7.3: What is Written, Where?
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Before presenting the data, I should acknowledge its arbitrary nature. I pass over the ‘Archanes Script’ (but see Flouda, this volume, Whittaker, this volume) and broader prepalatial seal use, which could well represent the beginnings of writing and administration on Crete (Schoep 1999 a : 268). Making sense of seal use throughout the Bronze Age is rather tricky; at times, it could arguably be considered a form of writing, particularly those seals bearing Cretan Hieroglyphic (hereafter CH) signs, while at others, seal use is better viewed as writing’s “quasi-complementary, quasi-supplementary and quasi-independent partner” (Palaima 1990: 83). While my focus is on those material supports which bear writing, it is necessary, as will become clear, to include in the discussion those sealing shapes which do not. I leave to the side more marginal, less well-understood, writing practices such as potters’ and masons’ marks.
The CH corpus comprises around 200 clay documents, 136 seals and 16 miscellaneous items (incised and painted pots, and an incised stone block, e.g. Figure 1 ). These are distributed widely across central and north-eastern Crete ( Figure 4 shows the key sites mentioned throughout), with seal impressions and a prism-shaped stamp seal found on Samothrace and Kythera respectively (Lebessi et al. 1995: 63; Olivier and Godart 1996: 20–21, 22; Tsipopoulou and Hallager 1996: 165). The four key deposits are: Quartier Mu, Malia (an elite residential complex); the Dépôt Hiéroglyphique , Palace of Malia; the Hieroglyphic Deposit, Knossos; and Petras (all palatial buildings). The clay documents comprise crescents (all terms are defined below), noduli , flat-based sealings, cones, medallions, labels, three- and four-sided bars, and tablets (Olivier and Godart 1996: 10–11; Younger 1996–1997: 396). There are also substantial numbers of direct object sealings, which show seal impressions but no incised writing (Krzyszkowska 2005: 99). One should note that, throughout the Bronze Age, while seals come in a huge range of shapes and materials, impressions are almost always made by stamping the seal on the clay, not rolling it. CH is used in the First Palace Period, Middle Minoan II, at Quartier Mu and Petras, and into the early Second Palace Period, Middle Minoan III, at Knossos and Malia Palaces (Olivier and Godart 1996: 27–28; Schoep 2001 a : 157–158).
There are around 1370 Linear A (LA) clay documents (Schoep 2002 a : 38; e.g. Figure 2 ). There are some 300 tablets, together with a few three- or four-sided bars, and a single ‘label’ comprised of a flat, oblong piece of clay, pierced at its pointed end (Hallager 1996: 33, 37; Schoep 2002 a : 16, 20–21). The sealings can be classified as noduli, flat-based nodules, roundels, and single-hole and two-hole hanging nodules (Hallager 1996: 35–37). Direct object sealings are restricted to the First Palace Period (Krzyszkowska 2005: 155). LA is also incised, engraved or painted on a range of other supports, including stone vessels, gold and silver pins and a ring, walls, pots and a terracotta figurine; these objects are found in religious and domestic contexts, and their distribution is mainly concentrated in central Crete (Schoep 2002 a : 13–14). LA is used during the First Palace Period at Phaistos; this use proliferates during the Second Palace Period, Middle Minoan III to Late Minoan IB, when it is widely distributed across Crete and on Thera, Melos and Kea (Karnava 2008: 418; Schoep 2002 a : 17–19, 21).
There are over 5000 inscribed clay documents in the Linear B (LB) corpus (e.g. Figure 3 ), the most numerous of which are tablets. The only LB sealing type which can bear an inscription is the gable-shaped hanging nodule, and there is also a very small number of clay ‘labels’ (Krzyszkowska 2005: 280). Sealing types without inscriptions are irregular hanging nodules, combination sealings, direct object sealings, stoppers and noduli (Krzyszkowska 2005: 280). The principal deposits are the palatial sites of Knossos and Chania on Crete, and Mycenae, Thebes, Tiryns and Pylos in mainland Greece; the Room of the Chariot Tablets, Knossos, is probably the earliest use of LB, in Late Minoan IIIA1, with the documents from Chania, Thebes and Mycenae dating from Late Minoan / Helladic IIIA2–late into IIIB1–late, and those from Tiryns, Midea and Pylos coming from Late Helladic IIIB / C (Driessen 2008: 76; Shelmerdine and Bennet 2008: 292).
There are also nearly 180 examples of Inscribed Stirrup Jars (ISJs), a type of large coarse-ware storage or transport jar on which an LB inscription is painted before firing, dated to roughly Late Minoan/Helladic IIIB (van Alfen 2008: 235, 238). They are found at several mainland sites, although ceramic analysis indicates that most originated from the Chania region in Western Crete (van Alfen 2008: 235). Finally, LB written on non-administrative objects is extremely rare (Palaima 1987 a : 502).


5.7: Sealed Bids Method
The Sealed Bids method provides a method for discrete fair division, allowing for the division of items that cannot be split into smaller pieces, like a house or a car. Because of this, the method requires that all parties have a large amount of cash at their disposal to balance out the difference in item values.
The method begins by compiling a list of items to be divided. Then:
1) Each party involved lists, in secret, a dollar amount they value each item to be worth. This is their sealed bid.
2) The bids are collected. For each party, the value of all the items is totaled, and divided by the number of parties. This defines their fair share.
3) Each item is awarded to the highest bidder.
4) For each party, the value of all items received is totaled. If the value is more than that party’s fair share, they pay the difference into a holding pile. If the value is less than that party’s fair share, they receive the difference from the holding pile. This ends the initial allocation.
5) In most cases, there will be a surplus, or leftover money, in the holding pile. The surplus is divided evenly between all the players. This produces the final allocation.
While the assumptions we made for fair division methods specified that an arbitrator should not be necessary, it is common for an independent third party to collect the bids and announce the outcome. While not technically necessary, since the method can be executed without a third party involved, this protects the secrecy of the bids, which can sometimes help avoid resentment or bad feelings between the players.
Sam and Omar have cohabitated for the last 3 years, during which time they shared the expense of purchasing several items for their home. Sam has accepted a job in another city, and now they find themselves needing to divide their shared assets.
Each records their value of each item, as shown below.
\(\begin{array}{|l|l|l|}
\hline & \textbf { Sam } & \textbf { Omar } \\
\hline \textbf { Couch } & \$ 150 & \$ 100 \\
\hline \textbf { TV } & \$ 200 & \$ 250 \\
\hline \textbf { Video game system } & \$ 250 & \$ 150 \\
\hline \textbf { Surround sound system } & \$ 50 & \$ 100 \\
\hline
\end{array}\)
Solution
Sam’s total valuation of the items is \(\$150+\$200+\$250+\$50 = \$650\), making a fair share for Sam \(\$650/2 = \$325\).
Omar’s total valuation of the items is \(\$100+\$250+\$150+\$100 = \$600\), making a fair share for Omar \(\$600/2 = \$300\).
Each item is now awarded to the highest bidder. Sam will receive the couch and video game system, providing \(\$150+\$250 = \$400\) of value to Sam. Since this exceeds his fair share, he has to pay the difference, $75, into a holding pile.
Omar will receive the TV and surround sound system, providing \(\$250+\$100=\$350\) in value. This is more than his fair share, so he has to pay the difference, $50, into the holding pile.
Thus, in the initial allocation, Sam receives the couch and video game system and pays $75 into the holding pile. Omar receives the TV and surround sound system and pays $50 into the holding pile. At this point, both players would feel they have received a fair share.
There is now $125 remaining in the holding pile. This is the surplus from the division. This is now split evenly, and both Sam and Omar are given back $62.50. Since Sam had paid in $75, the net effect is that he paid $12.50. Since Omar had originally paid in $50, the net effect is that he receives $12.50.
Thus, in the final allocation, Sam receives the couch and video game system and pays $12.50 to Omar. Omar receives the TV and surround sound system and receives $12.50. At this point, both players feel they have received more than a fair share.
Four small graphic design companies are merging operations to become one larger corporation. In this merger, there are a number of issues that need to be settled. Each company is asked to place a monetary value (in thousands of dollars) on each issue:
\(\begin{array}{|l|l|l|l|l|}
\hline & \textbf { Super Designs } & \textbf { DesignByMe } & \textbf { LayoutPros } & \textbf { Graphix } \\
\hline \textbf { Company name } & \$ 5 & \$ 3 & \$ 3 & \$ 6 \\
\hline \textbf { Company location } & \$ 8 & \$ 9 & \$ 7 & \$ 6 \\
\hline \textbf { CEO } & \$ 10 & \$ 5 & \$ 6 & \$ 7 \\
\hline \textbf { Chair of the board } & \$ 7 & \$ 6 & \$ 6 & \$ 8 \\
\hline
\end{array}\)
Solution
We can then calculate for each company:
\(\begin{array}{|l|l|l|l|l|}
\hline & \textbf { Super Designs } & \textbf { DesignByMe } & \textbf { LayoutPros } & \textbf { Graphix } \\
\hline \begin{array}{l}
\text { Total value of all } \\
\text { issues }
\end{array} & \$ 30 & \$ 23 & \$ 22 & \$ 27 \\
\hline \text { Fair share } & \$ 7.50 & \$ 5.75 & \$ 5.50 & \$ 6.75 \\
\hline
\end{array}\)
The items would then be allocated to the company that bid the most for each.
Company name would be awarded to Graphix
Company location would be awarded to DesignByMe
CEO would be awarded to Super Designs
Chair of the board would be awarded to Graphix
For each company, we calculate the total value of the items they receive, and how much they get or pay in the initial allocation
\(\begin{array}{|l|l|l|l|l|}
\hline & \textbf { Super Designs } & \textbf { DesignByMe } & \textbf { LayoutPros } & \textbf { Graphix } \\
\hline \text { Total value of issues awarded} & \$ 10 & \$ 9 & \$ 0 & \$ 14 \\
\hline \text { Amount they } & \$ 7.50-\$ 10 & \$ 5.75-\$ 9 & \$ 5.50-\$ 0 & \$ 6.75-\$ 14 \\
\text { pay/get } & \text { Pay } \$ 2.50 & \text { Pay } \$ 3.25 & \text { Get } \$ 5.50 & \text { Pay } \$ 7.25 \\
\hline
\end{array}\)
After the initial allocation, there is a total of \($2.50+\$3.25-\$5.50+\$7.25 = \$7.50\) in surplus. Dividing that evenly, each company gets $1.875 (approximately)
\(\begin{array}{|l|l|l|l|l|}
\hline & \textbf { Super Designs } & \textbf { DesignByMe } & \textbf { LayoutPros } & \textbf { Graphix } \\
\hline \text { Amount they } & \text { Pay \$2.50 } & \text { Pay \$3.25 } & \text { Get \$5.50 } & \text { Pay \$7.25 } \\
\text { pay/get } & \text { Get \$1.875 } & \text { Get \$1.875 } & \text { Get \$1.875 } & \text { Get \$1.875 } \\
\hline \text { After surplus } & \text { Pay \$0.625 } & \text { Pay \$1.375 } & \text { Get \$7.375 } & \text { Pay \$5.375 } \\
\hline
\end{array}\)
So in the final allocation,
Super Designs wins the CEO, and pays $625 ($0.625 thousand)
DesignByMe wins the company location and pays $1,375 ($1.375 thousand)
LayoutPros wins no issues, but receives $7,375 in compensation
Graphix wins the company name and chair of the board, and pays $5,375.
Jamal, Maggie, and Kendra are dividing an estate consisting of a house, a vacation home, and a small business. Their valuations (in thousands) are shown below. Determine the final allocation.
\(\begin{array}{|l|l|l|l|}
\hline & \textbf { Jamal } & \textbf { Maggie } & \textbf { Kendra } \\
\hline \textbf { House } & \$ 250 & \$ 300 & \$ 280 \\
\hline \textbf { Vacation home } & \$ 170 & \$ 180 & \$ 200 \\
\hline \textbf { Small business } & \$ 300 & \$ 255 & \$ 270 \\
\hline
\end{array}\)
- Answer
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Jamal’s total value is \(\$250 + \$170 + \$300 = \$720\). His fair share is $240 thousand.
Maggie’s total value is \(\$300 + \$180 + \$255 = \$735\). Her fair share is $245 thousand.
Kendra’s total value is \(\$280 + \$200 + \$270 = \$750\). Her fair share is $250 thousand.
In the initial allocation,
Jamal receives the business, and pays \(\$300 - \$240 = \$60\) thousand into holding.
Maggie receives the house, and pays \(\$300 - \$245 = \$55\) thousand into holding.
Kendra receives the vacation home, and gets \(\$250 - \$200 = \$50\) thousand from holding.
There is a surplus of \(\$60 + \$55 - \$50 = \$65\) thousand in holding, so each person will receive \(\$21,667\) from surplus. In the final allocation,
Jamal receives the business, and pays \(\$38,333\).
Maggie receives the house, and pays \(\$33,333\).
Kendra receives the vacation home, and gets \(\$71,667\).
Fair division does not always have to be used for items of value. It can also be used to divide undesirable items. Suppose Chelsea and Mariah are sharing an apartment, and need to split the chores for the household. They list the chores, assigning a negative dollar value to each item; in other words, the amount they would pay for someone else to do the chore (a per week amount). We will assume, however, that they are committed to doing all the chores themselves and not hiring a maid.
\(\begin{array}{|l|l|l|}
\hline & \textbf { Chelsea } & \textbf { Mariah } \\
\hline \textbf { Vacuuming } & -\$ 10 & -\$ 8 \\
\hline \textbf { Cleaning bathroom } & -\$ 14 & -\$ 20 \\
\hline \textbf { Doing dishes } & -\$ 4 & -\$ 6 \\
\hline \textbf { Dusting } & -\$ 6 & -\$ 4 \\
\hline
\end{array}\)
Solution
We can then calculate fair share:
\(\begin{array}{|l|l|l|}
\hline & \textbf { Chelsea } & \textbf { Mariah } \\
\hline \text { Total value } & -\$ 34 & -\$ 38 \\
\hline \text { Fair Share } & -\$ 17 & -\$ 19 \\
\hline
\end{array}\)
We award to the person with the largest bid. For example, we award vacuuming to Mariah since she dislikes it less (remember -8 > -10).
Chelsea gets cleaning the bathroom and doing dishes. Value: -$18
Mariah gets vacuuming and dusting. Value: -$12
Notice that Chelsea’s fair share is -$17 but she is doing chores she values at -$18. She should get $1 to bring her to a fair share. Mariah is doing chores valued at -$12, but her fair share is -$19. She needs to pay $7 to bring her to a fair share.
This creates a surplus of $6, which will be divided between the two. In the final allocation:
Chelsea gets cleaning the bathroom and doing dishes, and receives \(\$1 + \$3 = \$4\)/week.
Mariah gets vacuuming and dusting, and pays \(\$7 - \$3 = \$4\)/week.


The New Quantum Mechanics
Skills to Develop
- Describe some of the contributions of de Broglie and Heisenberg to "new quantum mechanics"
The "New Quantum Mechanics" was introduced a little after the "Old Quantum Mechanics" which was developed by Planck and Bohr. This is when quantum mechanics became very confusing to people. In the old quantum theory, we could think about particles moving in normal ways. The electrons orbit the nucleus just like planets orbit the sun. But this was wrong. The new quantum theory says that tiny particles behave in ways that are totally different from normal objects that we can see, like basketballs or planets. In fact, it is impossible to know exactly where they are or what they do.
Einstein had convinced physicists that light was a particle (now called photon) in 1905. Later in 1925, de Broglie proposed that particles like electrons could also be waves. De Broglie was a French prince who was initially interested in history, but during World War I he joined the army as a radio operator and became interested in waves. He was also a lover of music. These qualities lead him to interpret Bohr's atoms as "musical instruments." Bohr assumed that his energy levels were quantized, but didn't explain why. De Broglie proposed that quantization arose from the same effects as the quantized frequencies (fundamental and overtone) present in a guitar string, that make it work as an instrument. (Review standing waves in this section if needed.) De Broglie thought that electrons in "orbits" like in the Bohr model were like strings on instruments. A string has to have nodes at each end; an electron in an orbit also has to be a standing wave, with a whole number of wavelengths around its path. In other words,
\[n\lambda = 2\pi r\]
where n is an integer. Thus, the wave interferes constructively with itself when it makes a full circle.
According to his theory,
\[\lambda = \frac{h}{mv}\]
He derived this equation using the radii Bohr calculated for his orbits. You can see that the wavelength effect is only important for very small particles, like electrons (check this out for yourself!). He thought that the "wave-ness" of electrons would be present outside of atomic orbits as well, and this proved true. A beam of electrons accelerated by high voltage is predicted to have similar wavelength to X-rays, and like X-rays, can be diffracted by a crystal. The diffraction pattern let researchers measure the wavelength of electrons (whose velocity was known, because they were accelerated by a known voltage) just as Moseley had measured the wavelength of emitted X-rays to determine atomic numbers. Later the wavelengths of atoms were measured as well. The experimental wavelengths matched de Broglie's prediction, and the wave-particle duality for electrons was accepted, just like the wave-particle duality for light.
Heisenberg developed a way to work with the allowed energies of atoms using matrices. He believed that we must work with the quantities we can observe experimentally, like energies (via spectroscopy). This led him to the Uncertainty Principle which says that certain pairs of quantities (position and momentum, energy and time) can't be measured precisely at the same time. For instance, if you want to know exactly where a particle is, you can't also know exactly what its velocity is. This was part of the new "statistical" quantum mechanics. In classical mechanics, you can know everything about particles (like balls): exactly where they are, exactly where they are going, and predict exactly the results of collisions or other events. In the new quantum mechanics, it was argued that this perfect knowledge of particles was simply impossible. For instance, exactly where is a wave?
Although Heisenberg's matrix quantum mechanics is probably used more often because it works well with computer calculations, an equivalent description developed by Schrodinger gives more intuition, so we will describe it in detail. The methods give the same answers but use very different math. Schrödinger extended de Broglie's theory by applying well-known physics of waves (for instance, in instruments) to describe electrons. There's a detailed example in the next section .


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adverse childhood experience (ACE)
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a traumatic incident such as experiencing violence or abuse, witnessing violence at home or in the community, having a family member attempt suicide, or any event or environment that undermines a child’s sense of safety, security, and bonding
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built environment
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includes transportation access and roadways, the availability of green space, locations for community gathering, and the buildings or other physical structures within the neighborhood
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civic participation
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a range of activities that individuals can participate in that benefit the community as well as participant health
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collective efficacy
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a community’s ability to create change and influence behavior through social norms
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cost burdened
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describes households in which more than 30 percent of income is spent on housing
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discrimination
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unfair treatment of individuals and groups based on characteristics such as race, gender, age, sexual orientation, and ability
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economic insecurity
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a situation linked to poverty where an individual or family is unable to consistently afford housing, food, clothing, transportation, child care, or health care
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economic stability
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a situation where the individual has a steady source of income and consistent access to resources essential for a healthy life
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equality
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a situation where each individual is given the same resources or opportunities
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equity
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the distribution of the resources or opportunities each individual needs to reach an equal outcome among individuals
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food deserts
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areas where residents have limited or absent options for affordable and healthy foods
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food insecurity
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a condition in which individuals or families have limited access to adequate amounts of food
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generational poverty
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a persistent lack of a socially acceptable amount of money or possessions and disadvantage where children who grow up poor are more likely to be poor as adults
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health disparities
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preventable differences in health between groups of individuals, usually resulting from social or economic factors, geographic location, and environment
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health equity
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a state in which everyone has a fair and just opportunity to attain their highest level of health
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health inequities
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differences in health status or health resources between population groups arising from social conditions
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housing instability
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a situation where individuals and families have difficulty paying rent, live in overcrowded residences, move frequently, or spend a large part of their household income on housing
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living wage
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payment for employment that affords a modest standard of living for the individual and family
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poverty
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a state in which an individual lacks a socially acceptable amount of money or possessions
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social capital
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value gained from having positive interactions and connections between people that enables the community to function effectively and collaboratively in problem-solving efforts
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social cohesion
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the strengths of relationships within a community; an indicator is the amount of social capital, or shared group resources, within a community
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social determinants of health (SDOH)
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the conditions in the environments where people are born, live, learn, work, play, worship, and age that affect a wide range of health, functioning, and quality-of-life outcomes and risks
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social justice
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the view that everyone deserves equal rights and opportunities; also refers to overhauling social systems with the goal of sustainable equitable access into the future
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socioeconomic gradient
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the association between low socioeconomic position (SEP) and increased disease risk and premature death
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socioeconomic position (SEP)
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one’s position in society based on social and economic factors of income, education, and employment
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structural discrimination
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deeply rooted institutional policies, systems, laws, and practices that limit opportunities, resources, and power of individuals and populations based on race, ethnicity, gender, ability, SES, and religion


2.3: Functional groups containing sp3-hybridized heteroatom
- Recognize and assign IUPAC name to functional groups containing sp 3 -hybridized heteroatoms, including haloalkanes, alcohols, ethers, and amines.
- Predict the polarity of the bonds in the functional groups of haloalkanes, alcohols, ethers, and amines and recognize the creative sites based on the partial charges in the functional group regions.
- Identify primary, secondary, tertiary, and quaternary designation of haloalkanes, alcohols, ethers, and amines.
Halogenated hydrocarbons
When a \(\ce{H}\) in a hydrocarbon is replaced an \(\ce{X}\), where \(\ce{X }\) is a halogen, i.e., fluorine (\(\ce{F}\)), chlorine (\(\ce{Cl}\)), bromine (\(\ce{Br}\)), or iodine (\(\ce{I}\)), it is a halogenated hydrocarbon. A halogenated alkane is called haloalkane .
Nomenclature of halogenated hydrocarbons
Halogen is treated as a branch of a hydrocarbon. The parent chain is numbers following the same rules as for branched hydrocarbons, and the halogens are listed in alphabetic order, preceded by location number, like other alkyl branches.
What is the IUPAC name of the following compound?
Solution
- The longest chain is the parent name: the longest chain is 6-\(\ce{C}\) long, so the parent name is hexane.
- List the branches before the parent name in alphabetic order: the branches are chloro- and methyl-, so: chloromethylhexane
- Number the parent chain from the end that gives the lowest number to the first branch: \(\ce{Cl}\) is at #2 from the right, and \(\ce{CH3}\) is at #3 from the left, so number the parent chain from the right side that gives #2 to \(\ce{Cl}\) and #4 to methyl.
Answer: 2-chloro-4-methyl hexane
Wright the skeletal structure of 5-chlorohex-2-yne?
Solution
Find the parent name: hex-2-yne. Hex tells six \(\ce{C}\) chain and 2-yne tells there is a triple bond at \(\ce{C}\)#2, i.e. .
Attach the branches to the parent chain according to their location numbers: the only branch in this chase is chloro- at \(\ce{C}\)#5.
Answer:
What is the IUPAC name of the following compound?
Solution
- The longest chain is the parent name: the longest chain is 6-\(\ce{C}\) cycle with alternate double bonds, i.e., all the \(\ce{C's}\) are sp2-hybridized, i.e., an aromatic hydrocarbon with parent name benzene.
- List the branches before the parent name in alphabetic order: the branches are chloro- and bromo-, so: bromochlorobenzene
- Number the benzene ring starting from the 1st branch and going in the direction that gives the lowest number to the 2nd branch: \(\ce{Br}\) is at #1, and \(\ce{Cl}\) is at #3.
Answer: 1-bromo-3-chlorobenzene
Physical properties of halogenated hydrocarbons
The halogen atom is sp 3 -hybridized with three sp 3 orbitals occupied by lone pairs, and one makes bonds with the \(\ce{C}\), i.e., \(\ce{R-\overset{\Large{\cdot\cdot}}{\underset{\Large{\cdot\cdot}}{X}}\normalsize{:}}\), where \(\ce{X}\) is a halogen: \(\ce{F}\), \(\ce{Cl}\), \(\ce{Br}\), or \(\ce{I}\). The lone pairs are usually not shown except when needed. The \(\ce{C-X}\) bond is polar, i.e., \(\ce{\overset{\delta{+}}{C}{-}\overset{\delta{-}}{X}}\), because halogens are more electronegative than \(\ce{C}\). It makes the \(\ce{C}\) a \(\delta^{+}\), i.e., an electrophile in reactivity. The lone pairs also add to the polarity because these electrons are localized on one side of the nucleus in sp 3 orbitals. The bond polarity can be observed from electrostatic potential maps, shown in Table 5. Recall that green is neutral, red is \(\delta^{-}\) and blue is \(\delta^{+}\) in electrostatic potential map. Note that the listed electronegativity values of \(\ce{C}\) and \(\ce{I}\) are the same, but the bond is still polar. This is because the listed values are slightly different from the actual values. Further, the polarizability increases in this order: \(\ce{F}\)<\(\ce{Cl}\)<\(\ce{Br}\)<\(\ce{I}\), i.e., \(\ce{I}\) is the most polarizable.
Polarizability is the tendency of a matter to acquire dipole-moment when subjected to an electric field. Larger atoms have more loosely held outermost electrons, which makes them more polarizable. Therefore, the polarizability increases in going from top to bottom in a group, e.g., the polarizability order of halogen is: \(\ce{F}\)<\(\ce{Cl}\)<\(\ce{Br}\)<\(\ce{I}\).
Bond length increases in this order: \(\ce{F}\)<\(\ce{Cl}\)<\(\ce{Br}\)<\(\ce{I}\) and electronegativity has the opposite trend, i.e., decreases in this order: \(\ce{F}\)>\(\ce{Cl}\)>\(\ce{Br}\)>\(\ce{I}\). The dipole moment depends on both the electronegativity difference of the bonded atoms and the bond length; the two effects contradict each other. As a result, the dipole moment does not change much by changing the halogen. The bond dissociation energy decrease in this order: \(\ce{F}\)>\(\ce{Cl}\)>\(\ce{Br}\)>\(\ce{I}\), which makes \(\ce{C-F}\) the strongest and the least reactive and \(\ce{C-I}\) the weakest and the most reactive group.
| Halogen | \(\ce{H}\) for reference | \(\ce{F}\) | \(\ce{Cl}\) | \(\ce{Br}\) | \(\ce{I}\) |
|---|---|---|---|---|---|
| Electrostatic potential map | |||||
| \(\ce{C-X}\) bond length (pm) | 109 | 139 | 178 | 193 | 214 |
| \(\ce{C-X}\) bond dissociation energy (kJ/mol) | 440 | 464 | 355 | 309 | 228 |
| Electronegativity of \(\ce{X}\) or \(\ce{H}\) | 2.2 | 4.0 | 3.0 | 2.8 | 2.5 |
| Dipole moment (D) | 0.33 | 1.85 | 1.87 | 1.81 | 1.62 |
Solubility in water and boiling points of fluoro- and chloro-alkanes are comparable to alkanes of the same molar mass. Although the \(\ce{C-F}\) and \(\ce{C-Cl}\) are polar that tend to increase intermolecular forces due to dipole-dipole interaction, the volume of the molecule is small compared to the alkane of similar molar mass due to \(\ce{F}\) or \(\ce{Cl}\) being heavier atoms. Smaller volume means less contact area and fewer intermolecular forces. The opposing factors cancel each other.
Haloalkanes are denser than alkanes. Mono-fluoroalkanes and mono-chloroalkanes are less dense than water, but bromo- and iodo-alkanes are denser than water. For example, \(\ce{CH3CH2Cl}\) and \(\ce{CH3CH2Br}\) are liquids having densities of 0.891 g/mL and 1.354 g/mL at 25 o C. Although mono-chloroalkanes are less dense than water, di-, tri-, and tetra-chloroalkanes are denser than water. For example, \(\ce{CH3Cl}\) is a gas at room temperature with a density of 1.003 g/mL at its boiling point -23.8 o C, while \(\ce{CH2Cl2}\), \(\ce{CHCl3}\), and \(\ce{CCl4}\) are liquids with density higher than water, i.e., 1.327 g/mL, 1.483 g/mL, and 1.594 g/mL, respectively. So, hydrocarbons float on water while the denser haloalkanes sink in water.
Some uses of halogenated hydrocarbons
\(\ce{C-I}\), \(\ce{C-Br}\), and \(\ce{C-Cl}\) bond are polar and weaker than \(\ce{C-H}\) that makes them reactive functional groups. The raw organic material, i.e., alkanes, is usually first converted to haloalkanes intermediates to synthesize other organic compounds. The \(\ce{C-F}\) bond is stable, which is why Teflon, composed of \(\ce{-(CF2)_{n}{-}}\) chains, is one of the most inert polymers. Haloalkenes with all of the \(\ce{C-H}\) bonds replaced with \(\ce{C-F}\) or \(\ce{C-Cl}\), called chlorofluorocarbons (CFCs) are usually nontoxic, nonflammable, odorless, and noncorrosive liquids that make them ideal as solvents, degreasing agents, and heat transfer fluids in air condition and refrigerator systems. For example, \(\ce{CCl2F2}\) (Freon-12) and \(\ce{CCl3F}\) (Freon-11) were used as refrigerants, and \(\ce{CCl4}\) common name carbon tetrachloride was used as a solvent, cleaning and degreasing agent. They are being phased out because Freons destroy the stratosphere's ozone layer, and \(\ce{CCl4}\) is toxic and carcinogenic.
The CFCs like \(\ce{CCl2F2}\) and \(\ce{CCl3F}\), being stable molecules, survive when released in the atmosphere, reach the stratosphere, and decompose when UV light shines on them in the stratosphere. Their by-products, particularly \(\ce{Cl2}\), catalyze the destruction of the ozone layer in the stratosphere that filters out UV light. It resulted in an ozone-depleted region over the southern hemisphere called the ozone hole. Without the ozone layer, UV light reaches the earth's surface and causes damage to living things. Therefore, the CFCs are being replaced with haloalkanes with some \(\ce{C-H}\) bonds left to make them less stable, so they may decompose before reaching the stratosphere. As a result of this worldwide effort, the ozone hole is recovering over time, as shown in Figure \(\PageIndex{1}\).
The most promising replacements include hydrofluorocarbons (HFCs) and hydrochlorofluorocarbons (HCFCs), like \(\ce{CF3-CH2F}\) called HFC-134a and \(\ce{CH3-CCl2F}\) called HCFC-141b. Carbon tetrachloride \(\ce{CCl4}\), which is toxic and carcinogenic, is also being replaced with dichloromethane \(\ce{CH2Cl2}\) solvent.
Alcohols
An alkyl group attached with an alcohol (\(\ce{-OH}\)) group, i.e., \(\ce{R-OH}\), is an alcohol. A benzene ring attached with an alcohol group is a phenol.
Nomenclature of alcohols and phenols
The nomenclature of hydrocarbons is followed for IUPAC names of alcohols and phenols with the following changes.
- The longest chain containing the \(\ce{-OH}\) group is chosen for the parent name, and the 'e' in the suffix -ane, -ene, or -yne is replaced with -ol. For example, \(\ce{CH3-OH}\) is methanol, and \(\ce{CH3CH2-OH}\) is ethanol.
- Numbering starts from the end, giving the \(\ce{-OH}\) group the lowest number in preference over hydrocarbon or halogen branches. This is demonstrated in the following example, where the parent chain is not heptane; it is hexane that contains \(\ce{-OH}\) group, and the number starts from the end that gives \(\ce{-OH}\) lowest number 2 not from the other end that gives \(\ce{-Cl}\) the lowest number 1.
- In a cyclic compound containing only \(\ce{-OH}\) group, numbering is not needed, e.g, number is not used in naming cyclohexanone:
- In a cyclic compound containing \(\ce{-OH}\) group and a hydrocarbon or halogen group, numbering begins from \(\ce{-OH}\) group, e.g.:
- A benzene ring containing an \(\ce{-OH}\) group is given the parent name phenol. If there is another group present, the numbering starts from the \(\ce{-OH}\) group, and phenol is used as the parent name, e.g.:
What is the IUPAC name of this compound?
Solution
- Find the parent name: four \(\ce{C}\) chain is but, and double bond makes it an alkene, i.e., butene.
- Change the 'e' of the suffix with -ol that tells it is an alcohol, i.e., butenol.
- Start number from the end that gives \(\ce{-OH}\) group the lowest number: \(\ce{-OH}\) receives #1 and \(\ce{C=C}\) receives #3.
Answer: but-3-en-1-ol
What is the IUPAC name of the following alcohol?
Solution
- Find the parent name: two \(\ce{C}\) chain is ethane.
- Replace the last 'e' of the suffix with -ol, which tells it is an alcohol; in this case, there are two \(\ce{-OH}\) groups, so add di, i.e., ethandiol
- The number is the same from either end, i.e., #1 and #2 for the two \(\ce{-OH}\) groups.
Answer: ethane-1,2-diol
Ethane-1,2-diol ( ) is known by its common name ethylene glycol. Similarly, propane-1,2,3-triol ( ) is known by its common name, glycerol or glycerine.
What is the IUPAC name of this compound?
Solution
- Find the parent name: six \(\ce{C}\) cycle is cyclohexane.
- Replace the last 'e' of the suffix with -ol which means it is an alcohol, i.e., cyclohexanol.
- A cycle with only \(\ce{-OH}\) group do not need numbering.
Answer: cyclohexanol
What is the IUPAC name of this compound?
Solution
- Find the parent name: six \(\ce{C}\) chain with all sp2-hybridized \(\ce{C's}\) is benzene, and benzene with an \(\ce{-OH}\) group is phenol.
- List the substituent other than \(\ce{OH}\) as prefix: ethylphenol.
- Start numbering form the \(\ce{-OH}\) group and continue in the direction that gives the lowest overall number to the other substituents: ethyl receives #3.
Answer: 3-ethylphenol
Physical properties of alcohol and phenols
Alcohols contain a sp 3 -hybridized \(\ce{O}\) bonded to a \(\ce{C}\) and a \(\ce{H}\) and the remaining two sp 3 -orbitals occupied by lone pairs, i.e.,: \(\ce{R-\overset{\Large{\cdot\cdot}}{\underset{\Large{\cdot\cdot}}{O}}-H}\). The \(\ce{O}\) atom is more electronegative than \(\ce{C}\) and \(\ce{H}\), which means both the bonds are polar, i.e., \(\ce{\overset{\delta{+}}{C}{-}\overset{\delta{-}}{O}{-}\overset{\delta{+}}{H}}\), as observed in the electrostatic maps of some alcohols in Figure \(\PageIndex{2}\).
An alcohol group has three reactive points: an electrophilic \(\ce{\overset{\delta{+}}{C}}\), an acidic \(\ce{\overset{\delta{+}}{H}}\), and nucleophylic/basic \(\ce{\overset{\delta{-}}{O}}\). The acidity of an alcoholic \(\ce{H}\) is obvious from the fact that it has a pKa ~16 that is about the same acid strength as \(\ce{H2O}\).
Recall that \(\ce{\overset{\delta{+}}{C}}\) is called an electrophile. At the same time, proton exchange reactions are acid/base reactions, i.e., \(\ce{\overset{\delta{+}}{H}}\) is an acidic proton, and a \(\delta{-}\) site when reacts with a \(\ce{C}\) is called nucleophile, and when it donates its lone pair to a \(\ce{H}\), it is called a base.
Alcohols have significantly higher boiling points compared to alkanes of comparable molar mass.
For example, the boiling point of methanol (molar mass 32 g/mol) is 65 o C which is significantly higher than the -89 o C boiling point of ethane of comparable molar mass (30 g/mol). It is explained by the fact that, in addition to London dispersion forces, alcohols have hydrogen bonding between \(\ce{\overset{\delta{+}}{H}}\) of one molecule with \(\ce{\overset{\delta{-}}{O}}\) of the neighboring molecule as illustrated in Figure \(\PageIndex{3}\). It takes more thermal energy to break London dispersion forces + hydrogen bonding in alcohols than London dispersion forces alone in alkanes. The boiling points of alcohols increase as their molar mass increases, as shown in Table 6.
Smaller alcohols having up to three \(\ce{C'}\) are entirely soluble in water, i.e., miscible, one with four \(\ce{C'}\) is partially soluble, and those with longer than four \(\ce{C'}\) are almost insoluble as listed in Table 6. This trend of water solubility change is explained by a balance tween hydrophilic and hydrophobic components in an alcohol molecule, explained later.
| IUPAC name | Molecular formula | Molecular weight (g/mol) | Boiling point ( o C) | Solubility in water |
|---|---|---|---|---|
| Methanol | \(\ce{CH3OH}\) | 32 | 65 | Miscible |
| Ethanol | \(\ce{CH3CH2OH}\) | 46 | 78 | Miscible |
| Propan-1-ol | \(\ce{CH3CH2CH2OH}\) | 60 | 97 | Miscible |
| Butan-1-ol | \(\ce{CH3CH2CH2CH2OH}\) | 74 | 117 | Slightly soluble |
| Pentan-1-ol | \(\ce{CH3CH2CH2CH2CH2OH}\) | 88 | 138 | Insoluble |
| Hexan-1-ol | \(\ce{CH3CH2CH2CH2CH2CH2OH}\) | 102 | 157 | Insoluble |
Phenol has an \(\ce{-OH}\) attached to an sp 2 -hybridized \(\ce{C}\) of a benzene ring. Phenol is a volatile white crystalline solid that melts at 41 o C, and boils at 182 o C. Phenol is partially soluble in water.
When an atom with a lone pair of electrons on it is bonded to an sp 2 -hybridized \(\ce{C}\), it changes its hybridization from sp 3 to sp 2 for resonance to happen. The driving force is the resonance stabilization effect. The same happens to \(\ce{O}\) in phenol resulting in resonance that sends electrons from \(\ce{O}\) into the benzene ring as illustrated below. Consequently, the benzene ring of phenol is more electron-rich, i.e., more nucleophilic than benzene without such a group attached. The groups like \(\ce{-OH}\) that enhance the reactivity of the benzene ring by donating electrons to the ring are called activating groups .
Another consequence of this electron donation by \(\ce{O}\) is that the \(\ce{O}\) is more able to hold on to the electrons of \(\ce{O-H}\) bond when the proton leaves acting as an acid, i.e., phenolic \(\ce{O-H}\) is a stronger acid (pKa ~10) compared to an alcoholic \(\ce{O-H}\) (pKa ~16).
Polar functional groups like \(\ce{-OH}\) are attracted to water, i.e., water-loving or hydrophilic . This is based on the general principle "like dissolves like." On the other hand, Alkyl groups are nonpolar and repel water molecules, i.e., hydrophobic . Alkyl groups are more soluble in lipids and fats, which are also called lipophilic or nonpolar . Alcohols have both groups in them, and their properties depend on which group dominates. The hydrophilic character dominates when the alkyl chain is small, three (\(\ce{C's}\) or small, and the hydrophobic character dominates when the alkyl chain is long, i.e., more than four (\(\ce{C's}\) long making the compound slightly soluble or insoluble. In summary,
- increasing the alkyl-group increases hydrophobic character, e.g., propan-1-ol (\(\ce{CH3CH2CH2OH}\)) is water soluble, but butan-1-ol (\(\ce{CH3CH2CH2CH2OH}\)) is partially soluble, and
- increasing the polar group increases the hydrophilic character, e.g., butan-1-ol (\(\ce{CH3CH2CH2CH2OH}\)) with one (\(\ce{-OH}\)) group is partially soluble in water, but butane-1,2-diol (\(\ce{CH3CH2CH(OH)CH2OH}\)) with two (\(\ce{-OH}\)) groups is soluble.
This trend is generally applicable to all polar groups containing organic compounds.
Some examples of important alcohols and phenols
Methanol (\(\ce{CH3-OH}\)) is used as a solvent, paint remover, and fuel and converted to formaldehyde which is then used to make plastics. If methanol is ingested, it converts to formaldehyde, which causes headache and blindness and may cause death. That is why methanol is mixed with ethanol not meant for drinking.
Ethanol (\(\ce{CH3-OH}\)) is the active ingredient in alcoholic beverages. It is also used as a solvent for perfumes, varnishes, and medicines, e.g., in tincture iodine. The fermentation process obtains ethanol, but these days, it is mainly derived from ethene.
Isopropanol . 2-methylpropanol, commonly known as isopropanol, kills bacteria and viruses. Ethanol also has the same property. Ethanol or isopropanol are 60% to 80% fraction of hand sanitizers; the remaining is usually ethylene or glycerin to keep the skin soft. If ethanol is the active ingredient in the hand sanitizer, care is needed as ethanol is volatile and flammable.
Ethylene glycol , i.e., ethane-1,2-diol (\(\ce{HO-CH2CH2-OH}\)) is used as antifreeze in cooling systems of vehicles and as a solvent for paints, inks, and plastics. Ethylene glycol is poisonous because, if ingested, it converts to oxalic acid, which causes kidney stones.
Glycerin or glycerol , i.e., propane-1,2,3-triol (\(\ce{HO-CH2CH(OH)CH2-OH}\)) is obtained as a byproduct during the soap making from fats and oils. It is a viscous liquid used in skin lotions, cosmetics, shaving creams, and liquid soaps. Nitroglycerin, the active component of explosive dynamite, is derived from glycerol.
Bisphenol a (BPA) is used used in making polycarbonate that is used to manufacture beverage bottles. There is a concern that leaching out BPA from beverage bottles may cause harmful health effects.
Phenol is primarily used for the production of precursors of plastics. Phenol is used as an antiseptic and disinfectant. Phenol is also a part of the structure of essential oils responsible for plants' odor and flavor, e.g., isoeugenol from nutmeg, eugenol from clover, thymol from thyme, and vanillin from vanilla, shown in Table 1.
| Isoeugenol from nutmeg | Eugenol from clover | Thymol from thyme | Vanillin from vanilla |
|---|---|---|---|
| Isoeugenol | Eugenol | Thymol | Vanillin |
Thiol
Thiols are sulfur analogs of alcohols, i.e., they contain the thiol (\(\ce{-SH}\)) group. Like alcohols, thiols contain an electrophilic \(\ce{C}\) and acidic \(\ce{H}\) attached to \(\ce{S}\) atom as illustrated by the electrostatic potential map of methanethiol \(\ce{CH3-SH}\) in the figure on the right, where \(\ce{S}\) is shown as yellow, \(\ce{H}\) as white, and \(\ce{C}\) as gray sphere in the model.
The thiols are named following the rules of alcohols, except that suffix -thiol is added to the name of alkane, e.g., \(\ce{CH3-SH}\) is methanethiol and (\ce{CH3CH2-SH}\)) is ethanethiol.
One characteristic of thiols is that they have a strong odor, e.g., methanthiol (\(\ce{CH3-SH}\)) is an odor-causing compound present in oysters, cheddar cheese, onions, and garlic. The garlic odor also contains prop-2-ene-1-thiol (\ce{CH2=CH-CH2-SH}\). Onion odor is also due to propane-1-thiol ((\ce{CH3CH2CH2-SH}\), which is a lachrymator, i.e., a substance that makes eyes tear.
Ethers
Ethers have \(\ce{O}\) with two single bonds with two alkyl groups, i.e., \(\ce{R-O-R'}\) group.
Nomenclature of ethers
IUPAC naming of ether follows the rules of branched alkanes. The large alkyl group is named as parent chain, and the smaller alkyl groups with oxygen are named as a branch, i.e., an alkoxy (\(\ce{R-O{-}}\)) group. For example, \(\ce{CH3CH2-O-CH2CH3}\) is ethoxyethane and \(\ce{CH3CH2CH2-O-CH3}\) is 1-methoxypropane . However, trivial names for ethers are often used. A common name is formed by listing the two alkyl groups in alphabetic order, followed by the word ether. For example, common name of \(\ce{CH3CH2-O-CH2CH3}\) is diethylether and of \(\ce{CH3CH2CH2-O-CH3}\) is propyl methyl ether.
Physical properties of ethers
Ethers contain a sp 3 -hybridized \(\ce{O}\) bonded to two \(\ce{C's}\) and the remaining two sp 3 -orbitals occupied by lone pairs, i.e.,: \(\ce{R-\overset{\Large{\cdot\cdot}}{\underset{\Large{\cdot\cdot}}{O}}-R}\). The \(\ce{O}\) atom is more electronegative than \(\ce{C}\), which means both the bonds are polar, i.e., \(\ce{\overset{\delta{+}}{C}{-}\overset{\delta{-}}{O}{-}\overset{\delta{+}}{C}}\), as observed in the electrostatic maps of diethylether shown on in figure on the right.
Since there is no \(\ce{O-H}\) bond, pure ethers do not have hydrogen bonding; they have a weak dipole-dipole interaction and London dispersion forces. Therefore, the boiling points of ether are comparable to alkanes of the same molecular mass, e.g., diethyl ether (molar mass 74 g/mol) has a boiling point of 35 o C, which is almost the same as the boiling point of pentane (molar mass 72 g/mol, boiling point 36 o C). However, ethers do have \(\ce{\overset{\delta{-}}{O}}\) and that can establish hydrogen bonding with \(\ce{\overset{\delta{+}}{H}}\) of water molecule. So, ethers have solubility in water comparable to alcohols of the same molar mass, i.e., ethers with three or less \(\ce{C's}\) are miscible in water, and those with four or more \(\ce{C's}\) are slightly soluble or insoluble in water.
Ethers are less reactive than alcohols because alcohols have an acidic \(\ce{\overset{\delta{+}}{H}}\), which is missing in ethers. Because of relatively low chemical reactivity, ethers are used as solvents to conduct other compounds' reactions.
Epoxides are cyclic ethers in which one atom in a three-membered ring is \(\ce{O}\). Epoxides take the parent name oxirane, and numbering, if needed, starts from the \(\ce{O}\) atom. Three examples are shown below.
Epoxides are unstable and highly reactive due to angle strain as the bonds bent from a regular angle of 109.5 o to 60 o . They are used as intermediates in organic synthesis. Oxirane is a room-temperature gas that reacts fast with the compounds in microorganisms, causing their death. So, it is used as a fumigant in foodstuffs and textiles and as a hospital sterilizer for surgical instruments.
Diethyl ether has been used as an anesthetic agent for a long time. Its use in anesthesia has been discontinued because it is a volatile and flammable gas posing a fire or explosion risk in the surgery room. Further, it has an irritating effect on the respiratory passages and causes nausea. Fluorinated or chlorofluorinated ethers, such as desflurane, sevoflurane, and isoflurane listed below, are used as anesthetic agents because they are less volatile, less flammable, and have fewer other undesirable side effects.
Peroxides, sulfides, and disulfides
Peroxides have \(\ce{O-O}\) bond, which is one of the weak bonds with bond dissociation energy ~200 kJ/mol, about half of the strength of a \(\ce{C-C}\) or a \(\ce{C-H}\) bond. This bond is found in hydroperoxy group (\(\ce{R-O-O-H}\)) and peroxy group (\(\ce{R-O-O-R'}\). The peroxides are less common but more reactive due to weak \(\ce{O-O}\) bond.
Sulfides are \(\ce{S}\) analogs of ethers, i.e, they have sulfide (\(\ce{R-\overset{\Large{\cdot\cdot}}{\underset{\Large{\cdot\cdot}}{S}}-R}\)) group. The names of sulfides are similar to those of ethers, i.e., list the two alkyl groups alphabetically, followed by the word sulfide. For example, \(\ce{CH3CH2-S-CH2CH3}\) is diethyl sulfide and \(\ce{CH3CH2CH2-S-CH3}\) is propyl methyl sulfide. Sulfides are less common in nature. Disulfide \(\ce{R-S-S-R}\) group, i.e., a sulfur analog of peroxides found in proteins. The \(\ce{S-S}\) is a relatively weak bond with a bond dissociation energy of ~250 kJ/mol.
Primary ( p ), secondary ( s ), tertiary ( t ), and quaternary classification
The p , s, t - classification of \(\ce{C's}\) and \(\ce{H's}\)
In an organic compound:
- if a \(\ce{C}\) is not bonded with any other \(\ce{C}\) or bonded with only one other \(\ce{C}\), it is a primary \(\ce{C}\) ;
- if bonded with two other \(\ce{C's}\), it is a secondary \(\ce{C}\) ;
- if bonded with three other \(\ce{C's}\}, it is a tertiary \(\ce{C}\) ; and
- if bonded with four other \(\ce{C's}\), it is a quaternary \(\ce{C}\) .
The \(\ce{H's}\) on a primary \(\ce{C's}\) are primary \(\ce{H's}\) , those on a secondary \(\ce{C's}\) are secondary \(\ce{H's}\) , and those on tertiary \(\ce{C's}\) are tertiary \(\ce{H's}\). Quaternary \(\ce{C's}\) do not have any \(\ce{H}\) on them. The figure on the right illustrates the primary, secondary, tertiary, and quaternary \(\ce{C's}\) and \(\ce{H's}\) marked in different colors and pointed by arrows. Table 1 illustrates the primary, secondary, and tertiary haloalkanes, alcohols, alkyl groups in ethers, and amine.
| Classification | Primary | Secondary | Tertiary |
|---|---|---|---|
| Haloalkanes |
ethyl chloride |
iso propyl chloride |
t -butyl chloride |
| Alcohols |
ethanol |
sec -butanol |
t -butanol |
| Alkyl groups in ethers |
di ethyl ether |
diiso propyl ether |
ditertiary butyl ether |
| Amines |
t -butyl amine |
ethyl(methyl)amine |
ethyl(methyl)propylamine |
The p , s, t - classification of haloalkanes and alcohols
A halogen or an \(\ce{-OH}\) group:
- if bonded with a primary \(\ce{C}\), it is a primary haloalkane, or primary alcohol ;
- if bonded with a secondary \(\ce{C}\), it is a secondary haloalkane or secondary alcohol ;
- if bonded with a tertiary \(\ce{C}\}, it is a tertiary haloalkane or tertiary alcohol .
Quaternary \(\ce{C's}\) do not have any halogen or \(\ce{-OH}\) group on them.
The p , s, t - classification of alkyl groups
An alkyl group connected through:
- a primary \(\ce{C}\) to a parent chain or a functional group is a primary alkyl group (p-alkyl) ;
- a secondary \(\ce{C}\) is a secondary alky group (s-alkyl) ; and
- a tertiary \(\ce{C}\) is a tertiary alkyl group (t-alkyl) .
The primary ( p ), secondary ( sec , or s ), and tertiary ( t ) designation of alkyl groups (except for iso propyl used in the place of sec-propyl) are also accepted in IUPAC nomenclature, and these designations are often used in common names of organic compounds, as shown by the commons in Table 1.
The p , s, t - classification of amines
Amines are organic compounds that have one or more \(\ce{H's}\) of ammonia (\(\ce{NH3}\)) replaced with an aliphatic hydrocarbon group.
- If there is only one alkyl group bonded with the \(\ce{N}\), it is a primary amine group , i.e., \(\ce{RNH2}\);
- if two alkyl groups are bonded with the \(\ce{N}\), it is a secondary amine group, i.e., \(\ce{RR'NH}\);
- if three alkyl groups are bonded with the \(\ce{N}\), it is a tertiary amine group , i.e., \(\ce{RR'R"N}\), and
- if four alkyl groups are bonded with the \(\ce{N}\), it is not an amine but an ammonium ion, i.e., \(\ce{RR'R"R'''N^{+}}\), e.g., tetramethylammonium: .
Unlike haloalkenes and alcohols, the primary, secondary, and tertiary classification of amines is based on whether there are one, two, or three alkyl groups bonded with the \(\ce{N}\), irrespective of the \(\ce{C}\) bonded with the \(\ce{N}\) is primary, secondary, or tertiary. For example, as shown below, t -butylamine is a primary, and trimethylamine is a tertiary amine.
Amines
Ammonia has an sp 3 -hybridized \(\ce{N}\) having three single bonds with nitrogen and one lone pair, i.e., \(\ce{:\!NH3}\). If one or more \(\ce{H's}\) of ammonia are replaced with an aliphatic hydrocarbon, it is an amine group, and the compound is an amine, i.e., \(\ce{RNH2}\), \(\ce{RR'NH}\), or \(\ce{RR'R''N}\) are amine groups. If one or more \(\ce{H's}\) of ammonia are replaced with a benzene ring, it is an aromatic amine.
Nomenclature of amines
IUPAC naming of amines follows the rules for naming alcohols with the following changes:
- The suffix's last letter 'e' is replaced with amine, e.g., \(\ce{CH3NH2}\) is methanamine. A few examples are the following.
- If more than one hydrocarbon group is attached with the \(\ce{N}\), the longest one is chosen as the parent name, and the smaller ones are listed alphabetically as substituents preceded by N-, where N tells the branch is attached to \(\ce{N}\). For example, \(\ce{CH3CH2NHCH3}\) is N-methylethanamine.
- An alternate and easier way is to list the alkyl groups as substituents in alphabetic order (the middle one is enclosed in small brackets for easy reading), followed by the word amine. For example, \(\ce{CH3CH2NHCH3}\) can be named ethyl(methyl)amine. A few examples are listed below.
- A benzene ring containing an \(\ce{−NH2}\) group is given the parent name aniline. If there is another group present, the numbering starts from the \(\ce{−NH2}\) group, and aniline is used as the parent name, e.g.:
What is the IUPAC name of the compound shown in the figure on the right?
Solution
- List the alkyl groups in alphabetic order (place the middle name in brackets) and end with the word amine: ethyl(methyl)amine, or
- Choose the longest substituted as the parent name and the other as substitute preceded by N- to indicate the substituent is on \(\ce{N}\), i.e.,: N-methylethanamine.
What is the IUPAC name of the compound shown in the figure on the right?
Solution
Ethane is the only alkyl group, so the stem name is ethane. It has two amine groups, so use di-preceded by number to tell the location of the amine groups: ethan-1,2-diamine.
What is the IUPAC name of the compound shown in the figure on the right?
Solution
A benzene ring attached to an amine group has the parent name aniline. Methyl substituent is on \(\ce{N}\), so list it as a prefix branch name preceded by N-, i.e., N-methylaniline.
What is the IUPAC name of the compound shown in the figure on the right?
Solution
The alkyl group is cyclohexane; replace the last 'e' with amine: cyclohexanamine.
Physical properties of amines
Amines contain a sp 3 -hybridized \(\ce{N}\) bonded to \(\ce{C's}\) and \(\ce{H's}\) and one sp 3 -orbitals occupied by lone pairs, i.e.,: \(\ce{RR'R'N\!:}\). The \(\ce{N}\) atom is more electronegative than \(\ce{C}\), which means the \(\ce{C-N}\) and \(\ce{N-H}\) bonds are polar, i.e., \(\ce{\overset{\delta{+}}{C}{-}\overset{\delta{-}}{N}{-}\overset{\delta{+}}{H}}\), as observed in the electrostatic maps of methanamine shown in the figure on the right, where \(\ce{N}\) is blue, \(\ce{C}\) is gray, and \(\ce{H}\) is a white sphere.
The \(\ce{N}\) atom is less electronegative than \(\ce{O}\), so the \(\ce{C-N}\) bond (3.0-2.1 = 0.9) is less polar and less reactive than \(\ce{C-O}\) (3.5-2.1 = 1.4). Due to the less electronegativity of \(\ce{N}\), it is more willing to donate its lone pair to any proton around, i.e., amiens are basic compounds. Amines are also classified as organic bases.
Primary and secondary amines have hydrogen bonding, but the \(\ce{N-H}\) bond is less polar than the \(\ce{O-H}\) bond. Therefore, the boiling points of primary and secondary amines are higher than alkanes but lower than alcohol of comparable molar mass, as shown in Table 8.
| Name | Condensed | Molar mass | Boiling point |
| Ethane | \(\ce{CH3CH3}\) | 30.1 g/mol | -88.6 o C |
| Methylamine | \(\ce{CH3NH2}\) | 31.1 g/mo | -6.3 o C |
| Methanol | \(\ce{CH3OH}\) | 32.0 g/mol | 65.0 o C |
| Butane | \(\ce{CH3CH2CH2CH3}\) | 58 g/mole | 0 o C |
| Ttrimetylamine | \(\ce{(CH3)3N}\) | 59 g/mole | 3 o C |
| Propylamine | \(\ce{CH3CH2CH2NH2}\) | 59 g/mol | 48.0 o C |
Tertiary amines have boiling points comparable with alkanes as they do not have \(\ce{N-H}\) bond and do not make hydrogen bonding with each other, as shown in Table 8.
The primary, secondary, and tertiary amines make a hydrogen bonds with water. Therefore, amines containing up to five \(\ce{C's}\) are soluble in water, and those with more than five \(\ce{C's}\) are slightly soluble or insoluble.
Amines are present in amino acids, the building blocks of proteins. Amines are present in several physiologically active compounds. The body releases the histamine in response to injury or allergic reactions. Histamine causes blood vessels to dilate, and redness and swelling occur in the area. Antihistamine administered to block the effects of histamine is another amine, diphenhydramine.
Epinephrine (adrenaline) and norepinephrine (noradrenaline) are released in a "fight-or-flight" situation. They increase blood glucose and move the blood to the muscles. Norepinephrine is a remedy for colds, hay fever, and asthma. It contracts the capillaries in the mucous membranes of the respiratory passage.
Tranquilizers are drugs that relieve the symptoms of anxiety and tension. These include diazepam (earlier name valium), Chlordiazepoxide (trade name Librium), and other benzodiazepines, which are sedative and hypnotic medicines that cause calming effects and drowsiness.
Amines can donate a lone pair to a proton relatively easily because \(\ce{N}\) is less electronegative than \(\ce{O}\), i.e., amines are basic compounds with pKa value ~10. Alkaloids are nitrogen-containing basic compounds extracted from plants. Most alkaloids are physiologically active and used in anesthetics, antidepressants, and stimulants; many are habit-forming. For example, coniine extracted from "poison hemlock" can cause weakness, fast respiration, paralysis, and death. Nicotine found in tobacco is addictive, but in large doses, it causes depression, nausea, vomiting, or death. The solution of nicotine in water is used as an insecticide. Cocaine extracted from the coca plant is a stimulant for the central nervous system. Piperidine is responsible for the pungent smell and taste of black pepper. The Chemical structures of these alkaloids are shown below.
|
(S)-Conine |
Nicotine |
Cocaine |
Piperidine |
Caffeine is found in coffee and tea. Caffeine increases alertness and is used in some pain relievers to counter the drowsiness caused by antihistamines. Quinine from the bark of the cinchona tree is used to treat malaria. Atropine from belladonna accelerates slow heart rates and is anesthesia for eye exams. Their structures are shown below.
|
Caffeine |
Quinine |
Atropine |
Morphine and codeine are obtained from the opium poppy plant and used as painkillers and cough syrup. Heroin, which is strongly addictive, is received by the chemical modification of morphine. OxyContin, which is a prescription drug for the relief of fever pain, is structurally similar to heroin and also has similar physiological effects. The structures of these alkaloids are shown below.
|
Seedhead of Opium Poppy |
Morphine |
Codeine |
Heroin |
OxyContin |


5.5: Net ecosystem production and Eddy-covariance studies
Net ecosystem production (NEP) in ecology, limnology, and oceanography, is the difference between gross primary production (GPP) and net ecosystem respiration. [1] Net ecosystem production represents all the carbon produced by plants in water through photosynthesis that does not get respired by animals, other heterotrophs, or the plants themselves.
Net ecosystem production describes the total carbon in an ecosystem that can be stored, exported, or oxidized back into carbon dioxide gas. NEP is written in units of mass of carbon per unit area per time, for example, grams carbon per square meter per year (g C m−2 yr−1). In a given ecosystem, carbon quantified as net ecosystem production can eventually end up: oxidized by fire or ultraviolet radiation, accumulated as biomass, exported as organic carbon to another system, or accumulated in sediments or soils. Carbon classified as NEP can be in the form of particles in the particulate organic carbon (POC) pool such as phytoplankton cells (living) and detritus (non-living), or it can be in the form of dissolved substances that have not yet been decomposed in the dissolved organic carbon (DOC) pool. [2] In any form, if the carbon gets respired or decomposed by any living organism (plant, animal, bacteria, or other microscopic organism) to release carbon dioxide, that carbon no longer counts as NEP. [1]
- NEP = GPP - respiration [by plants] - respiration [by animals and other heterotrophs]
Net ecosystem production is all the carbon not respired, including respiration by plants and heterotrophic organisms such as animals and microbes. In contrast, net primary production (NPP) is all the carbon taken up by plants (autotrophs) minus the carbon that the plants themselves respire through cellular respiration.
- NPP = GPP - respiration [by plants]
- Lovett, Gary M.; Cole, Jonathan J.; Pace, Michael L. (2006-02-01). "Is Net Ecosystem Production Equal to Ecosystem Carbon Accumulation?" . Ecosystems . 9 (1): 152–155. doi : 10.1007/s10021-005-0036-3 . ISSN 1435-0629 . S2CID 5890190 .
- ^ Chester, Roy; Jickells, Tim (2012). Marine Geochemistry . John Wiley & Sons. doi : 10.1002/9781118349083 . ISBN<PHONE_NUMBER>083 .
Eddy Covariance
The eddy covariance (also known as eddy correlation and eddy flux ) is a key atmospheric measurement technique to measure and calculate vertical turbulent fluxes within atmospheric boundary layers. The method analyses high-frequency wind and scalar atmospheric data series, gas, energy, and momentum, [1] which yields values of fluxes of these properties. It is a statistical method used in meteorology and other applications (micrometeorology, oceanography, hydrology, agricultural sciences, industrial and regulatory applications, etc.) to determine exchange rates of trace gases over natural ecosystems and agricultural fields, and to quantify gas emissions rates from other land and water areas. It is frequently used to estimate momentum, heat, water vapour, carbon dioxide and methane fluxes. [2] [3] [4] [5] [6] [7]
The technique is also used extensively for verification and tuning of global climate models, mesoscale and weather models, complex biogeochemical and ecological models, and remote sensing estimates from satellites and aircraft. The technique is mathematically complex, and requires significant care in setting up and processing data. To date, [ when? ] there is no uniform terminology or a single methodology for the eddy covariance technique, but much effort is being made by flux measurement networks (e.g., FluxNet , Ameriflux , ICOS , CarboEurope , Fluxnet Canada , OzFlux , NEON , and iLEAPS ) to unify the various approaches.
The technique has additionally proven applicable under water to the benthic zone for measuring oxygen fluxes between the sea floor and overlying water. [8] In these environments, the technique is generally known as the eddy correlation technique, or just eddy correlation. Oxygen fluxes are extracted from raw measurements largely following the same principles as used in the atmosphere, and they are typically used as a proxy for carbon exchange, which is important for local and global carbon budgets. For most benthic ecosystems, eddy correlation is the most accurate technique for measuring in-situ fluxes. The technique's development and its applications under water remains a fruitful area of research. [9] [10] [11] [12] [13]
General principles
Representation of the air flow in the atmospheric boundary layer
Air flow can be imagined as a horizontal flow of numerous rotating eddies, that is, turbulent vortices of various sizes, with each eddy having horizontal and vertical components. The situation looks chaotic, but vertical movement of the components can be measured from the tower.
Physical meaning
At one physical point on the tower, at time 1, eddy 1 moves parcel of air c 1 down at speed w 1 . Then, at time 2, eddy 2 moves parcel c 2 up at speed w 2 . Each parcel has gas concentration, pressure, temperature, and humidity. If these factors, along with the speed are known, we can determine the flux. For example, if one knew how many molecules of water went down with eddies at time 1, and how many molecules went up with eddies at time 2, at the same point, one could calculate the vertical flux of water at this point over this time. So, vertical flux can be presented as a covariance of the vertical wind velocity and the concentration of the entity of interest.
Summary
The 3D wind and another variable (usually gas concentration, temperature or momentum) are decomposed into mean and fluctuating components. The covariance is calculated between the fluctuating component of the vertical wind and the fluctuating component of gas concentration. The measured flux is proportional to the covariance.
The area from which the detected eddies originate is described probabilistically and called a flux footprint. The flux footprint area is dynamic in size and shape, changing with wind direction, thermal stability and measurements height, and has a gradual border.
The effect of sensor separation, finite sampling length, sonic path averaging, as well as other instrumental limitations, affect frequency response of the measurement system and may need a co-spectral correction, especially noticeable with closed-path instruments and at low heights below 1 to 1.5 m.
Major assumptions
- Measurements at a point can represent an upwind area
- Measurements are done inside the boundary layer of interest
- Fetch/flux footprint is adequate – fluxes are measured only at area of interest
- Flux is fully turbulent – most of the net vertical transfer is done by eddies
- Terrain is horizontal and uniformed: average of fluctuations is zero; density fluctuations negligible; flow convergence & divergence negligible
- Instruments can detect very small changes at high frequency, ranging from minimum of 5 Hz and to 40 Hz for tower-based measurements
- Liang, Shunlin; Li, Xiaowen; Wang, Jindi, eds. (2012-01-01), "Chapter 16 - Vegetation Production in Terrestrial Ecosystems" , Advanced Remote Sensing , Academic Press, pp. 501–531, doi : 10.1016/b978-0-12-385954-9.00016-2 , ISBN 978-0-12-385954-9 , retrieved 2020-03-12
- Baldocchi, D., B. Hicks, and T. Meyers. 1988. Measuring biosphere-atmosphere exchanges of biologically related gases with micrometeorological methods. Ecology 69, 1331-1340
- Verma, S.B.: 1990, Micrometeorological methods for measuring surface fluxes of mass and energy, Remote Sensing Reviews 5(1): 99-115
- Lee, X., W. Massman, and B. Law. 2004. Handbook of Micrometeorology. Kluwer Academic Publishers, The Netherlands, 250 pp.
- Burba, G., 2013. Eddy Covariance Method for Scientific, Industrial, Agricultural and Regulatory Applications: a Field Book on Measuring Ecosystem Gas Exchange and Areal Emission Rates. LI-COR Biosciences, Lincoln, USA, 331 pp.
- Aubinet, M., T. Vesala, D. Papale (Eds.), 2012. Eddy Covariance: A Practical Guide to Measurement and Data Analysis. Springer Atmospheric Sciences, Springer Verlag, 438 pp.
- Burba, George (2022-09-06). Eddy Covariance Method For Scientific, Regulatory, and Commercial Applications . LI-COR Biosciences. ISBN 978-0-578-97714-0 .
- Berg, P., H. Røy, F. Janssen, V. Meyer, B. B. Jørgensen, M. Hüttel, and D. de Beer. 2003. Oxygen uptake by aquatic sediments measured with a novel non-invasive eddy correlation technique. Marine Ecology Progress Series. 261:75-83.
- University of Virginia. Aquatic Eddy Covariance Research Lab . Retrieved: 22 June 2015.
- The Florida State University. Eddy Correlation - Further Development and Studies of Flow and Light driven dynamics of Benthic Oxygen Exchange Archived 2014-04-18 at the Wayback Machine . Retrieved: 22 June 2015.
- Leibniz-Institute of Freshwater Ecology and Inland Fisheries. Eddy Correlation in Natural Waters . Retrieved: 22 June 2015.
- Max Planck Institute for Marine Microbiology. Eddy Correlation System (ECS) . Retrieved: 22 June 2015.
- Centre for Coastal Biogeochemistry Research. Eddy Correlation Archived 2013-12-13 at the Wayback Machine . Retrieved: 22 June 2015.
Excerpted from:
Wikipedia, https://en.Wikipedia.org/wiki/Eddy_covariance , November 2023
Wikipedia, https://en.Wikipedia.org/wiki/Net_ec...tem_production , November 2023


31.3: Pepsin Inhibitors and Prostaglandin Analogues
By the end of this section, you should be able to:
- 31.3.1 Identify the characteristics of mucosal protectants and prostaglandin analogues used to treat gastrointestinal disorders.
- 31.3.2 Explain the indications, action, adverse reactions, and interactions of mucosal protectants and prostaglandin analogues used to treat gastrointestinal disorders.
- 31.3.3 Describe nursing implications of mucosal protectants and prostaglandin analogues used to treat gastrointestinal disorders.
- 31.3.4 Explain the client education related to mucosal protectants and prostaglandin analogues used to treat gastrointestinal disorders.
Mucosal Protectants
Pepsin is a gastric enzyme that breaks down and supports the digestion of proteins from foods. A low gastric pH of 1.5–2 activates pepsin. However, pepsin may cause erosive damage to the mucosal lining of the stomach. To mitigate the potential harm caused by pepsin, mucosal protectants are employed. These agents serve a dual purpose by reducing both gastric acid production and pepsin activity. Notably, when the gastric pH exceeds 5, mucosal protectants act as inhibitors of pepsin further safeguarding the delicate mucosal lining of the stomach.
Sucralfate
Sucralfate is classified as a mucosal protectant or coating agent. When ingested, it transforms into a viscous substance that adheres to ulcerated areas, creating a protective barrier on the mucosal lining of the stomach and duodenum. This protective paste-like barrier can adhere or stick to the ulcers for up to 6 hours, shielding them from the damaging effects of pepsin and hydrochloric acid. Sucralfate is especially effective with duodenal ulcers, ulcers due to aspirin use, and chemotherapy-induced mucositis. It may also be beneficial for stress ulcer prophylaxis.
Sucralfate is available in a tablet form or a liquid suspension. To maximize effectiveness, it should be taken 4 times a day, approximately 60 minutes before meals and at bedtime. When used alongside antacids in antiulcer therapy, it's advisable to administer antacids 30 minutes before or after sucralfate. Additionally, due to its adhesive properties, sucralfate should be taken at least 2 hours before or after medications such as quinolones, digoxin, phenytoin, and tetracycline to prevent any interference with their effectiveness.
Table 31.8 is a drug prototype table for sucralfate. It lists drug class, mechanism of action, adult dosage, indications, therapeutic effects, drug and food interactions, adverse effects, and contraindications.
|
Drug Class
Mucosal protectant, coating agent Mechanism of Action Locally reacts with hydrochloric acid and pepsin in the stomach to form an adherent protective paste-like substance capable of acting as an acid buffer |
Drug Dosage
1 g orally 4 times daily for 4–8 weeks unless healing has been demonstrated by x-ray or endoscopic exam. |
|
Indications
Short-term treatment of active duodenal ulcers Therapeutic Effects Protects damaged gastric mucosa from further damage caused by pepsin and hydrochloric acid |
Drug Interactions
Cimetidine Digoxin Fluroquinolones Ketoconazole Phenytoin Quinidine Ranitidine Tetracycline Theophylline Food Interactions No significant interactions Sucralfate should be taken on an empty stomach |
|
Adverse Effects
Diarrhea Dry mouth Flatulence Nausea Pruritus Rash Dizziness Insomnia Headache Back pain Hyperglycemia |
Contraindications
Hypersensitivity Caution: Chronic kidney failure or dialysis secondary to risk of aluminum accumulation |
Prostaglandin Analogues
Prostaglandins are naturally occurring compounds present in the GI tract that protect the mucosal lining of the stomach and duodenum. They do this by inhibiting gastric acid and pepsin secretions.
Synthetic prostaglandin analogues, in a similar manner, act as mucosal protectants by decreasing acid secretions while increasing the secretion of bicarbonate and protective mucus. This combined action serves to fortify the defense mechanisms of the GI tract, preserving its delicate mucosal integrity.
Misoprostol
Misoprostol acts as an endogenous prostaglandin in the GI tract. As an antiulcer agent for the prevention of NSAID-induced ulcers, it inhibits basal and nocturnal gastric acid and pepsin secretions and also increases the secretion of bicarbonate and protective gastric mucous. Misoprostol also promotes vasodilation to maintain submucosal gastric blood flow. These actions prevent gastric ulcers.
Misoprostol is available in tablets that are administered with food 4 times a day alongside NSAID therapy. Misoprostol is readily available from the GI tract with an extensive first-pass metabolism, reaching peak levels in 60–90 minutes, with a brief half-life of 20–40 minutes.
Table 31.9 is a drug prototype table for misoprostol. It lists drug class, mechanism of action, adult dosage, indications, therapeutic effects, drug and food interactions, adverse effects, and contraindications.
|
Drug Class
Prostaglandin analogue Mechanism of Action Acts as a synthetic prostaglandin E1 analogue with both antisecretory and mucosal protective properties in response to various gastric stimuli (meals, histamine, pentagastrin, and coffee) |
Drug Dosage
200 mcg 4 times daily with food; if dose is not tolerated, a dose of 100 mcg can be used. |
|
Indications
Prevention of NSAID-induced gastric ulcers Therapeutic Effects Inhibits both basal and nocturnal gastric acid secretions |
Drug Interactions
Oxytocin Food Interactions No significant interaction |
|
Adverse Effects
Diarrhea/constipation Abdominal pain Dysmenorrhea/spotting Uterine contractions Postmenopausal bleeding Flatulence Headache |
Contraindications
Hypersensitivity to prostaglandins Pregnancy Caution: Cardiovascular disease People of childbearing age |
Adverse Effects and Contraindications
Common adverse effects of mucosal protectants include hyperglycemia, dizziness, insomnia, and GI symptoms such as nausea, vomiting, diarrhea, and flatulence. Contraindications include hypersensitivity.
Typical adverse effects of prostaglandin analogues include headache, diarrhea or constipation, abdominal pain, flatulence, and gynecological symptoms such as dysmenorrhea , vaginal bleeding, uterine contractions, and postmenopausal bleeding. Contraindications include hypersensitivity to prostaglandins and pregnancy. Misoprostol may endanger pregnancy (may cause abortion) and cause harm to the fetus when administered to a pregnant client.
Nursing Implications
The nurse should do the following for clients who are taking mucosal protectants and prostaglandin analogues:
- Prior to administering, assess the client’s medical history, current drug list, and allergies.
- Educate the client regarding short-term duration of use for these medications, typically 4–8 weeks.
- Inform clients of childbearing age about the potential risk to the fetus, which could result in a miscarriage.
- Provide the client with teaching regarding the drug and when to call the health care provider. See below for additional client teaching guidelines.
Client Teaching Guidelines
The client taking a mucosal protectant and prostaglandin analogue should:
- Adhere to the dosing regimen prescribed by the health care provider, typically short-term administration of 4–8 weeks.
- When using misoprostol, report symptoms of abnormal vaginal bleeding or uterine cramping to the health care provider as these can represent serious adverse effects.
The client taking a mucosal protectant and prostaglandin analogue should not:
- Use prostaglandin analogues if pregnant due to the risk of fetal demise.
FDA Black Box Warning
Misoprostol
Misoprostol can cause spontaneous abortion, premature birth, or birth defects.


1.91: Reciprocal Space
The basis vectors a* , b* , c* of the reciprocal space are related to the basis vectors a , b , c of the direct space (or crystal space) through either of the following two equivalent sets of relations:
(1)
a* . a = 1; b* . b = 1; c* . c = 1;
a* . b = 0; a* . c = 0; b* . a = 0; b* . c = 0; c* . a = 0; c* . b = 0.
(2)
a* = ( b × c )/ ( a , b , c );
b* = ( c × a )/ ( a , b , c );
c* = ( b × c )/ ( a , b , c );
where ( b × c ) is the vector product of basis vectors b and c and ( a , b , c ) = V is the triple scalar product of basis vectors a , b and c and is equal to the volume V of the cell constructed on the vectors a , b and c .
The reciprocal and direct spaces are reciprocal of one another, that is the reciprocal space associated to the reciprocal space is the direct space. They are related by a Fourier transform and the reciprocal space is also called Fourier space or phase space .
The vector product of two direct space vectors, \({\bold r_1} = u_1 {\bold a} + v_1 {\bold b} + w_1 {\bold c}\) and \({\bold r_2} = u_2 {\bold a} + v_2 {\bold b} + w_2 {\bold c}\) is a reciprocal space vector,
Reciprocally, the vector product of two reciprocal vectors is a direct space vector.
As a consequence of the set of definitions (1), the scalar product of a direct space vector r = u a + v b + w c by a reciprocal space vector r* = h a* + k b* + l c* is simply:
r . r* = uh + vk + wl .
In a change of coordinate system , The coordinates of a vector in reciprocal space transform like the basis vectors in direct space and are called for that reason covariant . The vectors in reciprocal transform like the coordinates in direct space and are called contravariant .