive decay formula to find the mass of the sample
after 60 days:
N(t) = N01
2t
T1
2
where: - N(t) is the final mass of the sample, - N0is the initial mass of the
sample, - T1
2is the half-life of the material, and - tis the time passed.
Step 2: Substituting the given values into the formula:
N(60) = 100 ×1
260
20
Step 3: Simplifying the equation:
N(60) = 100 ×1
23
12
N(60) = 100 ×1
8
N(60) = 100
8
N(60) = 12.5 grams
Step 4: Rounding the final mass to the nearest whole number, the mass of
the sample after 60 days is approximately 13 grams.
Question 16
Question
A certain radioactive substance decays at a rate proportional to the amount
present. If initially there are 500 grams of the substance and the half-life is 10
days, find the amount of substance present after 30 days.
Solution
Step 1: Determine the decay constant kusing the half-life formula,
1
2=e−k·10
Solving for k,
1
2=e−10k
−10k= ln 1
2
k=ln 1
2
−10 ≈0.0693
Step 2: Use the exponential decay formula to find the amount of substance
after 30 days,
A(t) = A0·e−kt
A(30) = 500 ·e−0.0693·30 ≈172.9 grams
Therefore, after 30 days, there are approximately 172.9 grams of the sub-
stance remaining.
13
Question 17
Question
A sample of a radioactive substance has an activity of 50 Bq at t= 0. After 3
hours, the activity of the sample is measured to be 25 Bq. Find the half-life of
the substance.
Solution
Step 1: Let A0be the initial activity of the substance, Abe the activity after
time t, and t1/2be the half-life of the substance. Step 2: The formula for
radioactive decay is given by A(t) = A01
2t
t1/2. Step 3: Plugging in the given
values, we have 25 = 50 1
23
t1/2. Step 4: Simplifying, we get 1
2=1
23
t1/2. Step
5: Using the property of exponents, we have 2−1= 2
−3
t1/2. Step 6: Equating the
exponents, we get −1 = −3
t1/2. Step 7: Solving for t1/2, we find t1/2= 3 hours.
Step 8: Therefore, the half-life of the substance is 3 hours.
Question 18
Question
The half-life of a radioactive substance is 24 hours. If there are initially 100
grams of the substance, how many grams will remain after 3 days?
Solution
Step 1: First, we need to determine the decay constant, λ, which is related to
the half-life, T1
2, by the formula λ=ln(2)
T1
2
. Step 2: Substituting T1
2= 24 hours
into the formula, we find λ=ln(2)
24 . Step 3: The decay of the substance can
be modeled by the equation N(t) = N0·e−λt, where N0is the initial quantity
of substance, tis the time elapsed, and N(t) is the quantity remaining after
time t. Step 4: Substituting N0= 100 grams and t= 3 days (since 3 days is
equivalent to 72 hours), we have N(72) = 100 ·e−ln(2)
24 ·72. Step 5: Simplifying
the expression, we get N(72) = 100 ·e−3 ln(2). Step 6: Using the property of
logarithms e−aln(b)= (eln(b))−a=b−a, we find N(72) = 100 ·2−3. Step 7:
Therefore, after 3 days, there will be 12.5 grams of the substance remaining.
Question 19
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?
14
Solution
Step 1: Calculate the decay constant λusing the half-life formula.
1
2=e−λ·10
e−λ·10 =1
2
−λ·10 = ln 1
2
λ=ln(2)
10
Step 2: Use the exponential decay model to find the amount of substance
remaining after 30 days.
A(t) = A0·e−λt
A(30) = 100 ·e−ln(2)
10 ·30
A(30) = 100 ·e−3 ln(2)
A(30) = 100 ·eln(2)−3
A(30) = 100 ·eln(2)−3
A(30) = 100 ·2−3
A(30) = 100 ·1
8
A(30) = 12.5
Therefore, after 30 days, there will be 12.5 grams of the substance remaining.
Question 20
Question
A sample of a certain radioactive material has an initial mass of 400 grams.
After 5 hours, only 100 grams of the material remain. If the half-life of the
material is 12 hours, determine the decay constant (k) of the material.
15
Solution
Step 1: Calculate the fraction of the initial mass remaining after 5 hours. Step
2: Use the fraction remaining to find the decay constant.
Step 1: To find the fraction of the initial mass remaining after 5 hours, we
can use the radioactive decay formula:
N(t) = N0·e−kt
where: N(t) is the amount of material remaining after time t N0is the initial
amount of material kis the decay constant tis the time elapsed
Given that N0= 400 g, N(5) = 100 g, and the half-life T1
2= 12 hours, we
have:
100 = 400 ·e−5k
Step 2: To solve for k, we need to isolate kin the equation obtained in Step
1. Halving both sides of the equation gives:
100
400 =e−5k
1
4=e−5k
Taking the natural logarithm of both sides gives:
ln 1
4= lne−5k
ln 1
4=−5k
Therefore, the decay constant kis given by:
k=−ln 1
4
5≈0.1386 hours−1
Question 21
Question
A particular radioactive isotope has a half-life of 25 years. If a sample initially
contains 100 grams of the isotope, how many grams will remain after 100 years?
Solution
Step 1: We can use the formula for radioactive decay to find the amount of
substance remaining after a certain time period: A=A01
2
t
T1
2, where: - A
is the amount remaining after time t, - A0is the initial amount, - T1
2is the
half-life of the substance.
16
Step 2: Substituting the given values into the formula, we get: A= 100 1
2100
25
Step 3: Simplifying the expression inside the parentheses: A= 100 1
24=
100 ×1
16 = 6.25 grams
Therefore, after 100 years, there will be 6.25 grams of the radioactive isotope
remaining.
Question 22
Question
A sample of radioactive material has an initial mass of 200 grams. After 10
days, the mass of the sample is reduced to 150 grams. If the half-life of the
material is 4 days, what is the decay constant of the material?
Solution
Step 1: Calculate the fraction of the initial mass remaining after 10 days.
Let N(t) be the mass of the sample at time t,N0be the initial mass, and λ
be the decay constant. The fraction of the initial mass remaining after time t,
f(t), is given by the exponential decay formula:
f(t) = N(t)
N0
=e−λt
Given that the initial mass is 200 grams and the mass after 10 days is 150
grams, we can write:
f(10) = 150
200 =e−λ·10
Step 2: Express the decay constant in terms of the half-life and solve for it.
The relation between the decay constant and the half-life (T1/2) is given by:
λ=ln(2)
T1/2
For this problem, the half-life is 4 days. Substituting into the formula, we
get:
e−λ·10 =150
200 =e−ln(2)·10
4
Step 3: Solve for the decay constant.
Solving the equation for λ:3
4=e−5 ln(2)
4
ln 3
4=−5 ln(2)
4
ln(3) −ln(4) = −5 ln(2)
4
17
ln 3
4=−5
4ln(2)
ln 3
4= ln2−5/4
3
4= 2−5/4
3
4=1
√25
3
4=1
√32
3√32
4= 1
Therefore, the decay constant for the radioactive material is λ=ln(2)
4.
Question 23
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 time t,N0is the
initial amount of substance, and kis a positive constant. If 80% of the substance
decays in 30 days, determine the value of kfor this substance decay.
Solution
1. We are given that 80% of the substance decays, which means 20% remains
after 30 days. Therefore, we have:
N(30) = N0·e−30k= 0.2N0
2. To simplify this equation, divide both sides by N0:
e−30k= 0.2
3. To solve for k, take the natural logarithm of both sides:
lne−30k= ln(0.2)
−30k= ln(0.2)
4. Solve for kby dividing both sides by −30:
k=ln(0.2)
−30
18
5. Calculate the value of kto determine the rate of decay of the radioactive
substance:
k≈ln(0.2)
−30 ≈0.023
Therefore, the value of kfor this radioactive substance decay is approxi-
mately 0.023.
Question 24
Question
A sample of a radioactive substance has an initial mass of 200 grams. After
6 hours, the mass of the sample decreases to 175 grams. If the half-life of the
substance is 4 hours, what is the decay constant of the substance?
Solution
Step 1: Determine the fraction of the sample remaining after 6 hours. The
formula for radioactive decay is given by: N(t) = N0·e−kt, where: N(t) =
quantity of the substance at time t,N0= initial quantity of the substance, k=
decay constant, t= time.
Given N(0) = 200 grams and N(6) = 175 grams, we can write: 175 =
200 ·e−6k
Step 2: Solve for the decay constant k. Divide both sides by 200: 175
200 =e−6k
7
8=e−6k
ln 7
8=−6k
Step 3: Calculate the decay constant k. From Step 2, we have: k=−ln(7
8)
6≈
0.0298
Therefore, the decay constant of the substance is approximately 0.0298 per
hour.
Question 25
Question
A certain radioactive substance has a half-life of 10 days. If we start with an
initial amount of 100 grams, how much of the substance will remain after 30
days?
Solution
Step 1: Determine the decay constant λusing the half-life formula T1/2=ln(2)
λ.
T1/2= 10 days
19
λ=ln(2)
T1/2
=ln(2)
10
Step 2: Calculate the amount of substance remaining after 30 days using the
exponential decay formula A(t) = A0·e−λt.
A0= 100 grams
t= 30 days
A(30) = 100 ·e
−
ln(2)
10
·30
A(30) = 100 ·e−3 ln(2)
A(30) = 100 ·1
23
A(30) = 12.5 grams
After 30 days, there will be 12.5 grams of the radioactive substance remain-
ing.
Question 26
Question
A sample of a radioactive isotope has an initial mass of 100 grams. After 5
hours, only 25 grams remain. If the half-life of the isotope is 3 hours, what is
the decay constant of the isotope?
Solution
Step 1: Calculate the fraction of the original sample that remains after 5 hours.
Step 2: Use the formula for radioactive decay to find the decay constant.
Step 1: Calculate the fraction of the original sample that remains after 5
hours.
The fraction of the original sample that remains after time tcan be calculated
using the formula:
Fraction remaining = 1
2t
half-life
Given that the half-life of the isotope is 3 hours and 5 hours have passed,
we can plug in the values:
Fraction remaining = 1
25
3
=1
21.67
≈0.363
So, approximately 36.3
20
Step 2: Use the formula for radioactive decay to find the decay constant.
The formula for radioactive decay is given by:
Fraction remaining = e−kt
where kis the decay constant.
Substitute the values we know into the equation:
0.363 = e−k×5
ln(0.363) = −5k
Solving for k:
k=−ln(0.363)
5
k≈0.293
Therefore, the decay constant of the isotope is approximately 0.293 per hour.
Question 27
Question
A sample of a radioactive substance has an activity of 3000 decays per minute.
After 10 hours, the activity of the sample has reduced to 750 decays per minute.
Determine the half-life of the substance.
Solution
Step 1: Convert the time given in hours to minutes.
10 hours ×60 minutes/hour = 600 minutes
Step 2: Use the exponential decay formula A=A0(1
2)
t
T1
2, where: - Ais the
final activity (750 decays/min), - A0is the initial activity (3000 decays/min), -
tis the elapsed time (600 minutes), - T1
2is the half-life we want to find.
Step 3: Write the equation using the given values and the unknown half-life.
750 = 3000 1
2600
T1
2
Step 4: Solve the equation for T1
2.
750
3000 =1
2600
T1
2
21
1
4= 2
−600
T1
2
log21
4=−600
T1
2
log24 = 600
T1
2
2 = 600
T1
2
T1
2=600
2= 300 minutes
Therefore, the half-life of the substance is 300 minutes, or 5 hours.
Question 28
Question
A sample of a radioactive substance has an initial mass of 100 grams. After 4
days, only 15 grams of the substance remain. If the half-life of the substance is
2 days, what is the decay constant of the substance?
Solution
Step 1: Find the fraction of the substance that remains after 4 days. Step 2:
Use the half-life formula to find the decay constant.
Step 1:
Let N0be the initial mass of the substance and Nbe the mass of the substance
remaining after 4 days. We are given N0= 100 grams and N= 15 grams. The
fraction of the substance remaining after 4 days is given by:
N
N0
=15
100 = 0.15
Step 2:
The half-life formula relates the remaining mass of a radioactive substance to
the decay constant λ:
N
N0
=e−λt
where tis the time elapsed. Using the information provided, we have:
0.15 = e−λ×4
Taking the natural logarithm of both sides gives:
ln(0.15) = −4λ
22
Solving for λ, we have:
λ=−ln(0.15)
4≈0.3344 days−1
Therefore, the decay constant of the substance is approximately 0.3344
days−1.
Question 29
Question
A sample of a radioactive isotope has an initial mass of 20 grams. After 30 days,
only 2.5 grams of the isotope remain. If the half-life of the isotope is 15 days,
what is the decay constant for this isotope?
Solution
Step 1: Let’s first find the fraction of the isotope that remains after 30 days.
Given that the half-life of the isotope is 15 days, after 30 days, we have gone
through 2 half-lives. The fraction of the isotope remaining after 2 half-lives is
given by
1
22
=1
4
Step 2: Now, let’s find the decay constant, denoted by λ, using the formula
for exponential radioactive decay. The decay of a radioactive substance is given
by
N(t) = N0e−λt
where: N(t) = amount of substance remaining at time t,N0= initial amount
of the substance, λ= decay constant, t= time elapsed.
Step 3: We can rewrite the above equation in terms of the mass of the
substance. Substitute N(t) = m
m0, where mis the mass at time tand m0is the
initial mass. The equation becomes
m
m0
=e−λt
Step 4: Plug in the known values after 30 days into the equation.
2.5
20 =e−λ×30
1
8=e−30λ
Step 5: Taking the natural logarithm of both sides to solve for λgives
−30λ= ln 1
8
23
−30λ= ln8−1
−30λ=−ln(8)
λ=ln(8)
30
Step 6: Finally, calculate the value of λ.
λ=ln(8)
30 ≈2.079
30 ≈0.0693 days−1
Question 30
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 remaining at time t,N0is the
initial amount of the isotope, kis the decay constant, and tis the time in years.
Suppose a sample of a radioactive isotope has an initial amount of 400 grams
and decays to 100 grams after 300 years. Find the decay constant kfor this
isotope.
Solution
Let’s start by writing down the given information: - Initial amount, N0= 400
grams - Remaining amount after 300 years, N(300) = 100 grams
We are looking for the decay constant k. We can use the formula N(t) =
N0e−kt to set up our equation and solve for k.
Step 1: Plug in the given information into the formula
100 = 400e−300k
Step 2: Divide by 400 to isolate the exponential term
100
400 =e−300k
Step 3: Simplify the fraction
1
4=e−300k
Step 4: Take the natural logarithm of both sides to eliminate the exponential
ln 1
4= lne−300k
Step 5: Use the property of logarithms to bring down the exponent
ln 1
4=−300kln(e)
24
Step 6: Recall that ln(e) = 1 and simplify
ln 1
4=−300k
Step 7: Solve for k
−300k= ln 1
4
k=−1
300 ln 1
4
Therefore, the decay constant k≈0.00315.
Question 31
Question
A certain radioactive material decays at a rate proportional to the amount of
the material present. If 200 grams of the material are present initially and 180
grams are present after 10 days, find the half-life of the material.
Solution
Let A(t) be the amount of material present at time t. The rate of decay is pro-
portional to the amount of material present, so we have the differential equation:
dA
dt =−kA
where kis the proportionality constant.
Step 1: Solve the differential equation to find A(t).
1
AdA =−kdt
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) = 200 to find the specific solution.
200 = Ce0=C
So, A= 200e−kt.
Step 3: Use the fact that 180 grams are present after 10 days to find k.
A(10) = 180
25
200e−10k= 180
e−10k=180
200 =9
10
−10k= ln 9
10
k=−1
10 ln 9
10
Step 4: Find the half-life of the material. The half-life, denoted by T1
2, is
the time required for half of the material to decay. We can find it by solving
the equation:
A(T1
2) = 1
2A0
200e−kT 1
2=1
2·200
e−kT 1
2=1
2
−kT 1
2= ln 1
2
T1
2=−ln 1
2
k
Hence, the half-life of the material is T1
2=10
ln(10
9)days.
Question 32
Question
A sample of a radioactive substance has an initial mass of 100 grams. After
6 hours, only 25 grams remain. If the half-life of the substance is 2 hours,
determine the rate constant of decay and the amount of the substance that will
remain after 12 hours.
Solution
Step 1: Determine the rate constant of decay using the formula for radioactive
decay: N(t) = N0e−kt, where: - N(t) is the amount of the substance remaining
after time t, - N0is the initial amount of the substance, - kis the rate constant
of decay, - tis the time elapsed.
We are given: N0= 100 grams, N(t) = 25 grams, and t= 6 hours. We also
know that the half-life of the substance is 2 hours, meaning that when t= 2
hours, half of the substance remains.
Substitute these values into the formula to solve for k:
25 = 100e−6k
26
Step 2: Solve for the rate constant k.
25 = 100e−6k
25
100 =e−6k
0.25 = e−6k
ln(0.25) = lne−6k
−1.3863 = −6k
k=1.3863
6
k≈0.2311 hours−1
Therefore, the rate constant of decay is approximately 0.2311 hours−1.
Step 3: Find the amount of the substance remaining after 12 hours. Substi-
tute t= 12 hours into the formula N(t) = N0e−kt:
N(12) = 100 ·e−0.2311·12
Step 4: Calculate the amount of the substance remaining after 12 hours.
N(12) = 100 ·e−0.2311·12
= 100 ·e−2.7732
≈100 ·0.06257
≈6.257 grams
Thus, approximately 6.257 grams of the substance will remain after 12 hours.
Question 33
Question
A sample of radium-226 has an initial mass of 20 grams. The half-life of radium-
226 is 1600 years. If 80 years have passed, what is the mass of the radium-226
sample remaining?
Solution
Step 1: Determine the decay constant from the half-life of radium-226. The
decay constant, denoted by λ, for a radioactive substance is related to its half-
life, T1/2, by the formula:
λ=ln(2)
T1/2
For radium-226 with a half-life of 1600 years, we have:
λ=ln(2)
1600 ≈4.325 ×10−4years−1
27
Step 2: Use the exponential decay formula to find the remaining mass. The
exponential decay formula for radioactive decay is given by:
m(t) = m0×e−λt
where: - m(t) is the mass remaining at time t, - m0is the initial mass, - λis
the decay constant, - tis the time elapsed.
Substitute m0= 20 grams, t= 80 years, and λ= 4.325 ×10−4years−1into
the formula:
m(80) = 20 ×e−(4.325×10−4×80)
Step 3: Calculate the mass remaining.
m(80) = 20 ×e−0.0346 ≈20 ×0.965604 ≈19.3121 grams
Therefore, the mass of the radium-226 sample remaining after 80 years is
approximately 19.3121 grams.
Question 34
Question
A sample of radioactive material decays according to the equation N(t) =
N0e−0.02t, where N(t) represents the amount of the material present at time t
(in years) and N0represents the initial amount of material. If the initial amount
of material is 100 grams, how long will it take for the amount of material to
decrease to 40 grams?
Solution
Step 1: Start by substituting the given values into the equation.
40 = 100e−0.02t
Step 2: Divide both sides by 100 to isolate the exponential term.
40
100 =e−0.02t
0.4 = e−0.02t
Step 3: Take the natural logarithm of both sides to solve for t.
ln(0.4) = lne−0.02t
ln(0.4) = −0.02t
Step 4: Divide by −0.02 to solve for t.
t=ln(0.4)
−0.02
t≈51.53 years
Therefore, it will take approximately 51.53 years for the amount of material
to decrease to 40 grams.
28
Question 2
Question
A sample of a radioactive isotope decays according to the function N(t) =
N0e−kt, where N0is the initial quantity of the isotope, N(t) is the quantity
remaining after time t, and kis the decay constant. Suppose a sample of a
radioactive isotope has an initial quantity of 100 grams and decays to 25 grams
after 10 days. Find the decay constant kfor this isotope.
Solution
Step 1: Use the given information to set up an equation. Since we are given
that the initial quantity N0= 100 grams decays to a quantity of 25 grams after
10 days, we have:
25 = 100 ·e−10k
Step 2: Solve for the decay constant k. Dividing both sides by 100, we get:
0.25 = e−10k
Taking the natural logarithm of both sides:
ln(0.25) = lne−10k
ln(0.25) = −10k
Solving for k:
k=−ln(0.25)
10
k=ln(4)
10
Therefore, the decay constant kfor this isotope is ln(4)
10 .
Question 3
Question
A certain radioactive isotope has a half-life of 10 days. If you start with a sample
of 100 grams, how many grams will remain after 30 days?
Solution
Step 1: Determine the decay constant λusing the formula:
λ=ln(2)
T1
2
2
where T1
2is the half-life of the isotope. Substitute T1
2= 10 days into the
formula:
λ=ln(2)
10
Step 2: Calculate the remaining amount Ausing the formula:
A=A0·e−λt
where: - Ais the remaining amount after time t, - A0is the initial amount, - λ
is the decay constant, and - tis the time elapsed. Substitute A0= 100 grams,
λ=ln(2)
10 , and t= 30 days into the formula:
A= 100 ·e−ln(2)
10 ·30
Question 4
Question
A sample of a radioactive element has an activity of 800 decays per second. The
half-life of the element is 5 hours. Calculate the initial number of radioactive
nuclei in the sample.
Solution
Step 1: Determine the decay constant λusing the formula T1/2=ln(2)
λ.
Step 1: λ=ln(2)
T1/2
=ln(2)
5 hours
Step 2: Convert the half-life to seconds for consistency.
Step 2: T1/2= 5 hours ×3600 s/hour = 18000 s
Step 3: Substitute the values into the formula to calculate the decay con-
stant:
Step 3: λ=ln(2)
18000 s
Step 4: Calculate the decay constant λ.
Step 4: λ≈0.6931
18000 ≈3.846 ×10−5s−1
Step 5: Use the formula for the decay of radioactive nuclei N(t) = N0e−λt,
where N(t) is the number of nuclei at time t,N0is the initial number of nuclei,
tis the time elapsed, and λis the decay constant.
Step 6: We know that the activity Ais related to the number of decays
per second and is given by A=λN. Given that the activity is 800 decays per
second, we have:
800 = (3.846 ×10−5)N0
3
Step 7: Solve for N0, the initial number of radioactive nuclei:
N0=800
3.846 ×10−5
Step 8: Calculate N0.
N0≈800
3.846 ×10−5≈2.08 ×107nuclei
Therefore, the initial number of radioactive nuclei in the sample is approxi-
mately 2.08 ×107.
Question 5
Question
A sample of a radioactive isotope decays such that its mass decreases by 25
Solution
Step 1: Determine the fraction of the sample remaining after 4 years.
The fraction of the sample remaining after 4 years is given by 1−0.25 = 0.75
(since the mass decreases by 25
Step 2: Calculate the mass of the sample after 4 years.
The mass of the sample after 4 years is 0.75 ×50 g = 37.5 grams.
Step 3: Determine the fraction of the sample remaining after 8 years.
After 8 years, the mass of the sample would decrease by another 25
Step 4: Calculate the mass of the sample after 8 years.
The mass of the sample after 8 years is 0.5625 ×50 g = 28.125 grams.
Step 5: Determine the fraction of the sample remaining after 12 years.
After 12 years, the mass of the sample would decrease by another 25
Step 6: Calculate the mass of the sample after 12 years.
The mass of the sample after 12 years is 0.421875 ×50 g = 21.09375 grams.
Therefore, the mass of the sample after 12 years is 21.09375 grams.
Question 6
Question
A sample of a radioactive element has an activity of 600 decays per second.
After 5 hours, the activity of the sample has decreased to 200 decays per second.
Determine the half-life of the element.
4
Solution
Step 1: Calculate the decay constant λfrom the given information. Step 2: Use
the decay constant to find the half-life of the element.
Step 1: Let A0be the initial activity and Atbe the activity after time t.
The relationship between the two is given by:
At=A0·e−λt
Given that A0= 600 decays per second and At= 200 decays per second after
t= 5 hours, we have:
200 = 600 ·e−5λ
Solving for λ:
e−5λ=1
3
−5λ= ln 1
3
λ=ln(3)
5
Step 2: The half-life T1/2is related to the decay constant λby:
T1/2=ln(2)
λ
Substitute the value of λ:
T1/2=ln(2)
ln(3)
5
=5 ln(2)
ln(3) ≈2.73 hours
Therefore, the half-life of the radioactive element is approximately 2.73
hours.
Question 7
Question
A sample of a radioactive isotope has an initial mass of 20 grams and decays
exponentially. After 3 days, the mass of the sample has decreased to 15 grams.
If the half-life of the isotope is 2 days, what is the decay constant λof the
isotope?
5
Solution
Step 1: Recall the formula for radioactive decay:
N(t) = N0e−λt
where: - N(t) is the remaining mass at time t, - N0is the initial mass, - λis
the decay constant, - tis the time elapsed.
Step 2: We are given: - N0= 20 grams, - N(3) = 15 grams, and - the
half-life T1/2= 2 days.
Step 3: Since the half-life is 2 days, we know that after 2 days, the remaining
mass will be half of the initial mass. Therefore, after 2 days:
N(2) = N0e−λ(2) =N0
2
Step 4: Substitute the given values:
20e−2λ= 10
Step 5: Solve for λ:
e−2λ=10
20
e−2λ=1
2
−2λ= ln 1
2
λ=−1
2ln 2
Step 6: Approximate λ:
λ≈ −1
2×0.6931
λ≈ −0.3465
Step 7: Therefore, the decay constant λof the isotope is approximately
−0.3465.
Question 8
Question
A sample of a radioactive isotope has an initial mass of 500 grams. The isotope
decays at a rate of 10
1. Determine the mass of the isotope after 5 hours.
2. How long will it take for the mass to reduce to 100 grams?
6
Solution
Let m(t) be the mass of the isotope at time thours, where m(0) = 500 grams.
The rate of decay is 10
1. Step 1: Calculate the mass of the isotope after 5 hours.
After 1 hour, the mass decreases by 10After 2 hours, the mass decreases
by another 10Similarly, after 3 hours: 0.9×405 = 364.5 grams, 4 hours:
0.9×364.5 = 328.05 grams, 5 hours: 0.9×328.05 = 295.245 grams.
Therefore, the mass of the isotope after 5 hours is 295.245 grams.
Step 2: Determine the time when the mass reduces to 100 grams.
Let tbe the time in hours when the mass reduces to 100 grams.
We can set up the equation:
500 ×(0.9)t= 100
Solving for t:
(0.9)t=100
500
(0.9)t= 0.2
t=ln(0.2)
ln(0.9)
t≈−1.609
−0.105 ≈15.323
So, it will take approximately 15.323 hours for the mass to reduce to 100
grams.
Question 9
Question
A sample of a radioactive substance decays at a rate of 6
Solution
Step 1: Determine the decay factor per hour.
Let rbe the decay rate as a decimal. We know that the decay rate is 6
Step 2: Use the decay formula to find the amount of substance remaining
after 24 hours.
The amount of substance remaining after thours is given by the formula:
A(t) = A0(1 −r)t
7
where: - A(t) is the amount of substance remaining after thours, - A0is the
initial amount of the substance (100 grams), - ris the decay rate (0.06), and -
tis the time period in hours (24 hours).
Substitute these values into the formula:
A(24) = 100(1 −0.06)24
Step 3: Calculate the amount of substance remaining after 24 hours.
A(24) = 100(0.94)24
A(24) ≈32.53 grams
Therefore, after 24 hours, there will be approximately 32.53 grams of the
substance remaining.
Question 10
Question
A sample of a radioactive isotope decays at a rate of 2.5% per hour. If the
initial amount of the sample was 100 grams, determine the amount of the sample
remaining after 12 hours, to the nearest gram.
Solution
Step 1: Find the decay constant. Step 2: Use the decay constant to determine
the amount of the sample remaining after 12 hours.
Step 1: Let N(t) be the amount of the sample remaining after time t,N0
be the initial amount of the sample, and kbe the decay constant. The rate of
decay is given as 2.5% per hour, which means k= 0.025.
Step 2: The formula for radioactive decay is given by N(t) = N0·e−kt.
Substitute N0= 100, k= 0.025, and t= 12 into the formula:
N(12) = 100 ·e−(0.025·12)
N(12) = 100 ·e−0.3
N(12) ≈100 ·0.740818
N(12) ≈74.08 grams
Therefore, the amount of the sample remaining after 12 hours is approxi-
mately 74 grams.
8
Question 11
Question
A sample of a radioactive substance has an initial mass of 100 grams. If the
half-life of the substance is 20 days, determine the mass of the substance after
60 days. Assume that the decay follows an exponential model.
Solution
Step 1: Determine the decay constant. Step 2: Use the decay constant to
calculate the mass after 60 days.
Step 1: The decay constant, denoted by λ, is related to the half-life, T1/2,
by the formula:
λ=ln(2)
T1/2
Given that the half-life T1/2is 20 days:
λ=ln(2)
20 ≈0.0346 days−1
Step 2: The amount of substance present at any time t,m(t), is given by
the formula:
m(t) = m0e−λt
where - m0is the initial mass of the substance (100 grams in this case), - λis
the decay constant, - tis the time elapsed.
Substitute λ= 0.0346 and t= 60 into the formula:
m(60) = 100e−0.0346∗60 = 100e−2.076 ≈13.55 grams
Therefore, the mass of the substance after 60 days is approximately 13.55
grams.
Question 12
Question
A certain radioactive substance decays according to the equation N(t) = N0e−kt,
where N(t) is the quantity of the substance remaining at time t,N0is the initial
quantity of the substance, kis the decay constant, and tis the time elapsed.
Given that the half-life of this substance is 10 days, find the decay constant
kfor this substance.
9
Solution
Step 1: Recall that the decay constant kis related to the half-life T1/2of the
substance by the formula:
k=ln 2
T1/2
Step 2: We are given that the half-life T1/2of the substance is 10 days.
Substituting this into the formula, we get:
k=ln 2
10
Step 3: Calculating the value of k, we have:
k≈0.6931
10 ≈0.06931 days−1
Therefore, the decay constant kfor this substance is approximately 0.06931
days−1.
Question 13
Question
A radioactive substance decays according to the equation N(t) = N0·e−kt,
where N(t) is the amount remaining after tyears, N0is the initial amount, kis
the decay constant, and tis the time in years.
Suppose an initial amount of a radioactive substance decays to 75
Solution
Step 1: The half-life of a radioactive substance is the time it takes for half of
the substance to decay. Given that the half-life is 10 years, we can determine
the decay constant kusing this information. We know that after one half-life,
the amount remaining is 1
2N0, so we can set up the equation:
1
2N0=N0·e−k·10
Step 2: We can simplify the equation by cancelling out N0on both sides and
taking the natural logarithm of both sides to solve for k:
ln 1
2= ln e−10k
ln 1
2=−10k
Step 3: Since we know that after 8 years the amount remaining is 75
10
0.75N0=N0·e−8k
Step 4: Simplifying the equation using the same steps as before, we get:
ln (0.75) = ln e−8k
ln (0.75) = −8k
Step 5: Now, we have a system of two equations with two unknowns. Solving
for kin both equations, we get:
−10k= ln 1
2
−8k= ln (0.75)
Step 6: Solving these equations gives us:
k=ln 1
2
10
k=ln (0.75)
8
Step 7: Using a calculator to evaluate the natural logarithms, we find:
k≈0.0693
Therefore, the decay constant k≈0.0693.
Question 14
Question
A sample of a radioactive isotope decays at a rate proportional to the amount
of the isotope present at any time. Suppose we have a sample of a radioactive
isotope with a half-life of 5 days that initially weighs 100 grams. How much of
the isotope remains after 15 days?
Solution
Step 1: Determine the decay constant, k, using the formula:
half-life = ln(2)
k
Given that the half-life is 5 days, we have:
5 = ln(2)
k
11
Solving for kgives:
k=ln(2)
5
Step 2: Use the exponential decay model to find the amount of isotope
remaining after 15 days, A(15), where A(t) = A0e−kt and A0= 100 grams:
A(15) = 100e−ln(2)
5×15
Step 3: Calculate the final amount A(15).
A(15) = 100e−3 ln(2)
A(15) = 100eln(2−3)
A(15) = 100 ×2−3
A(15) = 12.5 grams
Therefore, after 15 days, there will be 12.5 grams of the radioactive isotope
remaining.
Question 15
Question
A sample of radioactive material has a half-life of 20 days. If the initial mass
of the sample is 100 grams, find the mass of the sample after 60 days. Round
your answer to the nearest whole number.
Solution
Step 1: We can use the radioactive decay formula to find the mass of the sample
after 60 days:
N(t) = N01
2t
T1
2
where: - N(t) is the final mass of the sample, - N0is the initial mass of the
sample, - T1
2is the half-life of the material, and - tis the time passed.
Step 2: Substituting the given values into the formula:
N(60) = 100 ×1
260
20
Step 3: Simplifying the equation:
N(60) = 100 ×1
23
12
N(60) = 100 ×1
8
N(60) = 100
8
N(60) = 12.5 grams
Step 4: Rounding the final mass to the nearest whole number, the mass of
the sample after 60 days is approximately 13 grams.
Question 16
Question
A certain radioactive substance decays at a rate proportional to the amount
present. If initially there are 500 grams of the substance and the half-life is 10
days, find the amount of substance present after 30 days.
Solution
Step 1: Determine the decay constant kusing the half-life formula,
1
2=e−k·10
Solving for k,
1
2=e−10k
−10k= ln 1
2
k=ln 1
2
−10 ≈0.0693
Step 2: Use the exponential decay formula to find the amount of substance
after 30 days,
A(t) = A0·e−kt
A(30) = 500 ·e−0.0693·30 ≈172.9 grams
Therefore, after 30 days, there are approximately 172.9 grams of the sub-
stance remaining.
13
Question 17
Question
A sample of a radioactive substance has an activity of 50 Bq at t= 0. After 3
hours, the activity of the sample is measured to be 25 Bq. Find the half-life of
the substance.
Solution
Step 1: Let A0be the initial activity of the substance, Abe the activity after
time t, and t1/2be the half-life of the substance. Step 2: The formula for
radioactive decay is given by A(t) = A01
2t
t1/2. Step 3: Plugging in the given
values, we have 25 = 50 1
23
t1/2. Step 4: Simplifying, we get 1
2=1
23
t1/2. Step
5: Using the property of exponents, we have 2−1= 2
−3
t1/2. Step 6: Equating the
exponents, we get −1 = −3
t1/2. Step 7: Solving for t1/2, we find t1/2= 3 hours.
Step 8: Therefore, the half-life of the substance is 3 hours.
Question 18
Question
The half-life of a radioactive substance is 24 hours. If there are initially 100
grams of the substance, how many grams will remain after 3 days?
Solution
Step 1: First, we need to determine the decay constant, λ, which is related to
the half-life, T1
2, by the formula λ=ln(2)
T1
2
. Step 2: Substituting T1
2= 24 hours
into the formula, we find λ=ln(2)
24 . Step 3: The decay of the substance can
be modeled by the equation N(t) = N0·e−λt, where N0is the initial quantity
of substance, tis the time elapsed, and N(t) is the quantity remaining after
time t. Step 4: Substituting N0= 100 grams and t= 3 days (since 3 days is
equivalent to 72 hours), we have N(72) = 100 ·e−ln(2)
24 ·72. Step 5: Simplifying
the expression, we get N(72) = 100 ·e−3 ln(2). Step 6: Using the property of
logarithms e−aln(b)= (eln(b))−a=b−a, we find N(72) = 100 ·2−3. Step 7:
Therefore, after 3 days, there will be 12.5 grams of the substance remaining.
Question 19
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?
14
Solution
Step 1: Calculate the decay constant λusing the half-life formula.
1
2=e−λ·10
e−λ·10 =1
2
−λ·10 = ln 1
2
λ=ln(2)
10
Step 2: Use the exponential decay model to find the amount of substance
remaining after 30 days.
A(t) = A0·e−λt
A(30) = 100 ·e−ln(2)
10 ·30
A(30) = 100 ·e−3 ln(2)
A(30) = 100 ·eln(2)−3
A(30) = 100 ·eln(2)−3
A(30) = 100 ·2−3
A(30) = 100 ·1
8
A(30) = 12.5
Therefore, after 30 days, there will be 12.5 grams of the substance remaining.
Question 20
Question
A sample of a certain radioactive material has an initial mass of 400 grams.
After 5 hours, only 100 grams of the material remain. If the half-life of the
material is 12 hours, determine the decay constant (k) of the material.
15
Solution
Step 1: Calculate the fraction of the initial mass remaining after 5 hours. Step
2: Use the fraction remaining to find the decay constant.
Step 1: To find the fraction of the initial mass remaining after 5 hours, we
can use the radioactive decay formula:
N(t) = N0·e−kt
where: N(t) is the amount of material remaining after time t N0is the initial
amount of material kis the decay constant tis the time elapsed
Given that N0= 400 g, N(5) = 100 g, and the half-life T1
2= 12 hours, we
have:
100 = 400 ·e−5k
Step 2: To solve for k, we need to isolate kin the equation obtained in Step
1. Halving both sides of the equation gives:
100
400 =e−5k
1
4=e−5k
Taking the natural logarithm of both sides gives:
ln 1
4= lne−5k
ln 1
4=−5k
Therefore, the decay constant kis given by:
k=−ln 1
4
5≈0.1386 hours−1
Question 21
Question
A particular radioactive isotope has a half-life of 25 years. If a sample initially
contains 100 grams of the isotope, how many grams will remain after 100 years?
Solution
Step 1: We can use the formula for radioactive decay to find the amount of
substance remaining after a certain time period: A=A01
2
t
T1
2, where: - A
is the amount remaining after time t, - A0is the initial amount, - T1
2is the
half-life of the substance.
16
Step 2: Substituting the given values into the formula, we get: A= 100 1
2100
25
Step 3: Simplifying the expression inside the parentheses: A= 100 1
24=
100 ×1
16 = 6.25 grams
Therefore, after 100 years, there will be 6.25 grams of the radioactive isotope
remaining.
Question 22
Question
A sample of radioactive material has an initial mass of 200 grams. After 10
days, the mass of the sample is reduced to 150 grams. If the half-life of the
material is 4 days, what is the decay constant of the material?
Solution
Step 1: Calculate the fraction of the initial mass remaining after 10 days.
Let N(t) be the mass of the sample at time t,N0be the initial mass, and λ
be the decay constant. The fraction of the initial mass remaining after time t,
f(t), is given by the exponential decay formula:
f(t) = N(t)
N0
=e−λt
Given that the initial mass is 200 grams and the mass after 10 days is 150
grams, we can write:
f(10) = 150
200 =e−λ·10
Step 2: Express the decay constant in terms of the half-life and solve for it.
The relation between the decay constant and the half-life (T1/2) is given by:
λ=ln(2)
T1/2
For this problem, the half-life is 4 days. Substituting into the formula, we
get:
e−λ·10 =150
200 =e−ln(2)·10
4
Step 3: Solve for the decay constant.
Solving the equation for λ:3
4=e−5 ln(2)
4
ln 3
4=−5 ln(2)
4
ln(3) −ln(4) = −5 ln(2)
4
17
ln 3
4=−5
4ln(2)
ln 3
4= ln2−5/4
3
4= 2−5/4
3
4=1
√25
3
4=1
√32
3√32
4= 1
Therefore, the decay constant for the radioactive material is λ=ln(2)
4.
Question 23
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 time t,N0is the
initial amount of substance, and kis a positive constant. If 80% of the substance
decays in 30 days, determine the value of kfor this substance decay.
Solution
1. We are given that 80% of the substance decays, which means 20% remains
after 30 days. Therefore, we have:
N(30) = N0·e−30k= 0.2N0
2. To simplify this equation, divide both sides by N0:
e−30k= 0.2
3. To solve for k, take the natural logarithm of both sides:
lne−30k= ln(0.2)
−30k= ln(0.2)
4. Solve for kby dividing both sides by −30:
k=ln(0.2)
−30
18
5. Calculate the value of kto determine the rate of decay of the radioactive
substance:
k≈ln(0.2)
−30 ≈0.023
Therefore, the value of kfor this radioactive substance decay is approxi-
mately 0.023.
Question 24
Question
A sample of a radioactive substance has an initial mass of 200 grams. After
6 hours, the mass of the sample decreases to 175 grams. If the half-life of the
substance is 4 hours, what is the decay constant of the substance?
Solution
Step 1: Determine the fraction of the sample remaining after 6 hours. The
formula for radioactive decay is given by: N(t) = N0·e−kt, where: N(t) =
quantity of the substance at time t,N0= initial quantity of the substance, k=
decay constant, t= time.
Given N(0) = 200 grams and N(6) = 175 grams, we can write: 175 =
200 ·e−6k
Step 2: Solve for the decay constant k. Divide both sides by 200: 175
200 =e−6k
7
8=e−6k
ln 7
8=−6k
Step 3: Calculate the decay constant k. From Step 2, we have: k=−ln(7
8)
6≈
0.0298
Therefore, the decay constant of the substance is approximately 0.0298 per
hour.
Question 25
Question
A certain radioactive substance has a half-life of 10 days. If we start with an
initial amount of 100 grams, how much of the substance will remain after 30
days?
Solution
Step 1: Determine the decay constant λusing the half-life formula T1/2=ln(2)
λ.
T1/2= 10 days
19
λ=ln(2)
T1/2
=ln(2)
10
Step 2: Calculate the amount of substance remaining after 30 days using the
exponential decay formula A(t) = A0·e−λt.
A0= 100 grams
t= 30 days
A(30) = 100 ·e
−
ln(2)
10
·30
A(30) = 100 ·e−3 ln(2)
A(30) = 100 ·1
23
A(30) = 12.5 grams
After 30 days, there will be 12.5 grams of the radioactive substance remain-
ing.
Question 26
Question
A sample of a radioactive isotope has an initial mass of 100 grams. After 5
hours, only 25 grams remain. If the half-life of the isotope is 3 hours, what is
the decay constant of the isotope?
Solution
Step 1: Calculate the fraction of the original sample that remains after 5 hours.
Step 2: Use the formula for radioactive decay to find the decay constant.
Step 1: Calculate the fraction of the original sample that remains after 5
hours.
The fraction of the original sample that remains after time tcan be calculated
using the formula:
Fraction remaining = 1
2t
half-life
Given that the half-life of the isotope is 3 hours and 5 hours have passed,
we can plug in the values:
Fraction remaining = 1
25
3
=1
21.67
≈0.363
So, approximately 36.3
20
Step 2: Use the formula for radioactive decay to find the decay constant.
The formula for radioactive decay is given by:
Fraction remaining = e−kt
where kis the decay constant.
Substitute the values we know into the equation:
0.363 = e−k×5
ln(0.363) = −5k
Solving for k:
k=−ln(0.363)
5
k≈0.293
Therefore, the decay constant of the isotope is approximately 0.293 per hour.
Question 27
Question
A sample of a radioactive substance has an activity of 3000 decays per minute.
After 10 hours, the activity of the sample has reduced to 750 decays per minute.
Determine the half-life of the substance.
Solution
Step 1: Convert the time given in hours to minutes.
10 hours ×60 minutes/hour = 600 minutes
Step 2: Use the exponential decay formula A=A0(1
2)
t
T1
2, where: - Ais the
final activity (750 decays/min), - A0is the initial activity (3000 decays/min), -
tis the elapsed time (600 minutes), - T1
2is the half-life we want to find.
Step 3: Write the equation using the given values and the unknown half-life.
750 = 3000 1
2600
T1
2
Step 4: Solve the equation for T1
2.
750
3000 =1
2600
T1
2
21
1
4= 2
−600
T1
2
log21
4=−600
T1
2
log24 = 600
T1
2
2 = 600
T1
2
T1
2=600
2= 300 minutes
Therefore, the half-life of the substance is 300 minutes, or 5 hours.
Question 28
Question
A sample of a radioactive substance has an initial mass of 100 grams. After 4
days, only 15 grams of the substance remain. If the half-life of the substance is
2 days, what is the decay constant of the substance?
Solution
Step 1: Find the fraction of the substance that remains after 4 days. Step 2:
Use the half-life formula to find the decay constant.
Step 1:
Let N0be the initial mass of the substance and Nbe the mass of the substance
remaining after 4 days. We are given N0= 100 grams and N= 15 grams. The
fraction of the substance remaining after 4 days is given by:
N
N0
=15
100 = 0.15
Step 2:
The half-life formula relates the remaining mass of a radioactive substance to
the decay constant λ:
N
N0
=e−λt
where tis the time elapsed. Using the information provided, we have:
0.15 = e−λ×4
Taking the natural logarithm of both sides gives:
ln(0.15) = −4λ
22
Solving for λ, we have:
λ=−ln(0.15)
4≈0.3344 days−1
Therefore, the decay constant of the substance is approximately 0.3344
days−1.
Question 29
Question
A sample of a radioactive isotope has an initial mass of 20 grams. After 30 days,
only 2.5 grams of the isotope remain. If the half-life of the isotope is 15 days,
what is the decay constant for this isotope?
Solution
Step 1: Let’s first find the fraction of the isotope that remains after 30 days.
Given that the half-life of the isotope is 15 days, after 30 days, we have gone
through 2 half-lives. The fraction of the isotope remaining after 2 half-lives is
given by
1
22
=1
4
Step 2: Now, let’s find the decay constant, denoted by λ, using the formula
for exponential radioactive decay. The decay of a radioactive substance is given
by
N(t) = N0e−λt
where: N(t) = amount of substance remaining at time t,N0= initial amount
of the substance, λ= decay constant, t= time elapsed.
Step 3: We can rewrite the above equation in terms of the mass of the
substance. Substitute N(t) = m
m0, where mis the mass at time tand m0is the
initial mass. The equation becomes
m
m0
=e−λt
Step 4: Plug in the known values after 30 days into the equation.
2.5
20 =e−λ×30
1
8=e−30λ
Step 5: Taking the natural logarithm of both sides to solve for λgives
−30λ= ln 1
8
23
−30λ= ln8−1
−30λ=−ln(8)
λ=ln(8)
30
Step 6: Finally, calculate the value of λ.
λ=ln(8)
30 ≈2.079
30 ≈0.0693 days−1
Question 30
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 remaining at time t,N0is the
initial amount of the isotope, kis the decay constant, and tis the time in years.
Suppose a sample of a radioactive isotope has an initial amount of 400 grams
and decays to 100 grams after 300 years. Find the decay constant kfor this
isotope.
Solution
Let’s start by writing down the given information: - Initial amount, N0= 400
grams - Remaining amount after 300 years, N(300) = 100 grams
We are looking for the decay constant k. We can use the formula N(t) =
N0e−kt to set up our equation and solve for k.
Step 1: Plug in the given information into the formula
100 = 400e−300k
Step 2: Divide by 400 to isolate the exponential term
100
400 =e−300k
Step 3: Simplify the fraction
1
4=e−300k
Step 4: Take the natural logarithm of both sides to eliminate the exponential
ln 1
4= lne−300k
Step 5: Use the property of logarithms to bring down the exponent
ln 1
4=−300kln(e)
24
Step 6: Recall that ln(e) = 1 and simplify
ln 1
4=−300k
Step 7: Solve for k
−300k= ln 1
4
k=−1
300 ln 1
4
Therefore, the decay constant k≈0.00315.
Question 31
Question
A certain radioactive material decays at a rate proportional to the amount of
the material present. If 200 grams of the material are present initially and 180
grams are present after 10 days, find the half-life of the material.
Solution
Let A(t) be the amount of material present at time t. The rate of decay is pro-
portional to the amount of material present, so we have the differential equation:
dA
dt =−kA
where kis the proportionality constant.
Step 1: Solve the differential equation to find A(t).
1
AdA =−kdt
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) = 200 to find the specific solution.
200 = Ce0=C
So, A= 200e−kt.
Step 3: Use the fact that 180 grams are present after 10 days to find k.
A(10) = 180
25
200e−10k= 180
e−10k=180
200 =9
10
−10k= ln 9
10
k=−1
10 ln 9
10
Step 4: Find the half-life of the material. The half-life, denoted by T1
2, is
the time required for half of the material to decay. We can find it by solving
the equation:
A(T1
2) = 1
2A0
200e−kT 1
2=1
2·200
e−kT 1
2=1
2
−kT 1
2= ln 1
2
T1
2=−ln 1
2
k
Hence, the half-life of the material is T1
2=10
ln(10
9)days.
Question 32
Question
A sample of a radioactive substance has an initial mass of 100 grams. After
6 hours, only 25 grams remain. If the half-life of the substance is 2 hours,
determine the rate constant of decay and the amount of the substance that will
remain after 12 hours.
Solution
Step 1: Determine the rate constant of decay using the formula for radioactive
decay: N(t) = N0e−kt, where: - N(t) is the amount of the substance remaining
after time t, - N0is the initial amount of the substance, - kis the rate constant
of decay, - tis the time elapsed.
We are given: N0= 100 grams, N(t) = 25 grams, and t= 6 hours. We also
know that the half-life of the substance is 2 hours, meaning that when t= 2
hours, half of the substance remains.
Substitute these values into the formula to solve for k:
25 = 100e−6k
26
Step 2: Solve for the rate constant k.
25 = 100e−6k
25
100 =e−6k
0.25 = e−6k
ln(0.25) = lne−6k
−1.3863 = −6k
k=1.3863
6
k≈0.2311 hours−1
Therefore, the rate constant of decay is approximately 0.2311 hours−1.
Step 3: Find the amount of the substance remaining after 12 hours. Substi-
tute t= 12 hours into the formula N(t) = N0e−kt:
N(12) = 100 ·e−0.2311·12
Step 4: Calculate the amount of the substance remaining after 12 hours.
N(12) = 100 ·e−0.2311·12
= 100 ·e−2.7732
≈100 ·0.06257
≈6.257 grams
Thus, approximately 6.257 grams of the substance will remain after 12 hours.
Question 33
Question
A sample of radium-226 has an initial mass of 20 grams. The half-life of radium-
226 is 1600 years. If 80 years have passed, what is the mass of the radium-226
sample remaining?
Solution
Step 1: Determine the decay constant from the half-life of radium-226. The
decay constant, denoted by λ, for a radioactive substance is related to its half-
life, T1/2, by the formula:
λ=ln(2)
T1/2
For radium-226 with a half-life of 1600 years, we have:
λ=ln(2)
1600 ≈4.325 ×10−4years−1
27
Step 2: Use the exponential decay formula to find the remaining mass. The
exponential decay formula for radioactive decay is given by:
m(t) = m0×e−λt
where: - m(t) is the mass remaining at time t, - m0is the initial mass, - λis
the decay constant, - tis the time elapsed.
Substitute m0= 20 grams, t= 80 years, and λ= 4.325 ×10−4years−1into
the formula:
m(80) = 20 ×e−(4.325×10−4×80)
Step 3: Calculate the mass remaining.
m(80) = 20 ×e−0.0346 ≈20 ×0.965604 ≈19.3121 grams
Therefore, the mass of the radium-226 sample remaining after 80 years is
approximately 19.3121 grams.
Question 34
Question
A sample of radioactive material decays according to the equation N(t) =
N0e−0.02t, where N(t) represents the amount of the material present at time t
(in years) and N0represents the initial amount of material. If the initial amount
of material is 100 grams, how long will it take for the amount of material to
decrease to 40 grams?
Solution
Step 1: Start by substituting the given values into the equation.
40 = 100e−0.02t
Step 2: Divide both sides by 100 to isolate the exponential term.
40
100 =e−0.02t
0.4 = e−0.02t
Step 3: Take the natural logarithm of both sides to solve for t.
ln(0.4) = lne−0.02t
ln(0.4) = −0.02t
Step 4: Divide by −0.02 to solve for t.
t=ln(0.4)
−0.02
t≈51.53 years
Therefore, it will take approximately 51.53 years for the amount of material
to decrease to 40 grams.
28
Question 35
Question
A certain radioactive isotope decays at a rate of 4% per day. If there are initially
500 grams of the isotope, how much of it will remain after 10 days? Round your
answer to the nearest gram.
Solution
Step 1: Calculate the decay constant. The decay constant, denoted by λ, can
be calculated using the formula:
λ=−ln(1 −decay rate)
time period
Given that the decay rate is 4% per day, we have:
λ=−ln(1 −0.04)
1≈ −0.0392 per day
Step 2: Find the amount of the isotope after 10 days. The amount of the
radioactive isotope remaining after a certain time period can be calculated using
the formula:
Amount remaining = Initial amount ×eλ×time period
Substitute the values into the formula:
Amount remaining = 500 ×e−0.0392×10 ≈354.1 grams
Therefore, approximately 354 grams of the radioactive isotope will remain
after 10 days.
29
