14.8: Biological Effects of Ionizing Radiation
Learning Objectives
- Define various units of radiation.
- Describe effects of ionizing radiation on living organisms.
We hear many seemingly contradictory things about the biological effects of ionizing radiation. It can cause cancer, burns, and hair loss, yet it is used to treat and even cure cancer. How do we understand these effects? Once again, there is an underlying simplicity in nature, even in complicated biological organisms. All the effects of ionizing radiation on biological tissue can be understood by knowing that ionizing radiation affects molecules within cells, particularly DNA molecules.
Let us take a brief look at molecules within cells and how cells operate. Cells have long, double-helical DNA molecules containing chemical codes called genetic codes that govern the function and processes undertaken by the cell. It is for unraveling the double-helical structure of DNA that James Watson, Francis Crick, and Maurice Wilkins received the Nobel Prize. Damage to DNA consists of breaks in chemical bonds or other changes in the structural features of the DNA chain, leading to changes in the genetic code. In human cells, we can have as many as a million individual instances of damage to DNA per cell per day. It is remarkable that DNA contains codes that check whether the DNA is damaged or can repair itself. It is like an auto check and repair mechanism. This repair ability of DNA is vital for maintaining the integrity of the genetic code and for the normal functioning of the entire organism. It should be constantly active and needs to respond rapidly. The rate of DNA repair depends on various factors such as the cell type and age of the cell. A cell with a damaged ability to repair DNA, which could have been induced by ionizing radiation, can do one of the following:
- The cell can go into an irreversible state of dormancy, known as senescence.
- The cell can commit suicide, known as programmed cell death.
- The cell can go into unregulated cell division leading to tumors and cancers.
Since ionizing radiation damages the DNA, which is critical in cell reproduction, it has its greatest effect on cells that rapidly reproduce, including most types of cancer. Thus, cancer cells are more sensitive to radiation than normal cells and can be killed by it easily. Cancer is characterized by a malfunction of cell reproduction, and can also be caused by ionizing radiation. Without contradiction, ionizing radiation can be both a cure and a cause.
To discuss quantitatively the biological effects of ionizing radiation, we need a radiation dose unit that is directly related to those effects. All effects of radiation are assumed to be directly proportional to the amount of ionization produced in the biological organism. The amount of ionization is in turn proportional to the amount of deposited energy. Therefore, we define a radiation dose unit called the rad , as 1/100 of a joule of ionizing energy deposited per kilogram of tissue, which is
\[1 \mathrm{~rad}=0.01 \mathrm{~J} / \mathrm{kg}. \nonumber \]
For example, if a 50.0-kg person is exposed to ionizing radiation over her entire body and she absorbs 1.00 J, then her whole-body radiation dose is
\[(1.00 \mathrm{~J}) /(50.0 \mathrm{~kg})=0.0200 \mathrm{~J} / \mathrm{kg}=2.00 \mathrm{~rad} . \nonumber \]
If the same 1.00 J of ionizing energy were absorbed in her 2.00-kg forearm alone, then the dose to the forearm would be
\[(1.00 \mathrm{~J}) /(2.00 \mathrm{~kg})=0.500 \mathrm{~J} / \mathrm{kg}=50.0 \mathrm{~rad}, \nonumber \]
and the unaffected tissue would have a zero rad dose. While calculating radiation doses, you divide the energy absorbed by the mass of affected tissue. You must specify the affected region, such as the whole body or forearm in addition to giving the numerical dose in rads. The SI unit for radiation dose is the gray (Gy) , which is defined to be
\[1 \mathrm{~Gy}=1 \mathrm{~J} / \mathrm{kg}=100 \mathrm{~rad}. \nonumber \]
However, the rad is still commonly used. Although the energy per kilogram in 1 rad is small, it has significant effects since the energy causes ionization. The energy needed for a single ionization is a few eV, or less than \(10^{-18} \mathrm{~J}\). Thus, 0.01 J of ionizing energy can create a huge number of ion pairs and have an effect at the cellular level.
The effects of ionizing radiation may be directly proportional to the dose in rads, but they also depend on the type of radiation and the type of tissue. That is, for a given dose in rads, the effects depend on whether the radiation is \(\alpha\),\(\beta\),\(\gamma\) X-ray, or some other type of ionizing radiation. In the earlier discussion of the range of ionizing radiation, it was noted that energy is deposited in a series of ionizations and not in a single interaction. Each ion pair or ionization requires a certain amount of energy, so that the number of ion pairs is directly proportional to the amount of the deposited ionizing energy. But, if the range of the radiation is small, as it is for \(\alpha\) s, then the ionization and the damage created is more concentrated and harder for the organism to repair, as seen in Figure \(\PageIndex{1}\). Concentrated damage is more difficult for biological organisms to repair than damage that is spread out, so short-range particles have greater biological effects. The relative biological effectiveness (RBE) or quality factor (QF) is given in Table \(\PageIndex{1}\) for several types of ionizing radiation—the effect of the radiation is directly proportional to the RBE. A dose unit more closely related to effects in biological tissue is called the roentgen equivalent man or rem and is defined to be the dose in rads multiplied by the relative biological effectiveness.
\[\text { rem }=\operatorname{rad} \times \text { RBE } \nonumber \]
So, if a person had a whole-body dose of 2.00 rad of \(\gamma\) radiation, the dose in rem would be \((2.00 \mathrm{~rad})(1)=2.00 \mathrm{~rem} \text { whole body }\). If the person had a whole-body dose of 2.00 rad of \(\alpha\) radiation, then the dose in rem would be \((2.00 \mathrm{~rad})(20)=40.0 \mathrm{~rem} \text { whole body }\). The \(\alpha\) s would have 20 times the effect on the person than the \(\gamma\) s for the same deposited energy. The SI equivalent of the rem is the sievert (Sv), defined to be \(\mathrm{Sv}=\mathrm{Gy} \times \mathrm{RBE}\), so that
\[1 \mathrm{~Sv}=1 \mathrm{~Gy} \times \mathrm{RBE}=100 \mathrm{~rem}. \nonumber \]
The RBEs given in Table \(\PageIndex{1}\) are approximate, but they yield certain insights. For example, the eyes are more sensitive to radiation, because the cells of the lens do not repair themselves. Neutrons cause more damage than \(\gamma\) rays, although both are neutral and have large ranges, because neutrons often cause secondary radiation when they are captured. Note that the RBEs are 1 for higher-energy \(\beta\) s, \(\gamma\) s, and X-rays, three of the most common types of radiation. For those types of radiation, the numerical values of the dose in rem and rad are identical. For example, 1 rad of \(\gamma\) radiation is also 1 rem. For that reason, rads are still widely quoted rather than rem. Table \(\PageIndex{2}\) summarizes the units that are used for radiation.
| Type and energy of radiation | RBE 1 |
|---|---|
| X-rays | 1 |
| \(\gamma\) rays | 1 |
| \(\beta\) rays greater than 32 keV | 1 |
| \(\beta\) rays less than 32 keV | 1.7 |
| Neutrons, thermal to slow (<20 keV) | 2–5 |
| Neutrons, fast (1–10 MeV) | 10 (body), 32 (eyes) |
| Protons (1–10 MeV) | 10 (body), 32 (eyes) |
| \(\alpha\) rays from radioactive decay | 10–20 |
| Heavy ions from accelerators | 10–20 |
MISCONCEPTION ALERT: ACTIVITY VS. DOSE
“Activity” refers to the radioactive source while “dose” refers to the amount of energy from the radiation that is deposited in a person or object. A high level of activity doesn’t mean much danger if a person is far away from the source, or if substantial shielding is placed between the source and the person. A radioactive source can have a high activity in two different ways—the type of radioactive isotope may have a short half-life, resulting in more number of decays per second for a given amount of material, or there may be more material, resulting in more total number of decays per second.
For a given radioactive source with given activity, the dose received by nearby personnel can be reduced by radiation safety practices of establishing minimum distance perimeters (intensity of radiation follows inverse-square law) and designing and placing shielding materials around the source.
The large-scale effects of radiation on humans can be divided into two categories: immediate effects and long-term effects. Table \(\PageIndex{3}\) gives the immediate effects of whole-body exposures received in less than one day. If the radiation exposure is spread out over more time, greater doses are needed to cause the effects listed. This is due to the body’s ability to partially repair the damage. Any dose less than 100 mSv (10 rem) is called a low dose , 0.1 Sv to 1 Sv (10 to 100 rem) is called a moderate dose , and anything greater than 1 Sv (100 rem) is called a high dose . There is no known way to determine after the fact if a person has been exposed to less than 10 mSv.
| Dose in Sv 2 | Effect |
|---|---|
| 0–0.10 | No observable effect. |
| 0.1 – 1 | Slight to moderate decrease in white blood cell counts. |
| 0.5 | Temporary sterility; 0.35 for women, 0.50 for men. |
| 1 – 2 | Significant reduction in blood cell counts, brief nausea and vomiting. Rarely fatal. |
| 2 – 5 | Nausea, vomiting, hair loss, severe blood damage, hemorrhage, fatalities. |
| 4.5 | LD50/32. Lethal to 50% of the population within 32 days after exposure if not treated. |
| 5 – 20 | Worst effects due to malfunction of small intestine and blood systems. Limited survival. |
| >20 | Fatal within hours due to collapse of central nervous system. |
Immediate effects are explained by the effects of radiation on cells and the sensitivity of rapidly reproducing cells to radiation. The first clue that a person has been exposed to radiation is a change in blood count, which is not surprising since blood cells are the most rapidly reproducing cells in the body. At higher doses, nausea and hair loss are observed, which may be due to interference with cell reproduction. Cells in the lining of the digestive system also rapidly reproduce, and their destruction causes nausea. When the growth of hair cells slows, the hair follicles become thin and break off. High doses cause significant cell death in all systems, but the lowest doses that cause fatalities do so by weakening the immune system through the loss of white blood cells.
The two known long-term effects of radiation are cancer and genetic defects. Both are directly attributable to the interference of radiation with cell reproduction. For high doses of radiation, the risk of cancer is reasonably well known from studies of exposed groups. Hiroshima and Nagasaki survivors and a smaller number of people exposed by their occupation, such as radium dial painters, have been fully documented. Chernobyl victims will be studied for many decades, with some data already available. For example, a significant increase in childhood thyroid cancer has been observed. The risk of a radiation-induced cancer for low and moderate doses is generally assumed to be proportional to the risk known for high doses. Under this assumption, any dose of radiation, no matter how small, involves a risk to human health. This is called the linear hypothesis and it may be prudent, but it is controversial. There is some evidence that, unlike the immediate effects of radiation, the long-term effects are cumulative and there is little self-repair. This is analogous to the risk of skin cancer from UV exposure, which is known to be cumulative.
There is a latency period for the onset of radiation-induced cancer of about 2 years for leukemia and 15 years for most other forms. The person is at risk for at least 30 years after the latency period. Omitting many details, the overall risk of a radiation-induced cancer death per year per rem of exposure is about 10 in a million, which can be written as \(10 / 10^{6} \mathrm{~rem} \cdot \mathrm{yr}\).
If a person receives a dose of 1 rem, his risk each year of dying from radiation-induced cancer is 10 in a million and that risk continues for about 30 years. The lifetime risk is thus 300 in a million, or 0.03 percent. Since about 20 percent of all worldwide deaths are from cancer, the increase due to a 1 rem exposure is impossible to detect demographically. But 100 rem (1 Sv), which was the dose received by the average Hiroshima and Nagasaki survivor, causes an additional 3 percent risk, which can be observed with statistical significance above the 20 percent natural incidence rate.
The incidence of genetic defects induced by radiation is about one-third that of cancer deaths, but is much more poorly known. The lifetime risk of a genetic defect due to a 1 rem exposure is about 100 in a million or \(3.3 / 10^{6} \mathrm{~rem} \cdot \mathrm{yr}\), but the normal incidence is 60,000 in a million. Evidence of such a small increase, tragic as it is, is nearly impossible to obtain. For example, there is no evidence of increased genetic defects among the offspring of Hiroshima and Nagasaki survivors. Animal studies do not seem to correlate well with effects on humans and are not very helpful. For both cancer and genetic defects, the approach to safety has been to use the linear hypothesis, which is likely to be an overestimate of the risks of low doses. Certain researchers even claim that low doses are beneficial . Hormesis is a term used to describe generally favorable biological responses to low exposures of toxins or radiation. Such low levels may help certain repair mechanisms to develop or enable cells to adapt to the effects of the low exposures. Positive effects may occur at low doses that could be a problem at high doses.
Even the linear hypothesis estimates of the risks are relatively small, and the average person is not exposed to large amounts of radiation. Table \(\PageIndex{4}\) lists average annual background radiation doses from natural and artificial sources for Australia, the United States, Germany, and world-wide averages. Cosmic rays are partially shielded by the atmosphere, and the dose depends upon altitude and latitude, but the average is about 0.40 mSv/yr. A good example of the variation of cosmic radiation dose with altitude comes from the airline industry. Monitored personnel show an average of 2 mSv/yr. A 12-hour flight might give you an exposure of 0.02 to 0.03 mSv.
Doses from the Earth itself are mainly due to the isotopes of uranium, thorium, and potassium, and vary greatly by location. Some places have great natural concentrations of uranium and thorium, yielding doses ten times as high as the average value. Internal doses come from foods and liquids that we ingest. Fertilizers containing phosphates have potassium and uranium.  Carbon-14 has about 66 Bq/kg radioactivity whereas fertilizers may have more than 3000 Bq/kg radioactivity. Medical and dental diagnostic exposures are mostly from X-rays. It should be noted that X-ray doses tend to be localized and are becoming much smaller with improved techniques. Table \(\PageIndex{4}\) shows typical doses received during various diagnostic X-ray examinations. Note the large dose from a CT scan. While CT scans only account for less than 20 percent of the X-ray procedures done today, they account for about 50 percent of the annual dose received.
Radon is usually more pronounced underground and in buildings with low air exchange with the outside world. Almost all soil contains some \({ }^{226} \mathrm{Ra}\) and \({ }^{222} \mathrm{Ra}\), but radon is lower in mainly sedimentary soils and higher in granite soils. Thus, the exposure to the public can vary greatly, even within short distances. Radon can diffuse from the soil into homes, especially basements. The estimated exposure for \({ }^{222} \mathrm{Ra}\) is controversial. Recent studies indicate there is more radon in homes than had been realized, and it is speculated that radon may be responsible for 20 percent of lung cancers, being particularly hazardous to those who also smoke. Many countries have introduced limits on allowable radon concentrations in indoor air, often requiring the measurement of radon concentrations in a house prior to its sale. Ironically, it could be argued that the higher levels of radon exposure and their geographic variability, taken with the lack of demographic evidence of any effects, means that low-level radiation is less dangerous than previously thought.
Radiation Protection
Laws regulate radiation doses to which people can be exposed. The greatest occupational whole-body dose that is allowed depends upon the country and is about 20 to 50 mSv/yr and is rarely reached by medical and nuclear power workers. Higher doses are allowed for the hands. Much lower doses are permitted for the reproductive organs and the fetuses of pregnant women. Inadvertent doses to the public are limited to \(1 / 10\) of occupational doses, except for those caused by nuclear power, which cannot legally expose the public to more than \(1 / 1000\) of the occupational limit or 0.05 mSv/yr (5 mrem/yr). This has been exceeded in the United States only at the time of the Three Mile Island (TMI) accident in 1979. Chernobyl is another story. Extensive monitoring with a variety of radiation detectors is performed to assure radiation safety. Increased ventilation in uranium mines has lowered the dose there to about 1 mSv/yr.
| Source | Dose (mSv/yr) 3 | |||
|---|---|---|---|---|
| Source | Australia | Germany | United States | World |
| Natural Radiation - external | ||||
| Cosmic Rays | 0.30 | 0.28 | 0.30 | 0.39 |
| Soil, building materials | 0.40 | 0.40 | 0.30 | 0.48 |
| Radon gas | 0.90 | 1.1 | 2.0 | 1.2 |
| Natural Radiation - internal | ||||
|
\({ }^{40} \mathrm{K}\),\({ }^{14} \mathrm{C}\),\({ }^{226} \mathrm{Ra}\) |
0.24 | 0.28 | 0.40 | 0.29 |
| Medical & Dental | 0.80 | 0.90 | 0.53 | 0.40 |
| TOTAL | 2.6 | 3.0 | 3.5 | 2.8 |
To physically limit radiation doses, we use shielding , increase the distance from a source, and limit the time of exposure .
Figure \(\PageIndex{2}\) illustrates how these are used to protect both the patient and the dental technician when an X-ray is taken. Shielding absorbs radiation, and the appropriate type of shielding depends on the type of radiation. The greater the distance from the source, the more the radiation spreads out. The less time a person is exposed to a given source, the smaller is the dose received by the person. Doses from most medical diagnostics have decreased in recent years due to faster films that require less exposure time.
Risk versus Benefit
Medical doses of radiation are also limited. Diagnostic doses are generally low and have further lowered with improved techniques and faster films. With the possible exception of routine dental X-rays, radiation is used diagnostically only when needed so that the low risk is justified by the benefit of the diagnosis. Chest X-rays give the lowest doses—about 0.1 mSv to the tissue affected, with less than 5 percent scattering into tissues that are not directly imaged. Other X-ray procedures range upward to about 10 mSv in a CT scan, and about 5 mSv (0.5 rem) per dental X-ray, again both only affecting the tissue imaged. Medical images with radiopharmaceuticals give doses ranging from 1 to 5 mSv, usually localized. One exception is the thyroid scan using \({ }^{131} \mathrm{I}\). Because of its relatively long half-life, it exposes the thyroid to about 0.75 Sv. The isotope \({ }^{123} \mathrm{I}\) is more difficult to produce, but its short half-life limits thyroid exposure to about 15 mSv.
Section Summary
- The biological effects of ionizing radiation are due to two effects it has on cells: interference with cell reproduction, and destruction of cell function.
-
A radiation dose unit called the rad is defined in terms of the ionizing energy deposited per kilogram of tissue:
\[1 \mathrm{~rad}=0.01 \mathrm{~J} / \mathrm{kg}. \nonumber\]
- The SI unit for radiation dose is the gray (Gy), which is defined to be \(1 \mathrm{~Gy}=1 \mathrm{~J} / \mathrm{kg}=100 \mathrm{~rad} \).
-
To account for the effect of the type of particle creating the ionization, we use the relative biological effectiveness (RBE) or quality factor (QF) given in
Table
\(\PageIndex{1}\) and define a unit called the roentgen equivalent man (rem) as
\[\text { rem }=\operatorname{rad} \times \text { RBE }. \nonumber\]
-
Particles that have short ranges or create large ionization densities have RBEs greater than 1. The SI equivalent of the rem is the sievert (Sv), defined to be
\[\mathrm{Sv}=\mathrm{Gy} \times \mathrm{RBE} \text { and } 1 \mathrm{~Sv}=100 \mathrm{~rem} . \nonumber\]
- Whole-body, single-exposure doses of 0.1 Sv or less are low doses while those of 0.1 to 1 Sv are moderate, and those over 1 Sv are high doses. Some immediate radiation effects are given in Table \(\PageIndex{3}\). Effects due to low doses are not observed, but their risk is assumed to be directly proportional to those of high doses, an assumption known as the linear hypothesis. Long-term effects are cancer deaths at the rate of \(10 / 10^{6} \mathrm{~rem} \cdot \mathrm{y}\) and genetic defects at roughly one-third this rate. Background radiation doses and sources are given in Table \(\PageIndex{4}\). World-wide average radiation exposure from natural sources, including radon, is about 3 mSv, or 300 mrem. Radiation protection utilizes shielding, distance, and time to limit exposure.
Footnotes
Glossary
- gray (Gy)
- the SI unit for radiation dose which is defined to be \(1 \mathrm{~Gy}=1 \mathrm{~J} / \mathrm{kg}=100 \mathrm{~rad}\)
- linear hypothesis
- assumption that risk is directly proportional to risk from high doses
- rad
- the ionizing energy deposited per kilogram of tissue
- sievert
- the SI equivalent of the rem
- relative biological effectiveness (RBE)
- a number that expresses the relative amount of damage that a fixed amount of ionizing radiation of a given type can inflict on biological tissues
- quality factor
- same as relative biological effectiveness
- roentgen equivalent man (rem)
- a dose unit more closely related to effects in biological tissue
- low dose
- a dose less than 100 mSv (10 rem)
- moderate dose
- a dose from 0.1 Sv to 1 Sv (10 to 100 rem)
- high dose
- a dose greater than 1 Sv (100 rem)
- hormesis
- a term used to describe generally favorable biological responses to low exposures of toxins or radiation
- shielding
- a technique to limit radiation exposure


1.2.8: Affinity for Spontaneous Chemical Reaction - Law of Mass Action
The differential change in Gibbs energy of a closed system \(\mathrm{dG}\) is related to the change in chemical composition – organisation using equation (a) where \(\mathrm{A}\) is the affinity for spontaneous chemical reaction such that \(\mathrm{A} \, \mathrm{d} \xi \geq 0\).
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi \label{a} \]
Spontaneous chemical reaction is driven by the affinity for spontaneous change, \(\mathrm{A}\). Eventually the system reaches a minimum in Gibbs energy \(\mathrm{G}\) where the affinity for spontaneous change is zero.
\[\text { In general terms, } \mathrm{A}^{\mathrm{eq}}=-\sum \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{\mathrm{eq}}=0 \label{b} \]
We identify chemical equilibrium as the state where the chemical potentials driving the chemical flow from reactants to products are balanced by the chemical potentials driving chemical flow from products to reactants [1]. The condition given in Equation \ref{b} is based on the first and second laws of thermodynamics [1]. These two laws do not lead to quantitative statements concerning the rate of change of chemical composition; i.e. the dependence of the concentration of reactants and products on time.
At a given time \(\mathrm{t}\), the rate of change of composition \(v\) is defined by equation (c).
\[\mathrm{v}=\mathrm{d} \xi / \mathrm{dt} \nonumber \]
\[\text { Hence, with } A \, d \xi>0, A \, V>0 \nonumber \]
Therefore for chemical reaction in a closed system, the signs of \(\mathrm{A}\) and \(\mathrm{v}\) are identical. [1] Moreover if the system is at thermodynamic equilibrium such that \(\mathrm{A}\) is zero, then \(\mathrm{v}\) is zero. The latter sentence establishes a crucial link between chemical kinetics and chemical thermodynamics. However away from equilibrium we have no information concerning the rate of change of composition. The required property is the ratio \(\mathrm{d} \xi / \mathrm{dt}\) at time \(\mathrm{t}\) characterising the rate of change of the extent of chemical reaction. Intuitively one might argue that the rate depends on the affinity for spontaneous reaction—the greater the affinity for reaction the faster the reaction. The key equation might take the following form.
\[\mathrm{d} \xi / \mathrm{dt}=\mathrm{L} \, \mathrm{A} \nonumber \]
Here \(\mathrm{L}\) is a phenomenological parameter, describing the phenomenon of chemical reaction. Unfortunately no further progress can be made because we have no way of measuring the affinity \(\mathrm{A}\) for chemical reaction; no affinity meter is available which we can plunge into the reacting system and ‘read off’ the affinity.[2] In these terms the analysis comes to a halt.
In the context of chemical kinetics, the rate of chemical reaction is defined by equation (f) where \(n_{j}\) refers to the amount of chemical substance \(j\) as either product or reactant; positive for product
\[\mathrm{v}=\pm \mathrm{V}^{-1} \, \mathrm{dn} \mathrm{n}_{\mathrm{j}} / \mathrm{dt} \nonumber \]
For chemical reaction involving solutes in dilute solution the volume \(\mathrm{V}\) of the system (at fixed \(\mathrm{T}\) and \(p\)) is effectively independent of time.
\[\text { Then, } v=\pm \mathrm{dc}_{\mathrm{j}} / \mathrm{dt} \nonumber \]
The kinetics of chemical reactions in solution are simpler than those for reactions in the gas phase [3] and we confine comment to the former.
The relationship between velocity \(v\) and the chemical composition of a solution is described by the Law of Mass Action as proposed by Guldberg and Waage in 1867 [4]. The developments reported by Harcourt and Essen (1867), Bredig and Stern and by Lapworth in 1904 and by Goldschmidt in 1930 were important. Hammett [3] comments that by the 1930s the subject had emerged from the ‘dark ages’. Hammett draws attention to the contributions made by Bartlett, Ingold and Pedersen in the decades of 1920 and 1930. Effectively these authors showed that the rate of chemical reaction at time \(t\) is a function of the concentrations of substances in the systems at that time, \(t\). In the textbook case the spontaneous chemical reaction [5,6] between two chemical substances \(\mathrm{X}\) and \(\mathrm{Y}\) at fixed \(\mathrm{T}\) and \(p\) in aqueous solution, has the following form.
\[\mathrm{x} . \mathrm{X}(\mathrm{aq})+\mathrm{y} . \mathrm{Y}(\mathrm{aq})->\text { products } \nonumber \]
\[\text { Rate of reaction }=\mathrm{k} \,[\mathrm{X}]^{\alpha} \,[\mathrm{Y}]^{\beta} \nonumber \]
Here \(\alpha\) and \(\beta\) are orders of reaction with respect to substances \(\mathrm{X}\) and \(\mathrm{Y}\). These orders have to be determined from the experimental kinetic data because they do not necessarily correspond to the stoichiometric coefficients in the chemical equation [7].
We develop the argument by considering a chemical reaction in solution. An aqueous solution is prepared containing n1 moles of water(l) and \(\mathrm{n}_{\mathrm{X}}^{0}\) moles of chemical substance \(\mathrm{X}\) at time, \(t = 0\) where \(n_{X}^{0}>>n_{1}\). Spontaneous chemical reaction leads to the formation of product \(\mathrm{Y}\), where at ‘\(t=0\)’, \(\mathrm{n}_{\mathrm{Y}}^{0}\) is zero. Chemical reaction is described using equation (j).
\[\begin{array}{llll}
& \mathrm{X}(\mathrm{aq}) & \rightarrow & \mathrm{Y}(\mathrm{aq}) \\
\mathrm{At} \mathrm{t}=0 & \mathrm{n}_{\mathrm{X}}^{0} & \mathrm{n}_{\mathrm{y}}^{0}=0 & \mathrm{~mol}
\end{array} \nonumber \]
The convention is for the chemical reaction to be written in the form ‘reactants → products’, such that the affinity for reaction \(\mathrm{A}\) is positive and hence \(\mathrm{d}\xi\) and \(\mathrm{d} \xi / \mathrm{dt}\) are positive. Many chemical reactions of the form shown in equation (j) go to completion.
\[\text { Thus } \lim \mathrm{it}(\mathrm{t} \rightarrow \infty) \mathrm{n}_{\mathrm{X}}=0 ; \mathrm{n}_{\mathrm{Y}}=\mathrm{n}_{\mathrm{X}}^{0} \nonumber \]
The minimum in Gibbs energy (where \(\mathrm{A} = 0\)) is attained when all reactant has been consumed.
First Order Reactions
A given closed system contains \(\mathrm{n}_{\mathrm{X}}^{0}\) moles of chemical substance \(\mathrm{X}\) which \(\mathrm{X}\) decomposes to form chemical substance \(\mathrm{Z}\). At time \(t\), \(\xi\) moles of reactant \(\mathrm{X}\) have formed product \(\mathrm{Z}\).
\[\begin{array}{lcc}
\multicolumn{1}{c}{\text { Chemical Reaction }} & \mathrm{X} & \mathrm{Z} \\
\text { Amounts at } \mathrm{t}=0 ; & \mathrm{n}_{\mathrm{X}}^{0} & 0 \mathrm{~mol} \\
\text { Concentrations }(\mathrm{t}=0) & \mathrm{n}_{\mathrm{X}}^{0} / \mathrm{V} & 0 \mathrm{~mol} \mathrm{~m}^{-3} \\
\text { At time } \mathrm{t} & \mathrm{n}_{\mathrm{X}}^{0}-\xi & \mathrm{mol} \\
\text { Amounts } & \left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V} & \xi / \mathrm{V} \mathrm{mol} \mathrm{m}
\end{array} \nonumber \]
The law of mass action is the extra-thermodynamic assumption which relates the rate of change of concentration t o the composition of the system.
\[\text {Then, } -\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=\frac{\mathrm{d}[\xi / \mathrm{V}]}{\mathrm{dt}}=\mathrm{k} \, \frac{\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right]}{\mathrm{V}} \nonumber \]
The constant of proportionality, rate constant \(\mathrm{k}\), in this case has units of \(\mathrm{s}^{-1}\).
\[\text { Then in terms of reactant } \mathrm{X}, \frac{\mathrm{d} \xi}{\mathrm{dt}}=\mathrm{k} \,\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right] \nonumber \]
Chemical reaction proceeds leading to a decrease in the Gibbs energy of the system until \(\mathrm{n}_{\mathrm{X}}^{0}=\xi\) such that all reactant has been consumed. At this point \((\mathrm{d} \xi / \mathrm{dt})\) is zero, the system is at chemical equilibrium and the Gibbs energy is a minimum. Further from equation (m),
\[\int_{\xi=0}^{\xi} \frac{d \xi}{\left(n_{X}^{0}-\xi\right)}=\int_{t=0}^{t} k \, d t \nonumber \]
The Law of Mass Action is the most important extra-thermodynamic equation in chemistry.
Second Order Reactions
A given closed system contains \(\mathrm{n}_{\mathrm{X}}^{0}\) and \(\mathrm{n}_{\mathrm{Y}}^{0}\) moles of chemical substances \(\mathrm{X}\) and \(\mathrm{Y}\) respectively at fixed \(\mathrm{T}\) and \(p\). Spontaneous chemical reaction produces chemical substance \(\mathrm{Z}\). At time \(t\), \(\xi\) moles of product \(\mathrm{Z}\) are formed from chemical substances \(\mathrm{X}\) and \(\mathrm{Y}\).
\[\begin{array}{lccc}
{c}{\text { Chemical Reaction }} & \mathrm{X} & +\mathrm{Y} & \mathrm{Z} \\
\text { Amounts at } \mathrm{t}=0 ; & \mathrm{n}_{\mathrm{X}}^{0} & \mathrm{n}_{\mathrm{Y}}^{0} & 0 \mathrm{~mol} \\
\text { Concentrations }(\mathrm{t}=0) & \mathrm{n}_{\mathrm{X}}^{0} / \mathrm{V} & \mathrm{n}_{\mathrm{Y}}^{0} / \mathrm{V} & 0 \mathrm{~mol} \mathrm{~m} \\
\text { At time t } & & & \\
\text { Amounts } & \mathrm{n}_{\mathrm{X}}^{0}-\xi & \mathrm{n}_{\mathrm{Y}}^{0}-\xi & \xi \mathrm{mol} \\
\text { Concentrations } & \left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V} & \left(\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right) / \mathrm{V} & \xi / \mathrm{V} \mathrm{mol} \mathrm{m}
\end{array} \nonumber \]
The Law of Mass Action is the extra-thermodynamic assumption, relating the rate of change of concentration to the composition of the system. Then,
\[-\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=-\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=\frac{\mathrm{d}[\xi / \mathrm{V}]}{\mathrm{dt}}=\mathrm{k} \, \frac{\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right]}{\mathrm{V}} \, \frac{\left[\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right]}{\mathrm{V}} \nonumber \]
The unit of rate constant \(\mathrm{k}\) is ‘\(\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}\)’.
\[\text { Then, } \frac{\mathrm{d} \xi}{\mathrm{dt}}=\mathrm{k} \, \mathrm{V}^{-1} \,\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right] \,\left[\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right] \nonumber \]
\[\text { Or, } \quad \int_{\xi=0}^{\xi} \frac{\mathrm{d} \xi}{\left[n_{X}^{0}-\xi\right] \,\left[n_{Y}^{0}-\xi\right]}=\int_{t=0}^{t} k \, V^{-1} \, d t \nonumber \]
In applications of equation (q), rate constant \(\mathrm{k}\) and volume \(\mathrm{V}\) are usually treated as independent of time.[8]
The foregoing analysis of kinetics of chemical reactions illustrates the application of the variable \(\xi\) in describing the composition of a closed system. Most accounts of chemical kinetics start out with a consideration of concentrations of chemical substances in a given system. [9,10]
Nevertheless for each and every chemical reaction, the form of the relevant ‘Law of Mass Action’ has to be determined from the observed dependence of composition on time. The latter sentence does not do justice to the skills of chemists in this context.
Reaction to Chemical Equilibrium
In the previous section we considered those cases where chemical reaction goes to completion in that one or more of the reactants are consumed. For many cases this is not the case. Here we imagine that a dilute solution has been prepared using \(\mathrm{n}_{\mathrm{X}}^{0}\) and \(\mathrm{n}_{\mathrm{Y}}^{0}\) moles of solute reactants \(\mathrm{X}\) and \(\mathrm{Y}\). Chemical reaction at fixed \(\mathrm{T}\) and \(p\) proceeds spontaneously. The Gibbs energy of the system decreases reaching a minimum where the affinity for spontaneous reaction is zero. Chemical analysis shows that the resulting system contains product \(\mathrm{Z}\) together with reactants \(\mathrm{X}\) and \(\mathrm{Y}\), and that the chemical composition is independent of time; i.e. chemical kinetic equilibrium. Thus,
\[\begin{gathered}
\mathrm{X}(\mathrm{aq}) \quad+\quad \mathrm{Y}(\mathrm{aq}) \quad \Leftrightarrow \quad \mathrm{Z}(\mathrm{aq}) \\
\mathrm{At} \mathrm{} \mathrm{t}=0 \quad \mathrm{n}_{\mathrm{X}}^{0} \quad \mathrm{n}_{\mathrm{Y}}^{0} \quad \mathrm{n}_{\mathrm{Y}}^{\mathrm{e}}=\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}} \quad \mathrm{n}_{\mathrm{eq}}^{\mathrm{eq}}=\xi^{\mathrm{eq}} \quad \mathrm{mol} \\
\mathrm{At} \mathrm{} \mathrm{t}=\infty \quad \mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}} \quad \mathrm{n}_{\mathrm{eq}} \\
\text { or, } \quad \mathrm{c}_{\mathrm{X}}^{\mathrm{eq}}=\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \quad \mathrm{c}_{\mathrm{eq}}^{\mathrm{eq}}=\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \quad \mathrm{c}_{\mathrm{Z}}^{\mathrm{eq}}=\left(\frac{\xi^{\mathrm{eq}}}{\mathrm{V}}\right)
\end{gathered} \nonumber \]
A number of assumptions are based on the Law of Mass Action. \(\text { At time } t \text {, rate of forward reaction }=k_{f} \, c_{X}(t) \, c_{Y}(t) \text { and rate of the reverse reaction }=\mathrm{k}_{\mathrm{r}} \, \mathrm{c}_{\mathrm{Z}}(\mathrm{t})\)
Rate constants \(\mathrm{k}_{\mathrm{f}}\) and \(\mathrm{k}_{\mathrm{r}}\) are initially assumed to be independent of the extent of reaction. A key conclusion is now drawn. Because at ‘\(t \rightarrow\) infinity’, the properties of the system are independent of time, the system is ‘at chemical equilibrium where the rates of forward and reverse reactions are balanced.
\[\text { Then, } \mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{X}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{Y}}^{\mathrm{eq}}=\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{z}}^{\mathrm{eq}} \nonumber \]
\[\text { Hence, } \quad \mathrm{d} \xi^{\mathrm{eq}} / \mathrm{dt}=0 \nonumber \]
\[\text { and } \mathrm{k}_{\mathrm{eq}}^{\mathrm{eq}} \,\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right)=\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}} \, \frac{\xi^{\mathrm{eq}}}{\mathrm{V}} \nonumber \]
At this point we encounter a key problem -- we cannot determine \(\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}\) and \(\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}\) because at equilibrium the composition of the system is independent of time. Nevertheless we can express the ratio of rate constants as a function of the composition at equilibrium.
\[\left(\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathrm{k}_{\mathrm{r}}^{\mathrm{eq}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right)=\frac{\xi^{\mathrm{eq}}}{\mathrm{V}} \nonumber \]
The ratio \(\mathbf{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathbf{k}_{\mathrm{r}}^{\mathrm{eq}}\) is characteristic of the system (at defined \(\mathrm{T}\) and \(p\)), defining what we might call a ‘Law of Mass Action equilibrium constant’, \(\mathrm{K}(\operatorname{lma})\).
\[\text { Thus } \mathrm{K}(\operatorname{lm} a)=\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathrm{k}_{\mathrm{r}}^{\mathrm{eq}} \nonumber \]
In other words, the property \(\mathrm{K}(\operatorname{lma})\) is based on a balance of reaction rates whereas the thermodynamic equilibrium constant is based on a balance of chemical potentials. With reference to equation (v), at fixed \(\mathrm{T}\) and \(p\), the thermodynamic condition is given in equation (w).
\[\mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq})+\mu_{\mathrm{Y}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{Z}}^{\mathrm{eq}}(\mathrm{aq}) \nonumber \]
\[\text { For } \mathrm{p} \cong \mathrm{p}^{0}, \mathrm{~K}^{0}(\mathrm{~T})=\frac{\left(\mathrm{m}_{\mathrm{Z}} \, \gamma_{\mathrm{Z}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}}{\left(\mathrm{m}_{\mathrm{X}} \, \gamma_{\mathrm{X}} / \mathrm{m}^{0}\right)^{\mathrm{eq}} \,\left(\mathrm{m}_{\mathrm{Y}} \, \gamma_{\mathrm{Y}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}} \nonumber \]
Therefore the question is raised--- how is \(\mathrm{K}(\operatorname{lma})\) related to \(\mathrm{K}^{0}(\mathrm{T})\)? In the absence of further information, a leap of faith by chemists sets \(\mathrm{K}(\operatorname{lma})\) equal to \(\mathrm{K}^{0}(\mathrm{T})\). We avoid debating the meaning of the phrase ‘rate constants at equilibrium’ [7].
Energy of Activation
Spontaneous chemical reaction involving a single solute \(\mathrm{X}(\mathrm{aq}\)) can be described using a rate constant \(\mathrm{k}\) for a solution at fixed \(\mathrm{T}\) and \(p\) where the concentration of solute \(\mathrm{X}\) at time \(t\) is \(\mathrm{c}_{\mathrm{X}}(\mathrm{aq})\).
If we can assume that in solution there are no solute-solute interactions, rate constant \(\mathrm{k}\) is not dependent on the composition of the solution. In thermodynamic terms, the thermodynamic properties of solute \(\mathrm{X}\) are ideal; for such a system rate constant \(\mathrm{k}\) is independent of time and initial concentration of solute \(\mathrm{X}\). In other words experiment yields the property \(\mathrm{k}(\mathrm{T}, p)\) for a given solution indicating that the rate constant is a function of temperature and pressure. For nearly all chemical reactions in solution rate constants increase with increase in temperature, a dependence described by the Arrhenius equation. [11]
\[\mathrm{k}=\mathrm{A} \, \exp \left(-\mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}\right) \nonumber \]
Then rate constant, \(\mathrm{k}\) increases with increase in temperature. The idea emerges that spontaneous conversion of reactant to products is inhibited by an ‘energy’ barrier. Further in the \(\operatorname{limit}(\mathrm{T} \rightarrow \infty) \ln (\mathrm{k})=\mathrm{A}\), the pre-exponential factor which has the same units as rate constant \(\mathrm{k}\).
The assumption in the foregoing comments is that \(\mathrm{E}_{\mathrm{A}}\) is independent of temperature such that for example a first order rate constant \(\ln \left(\mathrm{k} / \mathrm{s}^{-1}\right)\) is a linear function of \(\mathrm{T}^{-1}\).
\[\text { Thus, } \quad \ln \left(\mathrm{k} / \mathrm{s}^{-1}\right)=\ln \left(\mathrm{A} / \mathrm{s}^{-1}\right)-\left(\mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}\right) \nonumber \]
The latter pattern is generally observed but there are many well-documented cases where the plot is not linear. In other words it is incorrect to conclude that equation (y) somehow predicts how rate constants depend on temperature. There is no substitute for actually measuring this dependence.
Transition State Theory
The law of mass action and the concept of an activation energy for a given chemical reaction are extrathermodynamic. This conclusion is unfortunate, implying that the treatment of kinetic data for reactions in solution is completely divorced from the thermodynamic treatment of the properties of solution. One can understand therefore why Transition Sate Theory (TST) attracts so much interest.[12,13] At the very least, this theory offers analysis of kinetic data a patina of thermodynamic respectability. We describe TST with respect to a chemical reaction where the dependence of composition on time is described using a first order rate constant.
For chemical reactions in the gas phase, statistical thermodynamics offers a reasonably straightforward approach to the description of both reactants and a transition state in which one vibrational mode for the transition state is transposed into translation along the reaction co-ordinate. The theory was re-expressed in terms of equations which could be directly related to the thermodynamics of the process of reaction in solutions.
Chemical reaction proceeds from reactant \(\mathrm{X}(\mathrm{aq})\) to products through a transition state \(\mathrm{X}^{\neq}(\mathrm{aq})\). As the reaction proceeds the amount of solute \(\mathrm{X}\), \(n_{\mathrm{X}}(\mathrm{aq})\) decreases but at all times reactant \(\mathrm{X}(\mathrm{aq})\) and transition state \(\mathrm{X}^{\neq}(\mathrm{aq})\) are in chemical equilibrium.
\[\text { Thus } \mathrm{X}(\mathrm{aq}) \Leftrightarrow=\mathrm{X}^{7} \rightarrow \text { products } \nonumber \]
The condition ‘chemical equilibrium’ is quantitatively expressed in terms of chemical potentials.
\[\mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\neq}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \nonumber \]
Conventionally where kinetics of reactions in solution are addressed, the composition of solutions is expressed in terms of concentrations using the unit, \(\mathrm{mol dm}^{-3}\). Then equation (zc) is formed assuming that ambient pressure is close to the standard pressure, \(p^{0}\); \(\mathrm{c}_{\mathrm{r}} =1 \mathrm{~mol dm}^{-3}\). Hence,
\[\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}}=\mu_{\neq}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\neq} \, \mathrm{y}_{\neq} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}} \nonumber \]
The standard Gibbs energy of activation \(\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})\) is given by equation (zd) leading to the definition of an equilibrium constant \({ }^{*} \mathrm{~K}^{0}(\mathrm{aq} ; \mathrm{T}) ; \text { with } \mathrm{p} \approx \mathrm{p}^{0}\).
\[\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=\mu_{\neq}^{0}(\mathrm{aq} ; \mathrm{T})-\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T}) \nonumber \]
\[\text { Then } \quad \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\neq} \, \mathrm{y}_{\neq} / \mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}}\right)^{\mathrm{eq}} \nonumber \]
\[\text { By definition, } \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=-\mathrm{R} \, \mathrm{T} \, \ln \left[{ }^{\neq} \mathrm{K}^{0}(\mathrm{aq} ; \mathrm{T})\right] \nonumber \]
\[\text { Therefore, } \mathrm{c}_{\neq}={ }^{\neq} \mathrm{K}^{0}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}} / \mathrm{y}_{\neq} \nonumber \]
Through the course of chemical reaction, as the concentration of reactant \(\mathrm{X}(\mathrm{aq})\) decreases, the condition given in equation (zb) holds. Chemical reaction is not instantaneous because \(\mu_{z}^{0}(\mathrm{aq} ; \mathrm{T})>\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T})\); a barrier exists to chemical reaction. Consequently the concentration \(\mathrm{c}_{\neq}\) is small. The analysis up to equation (zf) is based on a thermodynamic description of equilibrium between reactant and transition states. In the limit that the solution is very dilute, \(y_{X}=y_{\neq}=1\) at all time \(t\), \(\mathrm{T}\) and \(p\).
\[\text { Thus at given } T \text { and } p,[14] \quad k=\frac{k_{B} \, T}{h} \, \exp \left(\frac{-\Delta^{\neq} G^{0}}{R \, T}\right) \nonumber \]
\[\text { Then, } \mathrm{k}=\frac{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}} \, \exp \left(-\frac{\Delta^{\neq} \mathrm{H}^{0}}{\mathrm{R} \, \mathrm{T}}+\frac{\Delta^{\neq} \mathrm{S}^{0}}{\mathrm{R}}\right) \nonumber \]
\(\Delta^{\neq} \mathrm{G}^{0}\) is the standard Gibbs energy of activation defined in terms of reference chemical potentials of transition state and reactants. Here \(\mathrm{k}_{\mathrm{B}}\) is the Boltzmann constant and \(\mathrm{h}\) is Planck’s constant [15].
\[\Delta^{\neq} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\frac{\mathrm{h} \, \mathrm{k}(\mathrm{T})}{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}\right) \nonumber \]
At temperature \(\mathrm{T}\), \(\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})\) is re-expressed in terms of standard enthalpy and standard entropy of activation [16].
\[\text { Then, } \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=\Delta^{\neq} \mathrm{H}^{0}(\mathrm{aq} ; \mathrm{T})-\mathrm{T} \, \Delta^{\neq} \mathrm{S}^{0}(\mathrm{aq} ; \mathrm{T}) \nonumber \]
The standard isobaric heat capacity of activation [17,18],
\[\Delta^{\neq} \mathrm{C}_{\mathrm{p}}^{0}(\mathrm{aq})=\left(\frac{\partial \Delta^{\neq} \mathrm{H}^{0}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
In summary transition state theory allows kinetic data to be analysed using the protocols and equations of thermodynamics.
Analysis of the dependence of rate constants at fixed temperature as a function of pressure yields standard volumes of activation \(\Delta^{\neq} \mathrm{V}^{0}(\mathrm{aq} ; \mathrm{T})\). In further exercises this volumetric parameter is measured as functions of pressure at temperature \(\mathrm{T}\) and of temperature at fixed pressure [20].
Nevertheless a word of caution is in order. Johnston [21] points out that diagrams describing the progress of chemical reaction through several intermediates are often misleading. Such diagrams should be based on reference chemical potentials otherwise misleading conclusions can be drawn; see also [22 - 28].
Footnotes
[1] I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London , 1953.
[2] In fact in the treatment of data obtained using fast reaction techniques (e.g. temperature jump and pressure-jump) where the displacement from equilibrium is small, it is assumed that the rate of response is linearly related to the affinity for chemical reaction. (a) E. F. Caldin, Fast Reactions in Solution, Blackwell, Oxford, 1964. (b) M. J. Blandamer, Introduction to Chemical Ultrasonics, Academic Press, London, 1973.
[3] L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 1970, 2nd. Edition.
[4] K. J. Laidler, [The World of Physical Chemistry, Oxford University Press, Oxford, 1993,page 115] comments
‘Neither did they ( i.e. Guldberg and Waage] make any contributions to kinetics since they worked in terms of forces and not of rates, although they did tentatively suggest that rates might be proportional to forces’.
[5] The condition ‘spontaneous change‘ signals a ‘natural’ direction but that does mean that the process is instantaneous. The properties of the system are dependent on time. We make this point to counter such statements as ‘Thermodynamics … deals exclusively with system showing no temporal change; reacting systems are outside its province’; cf. E. A. Moelwyn-Hughes, Kinetics of Reactions in Solution, Clarendon Press, Oxford, 1947, page 162.
[6] Thermodynamics in the form of the first and second laws offers no way forward. Intuitively one might argue that the rate of change of composition would be directly related to the affinity for spontaneous reaction, \(\mathrm{A}\) where \(\pi\) is a proportionality factor characteristic of the system, temperature and pressure.
\[\text { Thus, } \mathrm{d} \xi / \mathrm{dt}=\pi \, \mathrm{A} \quad \text { where } \quad \lim \mathrm{it}(\mathrm{A} \rightarrow 0) \mathrm{d} \xi / \mathrm{dt}=0 \nonumber \]
In fact we might draw an analogy with Ohm’s law whereby electric current \(\mathrm{i}\) (= rate of flow of charge) is proportional to the electric potential gradient, \(\Delta \phi\), the constant of proportionality being the electrical conductivity, \(\mathrm{L}\); \(\mathrm{i}=\mathrm{L} \, \Delta \phi\) where \(\mathrm{L}\) is characteristic of the system, temperature and pressure. Indeed such a kinetic force-flow link might be envisaged. Indeed an link emerges with the phenomenon of electric potential driving an electric current through an electrical circuit.
Electric current = rate of flow of charge, unit = Ampere.
Electric potential has the unit, volt.
The product, \(\mathrm{I}. \(\mathrm{V}=[\mathrm{A}] \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]=[\mathrm{W}]\)
Here \(\mathrm{W}\) is the SI symbol for the unit of
power
, watt.
Rate of chemical reaction \(\mathrm{d} \xi / \mathrm{dt}=\left[\mathrm{mol} \mathrm{s} \mathrm{s}^{-1}\right]\)
Affinity \(\mathrm{A}=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\)
Then \(\mathrm{A} \, \mathrm{d} \xi / \mathrm{dt}=\left[\mathrm{J} \mathrm{mol} \mathrm{m}^{-1}\right] \,\left[\mathrm{mol} \mathrm{s}^{-1}\right]=[\mathrm{W}]\)
Interestingly chemists rarely refer to the 'wattage' of a chemical reaction.
[7] One cannot help be concerned with accounts which describe chemical equilibrium in terms of rates of chemical reaction. As we understand the argument runs along the following lines. For a chemical equilibrium having the following stochiometry, \(\mathrm{X}+\mathrm{Y} \Leftrightarrow\mathrm{Z}\) at equilibrium the rates of forward and reverse reactions are balanced.
\[\mathrm{k}_{\mathrm{f}} \,[\mathrm{X}] \,[\mathrm{Y}]=\mathrm{k}_{\mathrm{r}} \,[\mathrm{Z}] \nonumber \]
Here \(\mathrm{k}_{\mathrm{f}}\) and \(\mathrm{k}_{\mathrm{f}}\) are the forward and reverse rate constants.
\[\text { Then } \mathrm{K}=\mathrm{k}_{\mathrm{f}} / \mathrm{k}_{\mathrm{r}}=[\mathrm{Z}] /[\mathrm{X}] \,[\mathrm{Y}] \nonumber \]
The argument loses some of its force if one turns to accounts dealing with chemical kinetics when questions of order and molecularity are raised. In any case one cannot measure rates of chemical reactions 'at equilibrium' because at equilibrium 'nothing is happening'. Even in those cases where the rates of chemical reactions 'at equilibrium ' are apparently measured the techniques rely on following the return to equilibrium when the system is perturbed.
[8] In nearly all applications of the law of mass action to chemical reactions in solution a derived rate constant is based on on a description of the composition on time at fixed T and p. Therefore the calculated rate constant is an isobaric-isothermal property of the system. Nevertheless the volume is usually taken as independent of time. Certainly in most applications in solution chemistry the solutions are quite dilute and so throughout the course of the reaction the volume is effectively constant. An interesting point now emerges in that a given rate constant is an isothermal-isobaric-isochoric property.
[9] K. J. Laidler and N. Kallay, Kem. Ind.,1988, 37 ,183.
[10] M. J. Blandamer, Educ. Chemistry, 1999, 36 ,78.
[11] \(\mathrm{E}_{\mathrm{A}}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \text { such that } \mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=[1]\)
[12] The account given here stresses the link with classical thermodynamics. The key equations should be developed using statistical thermodynamics; see H. Eyring, J. Chem. Phys., 1935, 3 ,107.
[13] S. Glasstone, K. J. Laidler and H. Eyring , The Theory of Rate Processes, McGraw-Hill, New York, 1941.
[14] \(\frac{\mathrm{K}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}}=\frac{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}{[\mathrm{J} \mathrm{s}]}=\left[\mathrm{s}^{-1}\right]\)
[15] \(\frac{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}} \, \mathrm{c}_{\neq}=\left[\mathrm{s}^{-1}\right] \,\left[\mathrm{mol} \mathrm{dm}{ }^{-3}\right]=\left[\mathrm{mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}\right]\)
[16]
- K. J. Laidler, J. Chem. Educ.,1984, 61 ,497.
- S. R. Logan, J.Chem.Educ.,1982, 59 ,278.
- H. Maskill, Educ. Chem.,1985, 22 ,154.
- K. J. Laidler, J. Chem.Educ.,1988, 65 ,540.
- F. R. Cruikshank, A. J. Hyde and D. Pugh, J.Chem.Educ.,1977, 54 ,288.
[17] R. E. Robertson, Prog. Phys. Org. Chem.,1967, 4 , 213.
[18] M. J. Blandamer, J. M. W. Scott and R. E. Robertson, Prog. Phys. Org. Chem., 1985, 15 ,149.
[19] M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood, PTR Prentice Hall, New York,, 1992.
[20] H. S. Johnston, Gas Phase Reaction Rate Theory, Ronald Press, New York, 1966.
[21] R. B. Snadden, J.Chem.Educ.,1985, 62 ,653.
[22] M. I. Page, Educ. Chem.,1981, 18 ,52.
[23] R. D. Levine, J.Phys.Chem.,1979, 83 ,159.
[24] I. H. Williams, Chem. Soc. Rev.,1993, 22 ,277.
[25] A. Williams, Chem. Soc. Rev.,1994, 23 ,93.
[26] A. Drljaca, C. D. Hubbard, R. van Eldik, T. Asano, M. V. Basilevsky and W. J. Le Noble, Chem. Rev.,1998, 98 ,2167.
[27] R. K. Boyd, Chem. Rev.,1977, 77 ,93.


8.1: Why It Matters- Analysis and Synthesis
Why think and write analytically?
In college courses, you will be asked to read, reason, and write analytically. Effective analysts can distinguish the whole, identify parts, infer relationships, and make generalizations. Those skills enable individuals to connect ideas, detect inconsistencies, and solve problems in a systematic fashion. Understanding what analysis is, how to apply it, and how to convey the results effectively will be invaluable to you throughout your college and professional careers.
Contributors and Attributions
CC licensed content, Original
- Why It Matters: Analysis. Provided by : University of Mississippi. License : CC BY: Attribution
CC licensed content, Shared previously
- look downstairs into stairwell whirl. Authored by : Karl-Ludwig Poggemann. Located at : www.flickr.com/photos/hinkelstone/2765597758/. License : CC BY: Attribution


9.3: Improper (Cauchy) Integrals
Cauchy extended R-integration to unbounded sets and functions as follows.
Given \(f : E^{1} \rightarrow E\) and assuming that the right-hand side R-integrals and limits exist, define (first for unbounded sets, then for unbounded functions)
(i) \(\int_{a}^{\infty} f=\int_{[a, \infty)} f=\lim _{x \rightarrow \infty} R \int_{a}^{x} f\);
(ii) \(\int_{-\infty}^{a} f=\int_{(-\infty, a]} f=\lim _{x \rightarrow-\infty} R \int_{x}^{a} f\).
If both
\[\int_{0}^{\infty} f \text { and } \int_{-\infty}^{0} f\]
exists, define
\[\int_{-\infty}^{\infty} f=\int_{(-\infty, 0)} f+\int_{[0, \infty)} f.\]
Now, suppose \(f\) is unbounded near some \(p \in A=[a, b],\) i.e., unbounded on \(A \cap G_{\neg p}\) for every deleted globe \(G_{\neg p}\) about \(p\) (such points \(p\) are called singularities ).
Then (again assuming existence of the R-integrals and limits), we define
- in case of a singularity \(p=a\), \[\int_{a+}^{b} f=\int_{(a, b]} f=\lim _{x \rightarrow a+} R \int_{x}^{b} f;\]
- if \(p=b,\) then \[\int_{a}^{b-} f=\int_{[a, b)} f=\lim _{x \rightarrow b-} R \int_{a}^{x} f;\]
- if \(a<p<b\) and if \[\int_{a}^{p-} f \text { and } \int_{p+}^{b} f\]
exist, then
\[\int_{a}^{b} f=\int_{a}^{p-} f+\int_{p}^{p} f+\int_{p+}^{b} f.\]
The term
\[\int_{p}^{p} f=\int_{[p, p]} f\]
is necessary if \(R S\)- or \(L S\)-integrals are used.
Finally, if \(A\) contains several singularities, it must be split into subintervals, each with at most one endpoint singularity; and \(\int_{a}^{b} f\) is split accordingly. We call all such integrals improper or Cauchy (C) integrals. A C-integral is said to converge iff it exists and is finite.
This theory is greatly enriched if in the above definitions, one replaces \(R\)-integrals by Lebesgue integrals, using Lebesgue or LS measure in \(E^{1}.\) (This makes sense even when a Lebesgue integral (proper) does exist; see Theorem 1.) Below, \(m\) shall denote such a measure unless stated otherwise.
C-integrals with respect to \(m\) will be denoted by
\[C \int_{a}^{\infty} f d m, \quad C \int_{[a, b)} f, \quad \text {etc. }\]
"Classical" notation:
\[C \int f(x) d m(x) \text { or } C \int f(x) d x\]
(the latter if \(m\) is Lebesgue measure). We omit the "C" if confusion with proper integrals \(\int_{a}^{x} f\) is unlikely.
Note 1. C-integrals are limits of integrals, not integrals proper. Yet they may equal the latter (Theorem 1 below) and then may be used to compute them.
Caution. "Singularities" in \([a, b]\) may affect the primitive used in computations (cf. Problem 4 in §1). Then \([a,b]\) must be split (see above), and \(C \int_{a}^{b} f\) splits accordingly. (Additivity applies to C-integrals; see Problem 9, below.)
(A) The integral
\[L \int_{-1}^{1 / 2} \frac{d x}{x^{2}}\]
has a singularity at \(0.\) By Theorem 1 below, we get
\[\begin{aligned} L \int_{-1}^{1 / 2} \frac{d x}{x^{2}} &=\int_{-1}^{0-} \frac{d x}{x^{2}}+\int_{0+}^{1 / 2} \frac{d x}{x^{2}} \\ &=\lim _{x \rightarrow 0^{-}}\left(-\frac{1}{x}-1\right)+\lim _{x \rightarrow 0+}\left(-2+\frac{1}{x}\right)=\infty+\infty=\infty. \end{aligned}\]
(B) We have
\[C \int_{1 / 2}^{\infty} \frac{d x}{x^{2}}=\lim _{x \rightarrow \infty}\left(-\frac{1}{x}+2\right)=2.\]
Hence
\[C \int_{-1}^{\infty} \frac{d x}{x^{2}}=C \int_{-1}^{1 / 2} \frac{d x}{x^{2}}+C \int_{1 / 2}^{\infty} \frac{d x}{x^{2}}=\infty+2=\infty.\]
(C) The integral
\[L \int_{-1}^{1} \frac{|x|}{x} d x\]
has no singularities (consider deleted globes about \(0\)). The primitive \(F(x)=|x|\) exists (example (b) in Chapter 5, §5); so
\[L \int_{-1}^{1} \frac{|x|}{x} d x=\left.|x|\right|_{-1} ^{1}=0.\]
In the rest of this section, we state our theorems mainly for
\[C \int_{a}^{\infty} f,\]
but they apply, with similar proofs, to
\[C \int_{-\infty}^{\infty} f, \quad C \int_{a}^{b-} f, \quad \text {etc. }\]
The measure \(m\) is as explained above.
Let \(A=[a, \infty), f : E^{1} \rightarrow E\) (\(E\) complete).

(ii) The map \(f\) is \(m\)-integrable on \(A\) iff
\[C \int_{a}^{\infty}|f|<\infty\]
and \(f\) is \(m\)-measurable on \(A;\) then again,
\[C \int_{a}^{\infty} f d m=\int_{A} f d m.\]
- Proof
-
(i) Let \(f \geq 0\) on \(A.\) By the rules of Chapter 8, §5, \(\int_{A} f\) is always defined for such \(f;\) so we may set
\[F(x)=\int_{a}^{x} f d m, \quad x \geq a.\]
Then by Theorem 1(f) in Chapter 8, §5, \(F \uparrow\) on \(A;\) for \(a \leq x \leq y\) implies
\[F(x)=\int_{a}^{x} f \leq \int_{a}^{y} f=F(y).\]
Now, by the properties of monotone limits,
\[\lim _{x \rightarrow \infty} F(x)=\lim _{x \rightarrow \infty} \int_{a}^{x} f=C \int_{a}^{\infty} f\]
exists in \(E^{*};\) so by Theorem 1 of Chapter 4, §2, it can be found by making \(x\) run over some sequence \(x_{k} \rightarrow \infty,\) say, \(x_{k}=k\).
Thus set
\[A_{k}=[a, k], \quad k=1,2, \ldots.\]
Then \(\left\{A_{k}\right\} \uparrow\) and
\[\bigcup A_{k}=A=[a, \infty),\]
i.e., \(A_{k} \nearrow A\).
Moreover, by Note 4 in Chapter 8, §5, the set function \(s=\int f\) is \(\sigma\)-additive and semifinite \((\geq 0).\) Thus by Theorem 2 of Chapter 7, §4 (left continuity)
\[\int_{A} f d m=\lim _{k \rightarrow \infty} \int_{A_{k}} f=\lim _{k \rightarrow \infty} \int_{a}^{k} f=C \int_{a}^{\infty} f,\]
proving (i).
(ii) By clause (i),
\[C \int_{a}^{\infty}|f|=\int_{A}|f| d m\]
exists, as \(|f| \geq 0.\) Hence
\[C \int_{a}^{\infty}|f|<\infty\]
plus measurability amounts to integrability (Theorem 2 of Chapter 8, §6).
Moreover,
\[C \int_{a}^{\infty}|f|<\infty\]
implies the convergence of \(C \int_{a}^{\infty} f\) (see Corollary 1 below). Thus as
\[\lim _{x \rightarrow \infty} \int_{a}^{x} f\]
exists, we proceed exactly as before (here \(s=\int f\) is finite), proving (ii) also.\(\quad \square\)
Note 2. If \(E \subseteq E^{*},\) formula (1) results even if \(f\) is not \(m\)-measurable.
Note 3. While \(f\) cannot be integrable unless \(|f|\) is (Corollary 2 of Chapter 8, §6), it can happen that
\[C \int f\]
converges even if
\[C \int|f|=\infty\]
(this is called conditional convergence). A case in point is
\[C \int_{0}^{\infty} \frac{\sin x}{x} d x;\]
see Problem 8.
Thus \(C\)-integrals may be finite where proper integrals are \(\infty\) or fail to exist (a great advantage!). Yet they are deficient in other respects (see Problem 9(c)).
For our next theorem, we need the previously "starred" Theorem 2 in Chapter 4, (Review it!) As we shall see, C-integrals resemble infinite series.
Let \(A=[a, \infty), f : E^{1} \rightarrow E, E\) complete.
Suppose
\[\int_{a}^{x} f d m\]
exists for each \(x \in A.\) (This is automatic if \(E \subseteq E^{*};\) see Chapter 8, §5.)

- Proof
-
By additivity (Chapter 8, §5, Theorem 2; Chapter 8, §7, Theorem 3),
\[\int_{a}^{x} f=\int_{a}^{v} f+\int_{v}^{x} f\]
if \(a \leq v \leq x<\infty.\) (In case \(E \subseteq E^{*},\) this holds even if \(f\) is not integrable; see Theorem 2, of Chapter 8, §5.)

Then for any \(\varepsilon>0,\) there is some
\[b \in[a, \infty)=A\]
such that
\[\left|\int_{a}^{x} f d m-r\right|<\frac{1}{2} \varepsilon \quad \text { for } x \geq b.\]
(Why may we use the standard metric here?)
Taking \(x=b,\) we get (2').
Hence by the triangle law, (2) follows also. Thus this \(b\) satisfies (2).
Conversely, suppose such a \(b\) exists for every given \(\varepsilon>0.\) Fixing \(b,\) we thus have (2) and (2'). Now, with \(A=[a, \infty),\) define \(F : A \rightarrow E\) by
\[F(x)=\int_{a}^{x} f d m,\]
so
\[C \int_{a}^{\infty} f=\lim _{x \rightarrow \infty} F(x)\]
if this limit exists.   The existence of
\[C \int_{a}^{\infty} f d m=\lim _{x \rightarrow \infty} F(x) \neq \pm \infty\]
now follows by Theorem 2 of Chapter 4, §2. (We shall henceforth presuppose this "starred" theorem.)
Thus all is proved.\(\quad \square\)
Under the same assumptions as in Theorem 2, the convergence of
\[C \int_{a}^{\infty}|f| d m\]
implies that of
\[C \int_{a}^{\infty} f d m.\]
Indeed,
\[\left|\int_{v}^{x} f\right| \leq \int_{v}^{x}|f|\]
(Theorem 1(g) of Chapter 8, §5, and Problem 10 in Chapter 8, §7).
Note 4. We say that \(C \int f\) converges absolutely iff \(C \int|f|\) converges.
If \(|f| \leq|g|\) a.e. on \(A=[a, \infty)\) for some \(f, g : E^{1} \rightarrow E,\) then
\[C \int_{a}^{\infty}|f| \leq C \int_{a}^{\infty}|g|;\]
so the convergence of
\[C \int_{a}^{\infty}|g|\]
implies that of
\[C \int_{a}^{\infty}|f|.\]
For as \(|f|,|g| \geq 0,\) Theorem 1 reduces all to Theorem 1(c) of Chapter 8, §5.
Note 5. As we see, absolutely convergent C-integrals coincide with proper (finite) Lebesgue integrals of nonnegative or \(m\)-measurable maps. For conditional (i.e., nonabsolute) convergence, see Problems 6-9, 13, and 14.
Iterated C-Integrals. Let the product space \(X \times Y\) of Chapter 8, §8 be
\[E^{1} \times E^{1}=E^{2},\]
and let \(p=m \times n,\) where \(m\) and \(n\) are Lebesgue measure or LS measures in \(E^{1}\). Let
\[A=[a, b], B=[c, d], \text { and } D=A \times B.\]

As usual, we write "\(d x\)" for "\(d m(x)\)" if \(m\) is Lebesgue measure in \(E^{1};\) similarly for \(n.\)

If the integral (3) is finite, we say that it converges. Again, convergence is absolute if it holds also with \(f\) replaced by \(|f|,\) and conditional otherwise.
Let \(f : E^{2} \rightarrow E^{*}\) be \(p\)-measurable on \(E^{2}\) (\(p, m, n\) as above).
As \(|f| \geq 0,\) clause (i*) easily follows from our present Theorem 1(i) and Theorem 3(i) of Chapter 8, §8.
Similarly, clause (ii*) follows from Theorem 3(ii) of the same section.\(\quad \square\)
Let \(p=\) Lebesgue measure in \(E^{2}.\) Suppose \(f : E^{2} \rightarrow E^{*}\) is \(p\)-measurable on \(E^{2}.\) Set
\[F(r, \theta)=f(r \cos \theta, r \sin \theta), \quad r>0.\]

- Proof Outline
-
First let \(f=C_{D},\) with \(D\) a "curved rectangle"
\[\left\{(r, \theta) | r_{1}<r \leq r_{2}, \theta_{1}<\theta \leq \theta_{2}\right\}\]
for some \(r_{1}<r_{2}\) in \(X=(0, \infty)\) and \(\theta_{1}<\theta_{2}\) in \(Y=[0,2 \pi).\) By elementary geometry (or calculus), the area
\[p D=\frac{1}{2}\left(r_{2}^{2}-r_{1}^{2}\right)\left(\theta_{2}-\theta_{1}\right)\]
(the difference between two circular sectors).
For \(f=C_{D},\) formulas (a) and (b) easily follow from
\[p D=L \int_{E^{2}} C_{D} d p.\]
(Verify!) Now, curved rectangles behave like half-open intervals
\[\left(r_{1}, r_{2}\right] \times\left(\theta_{1}, \theta_{2}\right]\]
in \(E^{2},\) since Theorem 1 in Chapter 7, §1, and Lemma 2 of Chapter 7, §2, apply with the same proof. Thus they form a semiring generating the Borel field in \(E^{2}\).
Hence show (as in Chapter 8, §8 that Theorem 4 holds for \(f=C_{D}(D \in \mathcal{B}\)). Then take \(D \in \mathcal{M}^{*}\). Next let \(f\) be elementary and nonnegative, and so on, as in Theorems 2 and 3 in Chapter 8, §8.\(\quad \square\)
(D) Let
\[J=L \int_{0}^{\infty} e^{-x^{2}} d x;\]
so
\[\begin{aligned} J^{2} &=\left(C \int_{0}^{\infty} e^{-x^{2}} d x\right)\left(C \int_{0}^{\infty} e^{-y^{2}} d y\right) \\ &=C \int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y. \quad \text {(Why?)} \end{aligned}\]
Set
\[f(x, y)=e^{-\left(x^{2}+y^{2}\right)}\]
in Theorem 4(b).  Thus
\[C \int_{0}^{\infty} e^{-x^{2}} d x=L \int_{0}^{\infty} e^{-x^{2}} d x=\sqrt{\frac{1}{4} \pi}=\frac{1}{2} \sqrt{\pi}.\]


5.4: Is there an association? Analysis of tables
Contingency tables
How do you compare samples of categorical data? These frequently are text only, there are have no numbers, like in classic “Fisher’s tea drinker” example\(^{[1]}\). A British woman claimed to be able to distinguish whether milk or tea was added to the cup first. To test, she was given 8 cups of tea, in four of which milk was added first:
Code \(\PageIndex{1}\) (R):
The only way is to convert it to numbers, and the best way to convert is to count cases, make contingency table :
Code \(\PageIndex{2}\) (R):
Contingency table is not a matrix or data frame, it is the special type of R object called “table”.
In R formula language, contingency tables are described with simple formula
~ factor(s)
To use this formula approach, run xtabs() command:
Code \(\PageIndex{3}\) (R):
(More than one factors have to be connected with + sign.)
If there are more than two factors, R can build a multidimensional table and print it as a series of two-dimensional tables. Please call the embedded Titanic data to see how 3-dimensional contingency table looks. A “flat” contingency table can be built if all the factors except one are combined into one multidimensional factor. To do this, use the command ftable() :
Code \(\PageIndex{4}\) (R):
The function table can be used simply for calculation of frequencies (including missing data, if needed):
Code \(\PageIndex{5}\) (R):
The function mosaicplot() creates a graphical representation of a contingency table (Figure \(\PageIndex{1}\)):
Code \(\PageIndex{6}\) (R):
(We used mosaicplot() command because apply() outputted a matrix. If the data is a “table” with more than one dimension, object, plot() command will output mosaic plot by default.)
Contingency tables are easy enough to make even from numerical data. Suppose that we need to look on association between month and comfortable temperatures in New York. If the temperatures from 64 to 86°F (from 18 to 30°C) are comfort temperatures, then:
Code \(\PageIndex{7}\) (R):
Now we have two categorical variables, comfort and airquality$Month and can proceed to the table:
Code \(\PageIndex{8}\) (R):
Spine plot (Figure \(\PageIndex{2}\)) is good for this kind of table, it looks like a visually advanced “hybrid” between histogram, barplot and mosaic plot:
Code \(\PageIndex{9}\) (R):
(Another variant to plot these two-dimensional tables is the dotchart() , please try it yourself. Dotchart is good also for 1-dimensional tables, but sometimes you might need to use the replacement Dotchart1() from asmisc.r —it keeps space for y axis label.)
Table tests
To find if there is an association in a table, one should compare two frequencies in each cell: predicted (theoretical) and observed. The serious difference is the sign of association. Null and alternative hypotheses pairs are typically:
-
Null: independent distribution of factors \(\approx\) no pattern present \(\approx\) no association present
-
Alternative: concerted distribution of factors \(\approx\) pattern present \(\approx\) there is an association
In terms of p-values:
Function chisq.test() runs a chi-squared test , one of two most frequently used tests for contingency tables. Two-sample chi-squared (or \(\chi^2\)) test requires either contingency table or two factors of the same length (to calculate table from them first).
Now, what about the table of temperature comfort? assocplot(comf.month) shows some “suspicious” deviations. To check if these are statistically significant:
Code \(\PageIndex{10}\) (R):
No, they are not associated. As before, there is nothing mysterious in these numbers. Everything is based on differences between expected and observed values:
Code \(\PageIndex{11}\) (R):
(Note how expected values calculated and how they look: expected (null) are equal proportions between both rows and columns. June and September have 30 days each, hence slight differences in values—but not in expected proportions.)
Let us see now whether hair color and eye color from the 3-dimensional embedded HairEyeColor data are associated. First, we can examine associations graphically with assocplot() (Figure \(\PageIndex{3}\)):
Code \(\PageIndex{12}\) (R):
(Instead of apply() used in the previous example, we employed margin.table() which essentially did the same job.)
Association plot shows several things: the height of bars reflects the contribution of each cell into the total chi-squared, this allows, for example, to detect outliers. Square of rectangle corresponds with difference between observed and expected value, thus big tall rectangles indicate more association (to understand this better, compare this current plot with assocplot(comf.month) ). Color and position of rectangle show the sign of the difference.
Overall, it is likely that there is an association. Now we need to check this hypothesis with a test:
Code \(\PageIndex{13}\) (R):
The chi-squared test takes as null hypothesis “no pattern”, “no association”. Therefore, in our example, since we reject the null hypothesis, we find that the factors are associated.
And what about survival on the “Titanic”?
Code \(\PageIndex{14}\) (R):
Yes (as reader might remember from the famous movie), survival was associated with being in the particular class.
General chi-squared test shows only if asymmetry presents anywhere in the table. This means that if it is significant, then at least one group of passengers has the difference in survival. Like ANOVA, test does not show which one. Post hoc , or pairwise table test is able do show this:
Code \(\PageIndex{15}\) (R):
From the table of p-values, it is apparent that 3rd class and crew members were not different by survival rates. Note that post hoc tests apply p-value adjustment for multiple comparisons ; practically, it means that because 7 tests were performed simultaneously, p-values were magnified with some method (here, Benjamini & Hochberg method is default).
The file seedlings.txt contains results of an experiment examining germination of seeds infected with different types of fungi. In all, three fungi were tested, 20 seeds were tested for each fungus, and therefore with the controls 80 seeds were tested. Do the germination rates of the infected seeds differ?
Let us examine now the more complicated example. A large group of epidemiologists gathered for a party. The next morning, many woke up with symptoms of food poisoning. Because they were epidemiologists, they decided to remember what each of them ate at the banquet, and thus determine what was the cause of the illness. The gathered data take the following format:
Code \(\PageIndex{16}\) (R):
(We used head() here because the table is really long.)
The first variable ( ILL ) tells whether the participant got sick or not (1 or 2 respectively); the remaining variables correspond to different foods.
A simple glance at the data will not reveal anything, as the banquet had 45 participants and 13 different foods. Therefore, statistical methods must be used. Since the data are nominal, we will use contingency tables:
Code \(\PageIndex{17}\) (R):
(First, we ran ILL variable against every column and made a list of small contingency tables. Second, we converted list into 3-dimensional array, just like the Titanic data is, and also made sensible names of dimensions.)
Now our data consists of small contingency tables which are elements of array:
Code \(\PageIndex{18}\) (R):
(Note two commas which needed to tell R that we want the third dimension of the array.)
Now we need a kind of stratified (with every type of food) table analysis. Since every element in the tox.2 is \(2\times2\) table, fourfold plot will visualize this data well (Figure \(\PageIndex{4}\)):
Code \(\PageIndex{19}\) (R):
(In fourfold plots, association corresponds with the difference between two pairs of diagonal sectors. Since we test multiple times, confidence rings are suppressed.)
There are some apparent differences, especially for CAESAR , BREAD and TOMATO . To check their significance, we will at first apply chi-squared test multiple times and check out p-values:
Code \(\PageIndex{20}\) (R):
(An apply() allows us not to write the code for the test 13 times. You may omit cbind() since it used only to make output prettier. There were multiple warnings, and we will return to them soon.)
The result is that two foods exhibit significant associations with illness—Caesar salad and tomatoes. The culprit is identified! Almost. After all, it is unlikely that both dishes were contaminated. Now we must try to determine what was the main cause of the food poisoning. We will return to this subject later.
Let us discuss one more detail. Above, we applied chi-squared test simultaneously several times. To account for multiple comparisons, we must adjust p-values , magnify them in accordance with the particular rule, for example, with widely known Bonferroni correction rule, or with (more reliable) Benjamini and Hochberg correction rule like in the following example:
Code \(\PageIndex{21}\) (R):
Now you know how to apply p-value corrections for multiple comparisons. Try to do this for our toxicity data. Maybe, it will help to identify the culprit?
The special case of chi-squared test is the goodness-of-fit test , or G-test . We will apply it to the famous data, results of Gregor Mendel first experiment. In this experiment, he crossed pea plants which grew out of round and angled seeds. When he counted seeds from the first generation of hybrids, he found that among 7,324 seeds, 5,474 were round and 1850 were angled. Mendel guessed that true ratio in this and six other experiments is 3:1\(^{[2]}\):
Code \(\PageIndex{22}\) (R):
Goodness-of-fit test uses the null that frequencies in the first argument (interpreted as one-dimensional contingency table) are not different from probabilities in the second argument. Therefore, 3:1 ratio is statistically supported. As you might note, it is not radically different from the proportion test explained in the previous chapter.
Without p parameter, G-test simply checks if probabilities are equal. Let us check, for example, if numbers of species in supergroups of living organisms on Earth are equal:
Code \(\PageIndex{23}\) (R):
Naturally, numbers of species are not equal between supergroups. Some of them like bacteria (supergroup Monera) have surprisingly low number of species, others like insects (supergroup Ecdysozoa)—really large number (Figure \(\PageIndex{5}\)).
Chi-squared test works well when the number of cases per cell is more then 5. If there are less cases, R gives at least three workarounds.
First, instead of p-value estimated from the theoretical distribution, there is a way to calculate it directly, with Fisher exact test . Tea drinker table contains less then 5 cases per cell so it is a good example:
Code \(\PageIndex{24}\) (R):
Fisher test checks the null if odds ratio is just one. Although in this case, calculation gives odds ratio \((3\mathbin{:}1)/(1\mathbin{:}3)=9\), there are only 8 observations, and confidence interval still includes one. Therefore, contrary to the first impression, the test does not support the idea that aforementioned woman is a good guesser.
Fourfold plot (please check it yourself) gives the similar result:
Code \(\PageIndex{25}\) (R):
While there is apparent difference between diagonals, confidence rings significantly intersect.
Fisher test is computationally intensive so it is not recommended to use it for large number of cases.
The second workaround is the Yates continuity correction which in R is default for chi-squared test on 22 tables. We use now data from the original Yates (1934)\(^{[3]}\) publication, data is taken from study of the influence of breast and artificial feeding on teeth formation:
Code \(\PageIndex{26}\) (R):
(Note the warning in the end.)
Yates correction is not a default for the summary.table() function:
Code \(\PageIndex{27}\) (R):
(Note different p-value: this is an effect of no correction. For all other kind of tables (e.g., non \(2\times2\)), results of chisq.test() and summary.table() should be similar.)
The third way is to simulate chi-squared test p-value with replication:
Code \(\PageIndex{28}\) (R):
(Note that since this algorithm is based on random procedure, p-values might differ.)
How to calculate an effect size for the association of categorical variables? One of them is odds ratio from the Fisher test (see above). There are also several different effect size measures changing from 0 (no association) to (theoretically) 1 (which is an extremely strong association). If you do not want to use external packages, one of them, \(\phi\) coefficient is easy to calculate from the \(\chi\)-squared statistic.
Code \(\PageIndex{29}\) (R):
\(\Phi\) coefficient works only for two binary variables. If variables are not binary, there are Tschuprow’s T and Cramer’s V coefficients. Now it is better to use the external code from the asmisc.r distributing with this book:
Code \(\PageIndex{30}\) (R):
R package vcd has function assocstats() which calculates odds ratio, \(\phi\), Cramer’s V and several other effect measures.
In the open repository, file cochlearia.txt contains measurements of morphological characters in several populations (locations) of scurvy-grass, Cochlearia . One of characters, binary IS.CREEPING reflects the plant life form: creeping or upright stem. Please check if numbers of creeping plants are different between locations, provide effect sizes and p-values.
There are many table tests. For example, test of proportions from the previous chapter could be easily extended for two samples and therefore could be used as a table test. There is also mcnemar.test() which is used to compare proportions when they belong to same objects ( paired proportions ). You might want to check the help (and especially examples) in order to understand how they work.
In the betula (see above) data, there are two binary characters: LOBES (position of lobes on the flower bract) and WINGS (the relative size of fruit wings). Please find if proportions of plants with 0 and 1 values of LOBES are different between location 1 and location 2.
Are proportions of LOBES and WING values different in the whole dataset?
The typical sequence of procedures related with analysis of tables is listed below:
-
Check the phenomenon of association:
table()
,
xtabs()
-
Plot it first:
mosaicplot()
,
spineplot()
,
assocplot()
-
Decide is association is statistically significant:
chisq.test()
,
fisher.test()
-
Measure how strong is an association:
VTCoeffs()
-
Optionally, if there are more then two groups per case involved, run
post hoc
pairise tests with the appropriate correction:
pairwise.Table2.test()
To conclude this “differences” chapter, here is the Table \(\PageIndex{1}\) which will guide the reader through most frequently used types of analysis. Please note also the much more detailed Table 6.1.1 in the appendix.
| Normal | Non-normal | ||
|---|---|---|---|
| measurement or ranked | nominal | ||
| \(\mathbf{=2}\) samples | Student’s test | Wilcoxon test |
Chi-squared test
(+ post-hoc test) |
| \(\mathbf{>2}\) samples |
ANOVA or one-way + some post hoc test |
Kruskall-Wallis + some post hoc test |
Table \(\PageIndex{1}\) : Methods, most frequently used to analyze differences and patterns. This is the simplified variant of Table 6.1.1.
References
1. Fisher R.A. 1971. The design of experiments. 9th ed. P. 11.
2. Mendel G. 1866. Versuche über Pflanzen-Hybriden. Verhandlungen des naturforschenden Vereines in Brünn. Bd. 4, Abhandlungen: 12. http://biodiversitylibrary.org/page/40164750
3. Yates F. 1934. Contingency tables involving small numbers and the \(x^2\) test. Journal of the Royal Statistical Society. 1(2): 217–235.


3.1.2: Scientific Knowledge
-
- Last updated
- Save as PDF
- Anonymous
- LibreTexts
The purpose of science is to create scientific knowledge. Scientific knowledge refers to a generalized body of laws and theories to explain a phenomenon or behavior of interest that are acquired using the scientific method. Laws are observed patterns of phenomena or behaviors, while theories are systematic explanations of the underlying phenomenon or behavior. For instance, in physics, the Newtonian Laws of Motion describe what happens when an object is in a state of rest or motion (Newton’s First Law), what force is needed to move a stationary object or stop a moving object (Newton’s Second Law), and what happens when two objects collide (Newton’s Third Law). Collectively, the three laws constitute the basis of classical mechanics – a theory of moving objects. Likewise, the theory of optics explains the properties of light and how it behaves in different media, electromagnetic theory explains the properties of electricity and how to generate it, quantum mechanics explains the properties of subatomic particles, and thermodynamics explains the properties of energy and mechanical work. An introductory college level text book in physics will likely contain separate chapters devoted to each of these theories. Similar theories are also available in social sciences. For instance, cognitive dissonance theory in psychology explains how people react when their observations of an event is different from what they expected of that event, general deterrence theory explains why some people engage in improper or criminal behaviors, such as illegally download music or commit software piracy, and the theory of planned behavior explains how people make conscious reasoned choices in their everyday lives.
The goal of scientific research is to discover laws and postulate theories that can explain natural or social phenomena, or in other words, build scientific knowledge. It is important to understand that this knowledge may be imperfect or even quite far from the truth. Sometimes, there may not be a single universal truth, but rather an equilibrium of “multiple truths.” We must understand that the theories, upon which scientific knowledge is based, are only explanations of a particular phenomenon, as suggested by a scientist. As such, there may be good or poor explanations, depending on the extent to which those explanations fit well with reality, and consequently, there may be good or poor theories. The progress of science is marked by our progression over time from poorer theories to better theories, through better observations using more accurate instruments and more informed logical reasoning.
We arrive at scientific laws or theories through a process of logic and evidence. Logic (theory) and evidence (observations) are the two, and only two, pillars upon which scientific knowledge is based. In science, theories and observations are interrelated and cannot exist without each other. Theories provide meaning and significance to what we observe, and observations help validate or refine existing theory or construct new theory. Any other means of knowledge acquisition, such as faith or authority cannot be considered science.


8.5: Library Catalog
When to Use It
Use the library catalog to search for items that you can use and check out in the library. Some items may require a university/college ID for use.
Note that the library catalog:
- May include a table of contents (especially for print materials) and a link to full-text digital content.
- Does not contain specific journal articles. The catalog can only tell you whether a journal or other periodical title is available.
How to Use It
To access the catalog you should go to the home page of the library and look for the term catalog, books, OneSearch, or a similar term. From there, you can do a search by resource type (books, video, course reserves, etc…) or other options.
Search Types
The catalog allows searching by author, title, subject, and keyword as well as specialty numbers such as the Library of Congress call number and ISBN (International Standard Book Number). There is also an option for an advanced search.
Additional tips:
- Keyword searches are the broadest search, as they search all information in an item record. (The search tips in Precision Searching, are based on using keywords.)
- Subjects are a very specific set of terms that are helpful for precision searches. Often, the easiest way to find subject terms is to do a keyword search first and then look at the results for the subject terms for those that are good matches for your topic. There is more about subject heading searching in Library Databases later in this section.
- The Advanced Search screen allows a few additional search capabilities, such as multiple search fields to narrow the scope of a search term. You can also limit by year range, language, location, or format.