/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Assume a constant \(1^{14} \math... [FREE SOLUTION] | 91影视

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Chapter 19: Problem 44

Assume a constant \(1^{14} \mathrm{C} /^{12} \mathrm{C}\) ratio of 13.6 counts per minute per gram of living matter. A sample of a petrified tree was found to give 1.2 counts per minute per gram. How old is the tree? (For \(^{14} \mathrm{C}, t_{1 / 2}=5730\) years.)

Short Answer

Expert verified
The petrified tree is approximately 18,350 years old.

Step by step solution

01

Find the decay constant

To find the decay constant, we will use the half-life formula: \[t_{1/2} = \frac{0.693}{\lambda}\] where \(t_{1/2}\) is the half-life and \(\lambda\) is the decay constant. We are given the half-life value for \(^{14}C\), which is 5730 years. We can find the decay constant as follows: \[\lambda = \frac{0.693}{5730}\] Now let's compute the decay constant: \[\lambda \approx 1.21 \times 10^{-4}\, \mathrm{year}^{-1}\]
02

Determine the ratio of \(^{14}C\) to \(^{12}C\) in the sample

In the problem, we are given that the \(^{14}C/^{12}C\) ratio in living matter is 13.6 counts per minute per gram, and that the ratio in the tree sample is 1.2 counts per minute per gram. We need to determine the ratio of the sample to that of living matter: \[\frac{\mathrm{ratio_{sample}}}{\mathrm{ratio_{living}}} = \frac{1.2}{13.6}\] Now let's calculate the ratio: \[\frac{\mathrm{ratio_{sample}}}{\mathrm{ratio_{living}}} \approx 0.0882\]
03

Use decay formula to find the age of the tree

Now we will use the decay formula: \[N_t = N_0 \times e^{-\lambda t}\] where \(N_t\) is the number of radioactive isotopes at time \(t\), \(N_0\) is the initial number of radioactive isotopes, \(e\) is the base of the natural logarithm, \(\lambda\) is the decay constant, and \(t\) is the time elapsed (in years) since the tree died. We found the decay constant \(\lambda\) in step 1 and the ratio of \(^{14}C\) in the sample to that in living matter in step 2. We can use this information to rewrite the decay formula in terms of the ratios: \[\frac{\mathrm{ratio_{sample}}}{\mathrm{ratio_{living}}} = e^{-\lambda t}\] Rearrange the formula to find \(t\): \[t = - \frac{\ln (\frac{\mathrm{ratio_{sample}}}{\mathrm{ratio_{living}}})}{\lambda}\] Now let's substitute the values we found in steps 1 and 2 to get the age of the tree: \[t = -\frac{\ln(0.0882)}{1.21 \times 10^{-4}\, \mathrm{year}^{-1}}\] Finally, calculate the age of the tree: \[t \approx 18,350\, \mathrm{years}\] So, the petrified tree is approximately 18,350 years old.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radioactive Decay
Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This occurs naturally in certain types of atoms, such as carbon-14, which are found in all living organisms.
As these atoms decay, they transform into more stable elements. The decay process follows a predictable pattern, allowing scientists to use it for dating ancient objects.
In the case of carbon-14, it decays into nitrogen-14, and this process can be tracked over time. This is the basis for radiocarbon dating, where the initial number of carbon-14 atoms decreases as the sample ages.
Understanding radioactive decay is crucial for interpreting the ratios in the given exercise and using them to calculate the age of the sample.
Half-Life
Half-life is the time it takes for half of the radioactive nuclei in a sample to decay. For carbon-14, the half-life is 5730 years, meaning that every 5730 years, half of the carbon-14 in a sample will have decayed into nitrogen-14.
This concept allows for the calculation of the age of artifacts by measuring the remaining concentration of carbon-14. In the exercise, the known half-life of carbon-14 is used to determine the decay constant necessary for calculating the sample's age.
By understanding half-life, one can predict how much of a substance will remain after a particular number of years and thus deduce the time elapsed since the death of the organism.
Radiocarbon Dating
Radiocarbon dating is a method used to determine the age of ancient biological materials. It relies on measuring the amount of carbon-14 remaining in a sample.
Living organisms constantly exchange carbon with their environment, maintaining a steady amount of carbon-14. However, once they die, this exchange stops, and the carbon-14 begins to decay.
The process involves comparing the present carbon-14 levels in a sample with the expected levels in living matter, as seen in the exercise. By calculating this ratio and using the decay constant, we can estimate how long ago the organism died.
Radiocarbon dating is essential in archaeology and other fields to date ancient specimens accurately.
Decay Constant
The decay constant is a crucial factor in the equation of radioactive decay, representing the probability of a single nucleus decaying per unit of time.
Mathematically, it is related to the half-life through the formula: \[ \lambda = \frac{0.693}{t_{1/2}} \] For carbon-14, substituting the half-life gives a decay constant of approximately \(1.21 \times 10^{-4}\, \mathrm{year}^{-1}\).
This constant helps determine how quickly a substance undergoes radioactive decay. In the exercise, it's used in the decay formula to calculate the age of the tree.
With the decay constant, scientists can link the ratio of remaining carbon-14 to the time elapsed, allowing for accurate dating of ancient materials.

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Most popular questions from this chapter

Zirconium is one of the few metals that retains its structural integrity upon exposure to radiation. The fuel rods in most nuclear reactors therefore are often made of zirconium. Answer the following questions about the redox properties of zirconium based on the half-reaction $$ \mathrm{ZrO}_{2} \cdot \mathrm{H}_{2} \mathrm{O}+\mathrm{H}_{2} \mathrm{O}+4 \mathrm{e}^{-} \longrightarrow \mathrm{Zr}+4 \mathrm{OH}^{-} \quad 8^{\circ}=-2.36 \mathrm{V} $$ a. Is zirconium metal capable of reducing water to form hydrogen gas at standard conditions? b. Write a balanced equation for the reduction of water by zirconium. c. Calculate \(\mathscr{G} \circ, \Delta G^{\circ},\) and \(K\) for the reduction of water by zirconium metal. d. The reduction of water by zirconium occurred during the accidents at Three Mile Island in \(1979 .\) The hydrogen produced was successfully vented and no chemical explosion occurred? If \(1.00 \times 10^{3} \mathrm{kg}\) Zreacts, what mass of \(\mathrm{H}_{2}\) is produced? What volume of \(\mathrm{H}_{2}\) at 1.0 \(\mathrm{atm}\) and \(1000 .^{\circ} \mathrm{C}\) is produced? e. At Chernobyl in \(1986,\) hydrogen was produced by the reaction of superheated steam with the graphite reactor core: $$ \mathrm{C}(s)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}(g)+\mathrm{H}_{2}(g) $$ It was not possible to prevent a chemical explosion at Chernobyl. In light of this, do you think it was a correct decision to vent the hydrogen and other radioactive gases into the atmosphere at Three Mile Island? Explain.

In addition to the process described in the text, a second process called the carbon-nitrogen cycle occurs in the sun: a. What is the catalyst in this process? b. What nucleons are intermediates? c. How much energy is released per mole of hydrogen nuclei in the overall reaction? (The atomic masses of \(_{1}^{1} \mathrm{H}\) and \(\frac{4}{2} \mathrm{He}\) are 1.00782 \(\mathrm{u}\) and \(4.00260 \mathrm{u},\) respectively.)

During the research that led to production of the two atomic bombs used against Japan in World War II, different mechanisms for obtaining a supercritical mass of fissionable material were investigated. In one type of bomb, a gun shot one piece of fissionable material into a cavity containing another piece of fissionable material. In the second type of bomb, the fissionable material was surrounded with a high explosive that, when detonated, compressed the fissionable material into a smaller volume. Discuss what is meant by critical mass, and explain why the ability to achieve a critical mass is essential to sustaining a nuclear reaction

Radioactive copper-64 decays with a half-life of 12.8 days. a. What is the value of \(k\) in \(\mathrm{s}^{-1} ?\) b. A sample contains 28.0 \(\mathrm{mg}^{64} \mathrm{Cu}\) . How many decay events will be produced in the first second? Assume the atomic mass of \(^{64} \mathrm{Cu}\) is 64.0 \(\mathrm{u} .\) c. A chemist obtains a fresh sample of \(^{64} \mathrm{Cu}\) and measures its radioactivity. She then determines that to do an experiment, the radioactivity cannot fall below 25\(\%\) of the initial measured value. How long does she have to do the experiment?

To determine the \(K_{\mathrm{sp}}\) value of \(\mathrm{Hg}_{2} \mathrm{I}_{2},\) a chemist obtained a solid sample of \(\mathrm{Hg}_{2} \mathrm{I}_{2}\) in which some of the iodine is present as radioactive 131 \(\mathrm{I}\) . The count rate of the \(\mathrm{Hg}_{2} \mathrm{I}_{2}\) sample is \(5.0 \times 10^{11}\) counts per minute per mole of I. An excess amount of \(\mathrm{Hg}_{2} \mathrm{I}_{2}(s)\) is placed into some water, and the solid is allowed to come to equilibrium with its respective ions. A \(150.0-\mathrm{mL}\) sample of the saturated solution is withdrawn and the radioactivity measured at 33 counts per minute. From this information, calculate the \(K_{\mathrm{sp}}\) value for \(\mathrm{Hg}_{2} \mathrm{I}_{2}\) $$ \mathrm{Hg}_{2} \mathrm{I}_{2}(s) \rightleftharpoons \mathrm{Hg}_{2}^{2+}(a q)+2 \mathrm{I}^{-}(a q) \qquad K_{\mathrm{sp}}=\left[\mathrm{Hg}_{2}^{2+}\right]\left[\mathrm{I}^{-}\right]^{2} $$
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