CHEM 132 - ADVANCED GENERAL
CHEMISTRY II - Radioactive decay
calculations
Question Bank - Set 4
Liberty University
Question 1
Question
A sample of a radioactive isotope decays according to the function N(t) =
N0·e−kt, where N(t) represents the amount of the isotope remaining after t
years, N0is the initial amount of the isotope, and kis the decay constant.
Suppose an initial sample of a radioactive isotope is 100 grams, and after 5
years only 25 grams remain. Find the decay constant kfor this isotope.
Solution
Step 1: Given that N(t) = N0·e−kt, we can plug in the initial conditions to get
the equation N(t) = 100 ·e−5kand N(5) = 25.
Step 2: Substituting N(t) = 25 and N(t) = 100 ·e−5kinto the equation, we
get 25 = 100 ·e−5k.
Step 3: Divide both sides by 100 to simplify the equation to 25
100 =e−5k.
Step 4: Further simplify the equation to 1
4=e−5k.
Step 5: Take the natural logarithm of both sides to solve for k: ln 1
4= ln e−5k.
Step 6: Using the property of logarithms, we have ln 1
4=−5k·ln e.
Step 7: Recall that ln e= 1, so the equation simplifies to ln 1
4=−5k.
Step 8: Solve for kby dividing both sides by −5, giving us k=−ln 1
4
5.
Step 9: Finally, calculate kto get the numerical value. This will give us
k≈0.1386.
Therefore, the decay constant kfor this radioactive isotope is approximately
0.1386 per year.
Question 2
Question
A sample of a radioactive substance has an initial mass of 200 grams. After 10
hours, only 25 grams of the substance remain. If the half-life of the substance
is 4 hours, what is the decay constant of the substance? Round your answer to
two decimal places.
Solution
Step 1: Determine the decay constant using the formula for radioactive decay:
The formula for radioactive decay is given by: A=A0·e−kt, where: - Ais the
final mass of the substance, - A0is the initial mass of the substance, - kis the
decay constant, and - tis the time elapsed.
Given that A= 25 grams, A0= 200 grams, and t= 10 hours, we can
substitute these values into the formula and solve for k:
25 = 200 ·e−10k
Step 2: Solve for the decay constant k: Divide both sides by 200 to isolate
the exponential term:
e−10k=25
200
e−10k= 0.125
Take the natural logarithm of both sides to solve for k:
lne−10k= ln(0.125)
−10k= ln(0.125)
k=ln(0.125)
−10
Step 3: Calculate the decay constant k:
k≈ln(0.125)
−10 ≈−2.0794
−10 ≈0.2079
Therefore, the decay constant of the substance is approximately 0.21 (rounded
to two decimal places).
Question 3
Question
A certain radioactive material decays according to the differential equation dN
dt =
−kN, where N(t) is the amount of radioactive material present at time tand
kis a positive constant. The initial amount of the material is 200 grams and
after 10 hours, the amount of material left is 50 grams. Find the half-life of the
material and the amount of material present after 20 hours.
2
Solution
Step 1: Solve the differential equation dN
dt =−kN to find the general solution:
Z1
NdN =Z−kdt
ln |N|=−kt +C
N(t) = Ce−kt
Step 2: Use the initial condition N(0) = 200 to find the specific solution:
200 = Ce0
C= 200
So, the specific solution is N(t) = 200e−kt.
Step 3: Use the information provided after 10 hours: when t= 10, N(10) =
50:
50 = 200e−10k
50
200 =e−10k
1
4=e−10k
e10k= 4
10k= ln 4
k=ln 4
10
Step 4: Calculate the half-life (T1/2): The half-life is the time it takes for
half of the radioactive material to decay. It can be found using the formula
N(t) = N01
2t
T1/2. Substitute N(0) = 200 and N(t) = 100 (half of the initial
amount) into the solution:
100 = 200 1
2
t1/2
T1/2
1
2=1
2
t1/2
T1/2
t1/2
T1/2
= 1
t1/2=T1/2
Step 5: Find the half-life:
e10k= 4
3
e10
10 ln 4 = 4
eln 10
√4= 4
10
√4=4
T1/2=ln 4
10
Step 6: Find the amount of material present after 20 hours:
N(20) = 200e−ln 4
10 ·20
N(20) = 200e−2·ln 4
N(20) = 200 eln 4−2
N(20) = 200 ·1
42
N(20) = 200 ·1
16
N(20) = 12.5
grams
Question 4
Question
A sample of a radioactive isotope decays according to the formula 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. Given that the initial amount of the isotope is 100 grams and that 60
grams of the isotope remain after 10 years, calculate the decay constant k.
Solution
Step 1: We are given that the initial amount of the isotope is 100 grams, so
N0= 100.
Step 2: We are also given that 60 grams of the isotope remain after 10 years,
so N(10) = 60.
Step 3: Substituting the given values into the formula N(t) = N0e−kt, we
get the equation 60 = 100e−10k.
Step 4: Divide both sides by 100 to simplify the equation: 0.6 = e−10k.
Step 5: Take the natural logarithm of both sides to solve for k: ln(0.6) =
lne−10k.
Step 6: Using the property of logarithms lnab=bln(a), we get −10kln(e) =
ln(0.6).
Step 7: Since ln(e) = 1, the equation simplifies to −10k= ln(0.6).
Step 8: Divide by -10 to solve for k:k=−ln(0.6)
10 ≈0.0408.
Step 9: Therefore, the decay constant k≈0.0408 per year.
4
Question 5
Question
A sample of a radioactive substance has an initial mass of 100 grams. After 20
days, only 25 grams of the substance remains. If the half-life of the substance
is 10 days, determine the decay constant and the age of the sample.
Solution
Step 1: Calculate the decay constant (k). Given that the half-life of the sub-
stance is 10 days, we can use the formula for exponential decay:
N
N0
=e−kt
Where: - Nis the final amount of the substance (25 grams), - N0is the
initial amount of the substance (100 grams), - kis the decay constant, - tis the
time (20 days).
Substitute the given values into the formula:
25
100 =e−10k
1
4=e−10k
ln 1
4=−10k
k=ln(4)
10
k=2 ln(2)
10
Step 2: Calculate the age of the sample. To find the age of the sample, we
will use the formula:
N=N0e−kt
We are given that N= 25, N0= 100, and we have already calculated k.
Substitute these values into the formula:
25 = 100e−2 ln(2)
10 ·20
1
4=e−4 ln(2)
1
4= (eln(2))−4
5
1
4= 2−4
1
4=1
16
Therefore, the age of the sample is 20 days.
Question 6
Question
A certain radioactive substance decays according to the formula A(t) = A0e−0.02t,
where A(t) represents the amount of substance remaining after tyears and A0
is the initial amount of the substance. If the initial amount of the substance is
100 grams, find the amount of substance remaining after 10 years to the nearest
gram.
Solution
Let’s first substitute the given values A0= 100 grams and t= 10 years into the
formula A(t) = A0e−0.02t.
A(10) = 100e−0.02×10
Step 1: Calculate the exponent.
−0.02 ×10 = −0.2
Step 2: Substitute the exponent back into the formula.
A(10) = 100e−0.2
Step 3: Evaluate the exponential function.
A(10) ≈100 ×e−0.2≈100 ×0.8187 ≈81.87 grams
After 10 years, the amount of substance remaining is approximately 81.87
grams. Rounding to the nearest gram, the amount is 82 grams.
Question 7
Question
A certain radioactive substance decays exponentially at a rate of 0.05 per year.
If the initial amount of the substance is 100 grams, determine the amount of
substance remaining after 20 years.
6
Solution
Step 1: The exponential decay formula is given by A(t) = A0·e−kt, where: -
A(t) is the amount of substance remaining after time t, - A0is the initial amount
of the substance, - kis the decay rate, - tis the time elapsed.
Step 2: Substituting A0= 100 grams, k= 0.05, and t= 20 years into the
formula, we get:
A(20) = 100 ·e−0.05·20
Step 3: Calculate the value of A(20):
A(20) = 100 ·e−1
Step 4: Simplify the expression:
A(20) = 100 ·1
e
Step 5: Use the approximate value of e≈2.71828 to evaluate the expression:
A(20) ≈100 ·1
2.71828 ≈36.7893
Step 6: Therefore, the amount of substance remaining after 20 years is
approximately 36.7893 grams.
Question 8
Question
A sample of a radioactive isotope has an initial mass of 50 grams. After 48
hours, the mass of the sample is reduced to 45 grams due to radioactive decay.
If the half-life of the isotope is 24 hours, what is the decay constant of the
isotope?
Solution
Step 1: Determine the fraction of the sample remaining after 48 hours. Given
that the half-life of the isotope is 24 hours, we know that after 1 half-life, the
remaining mass is reduced to half of the original mass. Therefore, after 2 half-
lives (48 hours), the remaining mass is reduced to 1
22=1
4of the original
mass.
Step 2: Calculate the fraction of the sample that has decayed after 48 hours.
The fraction that has decayed is 1 −1
4=3
4.
Step 3: Set up the decay equation in terms of the decay constant. The decay
equation for a radioactive isotope is given by:
N(t) = N0·e−λt
7
where: - N(t) is the amount of the sample remaining after time t, - N0is the
initial amount of the sample, - λis the decay constant, and - tis the time passed.
Step 4: Plug in the known values and solve for the decay constant. Given
that N(t) = 45 grams, N0= 50 grams, and t= 48 hours, we can plug these
values into the decay equation:
45 = 50 ·e−λ·48
Step 5: Solve for the decay constant. Solving for λ:
e−λ·48 =45
50 =9
10
−λ·48 = ln 9
10
λ=−ln 9
10
48 ≈0.01414 hours−1
Therefore, the decay constant of the isotope is approximately 0.01414 hours−1.
Question 9
Question
A certain radioactive substance decays according to the equation N(t) = N0e−λt,
where N(t) is the amount of substance at time t,N0is the initial amount of
substance, λis the decay constant, and tis the time in years.
Given that the half-life of the substance is 500 years, find the decay constant
λ.
Solution
Step 1: The half-life of a radioactive substance is the time it takes for half of the
substance to decay. We can express the half-life in terms of the decay constant
λusing the formula:
Half-life = ln(2)
λ
Given that the half-life of the substance is 500 years, we can plug in this
value to solve for λ:
500 = ln(2)
λ
Step 2: To solve for λ, we can rearrange the equation as follows:
λ=ln(2)
500
Thus, the decay constant λfor this radioactive substance is λ=ln(2)
500 .
8
Question 10
Question
A certain radioactive substance decays according to the equation N(t) = N0e−kt,
where N(t) represents the amount of substance at time t,N0is the initial
amount of substance, kis the decay constant, and eis the base of the natural
logarithm. Given that the half-life of the substance is 5 days, find the decay
constant k.
Solution
Step 1: The half-life of a radioactive substance is the time it takes for half of
the substance to decay. In this case, we have N(t) = 1
2N0when t= 5 days.
Step 2: Substituting N(t) = 1
2N0and t= 5 into the decay equation, we get
1
2N0=N0e−5k. Step 3: Divide both sides by N0to simplify the equation to
1
2=e−5k. Step 4: Take the natural logarithm of both sides to solve for k:
ln 1
2= lne−5k. Step 5: Use the property of logarithms lnab=bln(a) to
simplify the equation to ln 1
2=−5k. Step 6: Finally, solve for kby dividing
by −5: k=−ln(1
2)
5. Step 7: Since ln 1
2=−ln(2), we have k=ln(2)
5. Thus,
the decay constant kis ln(2)
5≈0.1386 days−1.
Question 11
Question
A sample of a radioactive isotope has an initial mass of 1 gram and a half-life
of 24 hours. After 72 hours, what is the remaining mass of the sample?
Solution
Step 1: Calculate the decay constant 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 in hours.
Step 2: Since the half-life (t1/2) is given as 24 hours, we can relate it to the
decay constant using the formula t1/2=ln(2)
λ.
Step 3: Substituting t1/2= 24 hours into the above formula, we find λ=
ln(2)
24 .
Step 4: Now, we substitute N0= 1 gram, t= 72 hours, and λ=ln(2)
24 into
the formula N(t) = N0e−λt to find the remaining mass of the sample after 72
hours.
Step 5: Therefore, the remaining mass of the sample after 72 hours is
N(72) = 1×e−ln(2)
24 ×72. Simplifying this expression will give us the final answer.
9
Question 12
Question
A certain radioactive substance has a half-life of 20 years. If you start with 100
grams of the substance, how many grams will remain after 60 years?
Solution
Step 1: Determine the decay constant.
The decay constant (λ) can be calculated using the formula:
λ=ln(2)
T1
2
where T1
2is the half-life of the substance. In this case, T1
2= 20 years.
λ=ln(2)
20 ≈0.03465 per year
Step 2: Use the decay model to find the amount remaining.
The amount of substance remaining at any time tcan be calculated using the
formula:
N(t) = N0·e−λt
where N0is the initial amount of substance.
N(60) = 100 ·e−0.03465·60
= 100 ·e−2.079
≈100 ·0.125
= 12.5 grams
So, after 60 years, there will be approximately 12.5 grams of the substance
remaining.
Question 13
Question
An unknown radioactive substance decays by 25
Solution
Step 1: Determine the decay constant. Let N(t) represent the amount of sub-
stance remaining after thours. Since the substance decays by 25
10
Step 2: Use the exponential decay formula. The formula for radioactive
decay is given by:
N(t) = N0·eλt
where N0is the initial amount of substance. Plugging in the known values:
N(t) = 1000 ·e−0.25t
Step 3: Calculate the amount remaining after 6 hours. Substitute t= 6 into
the formula:
N(6) = 1000 ·e−0.25·6
N(6) = 1000 ·e−1.5≈1000 ·0.2231
N(6) ≈223 grams
Therefore, after 6 hours, approximately 223 grams of the substance will
remain.
Question 14
Question
A sample of a radioactive substance has an initial mass of 50 grams. After 20
days, only 10 grams of the substance remain. If the half-life of the substance is
10 days, what is the decay constant λof the substance?
Solution
Step 1: Calculate the decay constant λusing the formula for radioactive decay:
λ=−ln(0.5)
t1
2
where t1
2is the half-life of the substance. Substitute t1
2= 10 days into the
formula:
λ=−ln(0.5)
10
Step 2: Calculate the remaining fraction of the substance after 20 days using
the radioactive decay formula:
N(t) = N0e−λt
where: - N(t) is the remaining mass of the substance after time t-N0is the
initial mass of the substance - tis the time passed Substitute N(t) = 10 grams,
N0= 50 grams, t= 20 days into the formula:
10 = 50e−20λ
11
Step 3: Now, solve the equation obtained in Step 2 for λ:
e−20λ=10
50
e−20λ= 0.2
−20λ= ln(0.2)
λ=ln(0.2)
−20
Step 4: Calculate the value of λ:
λ=ln(0.2)
−20
Question 15
Question
A certain radioactive substance has a half-life of 10 days. If there are initially
200 grams of the substance, how many grams will remain after 30 days?
Solution
Step 1: Determine the decay constant λ
The decay constant λcan be calculated using the formula:
λ=ln(2)
T1
2
where T1
2is the half-life of the substance. Substituting T1
2= 10 days:
λ=ln(2)
10
Step 2: Determine the amount of substance remaining after 30 days
The amount of substance remaining after time tcan be calculated using the
formula:
N(t) = N0e−λt
where N0is the initial amount of substance. Substituting N0= 200 grams,
λ=ln(2)
10 , and t= 30 days:
N(30) = 200e−30 ln(2)
10
Step 3: Calculate the amount of substance remaining after 30 days
N(30) = 200e−3 ln(2)
12
N(30) = 200eln(2−3)
N(30) = 200(2−3)
N(30) = 200 ×1
8
N(30) = 25
Therefore, after 30 days, only 25 grams of the substance will remain.
Question 16
Question
A sample of a radioactive isotope decays according to the equation 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 of the isotope is
100 grams, determine how long it will take for the amount of the isotope to
reduce to 50 grams.
Solution
Step 1: Substitute the given values into the equation.
Given N0= 100 grams and N(t) = 50 grams, we have:
50 = 100e−0.02t
Step 2: Solve for t.
Dividing both sides by 100: 50
100 =e−0.02t
0.5 = e−0.02t
Step 3: Take the natural logarithm of both sides.
ln(0.5) = lne−0.02t
ln(0.5) = −0.02t
Step 4: Solve for t.
t=−ln(0.5)
0.02
t≈34.66 years
Therefore, it will take approximately 34.66 years for the amount of the iso-
tope to reduce to 50 grams.
13
Question 17
Question
A certain radioactive substance has a half-life of 10 days. If the initial amount
of the substance is 100 grams, determine the amount of the substance remaining
after 30 days.
Solution
Let’s denote the amount of the radioactive substance remaining after tdays as
A(t).
Step 1: Determine the decay constant λusing the formula N(t) = N0·e−λt,
where N(t) is the amount of substance remaining at time t,N0is the initial
amount, and λis the decay constant.
Given that the half-life T1/2= 10 days, we can find λusing the formula
λ=ln(2)
T1/2.
λ=ln(2)
10 ≈0.0693 (rounded to four decimal places)
Step 2: Write the exponential decay model for the amount of substance
remaining.
A(t) = 100 ·e−0.0693t
Step 3: Calculate the amount of substance remaining after 30 days by
substituting t= 30 into the model.
A(30) = 100·e−0.0693·30 = 100·e−2.079 ≈7.454 grams (rounded to three decimal places)
After 30 days, there will be approximately 7.454 grams of the radioactive
substance remaining.
Question 18
Question
A sample of a radioactive material initially contains 200 grams of a certain
radionuclide with a half-life of 20 days. After 60 days, only 25 grams of the
radionuclide remain. What is the decay constant for this radionuclide?
Solution
Step 1: Find the decay constant using the formula for radioactive decay:
N(t) = N0·e−λt
14
where: - N(t) is the amount of radionuclide remaining at time t, - N0is the
initial amount of radionuclide, and - λis the decay constant.
Step 2: Substitute the given values into the formula: At t= 0, N(0) = 200
grams and at t= 60, N(60) = 25 grams. So, we have:
25 = 200 ·e−λ·60
Step 3: Solve for the decay constant λ:
25
200 =e−60λ
0.125 = e−60λ
Taking the natural log of both sides:
ln(0.125) = lne−60λ
ln(0.125) = −60λ
λ=ln(0.125)
−60
λ≈0.0183 days−1
Therefore, the decay constant for this radionuclide is approximately 0.0183
days−1.
Question 19
Question
A sample of a radioactive substance decays according to the equation N(t) =
N0e−0.05t, where N(t) represents the amount of substance present after tyears
and N0is the initial amount of substance. If the initial amount of the substance
is 100 grams, determine the amount of substance present after 20 years. Round
your answer to the nearest hundredth.
Solution
Step 1: Plug in the initial values into the equation.
N(20) = 100e−0.05×20
= 100e−1
= 100 ×1
e
≈100 ×0.3679
≈36.79 grams
Therefore, the amount of substance present after 20 years is approximately
36.79 grams.
15
Question 20
Question
A sample of radium-226 has an initial mass of 1 gram. If the half-life of radium-
226 is 1600 years, determine the mass of the sample after 4800 years.
Solution
Step 1: Calculate the number of half-lives that have passed. Given that the
half-life of radium-226 is 1600 years, we can find the number of half-lives that
have passed in 4800 years:
Number of half-lives = 4800 years
1600 years/half-life = 3 half-lives
Step 2: Calculate the remaining mass of the sample after 3 half-lives. Since
the mass decreases by half with each half-life, the remaining mass after 3 half-
lives is:
Remaining mass = 1 gram ×1
23
= 1 gram ×1
8=1
8grams
Step 3: Write the final answer. After 4800 years, the mass of the sample of
radium-226 would be 1
8grams.
Question 21
Question
A sample of a radioactive material has an initial mass of 10 grams. After 30
minutes, the mass of the sample has decreased to 6 grams. If the half-life of the
material is 45 minutes, determine the decay constant kand write the exponential
decay equation for the sample.
Solution
Step 1: Calculate the fraction of the sample left after 30 minutes. Step 2:
Calculate the decay constant kusing the half-life formula. Step 3: Write the
exponential decay equation for the sample.
Step 1: The fraction of the sample left after 30 minutes can be calculated
using the formula:
fraction left = final mass
initial mass
fraction left = 6
10
fraction left = 0.6
16
Step 2: The decay constant kcan be calculated using the half-life formula:
fraction remaining after time t=e−kt
Substitute the values:
0.6 = e−k×30
ln(0.6) = −k×30
k=ln(0.6)
−30
k≈0.0185
Step 3: The exponential decay equation for the sample is:
mass at time t= 10 ×e−0.0185t
Therefore, the decay constant kis approximately 0.0185 and the exponential
decay equation for the sample is 10 ×e−0.0185t.
Question 22
Question
A certain radioactive isotope decays at a rate proportional to the amount
present. If the initial amount of the isotope is 100 grams and its half-life is
10 days, determine the amount of the isotope remaining after 30 days.
Solution
Step 1: Let Q(t) represent the amount of the isotope remaining after tdays. We
are given that the isotope decays at a rate proportional to the amount present,
so we have the differential equation
dQ
dt =−kQ
where kis the decay constant.
Step 2: We know that half-life is the time it takes for half of the substance
to decay. In this case, the half-life is 10 days, so we have
1
2Q(10) = Q(0)
We can rewrite this as Q(10)
Q(0) = 1/2
e−10k= 1/2
−10k= ln(1/2)
17
k=ln(2)
10
Thus, the decay constant is k=ln(2)
10 .
Step 3: Now, let’s solve the differential equation. We have
dQ
dt =−ln(2)
10 Q
Separating variables gives us
dQ
Q=−ln(2)
10 dt
Integrating both sides yields
ln(Q) = −ln(2)
10 t+C
where Cis the constant of integration.
Step 4: We are given that Q(0) = 100, so substituting this into our equation
gives
ln(100) = C
C= ln(100)
C= ln102
C= 2 ln(10)
C= 2 ln(2) + 2 ln(5)
C= ln22+ ln52
C= ln(4) + ln(25)
C= ln(100)
So, the equation becomes
ln(Q) = −ln(2)
10 t+ ln(100)
ln(Q) = ln(100) −ln(2)
10 t
ln(Q) = ln 100
2t/10
Q=100
2t/10
Step 5: Finally, we can determine the amount of the isotope remaining after
30 days by substituting t= 30 into the equation:
Q(30) = 100
230/10
18
Q(30) = 100
23
Q(30) = 100
8
Q(30) = 12.5 grams
Therefore, after 30 days, there will be 12.5 grams of the isotope remaining.
Question 23
Question
A sample of a radioactive material has an initial mass of 200 g and a half-life of
10 days. After 30 days, what is the mass of the sample remaining?
Solution
Step 1: Calculate the decay constant λ. The decay constant is related to the
half-life T1/2by the formula λ=ln(2)
T1/2.
Step 2: Substitute the given half-life T1/2= 10 days into the formula to find
λ:
λ=ln(2)
10 ≈0.0693 days−1
Step 3: Use the exponential decay formula N(t) = N0e−λt to find the mass
of the sample remaining after 30 days. Here N(t) represents the mass at time
t,N0represents the initial mass, and tis the time elapsed.
N(t) = 200 ·e−0.0693·30
Step 4: Calculate the mass of the sample remaining after 30 days:
N(30) = 200 ·e−2.079 ≈42.74 g
Therefore, after 30 days, the mass of the sample remaining is approximately
42.74 g.
Question 24
Question
A sample of a radioactive isotope has an initial mass of 200 grams. After 10
days, only 25 grams of the isotope remain. If the half-life of the isotope is 5
days, what is the decay constant for this isotope?
19
Solution
Step 1: Calculate the fraction of the original sample that remains after 10 days.
Step 2: Use this information to find the decay constant.
Step 1: The fraction of the original sample that remains after 10 days can
be calculated using the formula for radioactive decay:
m
m0
=1
2t
T
where: mis the final mass, m0is the initial mass, tis the time elapsed, Tis the
half-life of the isotope.
Plugging in the known values, we get:
25
200 =1
210
5
Solving for the fraction of the original sample that remains:
25
200 =1
22
1
8=1
4
Step 2: The decay constant, denoted by λ, is related to the half-life Tby
the formula:
λ=ln(2)
T
Given that the half-life Tis 5 days, we can calculate the decay constant:
λ=ln(2)
5
λ=0.6931
5
λ≈0.1386 days−1
Therefore, the decay constant for this isotope is approximately 0.1386 days−1.
Question 25
Question
A sample of a radioactive material has an initial mass of 100 grams. After 10
minutes, only 75 grams of the material remain. If the half-life of the material is
5 minutes, determine the decay constant and the amount of material that will
remain after 30 minutes.
20
Solution
Step 1: Calculate the decay constant λusing the half-life formula N(t) = N0·
2−t
T1
2where N(t) is the remaining mass at time t,N0is the initial mass, tis
the time elapsed, and T1
2is the half-life of the material.
Given N(10) = 75, N0= 100, and T1
2= 5, we can plug these values into the
formula to solve for λ:
75 = 100 ·2−10
5
0.75 = 2−2
22=1
0.75
λ= ln 1
0.75≈0.2877 min−1
So, the decay constant is approximately 0.2877 min−1.
Step 2: Determine the remaining mass after 30 minutes using the exponential
decay formula N(t) = N0·e−λt where N(t) is the remaining mass at time t,N0
is the initial mass, λis the decay constant, and tis the time elapsed.
Given N0= 100, λ= 0.2877, and t= 30, we can plug these values into the
formula:
N(30) = 100 ·e−0.2877·30 ≈34.394 grams
Therefore, approximately 34.394 grams of the material will remain after 30
minutes.
Question 26
Question
A certain radioactive substance has a half-life of 30 minutes. If you start with
500 grams of the substance, how much will remain after 3 hours?
Solution
Step 1: Calculate the number of half-lives in 3 hours. Let nbe the number of
half-lives. Since each half-life is 30 minutes, there are 6 half-lives in 3 hours.
n=3 hours
0.5 hours per half-life = 6
21
Step 2: Determine the amount of material remaining after 6 half-lives. The
amount of material remaining after nhalf-lives can be calculated using the
formula:
Amount remaining = Initial amount ×1
2n
Plugging in the values, we have:
Amount remaining = 500 ×1
26
Amount remaining = 500 ×1
64
Amount remaining = 500
64
Amount remaining = 7.8125 grams
Therefore, after 3 hours, there will be approximately 7.81 grams of the ra-
dioactive substance remaining.
Question 27
Question
A certain radioactive isotope has a half-life of 5 days. If we start with an initial
amount of 100 grams, how much of the substance will remain after 20 days?
Solution
Step 1: Calculate the decay constant. The decay constant, denoted by λ, is
related to the half-life T1/2by the equation λ=ln(2)
T1/2.
λ=ln(2)
5=0.6931
5= 0.1386 per day
Step 2: Use the formula for radioactive decay to find the remaining amount
of substance after 20 days. The formula is given by N(t) = N0e−λt, where: -
N(t) is the amount of substance remaining after time t-N0is the initial amount
of substance - λis the decay constant - tis the time passed
Plugging in the values:
N(20) = 100 ·e−0.1386·20
N(20) ≈100 ·e−2.772 ≈100 ·0.0633 ≈6.33 grams
Therefore, approximately 6.33 grams of the substance will remain after 20
days.
22
Question 28
Question
A sample of a radioactive substance has an initial mass of 25 grams. It decays
according to the function m(t) = 25e−0.03t, where m(t) is the mass remaining
after tyears. Find the rate of decay of the substance after 10 years.
Solution
Step 1: To find the rate of decay after 10 years, we need to find the derivative
of the mass function m(t) with respect to time, t.
Step 2: We have m(t) = 25e−0.03t. To find dm
dt , we differentiate m(t) with
respect to tusing the chain rule.
dm
dt =d
dt (25e−0.03t)
Step 3: Applying the chain rule: dm
dt = 25(−0.03)e−0.03t
dm
dt =−0.75e−0.03t
Step 4: We want to find the rate of decay after 10 years, so substitute t= 10
into the derivative we just found.
dm
dt
t=10
=−0.75e−0.03(10)
dm
dt
t=10
=−0.75e−0.3
dm
dt
t=10 ≈ −0.679 grams per year
Therefore, the rate of decay of the substance after 10 years is approximately
0.679 grams per year.
Question 29
Question
A certain radioactive substance decays with a half-life of 10 hours. If you start
with 500 grams of the substance, how much will be left after 30 hours?
Solution
Step 1: Use the radioactive decay formula N(t) = N01
2t
T1/2, where: - N(t) is
the amount of substance remaining after time t, - N0is the initial amount of
substance, - T1/2is the half-life of the substance.
Step 2: Substitute the given values into the formula: N(30) = 500 1
230
10
Step 3: Simplify the expression: N(30) = 500 1
23N(30) = 500×1
8N(30) =
62.5 grams
Answer: After 30 hours, there will be 62.5 grams of the substance left.
23
Question 30
Question
A sample of a radioactive isotope has an initial mass of 10 grams. After 5 hours,
the mass of the sample has decreased to 7 grams. If the half-life of the isotope
is 3 hours, determine the decay constant (λ) and the age of the sample.
Solution
Step 1: Find the decay constant (λ) using the formula N(t) = N0e−λt, where
N(t) is the current mass, N0is the initial mass, λis the decay constant, and t
is the time.
Given that N(5) = 7 and N0= 10, we can rewrite the formula as 7 = 10e−5λ.
Solving for λ:7
10 =e−5λ
ln 7
10=−5λ
λ=−1
5ln 7
10
Step 2: Calculate the decay constant λ.
λ=−1
5ln 7
10≈0.067
Step 3: Determine the age of the sample using t=ln(2)
λ.
Given λ≈0.067, we can find:
t=ln(2)
0.067 ≈0.693
0.067 ≈10.34 hours
Therefore, the decay constant is approximately 0.067 and the age of the
sample is approximately 10.34 hours.
Question 31
Question
A sample of a radioactive substance has an initial mass of 100 grams. After 10
hours, the mass of the substance has decreased to 60 grams. If the half-life of
the substance is 5 hours, what is the decay constant for this substance?
24
Solution
Step 1: Determine the fraction of the substance remaining after 10 hours. Step
2: Use the formula for radioactive decay to find the decay constant.
Step 1: The decay constant (λ) can be calculated using the formula:
m
m0
=e−λt
where: - mis the final mass (60 grams), - m0is the initial mass (100 grams), -
tis the time elapsed (10 hours).
Substitute the given values into the formula:
60
100 =e−λ·10
0.6 = e−10λ
Step 2: To solve for the decay constant, we need to take the natural logarithm
of both sides, which gives:
ln(0.6) = lne−10λ
ln(0.6) = −10λ
Finally, divide by −10 to find the value of the decay constant:
λ=−ln(0.6)
10
Question 32
Question
A certain radioactive isotope decays according to the equation N(t) = N0e−kt,
where N(t) is the quantity of the isotope remaining at time t,N0is the initial
quantity of the isotope, and kis a decay constant. If 80% of a sample of this
isotope decays in 6 hours, what is the half-life of the isotope?
Solution
Step 1: Given information, we have N(t) = N0e−kt, and after 6 hours, 80%
of the sample remains. This means that N(6) = 0.2N0. Step 2: Substituting
t= 6 hours into the equation, we get 0.2N0=N0e−6k. Step 3: Simplifying the
equation, we have e−6k= 0.2. Step 4: Taking natural logarithms of both sides,
we get −6k= ln(0.2). Step 5: Solving for k, we have k=−ln(0.2)
6. Step 6:
The half-life of a radioactive isotope is the time it takes for half of the sample
to decay. So, we need to find the time t1
2such that N(t1
2) = 0.5N0. Step
7: Substituting N(t1
2) = 0.5N0into the equation, we get 0.5N0=N0e−kt 1
2.
25
Step 8: Simplifying the equation, we have e−kt 1
2= 0.5. Step 9: Taking natural
logarithms of both sides, we get −kt 1
2= ln(0.5). Step 10: Solving for t1
2, we
have t1
2=−ln(0.5)
k. Step 11: Finally, substituting the value of kfrom Step 5
into the equation in Step 10, we get the half-life of the isotope.
Question 33
Question
The half-life of a radioactive isotope is 35 hours. If the initial amount of the
isotope is 500 grams, determine the amount of the isotope remaining after 105
hours.
Solution
Step 1: Calculate the decay constant using the half-life formula Step 2: Use the
decay constant to calculate the remaining amount of the radioactive isotope at
105 hours.
Step 1: Given that the half-life of the isotope is 35 hours, we can use the
formula for half-life decay to find the decay constant (k):
Half-life = ln(2)
k
35 = ln(2)
k
k=ln(2)
35
Step 2: Now that we have the decay constant, we can use the exponential
decay formula to find the remaining amount of the isotope after 105 hours:
N(t) = N0·e−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 elapsed.
Substitute the given values:
N(105) = 500 ·e−ln(2)
35 ·105
N(105) = 500 ·e−3 ln(2)
N(105) = 500 ·eln(2−3)
N(105) = 500 ·2−3
N(105) = 500 ·1
8
N(105) = 62.5 grams
Therefore, the amount of the isotope remaining after 105 hours is 62.5 grams.
26
Question 34
Question
A radioactive substance decays according to the law A(t) = A0e−kt, where A(t)
is the amount of the substance remaining at time t(in years), A0is the initial
amount of the substance, and kis a positive constant. If 50% of the substance
decays in 300 years, find the value of kfor this substance.
Solution
Step 1: Given that A(t) = A0e−kt, we know that when t= 0, A(t) = A0. Thus,
A(0) = A0e−k(0) =A0, which implies e0= 1. Therefore, A0=A0, as expected.
Step 2: We are told that 50% of the substance decays in 300 years. This
means that A(300) = 0.5A0. Substituting these values into the decay equation,
we get:
0.5A0=A0e−k(300)
Step 3: By dividing both sides of the equation by A0, we can simplify to:
0.5 = e−300k
Step 4: Taking the natural logarithm of both sides of the equation to solve
for k, we have:
ln(0.5) = lne−300k=−300k
Step 5: Solving for k, we find:
k=ln(0.5)
−300 ≈0.00231
Therefore, the value of kfor this substance is approximately 0.00231.
Question 35
Question
An unknown radioactive substance has a half-life of 10 hours. If there are
initially 500 grams of the substance present, how much of it will remain after
30 hours?
Solution
Step 1: Determine the decay constant using the formula k=0.693
T1
2
, where T1
2is
the half-life.
Step 1: k=0.693
10 = 0.0693 hours−1
27
Question 2
Question
A sample of a radioactive substance has an initial mass of 200 grams. After 10
hours, only 25 grams of the substance remain. If the half-life of the substance
is 4 hours, what is the decay constant of the substance? Round your answer to
two decimal places.
Solution
Step 1: Determine the decay constant using the formula for radioactive decay:
The formula for radioactive decay is given by: A=A0·e−kt, where: - Ais the
final mass of the substance, - A0is the initial mass of the substance, - kis the
decay constant, and - tis the time elapsed.
Given that A= 25 grams, A0= 200 grams, and t= 10 hours, we can
substitute these values into the formula and solve for k:
25 = 200 ·e−10k
Step 2: Solve for the decay constant k: Divide both sides by 200 to isolate
the exponential term:
e−10k=25
200
e−10k= 0.125
Take the natural logarithm of both sides to solve for k:
lne−10k= ln(0.125)
−10k= ln(0.125)
k=ln(0.125)
−10
Step 3: Calculate the decay constant k:
k≈ln(0.125)
−10 ≈−2.0794
−10 ≈0.2079
Therefore, the decay constant of the substance is approximately 0.21 (rounded
to two decimal places).
Question 3
Question
A certain radioactive material decays according to the differential equation dN
dt =
−kN, where N(t) is the amount of radioactive material present at time tand
kis a positive constant. The initial amount of the material is 200 grams and
after 10 hours, the amount of material left is 50 grams. Find the half-life of the
material and the amount of material present after 20 hours.
2
Solution
Step 1: Solve the differential equation dN
dt =−kN to find the general solution:
Z1
NdN =Z−kdt
ln |N|=−kt +C
N(t) = Ce−kt
Step 2: Use the initial condition N(0) = 200 to find the specific solution:
200 = Ce0
C= 200
So, the specific solution is N(t) = 200e−kt.
Step 3: Use the information provided after 10 hours: when t= 10, N(10) =
50:
50 = 200e−10k
50
200 =e−10k
1
4=e−10k
e10k= 4
10k= ln 4
k=ln 4
10
Step 4: Calculate the half-life (T1/2): The half-life is the time it takes for
half of the radioactive material to decay. It can be found using the formula
N(t) = N01
2t
T1/2. Substitute N(0) = 200 and N(t) = 100 (half of the initial
amount) into the solution:
100 = 200 1
2
t1/2
T1/2
1
2=1
2
t1/2
T1/2
t1/2
T1/2
= 1
t1/2=T1/2
Step 5: Find the half-life:
e10k= 4
3
e10
10 ln 4 = 4
eln 10
√4= 4
10
√4=4
T1/2=ln 4
10
Step 6: Find the amount of material present after 20 hours:
N(20) = 200e−ln 4
10 ·20
N(20) = 200e−2·ln 4
N(20) = 200 eln 4−2
N(20) = 200 ·1
42
N(20) = 200 ·1
16
N(20) = 12.5
grams
Question 4
Question
A sample of a radioactive isotope decays according to the formula 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. Given that the initial amount of the isotope is 100 grams and that 60
grams of the isotope remain after 10 years, calculate the decay constant k.
Solution
Step 1: We are given that the initial amount of the isotope is 100 grams, so
N0= 100.
Step 2: We are also given that 60 grams of the isotope remain after 10 years,
so N(10) = 60.
Step 3: Substituting the given values into the formula N(t) = N0e−kt, we
get the equation 60 = 100e−10k.
Step 4: Divide both sides by 100 to simplify the equation: 0.6 = e−10k.
Step 5: Take the natural logarithm of both sides to solve for k: ln(0.6) =
lne−10k.
Step 6: Using the property of logarithms lnab=bln(a), we get −10kln(e) =
ln(0.6).
Step 7: Since ln(e) = 1, the equation simplifies to −10k= ln(0.6).
Step 8: Divide by -10 to solve for k:k=−ln(0.6)
10 ≈0.0408.
Step 9: Therefore, the decay constant k≈0.0408 per year.
4
Question 5
Question
A sample of a radioactive substance has an initial mass of 100 grams. After 20
days, only 25 grams of the substance remains. If the half-life of the substance
is 10 days, determine the decay constant and the age of the sample.
Solution
Step 1: Calculate the decay constant (k). Given that the half-life of the sub-
stance is 10 days, we can use the formula for exponential decay:
N
N0
=e−kt
Where: - Nis the final amount of the substance (25 grams), - N0is the
initial amount of the substance (100 grams), - kis the decay constant, - tis the
time (20 days).
Substitute the given values into the formula:
25
100 =e−10k
1
4=e−10k
ln 1
4=−10k
k=ln(4)
10
k=2 ln(2)
10
Step 2: Calculate the age of the sample. To find the age of the sample, we
will use the formula:
N=N0e−kt
We are given that N= 25, N0= 100, and we have already calculated k.
Substitute these values into the formula:
25 = 100e−2 ln(2)
10 ·20
1
4=e−4 ln(2)
1
4= (eln(2))−4
5
1
4= 2−4
1
4=1
16
Therefore, the age of the sample is 20 days.
Question 6
Question
A certain radioactive substance decays according to the formula A(t) = A0e−0.02t,
where A(t) represents the amount of substance remaining after tyears and A0
is the initial amount of the substance. If the initial amount of the substance is
100 grams, find the amount of substance remaining after 10 years to the nearest
gram.
Solution
Let’s first substitute the given values A0= 100 grams and t= 10 years into the
formula A(t) = A0e−0.02t.
A(10) = 100e−0.02×10
Step 1: Calculate the exponent.
−0.02 ×10 = −0.2
Step 2: Substitute the exponent back into the formula.
A(10) = 100e−0.2
Step 3: Evaluate the exponential function.
A(10) ≈100 ×e−0.2≈100 ×0.8187 ≈81.87 grams
After 10 years, the amount of substance remaining is approximately 81.87
grams. Rounding to the nearest gram, the amount is 82 grams.
Question 7
Question
A certain radioactive substance decays exponentially at a rate of 0.05 per year.
If the initial amount of the substance is 100 grams, determine the amount of
substance remaining after 20 years.
6
Solution
Step 1: The exponential decay formula is given by A(t) = A0·e−kt, where: -
A(t) is the amount of substance remaining after time t, - A0is the initial amount
of the substance, - kis the decay rate, - tis the time elapsed.
Step 2: Substituting A0= 100 grams, k= 0.05, and t= 20 years into the
formula, we get:
A(20) = 100 ·e−0.05·20
Step 3: Calculate the value of A(20):
A(20) = 100 ·e−1
Step 4: Simplify the expression:
A(20) = 100 ·1
e
Step 5: Use the approximate value of e≈2.71828 to evaluate the expression:
A(20) ≈100 ·1
2.71828 ≈36.7893
Step 6: Therefore, the amount of substance remaining after 20 years is
approximately 36.7893 grams.
Question 8
Question
A sample of a radioactive isotope has an initial mass of 50 grams. After 48
hours, the mass of the sample is reduced to 45 grams due to radioactive decay.
If the half-life of the isotope is 24 hours, what is the decay constant of the
isotope?
Solution
Step 1: Determine the fraction of the sample remaining after 48 hours. Given
that the half-life of the isotope is 24 hours, we know that after 1 half-life, the
remaining mass is reduced to half of the original mass. Therefore, after 2 half-
lives (48 hours), the remaining mass is reduced to 1
22=1
4of the original
mass.
Step 2: Calculate the fraction of the sample that has decayed after 48 hours.
The fraction that has decayed is 1 −1
4=3
4.
Step 3: Set up the decay equation in terms of the decay constant. The decay
equation for a radioactive isotope is given by:
N(t) = N0·e−λt
7
where: - N(t) is the amount of the sample remaining after time t, - N0is the
initial amount of the sample, - λis the decay constant, and - tis the time passed.
Step 4: Plug in the known values and solve for the decay constant. Given
that N(t) = 45 grams, N0= 50 grams, and t= 48 hours, we can plug these
values into the decay equation:
45 = 50 ·e−λ·48
Step 5: Solve for the decay constant. Solving for λ:
e−λ·48 =45
50 =9
10
−λ·48 = ln 9
10
λ=−ln 9
10
48 ≈0.01414 hours−1
Therefore, the decay constant of the isotope is approximately 0.01414 hours−1.
Question 9
Question
A certain radioactive substance decays according to the equation N(t) = N0e−λt,
where N(t) is the amount of substance at time t,N0is the initial amount of
substance, λis the decay constant, and tis the time in years.
Given that the half-life of the substance is 500 years, find the decay constant
λ.
Solution
Step 1: The half-life of a radioactive substance is the time it takes for half of the
substance to decay. We can express the half-life in terms of the decay constant
λusing the formula:
Half-life = ln(2)
λ
Given that the half-life of the substance is 500 years, we can plug in this
value to solve for λ:
500 = ln(2)
λ
Step 2: To solve for λ, we can rearrange the equation as follows:
λ=ln(2)
500
Thus, the decay constant λfor this radioactive substance is λ=ln(2)
500 .
8
Question 10
Question
A certain radioactive substance decays according to the equation N(t) = N0e−kt,
where N(t) represents the amount of substance at time t,N0is the initial
amount of substance, kis the decay constant, and eis the base of the natural
logarithm. Given that the half-life of the substance is 5 days, find the decay
constant k.
Solution
Step 1: The half-life of a radioactive substance is the time it takes for half of
the substance to decay. In this case, we have N(t) = 1
2N0when t= 5 days.
Step 2: Substituting N(t) = 1
2N0and t= 5 into the decay equation, we get
1
2N0=N0e−5k. Step 3: Divide both sides by N0to simplify the equation to
1
2=e−5k. Step 4: Take the natural logarithm of both sides to solve for k:
ln 1
2= lne−5k. Step 5: Use the property of logarithms lnab=bln(a) to
simplify the equation to ln 1
2=−5k. Step 6: Finally, solve for kby dividing
by −5: k=−ln(1
2)
5. Step 7: Since ln 1
2=−ln(2), we have k=ln(2)
5. Thus,
the decay constant kis ln(2)
5≈0.1386 days−1.
Question 11
Question
A sample of a radioactive isotope has an initial mass of 1 gram and a half-life
of 24 hours. After 72 hours, what is the remaining mass of the sample?
Solution
Step 1: Calculate the decay constant 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 in hours.
Step 2: Since the half-life (t1/2) is given as 24 hours, we can relate it to the
decay constant using the formula t1/2=ln(2)
λ.
Step 3: Substituting t1/2= 24 hours into the above formula, we find λ=
ln(2)
24 .
Step 4: Now, we substitute N0= 1 gram, t= 72 hours, and λ=ln(2)
24 into
the formula N(t) = N0e−λt to find the remaining mass of the sample after 72
hours.
Step 5: Therefore, the remaining mass of the sample after 72 hours is
N(72) = 1×e−ln(2)
24 ×72. Simplifying this expression will give us the final answer.
9
Question 12
Question
A certain radioactive substance has a half-life of 20 years. If you start with 100
grams of the substance, how many grams will remain after 60 years?
Solution
Step 1: Determine the decay constant.
The decay constant (λ) can be calculated using the formula:
λ=ln(2)
T1
2
where T1
2is the half-life of the substance. In this case, T1
2= 20 years.
λ=ln(2)
20 ≈0.03465 per year
Step 2: Use the decay model to find the amount remaining.
The amount of substance remaining at any time tcan be calculated using the
formula:
N(t) = N0·e−λt
where N0is the initial amount of substance.
N(60) = 100 ·e−0.03465·60
= 100 ·e−2.079
≈100 ·0.125
= 12.5 grams
So, after 60 years, there will be approximately 12.5 grams of the substance
remaining.
Question 13
Question
An unknown radioactive substance decays by 25
Solution
Step 1: Determine the decay constant. Let N(t) represent the amount of sub-
stance remaining after thours. Since the substance decays by 25
10
Step 2: Use the exponential decay formula. The formula for radioactive
decay is given by:
N(t) = N0·eλt
where N0is the initial amount of substance. Plugging in the known values:
N(t) = 1000 ·e−0.25t
Step 3: Calculate the amount remaining after 6 hours. Substitute t= 6 into
the formula:
N(6) = 1000 ·e−0.25·6
N(6) = 1000 ·e−1.5≈1000 ·0.2231
N(6) ≈223 grams
Therefore, after 6 hours, approximately 223 grams of the substance will
remain.
Question 14
Question
A sample of a radioactive substance has an initial mass of 50 grams. After 20
days, only 10 grams of the substance remain. If the half-life of the substance is
10 days, what is the decay constant λof the substance?
Solution
Step 1: Calculate the decay constant λusing the formula for radioactive decay:
λ=−ln(0.5)
t1
2
where t1
2is the half-life of the substance. Substitute t1
2= 10 days into the
formula:
λ=−ln(0.5)
10
Step 2: Calculate the remaining fraction of the substance after 20 days using
the radioactive decay formula:
N(t) = N0e−λt
where: - N(t) is the remaining mass of the substance after time t-N0is the
initial mass of the substance - tis the time passed Substitute N(t) = 10 grams,
N0= 50 grams, t= 20 days into the formula:
10 = 50e−20λ
11
Step 3: Now, solve the equation obtained in Step 2 for λ:
e−20λ=10
50
e−20λ= 0.2
−20λ= ln(0.2)
λ=ln(0.2)
−20
Step 4: Calculate the value of λ:
λ=ln(0.2)
−20
Question 15
Question
A certain radioactive substance has a half-life of 10 days. If there are initially
200 grams of the substance, how many grams will remain after 30 days?
Solution
Step 1: Determine the decay constant λ
The decay constant λcan be calculated using the formula:
λ=ln(2)
T1
2
where T1
2is the half-life of the substance. Substituting T1
2= 10 days:
λ=ln(2)
10
Step 2: Determine the amount of substance remaining after 30 days
The amount of substance remaining after time tcan be calculated using the
formula:
N(t) = N0e−λt
where N0is the initial amount of substance. Substituting N0= 200 grams,
λ=ln(2)
10 , and t= 30 days:
N(30) = 200e−30 ln(2)
10
Step 3: Calculate the amount of substance remaining after 30 days
N(30) = 200e−3 ln(2)
12
N(30) = 200eln(2−3)
N(30) = 200(2−3)
N(30) = 200 ×1
8
N(30) = 25
Therefore, after 30 days, only 25 grams of the substance will remain.
Question 16
Question
A sample of a radioactive isotope decays according to the equation 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 of the isotope is
100 grams, determine how long it will take for the amount of the isotope to
reduce to 50 grams.
Solution
Step 1: Substitute the given values into the equation.
Given N0= 100 grams and N(t) = 50 grams, we have:
50 = 100e−0.02t
Step 2: Solve for t.
Dividing both sides by 100: 50
100 =e−0.02t
0.5 = e−0.02t
Step 3: Take the natural logarithm of both sides.
ln(0.5) = lne−0.02t
ln(0.5) = −0.02t
Step 4: Solve for t.
t=−ln(0.5)
0.02
t≈34.66 years
Therefore, it will take approximately 34.66 years for the amount of the iso-
tope to reduce to 50 grams.
13
Question 17
Question
A certain radioactive substance has a half-life of 10 days. If the initial amount
of the substance is 100 grams, determine the amount of the substance remaining
after 30 days.
Solution
Let’s denote the amount of the radioactive substance remaining after tdays as
A(t).
Step 1: Determine the decay constant λusing the formula N(t) = N0·e−λt,
where N(t) is the amount of substance remaining at time t,N0is the initial
amount, and λis the decay constant.
Given that the half-life T1/2= 10 days, we can find λusing the formula
λ=ln(2)
T1/2.
λ=ln(2)
10 ≈0.0693 (rounded to four decimal places)
Step 2: Write the exponential decay model for the amount of substance
remaining.
A(t) = 100 ·e−0.0693t
Step 3: Calculate the amount of substance remaining after 30 days by
substituting t= 30 into the model.
A(30) = 100·e−0.0693·30 = 100·e−2.079 ≈7.454 grams (rounded to three decimal places)
After 30 days, there will be approximately 7.454 grams of the radioactive
substance remaining.
Question 18
Question
A sample of a radioactive material initially contains 200 grams of a certain
radionuclide with a half-life of 20 days. After 60 days, only 25 grams of the
radionuclide remain. What is the decay constant for this radionuclide?
Solution
Step 1: Find the decay constant using the formula for radioactive decay:
N(t) = N0·e−λt
14
where: - N(t) is the amount of radionuclide remaining at time t, - N0is the
initial amount of radionuclide, and - λis the decay constant.
Step 2: Substitute the given values into the formula: At t= 0, N(0) = 200
grams and at t= 60, N(60) = 25 grams. So, we have:
25 = 200 ·e−λ·60
Step 3: Solve for the decay constant λ:
25
200 =e−60λ
0.125 = e−60λ
Taking the natural log of both sides:
ln(0.125) = lne−60λ
ln(0.125) = −60λ
λ=ln(0.125)
−60
λ≈0.0183 days−1
Therefore, the decay constant for this radionuclide is approximately 0.0183
days−1.
Question 19
Question
A sample of a radioactive substance decays according to the equation N(t) =
N0e−0.05t, where N(t) represents the amount of substance present after tyears
and N0is the initial amount of substance. If the initial amount of the substance
is 100 grams, determine the amount of substance present after 20 years. Round
your answer to the nearest hundredth.
Solution
Step 1: Plug in the initial values into the equation.
N(20) = 100e−0.05×20
= 100e−1
= 100 ×1
e
≈100 ×0.3679
≈36.79 grams
Therefore, the amount of substance present after 20 years is approximately
36.79 grams.
15
Question 20
Question
A sample of radium-226 has an initial mass of 1 gram. If the half-life of radium-
226 is 1600 years, determine the mass of the sample after 4800 years.
Solution
Step 1: Calculate the number of half-lives that have passed. Given that the
half-life of radium-226 is 1600 years, we can find the number of half-lives that
have passed in 4800 years:
Number of half-lives = 4800 years
1600 years/half-life = 3 half-lives
Step 2: Calculate the remaining mass of the sample after 3 half-lives. Since
the mass decreases by half with each half-life, the remaining mass after 3 half-
lives is:
Remaining mass = 1 gram ×1
23
= 1 gram ×1
8=1
8grams
Step 3: Write the final answer. After 4800 years, the mass of the sample of
radium-226 would be 1
8grams.
Question 21
Question
A sample of a radioactive material has an initial mass of 10 grams. After 30
minutes, the mass of the sample has decreased to 6 grams. If the half-life of the
material is 45 minutes, determine the decay constant kand write the exponential
decay equation for the sample.
Solution
Step 1: Calculate the fraction of the sample left after 30 minutes. Step 2:
Calculate the decay constant kusing the half-life formula. Step 3: Write the
exponential decay equation for the sample.
Step 1: The fraction of the sample left after 30 minutes can be calculated
using the formula:
fraction left = final mass
initial mass
fraction left = 6
10
fraction left = 0.6
16
Step 2: The decay constant kcan be calculated using the half-life formula:
fraction remaining after time t=e−kt
Substitute the values:
0.6 = e−k×30
ln(0.6) = −k×30
k=ln(0.6)
−30
k≈0.0185
Step 3: The exponential decay equation for the sample is:
mass at time t= 10 ×e−0.0185t
Therefore, the decay constant kis approximately 0.0185 and the exponential
decay equation for the sample is 10 ×e−0.0185t.
Question 22
Question
A certain radioactive isotope decays at a rate proportional to the amount
present. If the initial amount of the isotope is 100 grams and its half-life is
10 days, determine the amount of the isotope remaining after 30 days.
Solution
Step 1: Let Q(t) represent the amount of the isotope remaining after tdays. We
are given that the isotope decays at a rate proportional to the amount present,
so we have the differential equation
dQ
dt =−kQ
where kis the decay constant.
Step 2: We know that half-life is the time it takes for half of the substance
to decay. In this case, the half-life is 10 days, so we have
1
2Q(10) = Q(0)
We can rewrite this as Q(10)
Q(0) = 1/2
e−10k= 1/2
−10k= ln(1/2)
17
k=ln(2)
10
Thus, the decay constant is k=ln(2)
10 .
Step 3: Now, let’s solve the differential equation. We have
dQ
dt =−ln(2)
10 Q
Separating variables gives us
dQ
Q=−ln(2)
10 dt
Integrating both sides yields
ln(Q) = −ln(2)
10 t+C
where Cis the constant of integration.
Step 4: We are given that Q(0) = 100, so substituting this into our equation
gives
ln(100) = C
C= ln(100)
C= ln102
C= 2 ln(10)
C= 2 ln(2) + 2 ln(5)
C= ln22+ ln52
C= ln(4) + ln(25)
C= ln(100)
So, the equation becomes
ln(Q) = −ln(2)
10 t+ ln(100)
ln(Q) = ln(100) −ln(2)
10 t
ln(Q) = ln 100
2t/10
Q=100
2t/10
Step 5: Finally, we can determine the amount of the isotope remaining after
30 days by substituting t= 30 into the equation:
Q(30) = 100
230/10
18
Q(30) = 100
23
Q(30) = 100
8
Q(30) = 12.5 grams
Therefore, after 30 days, there will be 12.5 grams of the isotope remaining.
Question 23
Question
A sample of a radioactive material has an initial mass of 200 g and a half-life of
10 days. After 30 days, what is the mass of the sample remaining?
Solution
Step 1: Calculate the decay constant λ. The decay constant is related to the
half-life T1/2by the formula λ=ln(2)
T1/2.
Step 2: Substitute the given half-life T1/2= 10 days into the formula to find
λ:
λ=ln(2)
10 ≈0.0693 days−1
Step 3: Use the exponential decay formula N(t) = N0e−λt to find the mass
of the sample remaining after 30 days. Here N(t) represents the mass at time
t,N0represents the initial mass, and tis the time elapsed.
N(t) = 200 ·e−0.0693·30
Step 4: Calculate the mass of the sample remaining after 30 days:
N(30) = 200 ·e−2.079 ≈42.74 g
Therefore, after 30 days, the mass of the sample remaining is approximately
42.74 g.
Question 24
Question
A sample of a radioactive isotope has an initial mass of 200 grams. After 10
days, only 25 grams of the isotope remain. If the half-life of the isotope is 5
days, what is the decay constant for this isotope?
19
Solution
Step 1: Calculate the fraction of the original sample that remains after 10 days.
Step 2: Use this information to find the decay constant.
Step 1: The fraction of the original sample that remains after 10 days can
be calculated using the formula for radioactive decay:
m
m0
=1
2t
T
where: mis the final mass, m0is the initial mass, tis the time elapsed, Tis the
half-life of the isotope.
Plugging in the known values, we get:
25
200 =1
210
5
Solving for the fraction of the original sample that remains:
25
200 =1
22
1
8=1
4
Step 2: The decay constant, denoted by λ, is related to the half-life Tby
the formula:
λ=ln(2)
T
Given that the half-life Tis 5 days, we can calculate the decay constant:
λ=ln(2)
5
λ=0.6931
5
λ≈0.1386 days−1
Therefore, the decay constant for this isotope is approximately 0.1386 days−1.
Question 25
Question
A sample of a radioactive material has an initial mass of 100 grams. After 10
minutes, only 75 grams of the material remain. If the half-life of the material is
5 minutes, determine the decay constant and the amount of material that will
remain after 30 minutes.
20
Solution
Step 1: Calculate the decay constant λusing the half-life formula N(t) = N0·
2−t
T1
2where N(t) is the remaining mass at time t,N0is the initial mass, tis
the time elapsed, and T1
2is the half-life of the material.
Given N(10) = 75, N0= 100, and T1
2= 5, we can plug these values into the
formula to solve for λ:
75 = 100 ·2−10
5
0.75 = 2−2
22=1
0.75
λ= ln 1
0.75≈0.2877 min−1
So, the decay constant is approximately 0.2877 min−1.
Step 2: Determine the remaining mass after 30 minutes using the exponential
decay formula N(t) = N0·e−λt where N(t) is the remaining mass at time t,N0
is the initial mass, λis the decay constant, and tis the time elapsed.
Given N0= 100, λ= 0.2877, and t= 30, we can plug these values into the
formula:
N(30) = 100 ·e−0.2877·30 ≈34.394 grams
Therefore, approximately 34.394 grams of the material will remain after 30
minutes.
Question 26
Question
A certain radioactive substance has a half-life of 30 minutes. If you start with
500 grams of the substance, how much will remain after 3 hours?
Solution
Step 1: Calculate the number of half-lives in 3 hours. Let nbe the number of
half-lives. Since each half-life is 30 minutes, there are 6 half-lives in 3 hours.
n=3 hours
0.5 hours per half-life = 6
21
Step 2: Determine the amount of material remaining after 6 half-lives. The
amount of material remaining after nhalf-lives can be calculated using the
formula:
Amount remaining = Initial amount ×1
2n
Plugging in the values, we have:
Amount remaining = 500 ×1
26
Amount remaining = 500 ×1
64
Amount remaining = 500
64
Amount remaining = 7.8125 grams
Therefore, after 3 hours, there will be approximately 7.81 grams of the ra-
dioactive substance remaining.
Question 27
Question
A certain radioactive isotope has a half-life of 5 days. If we start with an initial
amount of 100 grams, how much of the substance will remain after 20 days?
Solution
Step 1: Calculate the decay constant. The decay constant, denoted by λ, is
related to the half-life T1/2by the equation λ=ln(2)
T1/2.
λ=ln(2)
5=0.6931
5= 0.1386 per day
Step 2: Use the formula for radioactive decay to find the remaining amount
of substance after 20 days. The formula is given by N(t) = N0e−λt, where: -
N(t) is the amount of substance remaining after time t-N0is the initial amount
of substance - λis the decay constant - tis the time passed
Plugging in the values:
N(20) = 100 ·e−0.1386·20
N(20) ≈100 ·e−2.772 ≈100 ·0.0633 ≈6.33 grams
Therefore, approximately 6.33 grams of the substance will remain after 20
days.
22
Question 28
Question
A sample of a radioactive substance has an initial mass of 25 grams. It decays
according to the function m(t) = 25e−0.03t, where m(t) is the mass remaining
after tyears. Find the rate of decay of the substance after 10 years.
Solution
Step 1: To find the rate of decay after 10 years, we need to find the derivative
of the mass function m(t) with respect to time, t.
Step 2: We have m(t) = 25e−0.03t. To find dm
dt , we differentiate m(t) with
respect to tusing the chain rule.
dm
dt =d
dt (25e−0.03t)
Step 3: Applying the chain rule: dm
dt = 25(−0.03)e−0.03t
dm
dt =−0.75e−0.03t
Step 4: We want to find the rate of decay after 10 years, so substitute t= 10
into the derivative we just found.
dm
dt
t=10
=−0.75e−0.03(10)
dm
dt
t=10
=−0.75e−0.3
dm
dt
t=10 ≈ −0.679 grams per year
Therefore, the rate of decay of the substance after 10 years is approximately
0.679 grams per year.
Question 29
Question
A certain radioactive substance decays with a half-life of 10 hours. If you start
with 500 grams of the substance, how much will be left after 30 hours?
Solution
Step 1: Use the radioactive decay formula N(t) = N01
2t
T1/2, where: - N(t) is
the amount of substance remaining after time t, - N0is the initial amount of
substance, - T1/2is the half-life of the substance.
Step 2: Substitute the given values into the formula: N(30) = 500 1
230
10
Step 3: Simplify the expression: N(30) = 500 1
23N(30) = 500×1
8N(30) =
62.5 grams
Answer: After 30 hours, there will be 62.5 grams of the substance left.
23
Question 30
Question
A sample of a radioactive isotope has an initial mass of 10 grams. After 5 hours,
the mass of the sample has decreased to 7 grams. If the half-life of the isotope
is 3 hours, determine the decay constant (λ) and the age of the sample.
Solution
Step 1: Find the decay constant (λ) using the formula N(t) = N0e−λt, where
N(t) is the current mass, N0is the initial mass, λis the decay constant, and t
is the time.
Given that N(5) = 7 and N0= 10, we can rewrite the formula as 7 = 10e−5λ.
Solving for λ:7
10 =e−5λ
ln 7
10=−5λ
λ=−1
5ln 7
10
Step 2: Calculate the decay constant λ.
λ=−1
5ln 7
10≈0.067
Step 3: Determine the age of the sample using t=ln(2)
λ.
Given λ≈0.067, we can find:
t=ln(2)
0.067 ≈0.693
0.067 ≈10.34 hours
Therefore, the decay constant is approximately 0.067 and the age of the
sample is approximately 10.34 hours.
Question 31
Question
A sample of a radioactive substance has an initial mass of 100 grams. After 10
hours, the mass of the substance has decreased to 60 grams. If the half-life of
the substance is 5 hours, what is the decay constant for this substance?
24
Solution
Step 1: Determine the fraction of the substance remaining after 10 hours. Step
2: Use the formula for radioactive decay to find the decay constant.
Step 1: The decay constant (λ) can be calculated using the formula:
m
m0
=e−λt
where: - mis the final mass (60 grams), - m0is the initial mass (100 grams), -
tis the time elapsed (10 hours).
Substitute the given values into the formula:
60
100 =e−λ·10
0.6 = e−10λ
Step 2: To solve for the decay constant, we need to take the natural logarithm
of both sides, which gives:
ln(0.6) = lne−10λ
ln(0.6) = −10λ
Finally, divide by −10 to find the value of the decay constant:
λ=−ln(0.6)
10
Question 32
Question
A certain radioactive isotope decays according to the equation N(t) = N0e−kt,
where N(t) is the quantity of the isotope remaining at time t,N0is the initial
quantity of the isotope, and kis a decay constant. If 80% of a sample of this
isotope decays in 6 hours, what is the half-life of the isotope?
Solution
Step 1: Given information, we have N(t) = N0e−kt, and after 6 hours, 80%
of the sample remains. This means that N(6) = 0.2N0. Step 2: Substituting
t= 6 hours into the equation, we get 0.2N0=N0e−6k. Step 3: Simplifying the
equation, we have e−6k= 0.2. Step 4: Taking natural logarithms of both sides,
we get −6k= ln(0.2). Step 5: Solving for k, we have k=−ln(0.2)
6. Step 6:
The half-life of a radioactive isotope is the time it takes for half of the sample
to decay. So, we need to find the time t1
2such that N(t1
2) = 0.5N0. Step
7: Substituting N(t1
2) = 0.5N0into the equation, we get 0.5N0=N0e−kt 1
2.
25
Step 8: Simplifying the equation, we have e−kt 1
2= 0.5. Step 9: Taking natural
logarithms of both sides, we get −kt 1
2= ln(0.5). Step 10: Solving for t1
2, we
have t1
2=−ln(0.5)
k. Step 11: Finally, substituting the value of kfrom Step 5
into the equation in Step 10, we get the half-life of the isotope.
Question 33
Question
The half-life of a radioactive isotope is 35 hours. If the initial amount of the
isotope is 500 grams, determine the amount of the isotope remaining after 105
hours.
Solution
Step 1: Calculate the decay constant using the half-life formula Step 2: Use the
decay constant to calculate the remaining amount of the radioactive isotope at
105 hours.
Step 1: Given that the half-life of the isotope is 35 hours, we can use the
formula for half-life decay to find the decay constant (k):
Half-life = ln(2)
k
35 = ln(2)
k
k=ln(2)
35
Step 2: Now that we have the decay constant, we can use the exponential
decay formula to find the remaining amount of the isotope after 105 hours:
N(t) = N0·e−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 elapsed.
Substitute the given values:
N(105) = 500 ·e−ln(2)
35 ·105
N(105) = 500 ·e−3 ln(2)
N(105) = 500 ·eln(2−3)
N(105) = 500 ·2−3
N(105) = 500 ·1
8
N(105) = 62.5 grams
Therefore, the amount of the isotope remaining after 105 hours is 62.5 grams.
26
Question 34
Question
A radioactive substance decays according to the law A(t) = A0e−kt, where A(t)
is the amount of the substance remaining at time t(in years), A0is the initial
amount of the substance, and kis a positive constant. If 50% of the substance
decays in 300 years, find the value of kfor this substance.
Solution
Step 1: Given that A(t) = A0e−kt, we know that when t= 0, A(t) = A0. Thus,
A(0) = A0e−k(0) =A0, which implies e0= 1. Therefore, A0=A0, as expected.
Step 2: We are told that 50% of the substance decays in 300 years. This
means that A(300) = 0.5A0. Substituting these values into the decay equation,
we get:
0.5A0=A0e−k(300)
Step 3: By dividing both sides of the equation by A0, we can simplify to:
0.5 = e−300k
Step 4: Taking the natural logarithm of both sides of the equation to solve
for k, we have:
ln(0.5) = lne−300k=−300k
Step 5: Solving for k, we find:
k=ln(0.5)
−300 ≈0.00231
Therefore, the value of kfor this substance is approximately 0.00231.
Question 35
Question
An unknown radioactive substance has a half-life of 10 hours. If there are
initially 500 grams of the substance present, how much of it will remain after
30 hours?
Solution
Step 1: Determine the decay constant using the formula k=0.693
T1
2
, where T1
2is
the half-life.
Step 1: k=0.693
10 = 0.0693 hours−1
27
Step 2: Use the exponential decay formula N(t) = N0e−kt to find the amount
of substance remaining after 30 hours.
Step 2: N(30) = 500e−0.0693(30)
Step 2: N(30) = 500e−2.079
Step 2: N(30) ≈500(0.125) ≈62.5 grams
After 30 hours, approximately 62.5 grams of the radioactive substance will
remain.
28
