CHM 101 GENERAL CHEMISTRY
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FALL
QUARTER 2008 |
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Section
2 |
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Lecture
Notes – |
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(last revised: |
18.0 Review: Recall our discussions of atoms and nuclei in
Chapter 2.
- Comparing Atoms and Nuclei
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Property |
Atom |
Nucleus |
Electron |
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Diameter |
~10–8 cm |
~10–13 cm |
varies |
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Charge |
0 |
+1 to +118 |
–1 |
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Mass |
(1.67 – 400)x10–24 g |
9x10–28 g |
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Density |
(0.1 – 20) g/cm3 |
~2x1014 g/cm3 |
varies |
- Components of the Atom
|
Property |
Proton |
Neutron |
Electron |
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Charge |
+1 |
0 |
–1 |
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Mass (AMU) |
1 |
1 |
5x10–4 |
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Mass (g) |
1.67493x10–24 |
1.67262x10–24 |
9.11x10–28 |
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“Atomic”
Symbol |
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·
A Chemists’ Model of the Atom:
·
An atom
consists of a positively charged nucleus surrounded by electrons. The charges
on the electrons exactly balance the charge on the nucleus so that an atom has
no charge.
·
A nucleus
consists of 1 or more protons and 0 or more neutrons bound together by nuclear
forces (strong enough to overcome the mutual repulsions of the electrons).
·
The chemical
properties of atoms are determined by the number of protons in the nucleus, but
do not depend on the number of
neutrons.
- Definitions and Notation
·
Atomic Number: The Atomic Number of an
element is the number of protons in
the nuclei of each of its atoms. The symbol for atomic number is Z. (Of
necessity, the atomic number is equal to the charge on the nucleus.)
·
Nuclide: Nuclide refers to a particular combination of a number of protons with a number of neutrons
to form a nucleus. For example, the most common nuclide of carbon contains 6
protons and 6 neutrons in each of its nuclei, and the most common nuclide of
oxygen contains 8 protons and 8 neutrons.
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Mass Number: The mass number of a nuclide is the sum of its number of protons and
its number of neutrons. For example, the most common nuclide of carbon has a
mass number of 12 (6 protons + 6 neutrons), and the most common nuclide of
oxygen has a mass number of 16 (8 protons and 8 neutrons). The symbol for
mass number is A.
·
Neutron Count: By definition, this is simply the difference between the mass number
and the atomic number:
number of neutrons = A – Z
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Isotopes:
Any time an element has two or more kinds
of nuclides (i. e., nuclides with the same number of protons but different
numbers of neutrons) we refer to the set of nuclides as isotopes of that element. (We do not use the term, isotope, as a
synonym for a single nuclide.)
·
Atomic Symbols Incorporating the Mass Number and
the Atomic Number: If we have a
nuclide of an element with the atomic symbol, X, the atomic number, Z, and the
mass number, A, we write the atomic symbol for the nuclide as:
For example, the three
different nuclides that comprise the isotopes of carbon (i. e., carbon-12,
carbon-13, and carbon-14) have the symbols
, 
, & 
Since the atomic number
determines the atomic symbol for the element, and vice versa, the atomic
symbols for nuclides often omit the atomic number. Thus the symbols for the
isotopes of carbon are sometimes written:
, 
, & 
·
Deuterium and Tritium: The heavy isotopes of hydrogen, hydrogen-2 and hydrogen-3,
have special names:
|
Nuclide |
Name |
Symbol |
Alternate |
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Hydrogen–1 |
Hydrogen |
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Hydrogen–2 |
Deuterium |
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Hydrogen–3 |
Tritium |
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18.1 Nuclear
Stability and Radioactive Decay: Over
85% of all known nuclides are unstable, meaning they are subject to radioactive
decay.
- Thermodynamic Instability: Thermodynamic
instability refers to a condition
where some combination of the
constituents of a given unstable nuclide is more stable than the nuclide itself. In other words, the
reaction to convert the nuclide to the other combination would be
accompanied by a release of energy (ΔE < 0).
- Kinetic Instability: Kinetic
instability refers to the probability that a
thermodynamically unstable nucleus will
undergo decomposition. We refer to this decomposition process as radioactive decay.
- Zone of Stability: If we make a graph of the neutron count (A –
Z) versus the atomic number (Z) for all known nuclides, we get Figure
18.1. The slope of this plot gives us the neutron-to-proton ratio versus
atomic number:
The heavy red points
indicate the positions of the 279 known, stable nuclides; hence the zone they
occupy on the graph is called the zone of
stability, while the gray dots mark the positions of unstable nuclides.
Notice how the neutron-to-proton ratio is nearly 1:1 for the stable nuclides of
the lightest elements (4He, 6Li, 12C, 16O),
but that as the atomic number increases, the necessary neutron-to-proton ratio
also increases, reaching a value of 1.53 for 202Hg.
- Radioactive Decay: Some important observations:
·
Neutron-Proton Ratio: As the atomic number increases, the neutron-proton
ratio necessary to make stable nuclides also increases, as seen above.
·
Large Atomic Numbers: All known nuclides with atomic numbers exceeding
83 (the element, Bismuth) are radioactively
unstable.
·
Even Numbers: Nuclides with even numbers of protons and neutrons are more likely to
be stable than those with odd numbers of either, or especially both. See Table
18.1.
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Magic Numbers: There are certain specific numbers (called magic numbers) of protons or neutrons that make for especially
stable nuclides. These magic numbers are 2, 8, 20, 28, 50, 82, and 126. These
numbers are akin to the numbers of electrons (2, 10, 18, 36, 54, and 86) that
give the noble gases their special chemical unreactivity.
·
Unstable Nuclides Above the Zone of Stability: One can surmise that an unstable nuclide above the
zone of stability (i. e., whose neutron-to-proton ratio is “too high”) “would
like” to decay so as to reduce its number of neutrons and/or increase its
number of protons.
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Unstable Nuclides Below the Zone of Stability: One can also surmise that an unstable nuclide
below the zone of stability (i. e., whose neutron-to-proton ratio is “too low”)
“would like” to decay so as to increase its number of neutrons and/or decrease
its number of protons.
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Unstable Nuclides Above Atomic Number 83: Another surmise one could make is that a(n unstable) nuclide whose atomic number exceeds 83 “would
like” to decay by a process that reduces its number of protons, even if it also
causes a decrease in its number of neutrons.
- Types of Radioactive Decay Processes: The earliest radioactive decay processes to be
described were named, logically enough, alpha emission, beta emission, and
gamma emission. Later studies revealed the identities of the emitted
particles:
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Process |
Type of
Particle |
Atomic
Symbol |
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Alpha (α) emission |
helium nucleus |
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Beta (β) emission |
electron |
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Gamma (γ) emission |
high-energy photon |
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·
Alpha Particle Emission: In alpha particle emission, an unstable nuclide
emits a helium nucleus. Many nuclides of heavy radioactive elements decay
primarily by alpha particle emission. Some examples:
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Changes
in Numbers of |
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Nuclide |
Symbol |
Equation |
Nucleons (ΔA) |
Protons (ΔZ) |
Neutrons |
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-4 |
-2 |
-2 |
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Spontaneous Fission: Spontaneous fission is a decay process in which a
heavy nuclide divides into two lighter nuclides of similar mass numbers.
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Beta Particle Emission: In beta particle emission, an unstable nuclide
emits an electron. There is no change in mass number; effectively a neutron is
converted to a proton, and the neutron-to-proton ratio gets lowered. Thus it is
a common decay mode for nuclides above the zone of stability. Here are examples
of nuclei that undergo beta decay:
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Changes
in Numbers of |
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Nuclide |
Symbol |
Equation |
Nucleons (ΔA) |
Protons (ΔZ) |
Neutrons |
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0 |
+1 |
-1 |
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Gamma Ray Emission: Gamma rays are simply high-energy photons lacking
either mass or charge. Gamma rays often accompany other modes of decay. For
example, the alpha decay of uranium-238 emits two gamma ray photons along with
its other decay products:
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Nuclide |
Symbol |
Equation |
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Positron Emission: In positron emission, an unstable nuclide emits a positron
(positive electron). There is no change in mass number; effectively a proton is
converted to a neutron, and the neutron-to-proton ratio gets raised. Thus it is
a common decay mode for nuclides below the zone of stability. Sodium-22 is an
example of a nuclide that undergoes positron emission:
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Changes
in Numbers of |
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Nuclide |
Symbol |
Equation |
Nucleons (ΔA) |
Protons (ΔZ) |
Neutrons |
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0 |
-1 |
+1 |
The positron not only has
the opposite charge of an electron, it is actually the antiparticle of the
electron. This means if a positron collides with an electron, both particles
are annihilated and their mass is converted to high-energy photons:
·
Electron Capture: Electron capture changes the nucleus in the same manner as positron
emission – a proton gets converted top a neutron, and the neutron-to-proton
ratio gets raised. The essential difference is that one of the innermost electrons
gets captured by the nucleus and is a reactant in the decay process. An example
is the decay of mercury-201 to gold. Gamma rays are always produced in electron
capture processes.
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Changes
in Numbers of |
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Nuclide |
Symbol |
Equation |
Nucleons (ΔA) |
Protons (ΔZ) |
Neutrons |
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0 |
-1 |
+1 |
- Sample Exercise 18.1 (pp. 844-5): Write balanced equations for the
following nuclear decay processes:
·
(a) 
produces
a positron.
We write out the full
equation for the reaction, expressing the other product as 
:
The way to balance this
equation is to make the totals on either side of the equation the same for both
the A’s and the Z’s:
11 = 0 + A A
= 11
6 = 1 + Z Z
= 5
And since 5 is the atomic
number for boron, we substitute B in place of X:
·
(b) 
produces a β particle:
This time we will balance
the A’s and the Z’s as we write the equation:
Now we look up atomic
number 84 in the periodic table and find polonium (
·
(c) 
produces an α particle:
Since the production of the α
particle reduces the mass number (A) by 4 (to 233) and the atomic number (Z) by
2 (to 91 = Pa: protoactinium) we just write out the
result:
- Sample Exercise 18.2 (pp. 845-6): Supply the missing particle in
each of the following:
·
(a) 
First we balance the A’s
and the Z’s:
195 + A = 195 A
= 0
79 + Z = 78 Z
= –1
The missing particle is
an electron, and the process is electron capture, so the full equation is:
·
(b) 
We need a positron to
balance this reaction, so the process is positron decay:
- Radioactive Decay Series: Some radioactive nuclides are relatively
stable, compared to their decay products. Thus when one of these nuclides
decays, the product will itself decay faster than it can accumulate. Some
of these decay series (or decay chains). A good example is the decay of
thorium-232, which involves chain that ends by reaching the stable
nuclide, lead-208:
Another good example is
the decay of uranium-235 to lead-207:
And still another is the
decay chain for uranium-238, ending with lead-206:
Assuming that the decay
series involves only alpha and beta particle emission, let’s balance this
overall equation. We start by writing:
Here, nα represents the number of alpha particles and nβ represents the number of beta particles emitted during
the decay chain. Now we can balance the mass numbers (the A’s):
238 = 206 + 4 nα
238 – 206 = 4 nα
32 = 4 nα
nα = 8
And now we can balance
the charges (the Z’s):
92 = 82 + 8x2 – nβ
92 = 82 + 16 – nβ
92 = 98 – nβ
92 – 98 = – nβ
– 6 = – nβ
nβ =
6
The decay chain produces
8 alpha particles and 6 beta particles as by-products of the conversion of
uranium-238 to lead-206.
This last chain is
diagrammed in Figure 18.2, where the diagonal arrows represent alpha decay, and
the horizontal arrows represent beta decay:
18.2 Kinetics
of Radioactive Decay: Kinetics of
radioactive decay is the answer to the question, how fast does a sample of a
given kind of nuclide decay? and how does the amount
of this nuclide vary over time?
- Rate of Decay: We will state without proving that if a
certain amount of a given nuclide decays at the rate of 100 atoms per
second, then twice the amount of the same nuclide will decay at the rate
of 200 atoms per second. In general, the rate of decay is proportional to
the number of nuclide atoms in the sample. If N atoms of nuclide are present at a time, t, and N0 atoms were present at the (earlier) time, t0, then the rate of decay can be expressed as
minus the change in the number of nuclide atoms in one unit of time:
We abbreviate the
expression (N – N0) as ΔN. (Note that it has a
negative value, since the number of nuclide atoms is decreasing over time.)
Similarly we abbreviate the (positive) expression (t – t0) as Δt. Thus the rate of decay is a positive quantity, and
k is the rate constant for the decay process. This
equation can be called the differential first order rate law.
- Solving the Rate Equation: Solving the rate equation is an exercise in
calculus for which we will not hold you responsible. We will give you the
result both now and when we test you on the kinetics of radioactive
decay). The solution is called the integrated first-order rate law. In
logarithmic form, it is written:
Here, N is the number of nuclide atoms present at the
time, t, and N0
is the number that were present at time t=0. The symbol, ln,
represents taking the natural logarithm of the expression in parentheses.
This expression can be written in the exponential form:
- Calculations Involving Measured Decay Rates: In radioactive decay problems, one often does
not have any direct information about N, N0, or the ratio, N/N0, but one might have information about the rates of decay at two
different times. How can one use this information with the integrated
first-order rate law? The answer is that one must start with the
differential first order rate law and write separate expressions for the
two rates:
Now one can divide the
first expression by the second:
Thus N/N0 can be calculated from the rates at the two
different times, and the result can plugged into the integrated rate law to
complete the solution of the problem. We will use this kind of calculation in
Section 18.4 when we study radiocarbon dating.
- Half-Life: The half-life (written t1/2) of a nuclide is defined as the time required
for half of a sample of that nuclide to decay. (This is the time at which N = 0.5N0.) We can plug this information into the
logarithmic form of the rate law:
Now we can solve for the
half-life:
- Calculating the Rate Constant
Given the Half-Life: In many
of the radioactive decay problems you will encounter, you will be provided
with the half life of a radioactive decay process. Here is how to calculate the rate constant
from that information:
The half-life of a decay
process is constant. This is seen in the graph of the beta-particle decay
process for strontium-90, illustrated in Figure 18.3. This process has a
half-life of 28.8 years. Notice that the curve takes on the familiar shape of
exponential decay.
- Sample Exercise 18.3 (p. 847): Technicium-99m is used in medical
imaging to form pictures of the heart and other internal organs.
Technicium-99m decays to technicium-99 by gamma-emission:
The rate constant for
this decay process is k = 1.16
x 10–1 /h. What is the half life of technecium-99m?
We use the equation for t1/2:
Half of a sample of
technecium-99m sample will decay over a period of 5.98 hours.
- Sample Exercise 18.4 (p. 848): The half life of molybdenum-99 is
67.0 hr. How much of a 1.000 mg sample of molybdenum-99 will remain after
335 hr?
Any time you are given
the half life of something and some period of elapsed time, it is a good idea
to compute the number of half-lives represented by that period. In this case,
the number is:
335/67.0 = 5.00 half-lives
In most cases the number
of half-lives has a fractional part and yields only an approximate result, but
in this case we have an integer number of half-lives and we can calculate as
follows:
|
Half-Lives |
Fraction
Left |
Mass
Left (mg) |
|
0 |
1 |
1.000 |
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1 |
0.5 |
0.500 |
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2 |
0.25 |
0.250 |
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3 |
0.125 |
0.125 |
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4 |
0.0625 |
0.0625 |
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5 |
0.03125 |
0.0313 |
Figure 18.4 is a plot of these
data. As expected, the shape of the plot is exponential:
that we examined
in Figure 18.2 are listed in Table 18.3:
What is the half-life for
the overall transition from uranium-238 to lead-206?
One could approximate the
answer by adding all the individual half-lives together. Let’s start by adding
the two largest values together:
|
Nuclide |
Half-Life |
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uranium-238 |
4.51x109 years |
4,510,000,000 years |
|
uranium-234 |
2.48x105 years |
248,000 years |
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4,510,248,000 years |
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= |
4.51x109 years |
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We need not consider any
more of the steps. The half-life for the overall process is essentially the
same as that of the longest individual step.
18.3 Nuclear
Synthesis: In Section 18.1 we
studied one major kind of nuclear reaction, radioactive
decay. Radioactive decay always involves the reaction of a single nucleus,
as in spontaneous fission, alpha particle emission, or beta particle emission
(or the reaction of a single atom as in the case of electron capture). Nuclear
synthesis (The book calls it nuclear transformation) is another kind of nuclear
reaction process in which one element is transformed into another.
- The First Observed Nuclear Transformations: You may recall that Lord Rutherford discovered
the nuclear nature of the atom by bombarding gold foil with a stream of
alpha particles. When, several years later (1919), he bombarded nitrogen
with alpha particles, he discovered that oxygen-17 was produced:
The next discovery was
made in 1933 by Irene Curie and (her husband)
Phosphorous-30, with its
odd number of protons (15) and its odd number of neutrons (also 15) is not one
of the four known stable nuclides with odd numbers both of protons and
neutrons. It is a radioactive nuclide with a half-life of about 14 days, and it
is not found in nature. In fact, it is the first artificial nuclide ever to be
observed and characterized.
- Particle Accelerators: Most nuclear transformations are achieved
through the use of particle accelerators. Notice that the above two
examples are of collisions between pairs of positively charged particles.
In order for such a pair of particles to overcome their mutual
electrostatic repulsion, they must be set on a high-velocity collision
course. This is what particle accelerators are designed to do. The two
types of particle accelerators illustrated in your text are the cyclotron
and the linear accelerator.
·
The Cyclotron:
·
The Linear Accelerator
- Neutron Bombardment: Alpha particles are not the only type of
particle that can be used to achieve nuclear transformations. Neutrons
(discovered in 1932) have the advantage, over bombardment with alpha and
other charged particles, that they do not need to overcome electrostatic
repulsion to achieve collision with a charged nucleus. This also means
that that they cannot be accelerated by electric fields. But it turned out
that they don’t need to be. In fact, Enrico
Fermi discovered that if he slowed down the neutrons in the beams he
produced, they were more likely to react with their target nuclei.
When a target nuclide
absorbs a neutron, it becomes initially converted to a different nuclide of the
same element, but with a higher count of neutrons. This moves the target away
from the zone of stability into the area that favors radioactive decay by beta
emission and consequent formation of a nuclide of the element with the next
higher atomic number. For example:
- Transuranium
Elements through Neutron Bombardment: In 1934, Fermi hypothesized that he could subject uranium to
neutron bombardment and create trans-uranium
elements, i. e., elements with atomic numbers 93 and higher and not known
in nature. When he ran the experiment, he thought he had produced element
number 93 by the reaction:
However, natural uranium
contains about 0.7% uranium-235, and when that absorbs a neutron, it undergoes
nuclear fission to produce, among other product, a highly radioactive nuclide
of barium. (We’ll say more about this later.) It wasn’t until 1940 that Edwin
McMillan (1907-91) and Philip Abelson (1913-)
confirmed the generation of neptunium-239 by neutron bombardment of
uranium-238. Then the following year, Abelson and Glenn Seaborg (1912-) confirmed that neptunium-239 would undergo beta decay to
generate plutonium-239:
Plutonium-239 would soon
be generated on an industrial scale by neutron bombardment of uranium-238. It
would then be used to form the explosive heart of the atomic bomb that was
dropped on
- Transuranium
Elements through Alpha Bombardment: It also proved possible to produce transuranium
elements by alpha bombardment. For example, the conversion of curium-242
to californium-245:
- Heavy-Ion Bombardment: Neutron bombardment and even alpha particle
bombardment achieve relatively small increments in atomic numbers (and
atomic masses). Achievement of higher atomic numbers requires that several
steps be run in series. For example, the curium-242 to start the above
reaction had to be produced by alpha bombardment of plutonium-239. And the
plutonium-239 in turn was produced by neutron bombardment of uranium-238.
Since the half-lives of transuranium elements
tend to decrease as the atomic number increases, there comes a point where
further increases in atomic number become impractical without using nuclei
that are heavier than alpha particles. Fortunately, the development of
higher and higher energy particle accelerators makes this possible. For
example, the nuclear synthesis of californium-246:
Some further examples of
the production of transuranium elements are shown in
Table 18.4 from your text:
18.4 Detection
and Uses of Radioactivity
- Measuring Radioactivity: The two most familiar instruments for making
radioactivity measurements are the Geiger
– Müller
counter (also called simply Geiger
counter) and the scintillation
counter.
·
The Geiger – Müller Counter: The Geiger – Müller counter takes advantage of the
fact that radioactive decay produces high-energy particles that can ionize the
matter through which they pass. (Thus radiation like that produced by
radioactive decay is often called ionizing
radiation.) The probe of a Geiger – Müller counter is filled with argon gas,
whose atoms will ionize when struck by high-energy particles:
|
|
high energy |
|
electric |
|
|
Ar (g) |
——> |
Ar+
(g) + e– |
——> |
Ar (g) |
When the Geiger – Müller counter is
turned on, it applies an electric potential across the argon tube. Since
neutral argon gas does not conduct electricity, there is normally no flow of
current across the tube. However current will flow momentarily each time an ion
is produced by a high-energy particle. (The electric current converts the argon
ion back to a neutral atom, thus shutting itself off.) Each time when there is
current flow can be considered an “event,” and these events can be counted and
interpreted as the rate of radioactive decay of the decay source. Figure 18.7
illustrates how a Geiger – Müller
counter works.
·
The Scintillation Counter: The detector in a scintillation counter is zinc
sulfide, which emits light when struck by a high-energy particle. A photocell
counts each flash from the detector and thus measures the rate of radioactive
decay of the decay source.
- Radioactive Dating: Under certain conditions, measurements of
radioactivity and/or analytical measurements of the amounts of radioactive
materials in a sample can be used to date the sample, i. e., to determine
how old it is. We will discuss carbon-14
dating and dating by
radioactivity.
·
Carbon-14 Dating: Carbon-14 dating (also
known as radiocarbon dating) can be
used to determine the ages of ancient articles made from wood or fabric. The
technique is based on measuring the ratio of carbon-14 to carbon-12 in the
article. It works because carbon-14 is radioactive and decays with a half-life
of 5,730 years. In principle, articles up to 10,000 and more years old can be
dated by this technique.
How It Works: Carbon-14 is generated when high-energy neutrons from space (a
component of cosmic rays) collide with atmospheric nitrogen-14:
The resulting carbon-14
is radioactive, and it undergoes decay by beta particle emission:
Over the centuries, the
rates of these two reactions have become equal, and the amount of atmospheric
carbon-14 has reached a constant (so-called steady-state)
value.
The resulting atoms of
carbon-14 are chemically reactive and they readily combine with atmospheric
oxygen to make carbon-14 dioxide, a radioactive molecule with the same
chemistry as ordinary carbon dioxide. Through photosynthesis, it is
incorporated into biomass (including wood and plant fiber). As long as the
biomass remains alive, its ratio of carbon-14 to carbon-12 will match that of
the atmosphere through replenishment of its carbon content. However, when the
biomass dies, its carbon-14 will begin to decay without replenishment while its
carbon-12 content remains constant. Since the decay process of carbon-14 has a
half-life of 5,730 years, the carbon-14 to carbon-12 ratio of a sample of this
biomass will fall to half its initial value after 5,730 years have elapsed. The
decay rate of its carbon-14 content will also fall to half the initial value
over this 5,730 year period since the rate of decay is proportional to its
carbon-14 content.
Systematic Errors in Carbon Dating: The half-life of carbon-14 is a well-established
number, so its use does not introduce any systematic errors into radiocarbon
dating results. However, what if the atmospheric ratio of carbon-14 to
carbon-12 changes over time, and we try to use the current ratio in our dating
calculation? We get a systematic error.
Suppose the cosmic ray
neutron flux 5,730 years ago were 20% lower than it is now. This would have
made the carbon-14 to carbon-12 ratio 20% lower at that time. Our measurement
today of the carbon-14 to carbon-12 ratio in our 5,730 year old wood sample would
be 50% of that lower starting value, and it would be 40% of the current value
of the ratio. Thus we would overestimate the age of the sample.
And suppose that the
carbon-12 content of the atmosphere (i. e., 12CO2) were higher
today than it was 5,730 years ago. This is probably the case, due to mankind’s
burning of fossil fuels, whose carbon is so old that all its original carbon-14
has long since decayed. Suppose also, that the cosmic ray neutron flux has
remained constant. In this case we would underestimate the age of our sample.
Corrections for Systematic Errors: Fortunately, there are sources of ancient wood
whose ages can be determined by counting tree rings (a technique called dendrochronology).
Samples of the wood of giant sequoias and bristlecone pines can thus be dated
as far back as 5,000 years by tree-ring counting. Then radiocarbon dating of
the same samples can be used to compute correction factors that can be used to
calibrate other radiocarbon dating results.
·
Sample Exercise 18.5 (pp. 853-4): The remnants of an ancient fire
showed a carbon-14 decay rate of 3.1 counts per minute per gram of carbon.
Assuming that freshly-cut wood decays at a rate of 13.6 counts per minute
(after correcting for changes in the atmospheric carbon-14 to carbon-12 ratio
over time), calculate the age of the remains. The
half-life of carbon-14 is 5,730 years.
Recall that the decay
rate (Rate) of our ancient sample is proportional to the
number (N) of carbon-14 nuclides
it contains:
And similarly that the
decay rate (Rate0) is proportional to its number (N0) of carbon-14 nuclides:
Thus we can compute the
ratio of the two rates:
Now we can let the k’s
cancel out, and we can plug in our rate measurements:
This gives us the input
we need in order to use the integrated rate law:
We can compute the rate
constant, k, from the half-life:
Now we can solve the rate
law for the time, t, and
plug in our numbers:
We can check to see if
our result makes sense. Our quantity (N/N0) would be 0.25 after two half-lives (11,460 yr)
and 0.125 after 3 (17,190 yr). Its actual value is (3.1/13.6) = 0.23, so we
expect that the answer would be slightly more than 2 half lives, in good
agreement with our result of 12,000 years.
·
Radiocarbon Dating by Mass Spectrometry: “Conventional” radiocarbon dating requires that
samples be burned with recovery of carbon dioxide and measurement of the
radioactive decay rates. This requires relatively large samples (up to several
grams). Mass spectrometry has the advantages of requiring much smaller samples
(around 1 milligram) and yielding direct and accurate measurements of the
carbon-12 to carbon-14 ratios.
·
Dating by Radioactivity: Carbon-14 is not the only radioactive nuclide that
can be used for age measurements. For example, the decay of uranium-238,
eventually producing lead-206, is useful under some circumstances for making
estimates of the age of uranium-containing rocks. Since uranium-238 has a
half-life (4.5 billion years) that is nearly as long as the age of the earth,
it can be used to determine the ages of some really old rocks. And if an even
longer half-life is needed, lutecium-176 might be the answer, with a half-life
of over 37 billion years.
·
Sample Exercise 18.6 (pp. 854-5): A rock containing uranium-238 and
lead-206 was analyzed to determine its approximate age. The analysis showed that
the ratio of lead-206 atoms to uranium-238 atoms was 0.115. You may assume that
there was no lead originally present, that all of the lead generated by the
decay chain is still present in the sample, and that the content of the
intermediate nuclides in the decay chain is negligible. The half-life of
uranium-238 is 4.5x109 years. Estimate the age of the rock.
We can use the integrated
first order rate law:
And we can compute the
rate constant from the half-life:
Our measured lead-206 to
uranium-238 ratio is the amount of lead produced by uranium-238 decay divided
by the remaining uranium-238 that has not decayed.
The present number of
uranium-238 atoms is the value of N that we need for the rate law equation. But for N0 we need the number of uranium-238 atoms originally present. This
number can be written:
If we assume that we now
have 1,000 uranium-238 atoms, we can solve for the number of lead atoms:
Now we can compute the
number of uranium-238 atoms originally present:
Now we have:
and 
Now we have all the
numbers we need to plug into the rate equation to solve for t:
- Radiotracers in Medicine: Radioactive nuclides have proven to be quite
valuable in biological research and in medicine. Here are some examples:
·
Iodine-131: Iodine when ingested will concentrate in the thyroid gland. When a
patient drinks a solution of sodium iodide containing small amounts of
iodine-131, the uptake of iodine by the thyroid gland can be monitored by
imaging the radiation produced by decay of the iodine-131, as seen in Figure
18.8:
·
Thallium-201: When thallium is ingested, it concentrates in healthy heart tissue.
Thallium-201 thus will form an image in healthy heart tissue and show by its
relative absence those parts of the heart damaged by a heart attack.
·
Technicium-99m: Technecium-99m behaves similarly to thallium-201 and can also be used
to help assess damage to the heart by a heart attack.
·
Other Medically Useful Nuclides: Table 18.5 lists some other radioactive nuclides
useful as diagnostic tracers in medicine.
·
Characteristics of a Useful Radiotracer:
It must be chemically
non-toxic.
It needs to concentrate
in the tissue of interest and not in surrounding tissue.
Its decay must produce a
detectable signal.
The radiation produced by
its decay must not cause the organism undue harm.
The nuclides produced by
its decay must be chemically non-toxic.