6.4: Polar Reactions
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Objectives
After completing this section, you should be able to
- identify the positive and negative ends of the bonds present in the common functional groups.
- explain how bond polarity can be enhanced by the interaction of a functional group with a solvent, metal cation or acid.
- explain how the polarizability of an atom can be an important factor in determining the reactivity of a bond.
- describe the heterolytic bond-breaking process.
- use curved (curly) arrows to indicate the movement of electron pairs during bond breakage and bond formation.
- predict whether a given species (compound or ion) is likely to behave as a nucleophile or as an electrophile.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- electrophile
- nucleophile
- polar reaction
- polarizability
Study Notes
You may wish to review Section 2.1 before you begin this section. The relative electronegativities of the elements shown in the periodic table should already be familiar. Remember that it is the relative electronegativities that are important, not the actual numerical values.
Make sure that you understand the polarity patterns of the common functional groups. Do not try to memorize these polarities; rather, concentrate on why they arise. You will encounter these group polarities so frequently that they will soon become “second nature” to you.
Halogens and the Character of the Carbon-Halogen Bond
With respect to electronegativity, most halogens are more electronegative than carbon. This results in a carbon-halogen bond that is polarized. As shown in the image below, the carbon atom has a partial positive charge, while the halogen has a partial negative charge.
The following table shows the relationship between the halogens and electronegativity. Notice, as we move up the periodic table from iodine to fluorine, electronegativity increases.
| Electronegativity | I < Br < Cl < F |
|---|
The following table shows the relationships between bond length, bond strength, and molecular size. As we progress down the periodic table from fluorine to iodine, molecular size increases. As a result, we also see an increase in bond length. Conversely, as molecular size increases and we get longer bonds, the strength of those bonds decreases.
| Bond length | C–F < C–Cl < C–Br < C–I |
|---|---|
| Bond strength | C–I < C–Br < C–Cl < C–F |
| Molecular size | F < Cl < Br < I |
The influence of bond polarity
Of the four halogens, fluorine is the most electronegative and iodine the least. That means that the electron pair in the carbon-fluorine bond will be dragged most towards the halogen end. Looking at the methyl halides as simple examples:
The electronegativities of carbon and iodine are not very different, and so there will be little separation of charge on the bond. One of the important set of reactions of alkyl halides involves replacing the halogen by something else - substitution reactions . These reactions can involve the carbon-halogen bond breaking to give positive and negative ions. The ion with the positively charged carbon atom then reacts with something either fully or slightly negatively charged. Alternatively, something either fully or negatively charged is attracted to the slightly positive carbon atom and pushes off the halogen atom.
You might have thought that either of these would be more effective in the case of the carbon-fluorine bond with the quite large amounts of positive and negative charge already present. But that's not so - quite the opposite is true! The thing that governs the reactivity is the strength of the bonds which have to be broken. It is difficult to break a carbon-fluorine bond, but easy to break a carbon-iodine one.
The Carbonyl Group
C=O is prone to additions and nucleophillic attack because or carbon's positive charge and oxygen's negative charge. The resonance of the carbon partial positive charge allows the negative charge on the nucleophile to attack the Carbonyl group and become a part of the structure and a positive charge (usually a proton hydrogen) attacks the oxygen. Just a reminder, the nucleophile is a good acid therefore "likes protons" so it will attack the side with a positive charge.
Before we consider in detail the reactivity of aldehydes and ketones, we need to look back and remind ourselves of what the bonding picture looks like in a carbonyl. Carbonyl carbons are sp 2 hybridized, with the three sp 2 orbitals forming overlaps with orbitals on the oxygen and on the two carbon or hydrogen atoms. These three bonds adopt trigonal planar geometry. The remaining unhybridized 2p orbital on the central carbonyl carbon is perpendicular to this plane, and forms a ‘side-by-side’ pi bond with a 2p orbital on the oxygen.
The carbon-oxygen double bond is polar: oxygen is more electronegative than carbon, so electron density is higher on the oxygen side of the bond and lower on the carbon side. Recall that bond polarity can be depicted with a dipole arrow, or by showing the oxygen as holding a partial negative charge and the carbonyl carbon a partial positive charge.
A third way to illustrate the carbon-oxygen dipole is to consider the two main resonance contributors of a carbonyl group: the major form, which is what you typically see drawn in Lewis structures, and a minor but very important contributor in which both electrons in the pbond are localized on the oxygen, giving it a full negative charge. The latter depiction shows the carbon with an empty 2p orbital and a full positive charge.
Nucleophilic Addition to Aldehydes and Ketones
The result of carbonyl bond polarization, however it is depicted, is straightforward to predict. The carbon, because it is electron-poor, is an electrophile: it is a great target for attack by an electron-rich nucleophilic group. Because the oxygen end of the carbonyl double bond bears a partial negative charge, anything that can help to stabilize this charge by accepting some of the electron density will increase the bond’s polarity and make the carbon more electrophilic. Very often a general acid group serves this purpose, donating a proton to the carbonyl oxygen.
The same effect can also be achieved if a Lewis acid, such as a magnesium ion, is located near the carbonyl oxygen. Unlike the situation in a nucleophilic substitution reaction, when a nucleophile attacks an aldehyde or ketone carbon there is no leaving group – the incoming nucleophile simply ‘pushes’ the electrons in the pi bond up to the oxygen.
Alternatively, if you start with the minor resonance contributor, you can picture this as an attack by a nucleophile on a carbocation.
After the carbonyl is attacked by the nucleophile, the negatively charged oxygen has the capacity to act as a nucleophile. However, most commonly the oxygen acts instead as a base, abstracting a proton from a nearby acid group in the solvent or enzyme active site.
This very common type of reaction is called a nucleophilic addition . In many biologically relevant examples of nucleophilic addition to carbonyls, the nucleophile is an alcohol oxygen or an amine nitrogen, or occasionally a thiol sulfur. In one very important reaction type known as an aldol reaction, the nucleophile attacking the carbonyl is a resonance-stabilized carbanion. In this chapter, we will concentrate on reactions where the nucleophile is an oxygen or nitrogen.
- Nucleophilic Addition to Aldehydes and Ketones
- Nucleophilic Substitution of RCOZ (Z = Leaving Group)
- General reaction
- General mechanism
Polarity Patterns in Other Common Functional Groups
Nucleophile?
Nucleophilic functional groups are those which have electron-rich atoms able to donate a pair of electrons to form a new covalent bond. In both laboratory and biological organic chemistry, the most relevant nucleophilic atoms are oxygen, nitrogen, and sulfur, and the most common nucleophilic functional groups are water, alcohols, phenols, amines, thiols, and occasionally carboxylates.
More specifically in laboratory reactions, halide and azide (N 3 - ) anions are commonly seen acting as nucleophiles.
Of course, carbons can also be nucleophiles - otherwise how could new carbon-carbon bonds be formed in the synthesis of large organic molecules like DNA or fatty acids? Enolate ions (section 7.5) are the most common carbon nucleophiles in biochemical reactions, while the cyanide ion (CN - ) is just one example of a carbon nucleophile commonly used in the laboratory. Reactions with carbon nucleophiles will be dealt with in chapters 13 and 14, however - in this chapter and the next, we will concentrate on non-carbon nucleophiles.
When thinking about nucleophiles, the first thing to recognize is that, for the most part, the same quality of 'electron-richness' that makes a something nucleophilic also makes it basic: nucleophiles can be bases, and bases can be nucleophiles . It should not be surprising, then, that most of the trends in basicity that we have already discussed also apply to nucleophilicity.
| Neutral Nucleophiles | Charged Nucleophiles |
|---|---|
| H 2 O, NH 3 , RNH 2 , R 2 NH, R 3 N, ROH, RCOOH, RSH, and PR 3 | ⊝ OH, ⊝ OR, ⊝ NH 2 , ⊝ NHR, ⊝ NR 2 , ⊝ SH, ⊝ SR, ⊝ SeR, ⊝ Cl, ⊝ Br, ⊝ I, ⊝ F, ⊝ CN, RCOO ⊝ |
Electrophiles
In the vast majority of the nucleophilic substitution reactions you will see in this and other organic chemistry texts, the electrophilic atom is a carbon which is bonded to an electronegative atom, usually oxygen, nitrogen, sulfur, or a halogen. The concept of electrophilicity is relatively simple: an electron-poor atom is an attractive target for something that is electron-rich, i.e . a nucleophile. However, we must also consider the effect of steric hindrance on electrophilicity. In addition, we must discuss how the nature of the electrophilic carbon, and more specifically the stability of a potential carbocationic intermediate, influences the S N 1 vs. S N 2 character of a nucleophilic substitution reaction.
Consider two hypothetical S N 2 reactions: one in which the electrophile is a methyl carbon and another in which it is tertiary carbon.
Because the three substituents on the methyl carbon electrophile are tiny hydrogens, the nucleophile has a relatively clear path for backside attack. However, backside attack on the tertiary carbon is blocked by the bulkier methyl groups. Once again, steric hindrance - this time caused by bulky groups attached to the electrophile rather than to the nucleophile - hinders the progress of an associative nucleophilic (S N 2) displacement.
The factors discussed in the above paragraph, however, do not prevent a sterically-hindered carbon from being a good electrophile - they only make it less likely to be attacked in a concerted S N 2 reaction . Nucleophilic substitution reactions in which the electrophilic carbon is sterically hindered are more likely to occur by a two-step, dissociative (S N 1) mechanism. This makes perfect sense from a geometric point of view: the limitations imposed by sterics are significant mainly in an S N 2 displacement, when the electrophile being attacked is a sp 3 -hybridized tetrahedral carbon with its relatively ‘tight’ angles of 109.4 o . Remember that in an S N 1 mechanism, the nucleophile attacks an sp 2 -hybridized carbocation intermediate, which has trigonal planar geometry with ‘open’ 120 angles.
With this open geometry, the empty p orbital of the electrophilic carbocation is no longer significantly shielded from the approaching nucleophile by the bulky alkyl groups. A carbocation is a very potent electrophile, and the nucleophilic step occurs very rapidly compared to the first (ionization) step.
Exercise \(\PageIndex{1}\)
Label the following either an electrophile or a nucleophile.
- Answer
-
A = Electrophile
B = Nucleophile
C = Both (carbonyl carbon is electrophile and oxygen is nucleophile)


2.4C: Dehydration Synthesis
- Explain dehydration (or condensation) reactions
Most macromolecules are made from single subunits, or building blocks, called monomers. The monomers combine with each other via covalent bonds to form larger molecules known as polymers. In doing so, monomers release water molecules as byproducts. This type of reaction is known as dehydration synthesis, which means “to put together while losing water.” It is also considered to be a condensation reaction since two molecules are condensed into one larger molecule with the loss of a smaller molecule (the water.)
In a dehydration synthesis reaction between two un-ionized monomers, such as monosaccharide sugars, the hydrogen of one monomer combines with the hydroxyl group of another monomer, releasing a molecule of water in the process. The removal of a hydrogen from one monomer and the removal of a hydroxyl group from the other monomer allows the monomers to share electrons and form a covalent bond. Thus, the monomers that are joined together are being dehydrated to allow for synthesis of a larger molecule.
A dehydration synthesis reaction involving un-ionized moners.. : In the dehydration synthesis reaction between two molecules of glucose, a hydroxyl group from the first glucose is combined with a hydrogen from the second glucose, creating a covalent bond that links the two monomeric sugars (monosaccharides) together to form the dissacharide maltose. In the process, a water molecule is formed.
When the monomers are ionized, such as is the case with amino acids in an aqueous environment like cytoplasm, two hydrogens from the positively-charged end of one monomer are combined with an oxygen from the negatively-charged end of another monomer, again forming water, which is released as a side-product, and again joining the two monomers with a covalent bond.
A dehydration synthesis reaction involving ionized monomers. : In the dehydration synthesis reaction between two amino acids, with are ionized in aqueous environments like the cell, an oxygen from the first amino acid is combined with two hydrogens from the second amino acid, creating a covalent bond that links the two monomers together to form a dipeptide. In the process a water molecule is formed.
As additional monomers join via multiple dehydration synthesis reactions, the chain of repeating monomers begins to form a polymer. Different types of monomers can combine in many configurations, giving rise to a diverse group of macromolecules. Three of the four major classes of biological macromolecules (complex carbohydrates, nucleic acids, and proteins), are composed of monomers that join together via dehydration synthesis reactions. Complex carbohydrates are formed from monosaccharides, nucleic acids are formed from mononucleotides, and proteins are formed from amino acids.
There is great diversity in the manner by which monomers can combine to form polymers. For example, glucose monomers are the constituents of starch, glycogen, and cellulose. These three are polysaccharides, classified as carbohydrates, that have formed as a result of multiple dehydration synthesis reactions between glucose monomers. However, the manner by which glucose monomers join together, specifically locations of the covalent bonds between connected monomers and the orientation (stereochemistry) of the covalent bonds, results in these three different polysaccharides with varying properties and functions. In nucleic acids and proteins, the location and stereochemistry of the covalent linkages connecting the monomers do not vary from molecule to molecule, but instead the multiple kinds of monomers (five different monomers in nucleic acids, A, G, C, T, and U mononucleotides; 21 different amino acids monomers in proteins) are combined in a huge variety of sequences. Each protein or nucleic acid with a different sequence is a different molecule with different properties.
Key Points
- During dehydration synthesis, either the hydrogen of one monomer combines with the hydroxyl group of another monomer releasing a molecule of water, or two hydrogens from one monomer combine with one oxygen from the other monomer releasing a molecule of water.
- The monomers that are joined via dehydration synthesis reactions share electrons and form covalent bonds with each other.
- As additional monomers join via multiple dehydration synthesis reactions, this chain of repeating monomers begins to form a polymer.
- Complex carbohydrates, nucleic acids, and proteins are all examples of polymers that are formed by dehydration synthesis.
- Monomers like glucose can join together in different ways and produce a variety of polymers. Monomers like mononucleotides and amino acids join together in different sequences to produce a variety of polymers.
Key Terms
- covalent bond : A type of chemical bond where two atoms are connected to each other by the sharing of two or more electrons.
- monomer : A relatively small molecule which can be covalently bonded to other monomers to form a polymer.
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4.1: Financial Reports- Overview
As discussed previously, investors and creditors assess a company's overall financial health by using published financial statements. Recall that the previous chapter about financial reporting illustrated how the core financial statements link into a cohesive network of financial information. The illustration in that chapter showed how the ending cash balance from the statement of cash flows (SCF) is also the ending cash balance in the SFP/BS. This is the final link that completes the connection of the core financial statements.
For example, in the SFP/BS for Wellbourn Services Ltd. at December 31 (which we saw also in the previous chapter) note how the cash ending balance links the two statements.
Wellbourne Services Limited
Statement of Financial Position
December 31, 2020
|
Assets
|
|
Liabilities
|
|
|
|---|---|---|---|---|
|
Cash |
$135,500 |
Accounts payable |
$ 77,500 |
|
|
Accounts receivable (net) |
225,000 |
Accrued liabilities |
225,000 |
|
|
Inventory |
130,000 |
Total current liabilities |
302,500 |
|
|
Total current assets |
490,500 |
Bonds payable |
160,000 |
|
|
Investments |
100,000 |
Total liabilities |
462,500 |
|
|
Property, plant, and equipment (net) |
246,000 |
Equity |
|
|
|
Intangible assets |
15,000 |
Common shares |
210,000 |
|
|
Total assets |
$851,500 |
Contributed surplus |
25,000 |
|
|
|
|
Retained earnings |
105,500 |
|
|
|
|
AOCI |
48,500 |
|
|
|
|
Total equity |
389,000 |
|
|
|
|
Liabilities and equity |
$851,500 |
Wellbourne Services Limited
Statement of Cash Flows
December 31, 2020
|
Cash flows from operating activities
|
|
||
| Cash paid for supplies | (25,000) | ||
| Cash paid to employees | (51,200) | ||
| Net cash used by operating activities | (26,200) | ||
|
Cash flows from investing activities
Purchase of equipment |
(25,000) | ||
| Net cash used by investing activities | (25,000) | ||
|
Cash flows from financing activities
|
|
|
|
| Issued bonds | 160,000 | ||
|
Net cash received by financing activities |
|
110,000 |
|
|
Net increase in cash |
|
58,800 |
|
|
Cash balance, January 1 |
|
76,700 |
|
|
Cash balance, December 31 |
|
$ 135,500 |
The SFP/BS provides information about a company's resources (assets) at a specific point in time and whether these resources are financed mainly by debt (current and long-term liabilities) or equity (shareholders' equity). In other words, the SFP/BS provides the information needed to assess a company's liquidity and solvency. Combined, these represent a company's financial flexibility .
Recall the basic accounting equation:
The key issues to consider regarding the SFP/BS are the valuation and management of resources (assets) and the recognition and timing of debt obligations (liabilities). Reporting the results within the SFP/BS creates a critical reporting tool to assess a company's overall financial health.
The statement of cash flows, discussed later in this chapter, identifies how the company utilized its cash inflows and outflows over the reporting period. Since the SCF separates cash flows into those resulting from operating activities versus investing and financing activities, investors and creditors can quickly see where the main sources of cash originate. If cash sources are originating more from investing activities than from operations, this means that the company is likely selling off some of its assets to cover its obligations. This may be appropriate if these assets are idle and no longer contributing towards generating profit, but otherwise, selling off useful assets could trigger a downward spiral, with profits plummeting as a result. If cash sources are originating mainly from financing activities, the company is likely sourcing its cash from debt or from issuing shares. Higher debt requires more cash to make the principal and interest payments, and more shares means that existing investors' ownership is becoming diluted. Either scenario will be cause for concern for both investors and creditors. Even if most of the cash sources are mainly from operating activities, a large difference between net income and the total cash from operating activities is a warning sign that investors and creditors should be digging deeper.
The bottom line after reviewing the two statements is: if debt is high and cash balances are low, the greater the risk of failure.
The SFP/BS will now be discussed in detail.


2.4: Success After Class - Reviewing Material
Learning Objectives
At the end of this section, the learner will:
- Describe the importance of a review period prior to an exam.
- Create a Five Day Study plan for an upcoming exam.
Planning a review period prior to an exam
Many students, however, don’t start thinking about test taking, whether weekly exams, mid-terms, or finals, until the day before when they engage in an all-nighter, or cramming. Students wanting to be successful have to have the self-discipline to schedule time to study well in advance of the exam. They have to actually do the work: the preparation needed in order to have the best opportunity for success on the exam. Then, they have to be able to apply their preparation accordingly and perform well on the exam.
It would be beneficial to spread this preparation and practice out over time and prepare periodically rather than to wait until the last minute and binge study or cram. Planning for a review period over five days prior to the exam and healthy eating and sleep habits cannot be overemphasized. Without a plan, preparation would not be the same and this may affect your test score. Binge studying and cramming also are not healthy. Careful review of the material prior to each exam is required for optimal performance, and student reports tell us that the habit of “cramming” the night before the exam is not going to lead to positive, long-term results. Staying up late puts stress on our brain and body, and not getting adequate sleep places our bodies at risk for getting sick.
Students need to develop a mindset that every time they attend class or complete a reading assignment they are preparing for the next exam, because so much of their grade rides on those exams. In college, it is not uncommon for your entire course grade to be based on two or three exams with no options for a retake or extra credit. Instead of an exam being over one or two chapters, exams are more likely to cover 5+ chapters of material, and in the case of a comprehensive exam, an entire semester’s worth of material. Intentionally working through the process of preparing for, taking, and learning from the testing process will facilitate successful outcomes.
Often, students jump right in to taking an exam feeling stressed and rushed with a need to regurgitate information immediately. However, approaching a test in a careful and methodical way can help ensure the best results. You can best utilize your exam time by taking the time to survey the exam directions and point values and planning how to approach sections of the test. Strategy is particularly important for tests with mixed types of questions (i.e., multiple choice and essay) or tests with multiple essay questions. For example, if you were to spend too much time on the multiple choice questions and not allocate enough time to complete an essay question, you may have answered the majority of the total number of questions but still do poorly on the exam if the essay question was worth a larger portion of the exam’s total points. Plan your time carefully and manage your stress.
Activity 3.4 – Creating a Study Plan
Creating a Study Plan
Identify when your next substantial exam will be given in a class. Create a five-day study plan using the formula described in “Create a Five-Day Study Plan for Exams”. Once you have created this plan, incorporate the schedule into your time management calendar for implementation.
Create a Five-Day Study Plan for Exams
Making a plan that will allow for the careful review of all of the assigned and presented material leads to less stress the day of an exam and the best conditions for recall and performance. Here is a simple method for creating a study plan:
- Break the material for the exam into four manageable “chunks.” If material can be logically divided by chapters, use that method. If not, make up your own chunks based on the content of the material. Note: It is not a good idea to break up information by the method of delivery (i.e., one chunk equals notes, one chunk equals readings, etc.). You want to include all information on specific topics together in one chunk (e.g., information about the topic of osmosis from the lecture notes and the textbook in a chunk).
- Plan to spend about two hours studying on each of the five days.
- You should work with the material on the exam in two ways, by preparing that material for review and by reviewing that material. Preparation of material incorporates study techniques discussed earlier in this chapter such as combining and clarifying lecture and textbook notes and creating visual organizers for topics.
Example of the Schedule for a Five-Day Study Plan
Day One
Prepare 1st Chunk – 2 hours
Day Two
Prepare 2nd Chunk – 2 hours
Review 1st Chunk – 30 minutes
Day Three
Prepare 3rd Chunk – 1.5 hours
Review 2nd Chunk – 30 minutes
Review 1st Chunk – 15 minutes
Day Four
Prepare 4th Chunk – 1 hour
Review 3rd Chunk – 30 minutes
Review 2nd Chunk – 15 minutes
Review 1st Chunk – 10 minutes
Day Five
Review 4th Chunk – 1 hour
Review 3rd Chunk – 30 minutes
Review 2nd Chunk – 30 minutes
Review 1st Chunk – 30 minutes
Taking some time to plan out what material will go into each chunk will ensure that you do not miss anything.
This method typically works very well for courses that offer two or three tests throughout the semester; however, a similar method could be modified by the number of days or the amount of time spent each day if a course offers more exams with less material on each exam.


19.5: Alternating voltages and currents
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So far, we have modeled how current propagates through a resistor under a constant potential difference, \(\Delta V\) . This is called “direct current” (DC) as the charges move in a constant direction through the resistor. Batteries supply fixed voltages, and circuits with batteries will almost always have DC current. The voltage that is supplied between two of the sockets in a household electrical outlet is “alternating”, and leads to “alternating current” (AC), where charges move back and forth, with no net displacement.
The potential difference across a household outlet varies sinusoidally:
\[\begin{aligned} \Delta V (t) = \Delta V_0 \sin(\omega t)\end{aligned}\]
where \(\Delta V_0\) is the maximal amplitude of the voltage ( \(120\text{V}\) in North America, \(220\text{V}\) in Europe), and \(\omega = 2\pi f\) , is the angular frequency of the voltage ( \(f=60\text{Hz}\) in North America, \(f=50\text{Hz}\) in Europe). When a resistor with resistance, \(R\) , is connected to an AC voltage, the resulting current, given by Ohm’s Law, is also alternating:
\[\begin{aligned} I(t)=\frac{\Delta V(t)}{R}=\frac{\Delta V_0}{R}\sin(\omega t)=I_0\sin(\omega t)\end{aligned}\]
On average, the alternating current through a resistor is zero. However, this does not mean that zero energy is dissipated, since the electrons in the resistor will still collide with atoms as they oscillate back and forth. We can define the average power, \(\bar P\) , that is dissipated in the resistor as the power that is dissipated over one oscillation cycle (with period, \(T\) ). To obtain the latter, we calculate the total energy, \(E\) , dissipated in the resistor over one cycle so that the power is simply given by \(E/T\) . We divide the interval of time, \(T\) , into infinitesimally small intervals, \(dt\) , so that the infinitesimal energy, \(dE\) , dissipated in an infinitesimal time, \(dt\) , is given by:
\[\begin{aligned} dE=P(t) dt\end{aligned}\]
The total energy dissipated in one period is then given by:
\[\begin{aligned} E=\int dE = \int_0^T P(t)dt\end{aligned}\]
so that the power dissipated in one cycle is given by:
\[\begin{aligned} \bar P=\frac{E}{T}=\frac{1}{T}\int_0^T P(t)dt\end{aligned}\]
The instantaneous power, \(P(t)\) , can be described in terms of the instantaneous current, \(P(t)=I^2(t)R\) , so that the average power can be written as:
\[\begin{aligned} \bar P = \frac{1}{T}\int_0^TP(t)dt=\frac{1}{T}\int_0^TI(t)^2Rdt=RI_0^2\frac{1}{T}\int_0^T\sin^2(\omega t)dt=\frac{1}{2}RI_0^2\end{aligned}\]
where we used the fact that \(T=\frac{2\pi}{\omega}\) to evaluate the integral. In order to make the formula similar to the DC equivalent (without the additional factor of \(1/2\) ), we can define the “root mean square” current, \(I_{rms}\) , as an average current, from which we can calculate the average power that is dissipated in a resistor:
\[\begin{aligned} I_{rms}&=\frac{I_0}{\sqrt 2}\\[4pt] \therefore\bar P&=I_{rms}^2R\end{aligned}\]
Similarly, one can define the “root mean square” voltage, \(\Delta V_{rms}\) , so that the average power dissipated with alternating current can be written in the same form as for the DC case:
\[\begin{aligned} V_{rms}&=\frac{\Delta V_0}{\sqrt 2}\\[4pt] \therefore\bar P&=I_{rms}^2R =\frac{\Delta V_{rms}^2}{R}=I_{rms}\Delta V_{rms}\end{aligned}\]


7.4: Gibbs Free Energy for Non-Standard
- To know the relationship between free energy and the equilibrium constant.
We have identified three criteria for whether a given reaction will occur spontaneously:
- \(ΔS_{univ} > 0 \),
- \(ΔG_{sys} < 0\) (applicable under constant temperature and constant pressure conditions), and
- the relative magnitude of the reaction quotient \(Q\) versus the equilibrium constant \(K\).
Recall that if \(Q < K\), then the reaction proceeds spontaneously to the right as written, resulting in the net conversion of reactants to products. Conversely, if \(Q > K\), then the reaction proceeds spontaneously to the left as written, resulting in the net conversion of products to reactants. If \(Q = K\), then the system is at equilibrium, and no net reaction occurs. Table \(\PageIndex{1}\) summarizes these criteria and their relative values for spontaneous, nonspontaneous, and equilibrium processes.
| Spontaneous | Equilibrium | Nonspontaneous* |
|---|---|---|
| *Spontaneous in the reverse direction. | ||
| ΔS univ > 0 | ΔS univ = 0 | ΔS univ < 0 |
| ΔG sys < 0 | ΔG sys = 0 | ΔG sys > 0 |
| Q < K | Q = K | Q > K |
Because all three criteria are assessing the same thing—the spontaneity of the process—it would be most surprising indeed if they were not related. In this section, we explore the relationship between the standard free energy of reaction (\(ΔG^o\)) and the equilibrium constant (\(K\)).
Free Energy and the Equilibrium Constant
Because \(ΔH^o\) and \(ΔS^o\) determine the magnitude and sign of \(ΔG^o\) and also because \(K\) is a measure of the ratio of the concentrations of products to the concentrations of reactants, we should be able to express K in terms of \(ΔG^o\) and vice versa. "Free Energy", \(ΔG\) is equal to the maximum amount of work a system can perform on its surroundings while undergoing a spontaneous change. For a reversible process that does not involve external work, we can express the change in free energy in terms of volume, pressure, entropy, and temperature, thereby eliminating \(ΔH\) from the equation for \(ΔG\). The general relationship can be shown as follows (derivation not shown):
\[ \Delta G = V \Delta P − S \Delta T \label{18.29} \]
If a reaction is carried out at constant temperature (\(ΔT = 0\)), then Equation \(\ref{18.29}\) simplifies to
\[\Delta{G} = V\Delta{P} \label{18.30} \]
Under normal conditions, the pressure dependence of free energy is not important for solids and liquids because of their small molar volumes. For reactions that involve gases, however, the effect of pressure on free energy is very important.
Assuming ideal gas behavior, we can replace the \(V\) in Equation \(\ref{18.30}\) by \(nRT/P\) (where \(n\) is the number of moles of gas and \(R\) is the ideal gas constant) and express \(\Delta{G}\) in terms of the initial and final pressures (\(P_i\) and \(P_f\), respectively):
\[\begin{align} \Delta G &=\left(\dfrac{nRT}{P}\right)\Delta P \\[4pt] &=nRT\dfrac{\Delta P}{P} \\[4pt] &=nRT\ln\left(\dfrac{P_\textrm f}{P_\textrm i}\right) \label{18.31}\end{align} \]
If the initial state is the standard state with \(P_i = 1 \,atm\), then the change in free energy of a substance when going from the standard state to any other state with a pressure \(P\) can be written as follows:
\[G − G^° = nRT\ln{P} \nonumber \]
This can be rearranged as follows:
\[G = G^° + nRT\ln {P} \label{18.32} \]
As you will soon discover, Equation \(\ref{18.32}\) allows us to relate \(ΔG^o\) and \(K_p\). Any relationship that is true for \(K_p\) must also be true for \(K\) because \(K_p\) and \(K\) are simply different ways of expressing the equilibrium constant using different units.
Let’s consider the following hypothetical reaction, in which all the reactants and the products are ideal gases and the lowercase letters correspond to the stoichiometric coefficients for the various species:
\[aA+bB \rightleftharpoons cC+dD \label{18.33} \]
Because the free-energy change for a reaction is the difference between the sum of the free energies of the products and the reactants, we can write the following expression for \(ΔG\):
\[\begin{align} \Delta{G} &=\sum_m G_{products}−\sum_n G_{reactants} \\[4pt] &=(cG_C+dG_D)−(aG_A+bG_B) \label{18.34} \end{align} \]
Substituting Equation \(\ref{18.32}\) for each term into Equation \(\ref{18.34}\),
\[\Delta G = [(cG^o_C+cRT \ln P_C)+(dG^o_D+dRT\ln P_D)]−[(aG^o_A+aRT\ln P_A)+(bG^o_B+bRT\ln P_B)] \nonumber \]
Combining terms gives the following relationship between \(ΔG\) and the reaction quotient \(Q\):
\[\begin{align} \Delta G &=\Delta G^\circ+RT \ln\left(\dfrac{P^c_\textrm CP^d_\textrm D}{P^a_\textrm AP^b_\textrm B}\right) \\[4pt] &=\Delta G^\circ+RT\ln Q \label{18.35} \end{align} \]
where \(ΔG^o\) indicates that all reactants and products are in their standard states. For gases at equilibrium (\(Q = K_p\)), and as you’ve learned in this chapter, \(ΔG = 0\) for a system at equilibrium. Therefore, we can describe the relationship between \(ΔG^o\) and \(K_p\) for gases as follows:
\[ \begin{align} 0 &= ΔG^o + RT\ln K_p \label{18.36a} \\[4pt] ΔG^o &= −RT\ln K_p \label{18.36b} \end{align} \]
If the products and reactants are in their standard states and \(ΔG^o < 0\), then \(K_p > 1\), and products are favored over reactants when the reaction is at equilibrium. Conversely, if \(ΔG^o > 0\), then \(K_p < 1\), and reactants are favored over products when the reaction is at equilibrium. If \(ΔG^o = 0\), then \(K_p = 1\), and neither reactants nor products are favored when the reaction is at equilibrium.
For a spontaneous process under standard conditions, \(K_{eq}\) and \(K_p\) are greater than 1.
\(ΔG^o\) is −32.7 kJ/mol of N 2 for the reaction
\[\ce{N2(g) + 3H2(g) <=> 2NH3(g)} \nonumber \]
This calculation was for the reaction under standard conditions—that is, with all gases present at a partial pressure of 1 atm and a temperature of 25°C. Calculate \(ΔG\) for the same reaction under the following nonstandard conditions:
- \(P_{\textrm N_2}\) = 2.00 atm,
- \(P_{\textrm H_2}\) = 7.00 atm,
- \(P_{\textrm{NH}_3}\) = 0.021 atm, and
- \(T = 100 ^oC\).
Does the reaction proceed to the right, as written, or to the left to reach equilibrium?
Given : balanced chemical equation, partial pressure of each species, temperature, and ΔG°
Asked for : whether the reaction proceeds to the right or to the left to reach equilibrium
Strategy :
- Using the values given and Equation \(\ref{18.35}\), calculate \(Q\).
- Determine if \(Q\) is >, <, or = to \(K\)
- Substitute the values of \(ΔG^o\) and \(Q\) into Equation \(\ref{18.35}\) to obtain \(ΔG\) for the reaction under nonstandard conditions.
Solution :
A The relationship between \(ΔG^o\) and \(ΔG\) under nonstandard conditions is given in Equation \(\ref{18.35}\). Substituting the partial pressures given, we can calculate \(Q\):
\[ \begin{align*} Q &=\dfrac{P^2_{\textrm{NH}_3}}{P_{\textrm N_2}P^3_{\textrm H_2}} \\[4pt] &=\dfrac{(0.021)^2}{(2.00)(7.00)^3}=6.4 \times 10^{-7} \end{align*} \nonumber \]
B Because \(ΔG^o\) is −, K must be a number greater than 1
C Substituting the values of \(ΔG^o\) and \(Q\) into Equation \(\ref{18.35}\),
\[\begin{align*} \Delta G &=\Delta G^\circ+RT\ln Q \\[4pt] &=-32.7\textrm{ kJ}+\left[(\textrm{8.314 J/K})(\textrm{373 K})\left(\dfrac{\textrm{1 kJ}}{\textrm{1000 J}}\right)\ln(6.4\times10^{-7})\right] \\[4pt] &=-32.7\textrm{ kJ}+(-44\textrm{ kJ}) \\[4pt] &=-77\textrm{ kJ/mol of N}_2 \end{align*} \nonumber \]
Because \(ΔG < 0\) and \(Q < K\) (because \(Q < 1\)), the reaction proceeds spontaneously to the right, as written, in order to reach equilibrium.
Calculate \(ΔG\) for the reaction of nitric oxide with oxygen to give nitrogen dioxide under these conditions: T = 50°C, P NO = 0.0100 atm, \(P_{\ce{O_2}}\) = 0.200 atm, and \(P_{\ce{NO_2}} = 1.00 × 10^{−4} atm\). The value of \(ΔG^o\) for this reaction is −72.5 kJ/mol of O 2 . Are products or reactants favored?
- Answer
-
−92.9 kJ/mol of \(\ce{O2}\); the reaction is spontaneous to the right as written. The reaction will proceed in the forward direction to reach equilibrium.
Calculate \(K_p\) for the reaction of \(\ce{H_2}\) with \(\ce{N2}\) to give \(\ce{NH3}\) at 25°C. \(ΔG^o\) for this reaction is −32.7 kJ/mol of \(\ce{N2}\).
Given : balanced chemical equation from Example \(\PageIndex{1}\), \(ΔG^o\), and temperature
Asked for : \(K_p\)
Strategy:
Substitute values for \(ΔG^o\) and T (in kelvin) into Equation \(\ref{18.36b}\) to calculate \(K_p\), the equilibrium constant for the formation of ammonia.
Solution
In Example \(\PageIndex{1}\), we used tabulated values of ΔG ∘ f to calculate \(ΔG^o\) for this reaction (−32.7 kJ/mol of N 2 ). For equilibrium conditions, rearranging Equation \(\ref{18.36b}\),
\[\begin{align*} \Delta G^\circ &=-RT\ln K_\textrm p \\[4pt] \dfrac{-\Delta G^\circ}{RT} &=\ln K_\textrm p \end{align*} \nonumber \]
Inserting the value of \(ΔG^o\) and the temperature (25°C = 298 K) into this equation,
\[\begin{align*}\ln K_\textrm p &=-\dfrac{(-\textrm{32.7 kJ})(\textrm{1000 J/kJ})}{(\textrm{8.314 J/K})(\textrm{298 K})}=13.2 \\[4pt] K_\textrm p &=5.4\times10^5\end{align*} \nonumber \]
Thus the equilibrium constant for the formation of ammonia at room temperature is product-favored. However, the rate at which the reaction occurs at room temperature is too slow to be useful.
Calculate K p for the reaction of NO with O 2 to give NO 2 at 25°C. \(ΔG^o\) for this reaction is −70.5 kJ/mol of \(\ce{O2}\).
- Answer
-
2.3 × 10 12
Although \(K_p\) is defined in terms of the partial pressures of the reactants and the products, the equilibrium constant \(K\) is defined in terms of the concentrations of the reactants and the products. The numerical magnitude of \(K_p\) and \(K\) are related:
\[K_p = K(RT)^{Δn} \label{18.37} \]
where \(Δn\) is the number of moles of gaseous product minus the number of moles of gaseous reactant. For reactions that involve only solutions, liquids, and solids, \(Δn = 0\), so \(K_p = K\). For all reactions that do not involve a change in the number of moles of gas present, the relationship in Equation \(\ref{18.36b}\) can be written in a more general form:
\[ΔG° = −RT \ln K \label{18.38} \]
Only when a reaction results in a net production or consumption of gases is it necessary to correct Equation \(\ref{18.38}\) for the difference between \(K_p\) and \(K\).
Although we typically use concentrations or pressures in our equilibrium calculations, recall that equilibrium constants are generally expressed as unitless numbers because of the use of activities or fugacities in precise thermodynamic work. Systems that contain gases at high pressures or concentrated solutions that deviate substantially from ideal behavior require the use of fugacities or activities, respectively.
Combining Equation \(\ref{18.38}\) with \(ΔG^o = ΔH^o − TΔS^o\) provides insight into how the components of \(ΔG^o\) influence the magnitude of the equilibrium constant:
\[\begin{align} ΔG° &= ΔH° − TΔS° \\[4pt] &= −RT \ln K \label{18.39} \end{align} \]
Notice that \(K\) becomes larger as \(ΔS^o\) becomes more positive, indicating that the magnitude of the equilibrium constant is directly influenced by the tendency of a system to move toward maximum disorder. Moreover, \(K\) increases as \(ΔH^o\) decreases. Thus the magnitude of the equilibrium constant is also directly influenced by the tendency of a system to seek the lowest energy state possible.
The magnitude of the equilibrium constant is directly influenced by the tendency of a system to move toward maximum entropy and seek the lowest energy state possible.
Relating Grxn and Kp: Relating Grxn and Kp(opens in new window) [youtu.be]
Temperature Dependence of the Equilibrium Constant
The fact that \(ΔG^o\) and \(K\) are related provides us with another explanation of why equilibrium constants are temperature dependent. This relationship is shown explicitly in Equation \(\ref{18.39}\), which can be rearranged as follows:
\[\ln K=-\dfrac{\Delta H^\circ}{RT}+\dfrac{\Delta S^\circ}{R} \label{18.40} \]
Assuming \(ΔH^o\) and \(ΔS^o\) are temperature independent, for an exothermic reaction (\(ΔH^o < 0\)), the magnitude of \(K\) decreases with increasing temperature, whereas for an endothermic reaction (ΔH° > 0), the magnitude of \(K\) increases with increasing temperature. The quantitative relationship expressed in Equation \(\ref{18.40}\) agrees with the qualitative predictions made by applying Le Chatelier’s principle. Because heat is produced in an exothermic reaction, adding heat (by increasing the temperature) will shift the equilibrium to the left, favoring the reactants and decreasing the magnitude of \(K\). Conversely, because heat is consumed in an endothermic reaction, adding heat will shift the equilibrium to the right, favoring the products and increasing the magnitude of \(K\). Equation \(\ref{18.40}\) also shows that the magnitude of \(ΔH^o\) dictates how rapidly \(K\) changes as a function of temperature. In contrast, the magnitude and sign of \(ΔS^o\) affect the magnitude of \(K\) but not its temperature dependence.
If we know the value of \(K\) at a given temperature and the value of \(ΔH^o\) for a reaction, we can estimate the value of \(K\) at any other temperature, even in the absence of information on \(ΔS^o\). Suppose, for example, that \(K_1\) and \(K_2\) are the equilibrium constants for a reaction at temperatures \(T_1\) and \(T_2\), respectively. Applying Equation \(\ref{18.40}\) gives the following relationship at each temperature:
\[\ln K_1 =\dfrac{-\Delta H^\circ}{RT_1}+\dfrac{\Delta S^\circ}{R} \nonumber \]
and
\[\ln K_2 =\dfrac{-\Delta H^\circ}{RT_2}+\dfrac{\Delta S^\circ}{R} \nonumber \]
Subtracting \(\ln K_1\) from \(\ln K_2\),
\[\ln K_2-\ln K_1=\ln\dfrac{K_2}{K_1}=\dfrac{\Delta H^\circ}{R}\left(\dfrac{1}{T_1}-\dfrac{1}{T_2}\right) \label{18.41} \]
Thus calculating \(ΔH^o\) from tabulated enthalpies of formation and measuring the equilibrium constant at one temperature (\(K_1\)) allow us to calculate the value of the equilibrium constant at any other temperature (\(K_2\)), assuming that \(ΔH^o\) and \(ΔS^o\) are independent of temperature.
The equilibrium constant for the formation of \(\ce{NH3}\) from \(\ce{H2}\) and \(\ce{N2}\) at 25°C was calculated to be K p = 5.4 × 10 5 in Example \(\PageIndex{3}\). What is \(K_p\) at 500°C? (Use the data from Example \(\PageIndex{1}\).)
Given : balanced chemical equation, \(ΔH^o°\), initial and final \(T\), and \(K_p\) at 25°C
Asked for: \(K_p\) at 500°C
Strategy :
Convert the initial and final temperatures to kelvin. Then substitute appropriate values into Equation \(\ref{18.41}\) to obtain \(K_2\), the equilibrium constant at the final temperature.
Solution :
The value of \(ΔH^o\) for the reaction obtained using Hess’s law is −91.8 kJ/mol of N 2 . If we set T 1 = 25°C = 298.K and T 2 = 500°C = 773 K, then from Equation \(\ref{18.41}\) we obtain the following:
\[\begin{align*}\ln\dfrac{K_2}{K_1}&=\dfrac{\Delta H^\circ}{R}\left(\dfrac{1}{T_1}-\dfrac{1}{T_2}\right)
\\[4pt] &=\dfrac{(-\textrm{91.8 kJ})(\textrm{1000 J/kJ})}{\textrm{8.314 J/K}}\left(\dfrac{1}{\textrm{298 K}}-\dfrac{1}{\textrm{773 K}}\right)=-22.8
\\[4pt] \dfrac{K_2}{K_1}&=1.3\times10^{-10}
\\[4pt] K_2&=(5.4\times10^5)(1.3\times10^{-10})=7.0\times10^{-5}\end{align*} \nonumber \]
Thus at 500°C, the equilibrium strongly favors the reactants over the products.
In the exercise in Example \(\PageIndex{3}\), you calculated K p = 2.2 × 10 12 for the reaction of NO with O 2 to give NO 2 at 25°C. Use the \(ΔH^o_f\) values in the exercise in Example \(\PageIndex{1}\) to calculate \(K_p\) for this reaction at 1000°C.
- Answer
-
5.6 × 10 −4
The Van't Hoff Equation: The Van't Hoff Equation (opens in new window) [youtu.be]
Summary
For a reversible process that does not involve external work, we can express the change in free energy in terms of volume, pressure, entropy, and temperature. If we assume ideal gas behavior, the ideal gas law allows us to express ΔG in terms of the partial pressures of the reactants and products, which gives us a relationship between ΔG and K p , the equilibrium constant of a reaction involving gases, or K, the equilibrium constant expressed in terms of concentrations. If \(ΔG^o\) < 0, then K > 1, and products are favored over reactants at equilibrium. Conversely, if \(ΔG^o\) > 0, then K < 1, and reactants are favored over products at equilibrium. If \(ΔG^o\) = 0, then K=1, and neither reactants nor products are favored at equilibrium. We can use the measured equilibrium constant K at one temperature and ΔH° to estimate the equilibrium constant for a reaction at any other temperature.


14.3: South America
Ceramics in Early South America
The ceramic objects of the Paracas, Nazca, and Moche communities of Peru vary in artistic forms and were important cultural artifacts. Like the Tiwanaku and Waki people of Bolivia and their contemporaries, the Wari people of Peru produced pottery that was multifaceted in both aesthetics and utility.
Compare the aesthetic style, technique, and function of ceramics made by the Paracas, Nazcu, Moche, Tiwanaku, Waki, and Wari cultures
Key Points
- The Paracas culture (from 800 to 100 BCE) immediately preceded and heavily influenced the Nazca culture.
- The art of the Paracas culture has mainly been preserved in tombs and on mummies .
- The Nazca, like all Pre-Columbian societies in South America including the Inca , had no writing system, in contrast to the contemporary Maya of Mesoamerica . The iconography or symbols on their ceramics served as a means of communication.
- As in other cultures, Moche ceramics were probably used for educational purposes and communication.
- The Moche culture used mold technology to replicate ceramic forms .
- Irrigation and the passage of fluids in the human body are important themes in Moche culture and artwork.
- The realistic detail in Moche ceramics may have helped them serve as didactic models.
- The Tiwanaku state was an important precursor to the Inca Empire.
- Small decorative objects that held ritual religious meaning were used to spread the influence of the capital city of Tiwanaku to surrounding communities.
- The Tiwanaku and Wari cultures must have interacted, given the similarities in the artifacts of each culture, but whether their relationship was amicable or antagonistic is unknown.
Key Terms
- slip painting : A type of pottery decoration in which a liquid mixture of clay and/or other materials suspended in water is applied to wet clay before it is fired in a kiln.
- Huacas : In Quechua, a Native American language of South America, a huaca or waqa is an object that represents something revered, typically a monument of some kind. The term huaca can refer to natural locations, such as immense rocks.
- Middle Horizon : Cultural period of Peru and the Andean region lasting from 600 to 1000 CE.
- resin painting : A type of pottery decoration in which ceramics are painted after they are fired in a kiln with a sticky organic substance exuded by trees and other plants.
- Wari : A Middle Horizon civilization that flourished in the south-central Andes and coastal area of modern-day Peru, from about 500 to 1000 CE.
- kero : A type of wooden drinking vessel produced by the Incas and earlier Andean cultures.
- phytomorphic : Having the attributes of a plant.
- Moche : A civilization that flourished in northern Peru from about 100 to 800 AD, particularly noted for their elaborately painted ceramics, gold work, monumental constructions (huacas), and irrigation systems.
- Andean : Of or pertaining to the Andes mountains in South America.
Paracas and Nazca Cultures
The Paracas culture was an important Andean society between approximately 800 and 100 BCE, with an extensive knowledge of irrigation and water management. It developed in the Paracas Peninsula in the Ica Region of Peru.
The Nazca culture flourished from 100 to 800 CE beside the dry southern coast of Peru in the river valleys of the Rio Grande de Nazca drainage and the Ica Valley. Heavily influenced by the preceding Paracas culture, the Nazca produced an array of beautiful crafts and technologies such as ceramics, textiles, and geoglyphs (most commonly known as the Nazca lines ).
Paracas Ceramics
Many ceramics of the Paracas have been found in tombs, particularly in the Paracas Cavernas. These are shaft tombs set into the top of Cerro Colorado, each containing multiple burial sites. The associated ceramics include incised polychrome (the surface has been incised with a sharp tool and painted multiple colors), “negative” resist decoration (pottery is covered in material, then painted and uncovered to reveal a pattern of negative and positive space), and other techniques of the Paracas tradition.
Nazca Ceramics
The Nazca culture is characterized by its beautiful polychrome pottery, painted with at least 15 distinct colors. The shift from post-fire resin painting to pre-fire slip painting marked the end of Paracas- style pottery and the beginning of Nazca-style pottery. Archaeologists have excavated highly valued polychrome pottery among all classes of Nazca society, illustrating that it was not just the elite that had access to these pieces.
The Nazca pottery sequence has been divided into nine phases, progressing from realistic subject matter such as fruits, plants, people, and animals to motifs that included abstract elements as part of the design and geometric iconography. The Nazca, like all Pre-Columbian societies in South America including the Inca, had no writing system, in contrast to the contemporary Maya of Mesoamerica. The iconography or symbols on their ceramics served as a means of communication.
Moche Ceramics
From 100 to 800 CE, Moche civilization flourished in northern Peru with its capital, Huacas del Sol y de la Luna, located near present-day Trujillo.
Traditional North Coast Peruvian ceramic art uses a limited palette , relying primarily on red and white, fine line painting, fully modeled clay, naturalistic figures, and stirrup spouts (in which the stirrup handle forms part of the spout, which emanates from the top of the stirrup). Moche ceramics created between 150 and 800 AD epitomize this style. These realistic pots have been found not only in major North Coast archaeological sites, such as Huaca de la luna, Huaca del Sol, and Sipan, but at small villages and unrecorded burial sites as well.
The realistic detail in Moche ceramics may have helped them serve as didactic models. Older generations could pass down general knowledge about reciprocity and embodiment to younger generations through such portrayals. Important social activities are documented in Moche pottery, including war, sex, metalwork , and weaving. Moche ceramics vary widely in shape and theme and are not generally uniform, although the use of mold technology did enable for mass production.
Because irrigation was the source of wealth and foundation of the empire, Moche culture emphasized the importance of circulation and flow. Expanding upon this, the Moche focused on the passage of fluids in their artwork, particularly life fluids through vulnerable human orifices.
The coloration of Moche pottery is often simple and follows the Peruvian tradition with yellowish cream and rich red used almost exclusively on elite pieces and with white and black used rarely. Their adobe buildings have been mostly destroyed by looters and natural forces over the last 1300 years, but the huacas that remain show that their murals featured vibrant colors.
Tiwanaku and Waki Cultures
Tiwanaku is an important Pre-Columbian archaeological site in western Bolivia, South America. Tiwanaku is recognized by Andean scholars as one of the most important precursors to the Inca Empire, flourishing as the ritual and administrative capital of a major state power for approximately 500 years.
The city and its inhabitants left no written history, and the modern locals know little about the ancient city and its activities. An archaeologically based theory asserts that around 400 CE, Tiwanaku went from a locally dominant force to a predatory state. Tiwanaku expanded its reaches into the Yungas and brought its culture and way of life to many other cultures in Peru, Bolivia, and Chile. However, Tiwanaku was not exclusively a violent culture. To expand its reach, Tiwanaku used politics to create colonies, negotiate trade agreements (which made the other cultures rather dependent), and establish state cults. Many others were drawn into the Tiwanaku empire due to religious beliefs.
Ceramics & Textiles
Tiwanaku ceramics and textiles were composed of bright colors and stepped patterns. An important ceramic artifact is the kero , a drinking cup that was ritually smashed after ceremonies and placed in burials. Tapestries and tunics provide examples of textiles found at Tiwanaku.
These decorative objects typically depicted herders, effigies , trophy heads, sacrificial victims, and felines. Religion and influence from the main site to the satellite centers were spread through small portable objects that held ritual religious meaning. They were created in wood, engraved bone, and cloth and depicted puma and jaguar effigies, incense burners, carved wooden hallucinogenic snuff tablets, and human portrait vessels .
The Wari
Throughout their imperial reign, the Tiwanaku shared domination of the Middle Horizon with the Wari, whose culture rose and fell around the same time and was centered 500 miles north in the southern highlands of Peru. It is unknown whether the relationship between the two empires was cooperative or antagonistic. Definite interaction between the two is proved by their shared iconography in art. Significant elements of both styles (the split eye, trophy heads, and staff-bearing profile figures, for example) seem to have been derived from the earlier Pukara culture in the northern Titicaca Basin.
Textiles in Early South America
The intricate, complex textiles of the Paracas and Nazca cultures were often associated with a burial ritual.
Discuss the distinguishing characteristics of textiles in pre-Colombian Andean society
Key Points
- Both the Nazca and preceding Paracas culture created intricate textiles, most likely produced by women using a backstrap loom .
- Like the two cultures’ ceramics , many of their textiles were associated with burial rituals .
- Because of the dry climate in southern Peru, many Nazca and Paracas textiles have been well-preserved.
- The so-called “Paracas Textile” tells modern scholars a great deal about the inhabitants of ancient Peru. For instance, they traded extensively with people who lived well outside their territory.
Key Terms
- Andean : Of or pertaining to the Andes mountains in South America.
- backstrap loom : A simple loom with roots in ancient civilizations, consisting of two sticks or bars between which the warps are stretched. One bar is attached to a fixed object, and the other to the weaver through a strap around the back.
Textiles of Andean Cultures
The Paracas culture was an important Andean society between approximately 800 and 100 BCE, with an extensive knowledge of irrigation and water management. It developed in the Paracas Peninsula in the Ica Region of Peru.
Many ceramics and textiles of the Paracas have been found in tombs, particularly in the Paracas Cavernas. These shaft tombs, set into the top of Cerro Colorado, each contain multiple burial sites. The associated ceramics include incised polychrome, “negative” resist decoration, and other wares of the Paracas tradition. The associated textiles include many complex weave structures and elaborate plaiting and knotting techniques.
The later Nazca people also produced technically complex textiles. The Nazca flourished from 100 to 800 CE beside the dry southern coast of Peru in the river valleys of the Rio Grande de Nazca and the Ica Valley. Heavily influenced by the preceding Paracas culture, the Nazca produced an array of beautiful crafts and technologies such as ceramics, textiles, and geoglyphs (most commonly known as the Nazca lines).
Nazca textiles were most likely woven from spun cotton and wool by women at habitation sites. The textiles would have been made using a backstrap loom, similar to the way textiles are made in the region today. Textiles were woven with the common motifs before these appeared on painted pottery. The dry desert has preserved the textiles of both the Nazca and Paracas cultures, which comprise most of what is known about early textiles in the region. Shawls, dresses, tunics , belts, and bags have been found through excavations at Cahuachi and elsewhere. Many textiles associated with the Nazca culture are garments that were included with grave goods found at burial sites.
Mummy Bundles on the Paracas Peninsula
One of the most extraordinary masterpieces of Andean textiles is a nearly 2,000-year-old cloth from the South Coast of Peru. Despite its small size (about two by five feet), it contains a vast amount of information about the people who lived in ancient Peru, and despite its great age and delicacy, its colors are brilliant and tiny details amazingly intact. The arid environment of southern Peru along the Pacific shore allows organic material buried in the sand to remain well-preserved for hundreds or even thousands of years.
In the ancient cemeteries on the Paracas Peninsula, the dead were wrapped into “mummy bundles” with layers of cloth and clothing. The largest and richest mummy bundles contained hundreds of brightly embroidered textiles, feathered costumes, and fine jewelry, interspersed with food offerings such as beans. Early reports claimed that this cloth came from the Paracas peninsula, so it was called “THE Paracas textile,” to mark its excellence and uniqueness. Currently, scholars have revised this provenance and now attribute the cloth to the Nasca culture.
Like other very fine cloths, the Paracas Textile is finished so carefully on both sides that it is almost impossible to distinguish which is the correct side. Although the central cloth and its framing border are created by different techniques, both display perfect reversibility—except for three border figures. These three appear in back view on one side of the cloth, thereby designating a “front” and “back” to the textile.
Architecture in Early South America
Chavín de Huántar and Tiwanaku were important ceremonial centers in pre-Inca South America.
Discuss the multiple functions of architecture in early South America
Key Points
- During its heyday, Chavín de Huántar was used as a religious center for ceremonies and events or consultation with an oracle.
- The temple at Chavín de Huántar is a massive flat-topped pyramid surrounded by lower platforms, along with a U-shaped plaza with a sunken circular court in the center.
- Tiwanaku is recognized as one of the most important precursors to the Inca Empire.
- Archaeologists still struggle to understand how the megaliths used to construct Tiwanaku were transported to the site.
Key Terms
- Chavin : A civilization that developed in the northern Andean highlands of Peru from 900 to 200 BCE. Their influence extended to other civilizations along the coast.
- megalith : A gigantic stone, often weighing several tons.
- trilithon : A megalithic post-and-lintel structure.
Chavín de Huántar and Tiwanaku were important ceremonial centers in pre-Inca South America.
Chavín de Huántar
Chavín de Huántar is an archaeological site containing ruins and artifacts , constructed circa 1200 BCE and occupied until around 400-500 BCE by the Chavín, a major pre-Inca culture . The site is located 160 miles north of Lima, Peru at an elevation of 10,000 feet, on the edge of the Conchucos Valley.
Occupation at Chavín de Huántar has been carbon dated to at least 3000 BCE, with ceremonial activity occurring primarily toward the end of the second millennium and through the middle of the first millennium BCE. While the fairly large population was based on an agricultural economy, the city’s location at the headwaters of the Marañn River, between the coast and the jungle, made it ideal for the dissemination and collection of both ideas and material goods. This archaeological site has revealed a great deal about the Chavín culture. The transformation of the center into a valley-dominating monument had a complex effect. Chavín de Huántar became a pan-regional place of importance. People used it to gather, attend and participate in rituals , and consult with oracles.
Findings at Chavín de Huántar indicate that social instability and upheaval began between 500 and 300 BCE, at the same time the larger Chavín civilization began to decline. Large ceremonial sites were abandoned, some unfinished, and were replaced by villages and agricultural land.
The temple at Chavín de Huántar was the religious center of the Chavín people and the capital of the Chavín culture. This massive flat-topped pyramid is surrounded by lower platforms and located in a U-shaped plaza with a sunken circular court in the center. The inside of the temple walls are decorated with sculptures and carvings. Chavín de Huántar was constructed over many stages, starting prior to 1200 BCE, with most major construction over by 750 BCE. The site continued as a ceremonial center until around 500 BCE.
Tiwanaku
Tiwanaku is an important Pre-Columbian archaeological site in western Bolivia. It is recognized by Andean scholars as one of the most important precursors to the Inca Empire, flourishing as the ritual and administrative capital of a major state power for approximately 500 years.
The city and its inhabitants left no written history, and the modern locals know little about the ancient city and its activities. However, the site might have been inhabited as early as 1500 BCE. An archaeological theory asserts that around 400 CE, Tiwanaku went from a locally dominant force to a predatory state. It expanded its reaches into the Yungas and brought its culture and way of life to many cultures in Peru, Bolivia, and Chile. However, Tiwanaku was not exclusively a violent culture. To expand its reach, Tiwanaku used politics to create colonies, negotiate trade agreements (which made the other cultures rather dependent), and establish state cults. Many were drawn into the Tiwanaku empire by their religious beliefs.
Architecture
Tiwanaku monumental architecture is characterized by megaliths of exceptional workmanship. The main architectural appeal of the site comes from the carved images and designs on carved doorways and megalithic constructions such as the Gate of the Sun. Tiwanaku’s architecture and skill in stone-cutting reveals a knowledge of descriptive geometry.
The Gate of the Sun is a trilithon that stands nearly 10 feet tall and 13 feet wide. Its weight is estimated at approximately 10 tons. Although there have been various modern interpretations of its mysterious inscriptions, the carvings that decorate the gate are believed to possess astronomical significance and may have served a calendrical purpose.
The Gate of the Sun shares its location with the Kalasasaya, a temple in a megalithic courtyard more than 300 feet long. Since the late 20th century, researchers have theorized that this was not the gateway’s original location. The walls are covered with tenon heads of many styles , suggesting that the structure was reused for different purposes over time. What stands today is not the original configuration of the megaliths that comprise the Kalasasaya. Scholars believe it was originally constructed in a similar fashion as Stonehenge, with its stones spaced evenly apart and standing vertically.
The quarries from which the stone blocks used in the construction of structures at Tiwanaku came lie at significant distances from this site, which has led scholars to speculate on how they could have been moved. One theory is that giant andesite stones weighing more than 40 tons were transported some 90 kilometers across Lake Titicaca on reed boats and then laboriously dragged another 10 kilometers to the city.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- aca-del-sol-ao-c3-bbt-2007.jpeg. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Moche. License : CC BY-SA: Attribution-ShareAlike
- Erotic Moche Pot. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Moche. License : CC BY-SA: Attribution-ShareAlike
- Wari earthenware pot with painted design, 650-800 CE (Middle Horizon).. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Huari_-_Head_Pot_with_Painted_Design_-_Walters_482849_-_Three_Quarter_Right.jpg. License : CC BY-SA: Attribution-ShareAlike
- zonnepoort-tiwanaku.jpeg. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Zonnepoort_tiwanaku.jpg. License : CC BY-SA: Attribution-ShareAlike
- Moche. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Moche. License : CC BY-SA: Attribution-ShareAlike
- Nazca culture. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Nazca_culture. License : CC BY-SA: Attribution-ShareAlike
- Paracas culture. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Paracas_culture. License : CC BY-SA: Attribution-ShareAlike
- Nazca culture. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Nazca_culture%23Arts_and_technology. License : CC BY-SA: Attribution-ShareAlike
- Andean. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/Andean. License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//art-history/definition/phytomorphic. License : CC BY-SA: Attribution-ShareAlike
- geoglyph. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/geoglyph. License : CC BY-SA: Attribution-ShareAlike
- The Paracas Textile. Provided by : Khan Academy. Located at : www.khanacademy.org/humanities/art-americas/south-america-early/paracas-nasca/a/the-paracas-textile. License : CC BY-SA: Attribution-ShareAlike
- Backstrap Loom. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Loom#/media/File:Backstrap_loom.jpg. License : CC BY-SA: Attribution-ShareAlike
- Detail of Paracas Textile. Provided by : Khan Academy. Located at : www.khanacademy.org/humanities/art-americas/south-america-early/paracas-nasca/a/the-paracas-textile. License : CC BY-SA: Attribution-ShareAlike
- Andean. Provided by : Wikipedia. Located at : en.wiktionary.org/wiki/Andean. License : CC BY-SA: Attribution-ShareAlike
- Loom. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Loom. License : CC BY-SA: Attribution-ShareAlike
- Nazca Culture. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Nazca_culture%23Arts_and_technology. License : CC BY-SA: Attribution-ShareAlike
- Paracas Culture. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Paracas_culture. License : CC BY-SA: Attribution-ShareAlike
- kalasasaya3.jpeg. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Tiwanaku. License : CC BY: Attribution
- zonnepoort-tiwanaku.jpeg. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Zonnepoort_tiwanaku.jpg. License : Public Domain: No Known Copyright
- Chavin circular plaza cyark. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Chavin_circular_plaza_cyark.jpg. License : CC BY-SA: Attribution-ShareAlike
- Chavin de Huantar. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Chavin_de_Huantar.JPG. License : Public Domain: No Known Copyright
- Tiwanaku. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Tiwanaku. License : CC BY-SA: Attribution-ShareAlike
- Tiwanaku and Wari. Provided by : Boundless Learning. Located at : www.boundless.com/atoms/7949. License : CC BY-SA: Attribution-ShareAlike
- Chavin Culture. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Chavin_culture. License : CC BY-SA: Attribution-ShareAlike
- Gate of the Sun. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Gate_of_the_Sun. License : CC BY-SA: Attribution-ShareAlike
- Chavin de Huantar. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Chavin_de_Huantar. License : CC BY-SA: Attribution-ShareAlike
- Chavin. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Chavin. License : CC BY-SA: Attribution-ShareAlike


There are occasions in this course where you will need to compute elasticities. There are two formulas used to do this. One is the point formula. The point formula will be most important for this course and is also most commonly in published studies on food demand. The other formula is called the arc or average formula. Let us spend some time on each.
The Point Formula for Demand Elasticities
The general formula for an elasticity can be rearranged algebraically to arrive at the point formula. To illustrate, rewrite the own-price elasticity formula shown above as follows:
\[\epsilon_{11} = \dfrac{\% \Delta Q_{1}}{\% \Delta P_{1}} = \dfrac{(\Delta Q_{1} / Q_{1}) \times 100}{(\Delta P_{1} / P_{1}) \times 100} = \dfrac{\Delta Q_{1}}{\Delta P_{1}} \times \dfrac {P_{1}}{Q_{1}}\]
Point formulas for different kinds of demand elasticities are reported in Table \(\PageIndex{1}\). Note the similarity in the formula for each kind of elasticity. As a practical matter, you need three terms (numbers) to compute an elasticity using the point formula.
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Depending on the type of elasticity, the first number you need is \(\dfrac{\Delta Q_{i}}{\Delta P_{i}}, \dfrac{\Delta Q_{i}}{\Delta P_{j}}, \dfrac{\Delta Q_{i}}{\Delta M}, \: or \: \dfrac{\Delta Q_{i}}{\Delta X}\). These numbers are slope terms from a direct demand relationship. You might remember from algebra class that the slope is the rise over the run. In this case the rise is \(\Delta Q_{i}\) and the run is \(\Delta P_{i}, \Delta P_{j}, \Delta M, \: or \: \Delta X\).
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The second number you need is the value of the demand variable in question \(P_{i}, P_{j}, M, \: or \: X\).
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The third and final number you need is the value of \(Q_{i}\).
Table \(\PageIndex{1}\). Point formulas for demand elasticities.
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Type
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Formula
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Own-price elasticity
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\(\epsilon_{ii} = \dfrac{\Delta Q_{i}}{\Delta P_{i}} \times \dfrac{P_{i}}{Q_{i}}\)
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Cross-price elasticity
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\(\epsilon_{ij}= \dfrac{\Delta Q_{i}}{\Delta P_{j}} \times \dfrac{P_{j}}{Q_{i}}, i \neq j\)
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Income elasticity
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\(\epsilon_{iM} = \dfrac{\Delta Q_{i}}{\Delta M} \times \dfrac{M}{Q_{i}}\)
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Elasticity for other demand shift variable \(X\)
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\(\epsilon_{iX} = \dfrac{\Delta Q_{i}}{\Delta X} \times \dfrac{X}{Q_{i}}\)
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To illustrate the implementation of the point formula, consider the following direct demand equation for good 1:
\[Q_{1} = 1.5A + 0.01M + 2P_{2} -4P_{1}\]
In this equation, \(Q_{1}\) is the quantity of good 1 in thousands of units, \(A\) is advertising, \(M\) is disposable income in dollars, \(P_{2}\) is the price of good 2 in dollars, and \(P_{1}\) is the price of good 1 in dollars. Let us first compute the quantity demanded for the following data point: \((A^{0} = 40, M^{0} = 30000, \(P_{2}^{0} = 20, \: and \: P_{1}^{0} = 60)\). Entering these values into the demand equation provides a quantity demanded of
\[Q_{1}^{0} = 1.5 \times 40 + 0.01 \times 30000 + 2 \times 20 \times 60 = 160 \: thousand \: units\]
You can now use the point formula to obtain elasticity measures. These elasticities are reported in Table \(\PageIndex{2}\) under the heading “Data Point 0”. Take some time to look at the elasticity computations. Make sure you see where each value used in the elasticity computation originates.
Now, repeat the elasticity calculation at another, slightly different data point: \((A^{1} = 40, M^{1} = 30000, P_{2}^{1} = 20, \: and \: P_{1}^{1} = 50)\). At this new data point, let us call it “Data Point 1”, we get a new quantity demanded of
\(Q_{1}^{1} = 1.5 \times 40 + 0.01 \times 30000 + 2 \times 20 - 4 \times 50 = 200 \: thousand \: units\)
Elasticities for this new data point are also reported in Table \(\PageIndex{2}\). Notice that all the point elasticities are different even though the data points themselves differ only in the value of \(P_{1}\). The takeaway here is that the point formula gives an elasticity number at a specific data point and elasticities can depend on point of location along the demand equation. This is one of the reason why caution should be used when using an elasticity to forecast the effects of a large change in one of the demand variables. The elasticity values could be very different at the new point and the old.
Table \(\PageIndex{2}\). Computing elasticities with the point formula.
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Type
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Data Point 0
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Data Point 1
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Own-price elasticity
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\(\epsilon_{11} = -4 \times \dfrac{60}{160} = -1.5\)
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\(\epsilon_{11}= -4 \times \dfrac{50}{200} = 1.0\)
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Cross-price elasticity
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\(\epsilon_{12} = 2 \times \dfrac{20}{160} = 0.25\)
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\(\epsilon_{12} = 2 \times \dfrac{20}{200} = 0.20\)
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Income elasticity
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\(\epsilon_{1M} = 0.01 \times \dfrac{30000}{160} = 1.875\)
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\(\epsilon_{1M} = 0.01 \times \dfrac{30000}{200} = 1.5\)
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Advertising elasticity
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\(\epsilon_{1A} = 1.5 \times \dfrac{40}{160} = 0.375\)
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\(\epsilon_{1A} = 1.5 \times \dfrac{40}{200} = 0.30\)
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In this course, you will primarily be using linear demand specifications, such as the one we used to get the elasticity numbers in Table 3. With linear demands, the elasticity will depend on the point of location along the demand schedule. This is illustrated below in Panel A of Demonstration 3. Panel A shows that demand is elastic at high prices and inelastic at low prices.
The idea of demand being more elastic at higher prices makes intuitive sense and is probably true for most real-world cases. For instance, consumers will be more responsive when the gasoline price increases from $4.00 per gallon to $4.40 per gallon than when the gasoline price increases from $2.00 to $2.20 per gallon. In each case, there is a 10 percent increase in price, but the increase causes more pain at $4.00 than at $2.00. Nevertheless, the fact that elasticities vary along a linear demand schedule, such as that presented in Panel A of Demonstration \(\PageIndex{1}\), is a consequence of using a linear specification for demand.
Panel B of Demonstration \(\PageIndex{1}\), shows an alternative demand specification that provides a constant elasticity. This is a log-log demand specification. The term “log-log”" is used because this type of demand schedule is linear if we take its logarithmic transformation (it is linear in logarithms). With the linear demand curve, the slope does not change. In other words, \(\dfrac{\Delta Q_{1}}{\Delta P_{1}}\) is always the same regardless of location on the demand schedule. However, as shown in Panel A of Demonstration \(\PageIndex{1}\), the own-price elasticity is not constant and will vary between being elastic, unitary elastic, and inelastic as we move from high to low prices. With log-log demand, the slope of the demand schedule changes as we move from high to low prices, but the elasticity is always constant.
The Arc Formula for Elasticities
An alternative formula to compute elasticities is the the arc formula. The arc formula returns the elasticity at the average of two points on the demand equation. Table 4 provides arc elasticity formulas for different demand elasticities.
Table \(\PageIndex{3}\). Arc formulas for demand elasticities.
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Type
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Formula
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|
Own-price elasticity
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\(\epsilon_{ii} = \dfrac{Q_{i}^{1} - Q_{i}^{0}}{P_{i}^{1} - P_{i}^{0}} \times \dfrac{P_{i}^{1} + P_{i}^{0}}{Q_{i}^{1} + Q_{i}^{0}}\)
|
|
Cross-price elasticity
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\(\epsilon_{ij} = \dfrac{Q_{i}^{1} - Q_{i}^{0}}{P_{j}^{1} - P_{j}^{0}} \times \dfrac{P_{j}^{1} + P_{j}^{0}}{Q_{i}^{1} + Q_{i}^{0}}, i \neq j\)
|
|
Income elasticity
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\(\epsilon_{iM} = \dfrac{Q_{i}^{1}-Q_{i}^{0}}{M^{1}-M^{0}} \times \dfrac{M^{1} + M^{0}}{Q_{i}^{1} + Q_{i}^{0}}\)
|
|
Elasticity for other demand shift variable \(X\)
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\(\epsilon_{iX} = \dfrac{Q_{i}^{1}- Q_{i}^{0}}{X^{1}-X^{0}} \times \dfrac{X^{1} + X^{0}}{Q_{i}^{1} + Q_{i}^{0}}\)
|
Let us compute the own-price arc elasticity over the two data points we have already considered from the demand equation above. For the own price elasticity, the two price quantity pairs are (60,160) and (50,200), respectively. Using these, the arc elasticity formula provides an elasticity at the average of these two points on the demand schedule
\(\epsilon_{11} = \dfrac{200-160}{50-60} \times \dfrac{50 + 60}{200 + 160} = -1.22.\)
Notice that it does not matter which point is designated with a superscript 1 or superscript 0, but one must keep the points straight in the first term of the arc formula. Reverse the ordering of the points so that (60,160) is assigned the superscript 0 and (50,200) is assigned the superscript 1, you get
\(\epsilon_{11} = \dfrac{160-200}{60-50} \times \dfrac{60 + 50}{160 + 200} = -1.22.\)
Using the point formula, you found the own-price elasticity at \((P_{1}= 60, Q_{1} = 160)\) to be \(\epsilon_{11} = 1.5\). At \((P_{1} = 50, Q_{1} = 200)\), it was \(\epsilon_{11} =-1\). The average of these two points is \((P_{1}= 55, Q_{1}=180)\). Using the point formula to measure elasticity at this average point provides \(\epsilon_{11}=-1.22\), which is what was obtained from the arc formula. This should help you get some intuition about the arc formula. The arc formula provides an elasticity at the average of the two points used in its computation.
A compelling feature of the arc formula is that you only need to know two points to compute an elasticity. However, it is important to emphasize the “all else held constant” provision. In the example above, this provision was met because income, the price of good 2, and advertising were held constant. The only difference in the two data points was the price of good 1. Application of the arc formula requires you to assume that the entire change from \(Q_{1}^{0}\) to \(Q_{1}^{1}\) is attributable only to an observed change in \(P_{1}\). In the real world, it may be hard to defend this assumption.
As an exercise, use the arc formula to replicate the elasticity ranges reported above in Demonstration 3.2.2 after you increase or decrease the price. In this case you know that all else has remained constant because you are comparing two points on the same demand schedule. There may be a small bit of rounding error in the demonstration, but the arc formula should give you something very close to the same number when you compute it from a price increase or decrease.


2.6: Checklist for General Survey
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Use this checklist to perform a “General Survey.” Checklists for hand washing, using hand sanitizer, and obtaining vital signs are included in Appendix A .
Steps
Disclaimer: Always review and follow agency policy regarding this specific skill.
- Knock, enter the room, greet the patient, and provide for privacy.
- Introduce yourself, your role, the purpose of your visit, and an estimate of the time it will take.
- Perform hand hygiene.
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Ask the patient their legal name and date of birth for the first identifier. Verify information provided in their chart or wristband, if present. Use one of the following for the second identifier:
- Scan wristband
- Compare name/DOB to MAR
- Ask staff to verify patient (in settings where wristbands are not worn)
- Compare picture on MAR to patient
- Address patient needs (pain, toileting, glasses/hearing aids) prior to starting assessment. Note if the patient has signs of distress such as difficulty breathing or chest pain. If signs are present, defer general survey and obtain emergency assistance per agency policy.
- Explain the procedure to the patient; ask if he/she has any questions. Obtain an interpreter as needed if English is not the patient’s primary language.
- Pause and explain to the instructor what you would purposefully observe and assess during a general survey assessment.
- Upon completion of the survey, thank the patient and ask if anything is needed.
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Ensure five safety measures before leaving the room:
- CALL LIGHT: Within reach
- BED: Low and locked (in lowest position and brakes on)
- SIDE RAILS: Secured
- TABLE: Within reach
- ROOM: Risk-free for falls (scan room and clear any obstacles)
- Perform hand hygiene and clean stethoscope.
- Follow agency policy for reporting findings outside of normal range.
- Document the assessment.


5.2: Defining the Genus Homo
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- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
- OpenStax
Learning Objectives
By the end of this section, you will be able to:
- Describe the time periods and geological context of the genus Homo .
- Identify some key differences between the genus Homo and Australopithecus.
- Define some of the limitations of and challenges in the classification of hominin species in the genus Homo .
- Explain the concept of encephalization as it relates to early hominin evolutionary development and as a tool for hominin classification.
Putting Homo into Context
Before learning about the hominin species that make up the category genus Homo , it will be helpful to become familiar with the key archaeological time periods with which Homo is associated. The species and cultural developments mentioned below will be explored in greater detail in the sections that follow.
- Lower Paleolithic (from roughly 3 million years ago to approximately 300,000 BCE): This period includes H. habilis and the Oldowan tool industry, followed by H. ergaster and the Acheulean tool industry.
- Middle Paleolithic (approximately 300,000–40,000 BCE): This period includes continued use of Acheulean tools by H. heidelbergensis, followed by H. neanderthalensis and the Mousterian tool industry.
- Upper Paleolithic (approximately 43,000–26,000 BCE): The Upper Paleolithic saw the emergence of cave art like that found in the famous Chauvet Cave in France (Figure 5.29), Venus figurines (Figure 5.28), and an increased use of bone and antler in tools and jewelry. The most recent ice age occurred during this time, with glaciers covering huge parts of the planet. The emergence of Paleoindians and the use of Clovis points, which were used to kill large game such as mastodons and mammoths, occurred near the end of this time period.
- Neolithic (Agricultural Age) (8,000–3,000 BCE): New innovations appear during the agricultural age, or “Neolithic revolution,” as H. sapiens set up permanent settlements. Humans begin to transition from being hunters and gatherers to growing grow crops, owning land, and domesticating animals.
The Challenge of Defining the Genus Homo
The previous chapter introduced the australopithecines, who were diverse in their physical characteristics (gracile and robust), with large jaws and teeth and small brain size. A key characteristic shared by both the australopithecines and the genus Homo is bipedalism. The transition to bipedalism is linked with various anatomical changes, including longer legs, changes in spinal curvature, and the development of arches in the feet to conserve energy and increase balance when walking.
What criteria other than bipedalism might be used to classify a species under the genus Homo ? Many anthropologists have attempted to establish specific criteria to use in determining a classification of Homo. Paleoanthropologists Mary Leakey, Louis Leakey, and John Napier, as well as primatologist Phillip Tobias, were among the first to extensively study the fossils of Homo habilis , considered to be one of the earliest species in the genus Homo . H. habilis had a brain size of around 661–700 cc, which was larger than the australopithecines’, with hands that were capable of the dexterity needed for making tools, due to bone structure changes and a repositioning of the thumb, which allowed for better grip.
The type specimen OH 7 of H. habilis dated between 2 and 1.7 MYA and was found in 1960 at Olduvai Gorge by Jonathan and Mary Leakey. It was described by Louis Leakey in 1964. Type specimen refers to a specimen that serves as the standard for the taxon or classification group for that species. OH 7 is the identification or accession number of this specific specimen and stands for “Olduvai Hominid #7.” The specimen consisted of a partial juvenile skull, hand, and foot bones. It possessed teeth that were much smaller than those of any australopithecine and was possibly in coexistence with the robust australopithecines ( Paranthropus ). Based on an endocranial cast (an imprint of the interior of the brain case), it was determined that H. habilis may have possessed what is called a Broca’s area in the brain. Broca’s area , which includes two Brodmann areas (referred to as 44 and 45), is located in the middle of the left cerebral cortex of the brain and is especially important for speech development (Figure 5.2). Some scientists have suggested that H. habilis started to develop the neural networks necessary for human speech, while others argue that H. habilis probably already had speech.
The postcranial features (skeletal structures in the body other than the skull) of Homo habilis are not as well established, as is the case for many other early hominin fossils. This can be problematic, as many hominin species coexisted with overlapping traits. Likewise, it can be problematic to have postcranial material and not the cranium or skull. The skull often serves as a diagnostic tool when postcranial materials do not provide enough evidence or provide confusing evidence.
Based on their research on H. habilis , Mary Leakey, Louis Leakey, and John Napier proposed the following criteria for classifying Homo : a brain size over 600 cc; a round, globular skull; tool use; reduced prognathism (protrusion of the jaw) and smaller jaws and mandibles; humanlike postcranial features; and feet that are fully adapted for walking (Leakey, Tobias, and Napier 1964). While this list established specific and fairly comprehensive guidance, the diversity of traits and the ways in which they overlapped didn’t always line up with the criteria.
H. habilis has been at the center of several debates regarding their taxonomic position and relationship with other early archaic Homo species. For example, H. habilis was initially believed to have been a direct human ancestor through the lineage of Homo erectus and then modern humans. This viewpoint is now debated and has resulted in a scientific divide between those supporting H. habilis and those suggesting another Homo species, H. rudolfensis , as being the ancestor of H. erectus. H. rudolfensis is an archaic Homo dated to about 2 MYA, which coexisted with other Homo species during that time period. A cranium was discovered in 1972 along Lake Turkana in Kenya by Bernard Ngeneo, a local Kenyan. The specimen was later described by paleoanthropologist Richard Leakey. There is a lot that is not known about this species; scientists are missing postcranial materials, and as of yet no tools have been found. There are hypotheses that propose that H. rudolfensis might be a H. habilis male, exhibiting a larger cranium than that seen in a female H. habilis . Others suggest it is a completely different species. Another controversy centers on tool use. While Homo habilis was long regarded as the earliest hominin to use stone tools, it has been determined, based on evidence of cutmarks, that at least one australopithecine ( A. garhi ) used stone tools before H. habilis , at around 2.6 MYA (Semaw et al. 1997).
While there are still questions as to the phylogenetic relationship of H. habilis and H. rudolfensis , there is general agreement that Homo did evolve from Australopithecus . The timing and placement of the split between Australopithecus and Homo , however, is still debated. H. habilis was determined to not be an Australopithecus due to its smaller teeth, a humanlike foot, and hand bones that suggested an ability to manipulate objects with precision.
One of the main considerations in classifying H. habilis as a Homo and not an Australopithecus was its cranial capacity , which is a measurement that indicates brain size. With some exceptions, cranial capacity can serve as an indicator of where a hominin fossil might belong in the hominin phylogenetic tree. Encephalization refers to a progressive increase in brain size over time. In human evolution, we can observe encephalization beginning with Homo habilis and progressing more rapidly through H. erectus . Encephalization correlates with an increase in behavioral, cognitive, and cultural complexity. Cognitive developments correspond with our ability to construct and form ideas, including the ability think in and communicate via symbolic and abstract language, such as that used in storytelling, ritual, and art. There are always exceptions, however, such as the island-dwelling, small-brained H. floresiensis , who will be introduced later in this chapter. In spite of having a very small brain, H. floresiensis made and used tools and built fires. This discovery has challenged what we thought we knew about the correlation of brain size and cognitive development in human evolution.
The encephalization quotient (EQ) can serve as a good indicator (with some exceptions) of classification within the genus Homo .  Body size is a factor in these measurements as expected brain size reflects the relationship between brain and body size for a given taxonomic group (Jerison 1973). The larger the brain weight relative to the overall body weight, the more likely that the brain was used for more complex cognitive tasks. Harry J. Jerison (1973) was the first to develop EQ measurements. The formula he used for calculating EQ in birds and mammals is brain mass/0.12 × (body mass) 0.66 . Other formulas have also been proposed, such as EQ = brain mass (11.22 × body mass 0.76) (Martin 1981). While EQ is a strong tool for studying brain size in early hominins, there are always potential margins of error when dealing with fragmentary fossils, and increasingly alternative forms of measurements are being proposed. One study proposes that EQ should no longer be used as a tool in calculating brain size in primates and other vertebrate species, based on the premise that cognitive performance does not depend on body size and so body size should not be included in the formula (Schaik et al. 2021). Other theories consider the number of cortical neurons and neural connections as most important when considering cognitive ability (Roth and Dicke 2012). According to this approach, the density of the cortex is more associated with intelligence than is brain size. These alternate approaches would perhaps better explain those exceptions in the fossil records, such as H. floresiensis . Other interesting research is looking at potential levels of cognition and memory as it relates to levels of tool complexity (Read and van der Leeuw 2008).
In spite of these criticisms, many see EQ measurements as providing fairly consistent results. Modern humans ( Homo sapiens ) have an EQ of roughly 6.0–7.0 (meaning that their brain mass is six to seven times greater than what one would expect to find in a comparable mammal of the same body size). H. erectus has an EQ of 4.0, while for an australopithecine EQ is around 2.5 to 3.0 (Fuente 2012, 227). Figure 5.4 shows increases in average brain sizes for various species over time.