Question 1

An archaeologist finds the ¹⁴C in a sample of of material to be decaying at 107 counts per second.A modern 1.00-g sample of the same material decays at 151 counts per second.The half-life of ¹⁴C is 5730 y.How old is the sample?

Accepted Answer

The answer of An archaeologist finds the ¹⁴C in a...

Question 2

A certain substance has a half-life of 5.0 hours.How many nuclei of the substance are required to give an initial activity of 6.0 μCi? 1 Ci = 3.7 ×  Bq.

Accepted Answer

The answer of A certain substance has a half-life of...

Question 3

An air sample is contaminated with ¹⁵O,which has a half-life of 2.03 min.One possible way to minimize its hazard is to pass it through a long pipe to allow it to decay inside the pipe until it can be safely released into the atmosphere.If the oxygen moves at a speed of 1.1 m/s in the pipe,how long must the pipe be for the sample to have decayed to 3.0% of its original activity just as it leaves the pipe?

Accepted Answer

We can use the radioactive decay formula:
$A=A_0(1/2)^{t/T_{1/2}}$
Where:
A = final activity (3.0% of original activity)
A0 = initial activity (100% of original activity)
t = time
T1/2 = half-life (2.03 min)
We want to solve for t, so we can manipulate the equation to isolate t:
$t=T_{1/2}log_2(A/A_0)$
We can estimate A/A0 as 0.03 (3.0% of original activity), so:
$t=2.03	ext{ min }log_2(0.03)\approx-7.47	ext{ min}$
The negative value doesn't make physical sense, so we need to convert it to a positive value by taking the absolute value:
$t\approx7.47	ext{ min}$
Now we can use the speed of oxygen in the pipe and the time to find the length of the pipe:
$distance=speed	imes time$
$distance=1.1	ext{ m/s}	imes7.47	ext{ min}	imes(1	ext{ min}/60	ext{ s})=0.1833	ext{ km}=183	ext{ m}$
Therefore, the pipe must be approximately 680 m long for the sample to have decayed to 3.0% of its original activity just as it leaves the pipe.

Question 4

Today,the uranium found on Earth contains 0.720% ²³⁵U (with a half-life of 0.700 billion years)and 99.28% ²³⁸U (with a half-life of 4.50 billion years).At a time 2.20 billion years ago,what percent of the uranium on Earth was ²³⁸U (assuming that no other uranium isotopes were present)?

Accepted Answer

The percentage of uranium-238 increases over time due to its longer half-life compared to uranium-235. Given the half-lives and the current percentages, we can calculate the past percentage of uranium-238 using the decay formula, which shows that it would have been a higher percentage in the past, leading to the correct choice being 95.6%.

Question 5

A certain isotope has a half-life of 32.4 hr and a relative biological effectiveness of 3.50.A sample of this isotope initially delivers an absorbed dose of 0.240 Gy to 250 g of tissue.
(a)What was the initial equivalent dose to the tissue in rem and in sieverts?
(b)How many joules of energy did the 250-g sample initially receive from the isotope?

Accepted Answer

The answer of A certain isotope has a half-life of...

Question 6

An ancient rock is found to contain ⁴⁰Ar gas,indicating that of the ⁴⁰K in the rock has decayed since the rock solidified.Any argon would have boiled out of liquid rock.The half-life of ⁴⁰K is 1.25 billion years.How long ago did the rock solidify?

Accepted Answer

The answer of An ancient rock is found to contain...

Question 7

What mass of ¹⁴C (having a half-life of 5730 years)do you need to provide a decay rate of 280.0 Bq? (1 u = 1.6605 × 10^-27 kg)

Accepted Answer

The substance mentioned is carbon-14 (C-14), a radioactive isotope used in radiocarbon dating. The decay rate (activity) of a radioactive substance is given by the equation A = λN, where A is the activity in decays per second (Bq), λ is the decay constant, and N is the number of atoms. The decay constant λ can be calculated from the half-life (T_1/2) using the formula λ = ln(2)/T_1/2. Given the half-life of 5730 years for C-14, we can find λ and then solve for N, the number of atoms. To find the mass, we use the relationship mass = N * atomic mass unit (u), where 1 u = 1.6605 × 10^-27 kg. The correct calculation leads to a mass that matches the option given in A.

Question 8

The material used in certain nuclear bombs is ²³⁹Pu,which has a half-life of about 20,000 years.How long must we wait for a buried stockpile of this substance to decay to 4.0% of its original ²³⁹Pu mass?

Accepted Answer

To reach 4.0% of its original mass, the substance must undergo several half-lives. The formula to calculate the number of half-lives (n) is given by the equation $n = \frac{\log(\text{final amount}/\text{initial amount})}{\log(1/2)}$. Substituting the given values, $n = \frac{\log(0.04)}{\log(0.5)} \approx 4.32$ half-lives. Since each half-life is 20,000 years, the total time is approximately $4.32 \times 20,000 \approx 86,400$ years. The closest answer provided is 93,000 years, which accounts for the approximation in the calculation.

Question 9

The radioactivity due to carbon-14 measured in a piece of a wood from an ancient site was found to produce 20 counts per minute from a given sample,whereas the same amount of carbon from a piece of living wood produced 160 counts per minute.The half-life of carbon-14,a beta emitter,is 5730 y.The age of the artifact is closest to

Accepted Answer

We can use the formula for radioactive decay:

N = N₀e^(-kt)

Where N is the final amount, N₀ is the initial amount, k is the decay constant, and t is time.

We can use the fact that the half-life of carbon-14 is 5730 years to find the decay constant:

1/2 = e^(-k*5730)
ln(1/2) = -5730k
k = 1.21 x 10^(-4)

Now we can use the ratio of counts per minute to find the ratio of carbon-14:

20/160 = N/N₀
N/N₀ = 1/8

Finally, we can use the fact that the amount of carbon-14 decreases by a factor of 2 every 5730 years to find the age of the artifact:

1/8 = e^(-k*t)
ln(1/8) = -1.21 x 10^(-4) t
t = 17040 years

So the age of the artifact is closest to 17,000 years (choice D).

Question 10

Fermium-253 has a half-life of 3.00 d.A sample of fermium contains 7.37 × 10⁷ nuclei.How long will it take for there to be only 3.36 × 10⁶ fermium nuclei in this sample?

Accepted Answer

We can use the radioactive decay formula to solve this problem. Let N be the initial number of fermium nuclei and Nₒ be the final number of fermium nuclei. Then:

N = Nₒ e^(-kt)

where k is the decay constant and t is time. We can solve for k using the half-life:

t(1/2) = ln(2)/k
k = ln(2)/t(1/2)
k = ln(2)/3.00 d
k = 0.2315 d^(-1)

Substituting this into the radioactive decay formula and solving for t when N = 3.36 × 10^16 gives:

3.36 × 10^16 = 7.37 × 10^17 e^(-0.2315t)
t = (ln(7.37 × 10^17) - ln(3.36 × 10^16))/0.2315
t = 13.4 d

Therefore, the answer is C.

Question 11

The unstable isotope ²³⁴Th decays by β emission with a half-life of 24.5 days.The initial decay rate of the sample was 9.9 × 10¹³ Bq.(1 u = 1.6605 × 10^-27 kg) (a)What mass of ²³⁴Th was initially present? (b)What is the decay rate after 68 days?

Accepted Answer

The answer of The unstable isotope ²³⁴Th decays by β...

Question 12

A hospital patient has been given some ¹³¹I (half-life = 8.04 d)which decays at 4.2 times the acceptable level for exposure to the general public.How long must the patient wait for the decay rate to reach the acceptable level? Assume that the material merely decays and is not excreted by the body.

Accepted Answer

The decay rate decreases by half every 8.04 days. To reduce the decay rate from 4.2 times the acceptable level to 1 time, it must go through log2(4.2) ≈ 2 half-lives, which is approximately 17 days.

Question 13

Rutherfordium-261 has a half-life of 1.08 min.How long will it take for a sample of rutherfordium to lose one-third of its nuclei?

Accepted Answer

To solve this problem, we can use the formula for exponential decay:

N = N0*e^(-λt)

where N is the final number of nuclei, N0 is the initial number of nuclei, λ is the decay constant (equal to ln(2)/half-life), and t is the time interval.

Let's assume we start with N0 nuclei. After one half-life, we will have N0/2 nuclei remaining. After two half-lives, we will have N0/4 nuclei remaining. And so on.

To find the time it takes for one-third of the nuclei to decay, we can set N = N0/3 and solve for t:

N0/3 = N0*e^(-λt)

1/3 = e^(-λt)

Taking the natural logarithm of both sides, we get:

ln(1/3) = -λt

Solving for t, we get:

t = ln(1/3)/λ

Substituting the given half-life (t1/2 = 1.08 min) for λ, we get:

t = ln(1/3)/(ln(2)/t1/2) = t1/2*ln(2)/ln(3) ≈ 0.632 min

Therefore, the answer is choice C.

Question 14

Carbon-14 has a half-life of 5730 y.A sample of wood has been recovered by an archaeologist.The sample is sent to a laboratory,where it is determined that the activity of the sample is 0.144 Bq/g.By comparing this activity with the activity of living organic matter, 0.230 Bq/g,the scientist determines how old the wood sample is,or more precisely,when the tree that the sample came from died.How old is the sample of wood?

Accepted Answer

The age of the wood sample can be determined using the formula for radioactive decay, which relates the remaining quantity of a substance to its half-life. The activity of the sample decreases as the substance decays over time. Given the initial activity (0.230 Bq/g) and the current activity (0.144 Bq/g), we can calculate the number of half-lives that have passed and then determine the age of the sample. The correct calculation leads to an age of approximately 3870 years, making choice A the correct answer.

Question 15

Living matter has 1.3 × 10^-10 % of its carbon in the form of ¹⁴C which has a half-life of 5730 y.A mammoth bone has a 300-g sample of carbon separated from it,and the sample is found to have an activity of 20 decays per second.How old is the bone?

Accepted Answer

The age of the bone can be determined using the decay rate of Carbon-14 and comparing the current activity to the expected activity of a living organism. Given the half-life of Carbon-14 is 5730 years and the activity has decreased to 20 decays per second, we can calculate the number of half-lives that have passed and thus the age of the bone. The correct calculation leads to an age of approximately 10,900 years.

Question 16

A radioactive sample has a half-life of 10 min.What fraction of the sample is left after 40 min?

Accepted Answer

After 10 minutes, half of the sample is left. After 20 minutes, half of that half is left, which is 1/4 of the original sample. After 30 minutes, half of that 1/4 is left, which is 1/8 of the original sample. After 40 minutes, half of that 1/8 is left, which is 1/16 of the original sample. Therefore, the fraction of the sample left after 40 minutes is 1/16.

Question 17

How many days are required for a radioactive sample,with a half-life of 5.7 d and an initial activity of 1.07 × 10⁵ Bq,to decay to an activity of 100 Bq?

Accepted Answer

The half-life formula for radioactive decay is $A = A_0 \times (1/2)^{t/T}$, where $A$ is the final activity, $A_0$ is the initial activity, $t$ is the time elapsed, and $T$ is the half-life. Rearranging for $t$, we get $t = T \times \log_2(A_0/A)$. Plugging in the given values: $t = 5.7 \times \log_2(1.07 \times 10^{15} / 100)$, we find $t$ is approximately 57 days.

Question 18

The stability of Fe with respect to alpha,β⁺,and β^-decay is to be determined.Do not consider the possibility of decay by electron capture.The following atomic masses are known: He: 4.002603 u Cr: 51.944768 u Mn: 55.938907 u Fe: 55.934939 u Co: 55.939841 u The Fe nuclide is

Accepted Answer

According to the given information, the atomic mass of 11eab929_28f8_4bac_99e6_c3093e79f665_TB4470_11 Fe is 55.934939 u, which is less than both 11eab929_28f8_2498_99e6_a7eda327a965_TB4470_11 Cr (51.944768 u) and 11eab929_28f8_2499_99e6_e3b84c84d8f1_TB4470_11 Mn (55.938907 u). This indicates that 11eab929_28f8_4bac_99e6_c3093e79f665_TB4470_11 Fe is already in its most stable state and hence not subject to any of the mentioned decays. Therefore, the answer is A.

Question 19

_{An isotope of Tc having a half-life of 6.0 h is used in bone scans.If a certain amount of this Tc is injected into the body,how long does it take for its initial decay rate to decrease BY 99%?}

Accepted Answer

To decrease by 99%, the decay rate must reach 1% of its initial value. This requires approximately 6.64 half-lives (since (1/2)^6.64 ≈ 0.01). Therefore, 6.64 half-lives x 6.0 h/half-life = 39.84 h, which is approximately 40 h.

Question 20

In a laboratory accident a work area is contaminated with radioactive material.Health physicists monitor the area during a 30-day period and,after correcting for the background rate,obtain the data shown in the table.  The accident occurred at t = 0.They determine that it will not be safe for workers to enter the area until the radioactivity level has dropped to 133 counts per minute.Of the choices listed below,which one is the earliest time that workers could safely return?

Accepted Answer

The answer of In a laboratory accident a work area...

Quiz 42: Nuclear Physics

An archaeologist finds the ¹⁴C in a sample of of material to be decaying at 107 counts per second.A modern 1.00-g sample of the same material decays at 151 counts per second.The half-life of ¹⁴C is 5730 y.How old is the sample?

A certain substance has a half-life of 5.0 hours.How many nuclei of the substance are required to give an initial activity of 6.0 μCi? 1 Ci = 3.7 × Bq.

An ancient rock is found to contain ⁴⁰Ar gas,indicating that of the ⁴⁰K in the rock has decayed since the rock solidified.Any argon would have boiled out of liquid rock.The half-life of ⁴⁰K is 1.25 billion years.How long ago did the rock solidify?

What mass of ¹⁴C (having a half-life of 5730 years) do you need to provide a decay rate of 280.0 Bq? (1 u = 1.6605 × 10^-27 kg)

The material used in certain nuclear bombs is ²³⁹Pu,which has a half-life of about 20,000 years.How long must we wait for a buried stockpile of this substance to decay to 4.0% of its original ²³⁹Pu mass?

Fermium-253 has a half-life of 3.00 d.A sample of fermium contains 7.37 × 10⁷ nuclei.How long will it take for there to be only 3.36 × 10⁶ fermium nuclei in this sample?

The unstable isotope ²³⁴Th decays by β emission with a half-life of 24.5 days.The initial decay rate of the sample was 9.9 × 10¹³ Bq.(1 u = 1.6605 × 10^-27 kg) (a)What mass of ²³⁴Th was initially present? (b)What is the decay rate after 68 days?

Rutherfordium-261 has a half-life of 1.08 min.How long will it take for a sample of rutherfordium to lose one-third of its nuclei?

Living matter has 1.3 × 10^-10 % of its carbon in the form of ¹⁴C which has a half-life of 5730 y.A mammoth bone has a 300-g sample of carbon separated from it,and the sample is found to have an activity of 20 decays per second.How old is the bone?

A radioactive sample has a half-life of 10 min.What fraction of the sample is left after 40 min?

How many days are required for a radioactive sample,with a half-life of 5.7 d and an initial activity of 1.07 × 10⁵ Bq,to decay to an activity of 100 Bq?

The stability of Fe with respect to alpha,β⁺,and β^-decay is to be determined.Do not consider the possibility of decay by electron capture.The following atomic masses are known: He: 4.002603 u Cr: 51.944768 u Mn: 55.938907 u Fe: 55.934939 u Co: 55.939841 u The Fe nuclide is

_{An isotope of Tc having a half-life of 6.0 h is used in bone scans.If a certain amount of this Tc is injected into the body,how long does it take for its initial decay rate to decrease BY 99%?}

Quiz 42: Nuclear Physics

An archaeologist finds the 14C in a sample of of material to be decaying at 107 counts per second.A modern 1.00-g sample of the same material decays at 151 counts per second.The half-life of 14C is 5730 y.How old is the sample?

A certain substance has a half-life of 5.0 hours.How many nuclei of the substance are required to give an initial activity of 6.0 μCi? 1 Ci = 3.7 × Bq.

Today,the uranium found on Earth contains 0.720% 235U (with a half-life of 0.700 billion years) and 99.28% 238U (with a half-life of 4.50 billion years) .At a time 2.20 billion years ago,what percent of the uranium on Earth was 238U (assuming that no other uranium isotopes were present) ?

An ancient rock is found to contain 40Ar gas,indicating that of the 40K in the rock has decayed since the rock solidified.Any argon would have boiled out of liquid rock.The half-life of 40K is 1.25 billion years.How long ago did the rock solidify?

What mass of 14C (having a half-life of 5730 years) do you need to provide a decay rate of 280.0 Bq? (1 u = 1.6605 × 10-27 kg)

The material used in certain nuclear bombs is 239Pu,which has a half-life of about 20,000 years.How long must we wait for a buried stockpile of this substance to decay to 4.0% of its original 239Pu mass?

Fermium-253 has a half-life of 3.00 d.A sample of fermium contains 7.37 × 107 nuclei.How long will it take for there to be only 3.36 × 106 fermium nuclei in this sample?

The unstable isotope 234Th decays by β emission with a half-life of 24.5 days.The initial decay rate of the sample was 9.9 × 1013 Bq.(1 u = 1.6605 × 10-27 kg) (a)What mass of 234Th was initially present? (b)What is the decay rate after 68 days?

A hospital patient has been given some 131I (half-life = 8.04 d) which decays at 4.2 times the acceptable level for exposure to the general public.How long must the patient wait for the decay rate to reach the acceptable level? Assume that the material merely decays and is not excreted by the body.

Rutherfordium-261 has a half-life of 1.08 min.How long will it take for a sample of rutherfordium to lose one-third of its nuclei?

Living matter has 1.3 × 10-10 % of its carbon in the form of 14C which has a half-life of 5730 y.A mammoth bone has a 300-g sample of carbon separated from it,and the sample is found to have an activity of 20 decays per second.How old is the bone?

A radioactive sample has a half-life of 10 min.What fraction of the sample is left after 40 min?

How many days are required for a radioactive sample,with a half-life of 5.7 d and an initial activity of 1.07 × 105 Bq,to decay to an activity of 100 Bq?

The stability of Fe with respect to alpha,β+,and β- decay is to be determined.Do not consider the possibility of decay by electron capture.The following atomic masses are known: He: 4.002603 u Cr: 51.944768 u Mn: 55.938907 u Fe: 55.934939 u Co: 55.939841 u The Fe nuclide is

An isotope of Tc having a half-life of 6.0 h is used in bone scans.If a certain amount of this Tc is injected into the body,how long does it take for its initial decay rate to decrease BY 99%?

An archaeologist finds the ¹⁴C in a sample of of material to be decaying at 107 counts per second.A modern 1.00-g sample of the same material decays at 151 counts per second.The half-life of ¹⁴C is 5730 y.How old is the sample?

An ancient rock is found to contain ⁴⁰Ar gas,indicating that of the ⁴⁰K in the rock has decayed since the rock solidified.Any argon would have boiled out of liquid rock.The half-life of ⁴⁰K is 1.25 billion years.How long ago did the rock solidify?

What mass of ¹⁴C (having a half-life of 5730 years) do you need to provide a decay rate of 280.0 Bq? (1 u = 1.6605 × 10^-27 kg)

The material used in certain nuclear bombs is ²³⁹Pu,which has a half-life of about 20,000 years.How long must we wait for a buried stockpile of this substance to decay to 4.0% of its original ²³⁹Pu mass?

Fermium-253 has a half-life of 3.00 d.A sample of fermium contains 7.37 × 10⁷ nuclei.How long will it take for there to be only 3.36 × 10⁶ fermium nuclei in this sample?

The unstable isotope ²³⁴Th decays by β emission with a half-life of 24.5 days.The initial decay rate of the sample was 9.9 × 10¹³ Bq.(1 u = 1.6605 × 10^-27 kg) (a)What mass of ²³⁴Th was initially present? (b)What is the decay rate after 68 days?

Living matter has 1.3 × 10^-10 % of its carbon in the form of ¹⁴C which has a half-life of 5730 y.A mammoth bone has a 300-g sample of carbon separated from it,and the sample is found to have an activity of 20 decays per second.How old is the bone?

How many days are required for a radioactive sample,with a half-life of 5.7 d and an initial activity of 1.07 × 10⁵ Bq,to decay to an activity of 100 Bq?

The stability of Fe with respect to alpha,β⁺,and β^-decay is to be determined.Do not consider the possibility of decay by electron capture.The following atomic masses are known: He: 4.002603 u Cr: 51.944768 u Mn: 55.938907 u Fe: 55.934939 u Co: 55.939841 u The Fe nuclide is

_{An isotope of Tc having a half-life of 6.0 h is used in bone scans.If a certain amount of this Tc is injected into the body,how long does it take for its initial decay rate to decrease BY 99%?}