2.14: Water - High Heat Capacity
- Explain the biological significance of water’s high specific heat
Water’s High Heat Capacity
The capability for a molecule to absorb heat energy is called heat capacity, which can be calculated by the equation shown in the figure. Water’s high heat capacity is a property caused by hydrogen bonding among water molecules. When heat is absorbed, hydrogen bonds are broken and water molecules can move freely. When the temperature of water decreases, the hydrogen bonds are formed and release a considerable amount of energy. Water has the highest specific heat capacity of any liquid. Specific heat is defined as the amount of heat one gram of a substance must absorb or lose to change its temperature by one degree Celsius. For water, this amount is one calorie, or 4.184 Joules. As a result, it takes water a long time to heat and a long time to cool. In fact, the specific heat capacity of water is about five times more than that of sand. This explains why the land cools faster than the sea.
C=QΔT.C=QΔT.
The resistance to sudden temperature changes makes water an excellent habitat, allowing organisms to survive without experiencing wide temperature fluctuation. Furthermore, because many organisms are mainly composed of water, the property of high heat capacity allows highly regulated internal body temperatures. For example, the temperature of your body does not drastically drop to the same temperature as the outside temperature while you are skiing or playing in the snow. Due to its high heat capacity, water is used by warm blooded animals to more evenly disperse heat in their bodies; it acts in a similar manner to a car’s cooling system, transporting heat from warm places to cool places, causing the body to maintain a more even temperature.
Key Points
- Water has the highest heat capacity of all liquids.
- Oceans cool slower than the land due to the high heat capacity of water.
- To change the temperature of 1 gram of water by 1 degree Celsius, it takes 1.00 calorie.
Key Terms
- heat capacity : The capability of a substance to absorb heat energy
- specific heat : the amount of heat, in calories, needed to raise the temperature of 1 gram of water by 1 degree Celsius


14.7: Acid-Base Titrations
- Interpret titration curves for strong and weak acid-base systems
- Compute sample pH at important stages of a titration
- Explain the function of acid-base indicators
As seen in the chapter on the stoichiometry of chemical reactions, titrations can be used to quantitatively analyze solutions for their acid or base concentrations. In this section, we will explore the changes in the concentrations of the acidic and basic species present in a solution during the process of a titration.
Titration Curves
Previously, when we studied acid-base reactions in solution, we focused only on the point at which the acid and base were stoichiometrically equivalent. No consideration was given to the pH of the solution before, during, or after the neutralization.
A titration is carried out for 25.00 mL of 0.100 M HCl (strong acid) with 0.100 M of a strong base NaOH the titration curve is shown in Figure \(\PageIndex{1}\). Calculate the pH at these volumes of added base solution:
- 0.00 mL
- 12.50 mL
- 25.00 mL
- 37.50 mL
Solution
Since HCl is a strong acid, we can assume that all of it dissociates. The initial concentration of H 3 O + is \(\ce{[H3O+]_0}=0.100\:M\). When the base solution is added, it also dissociates completely, providing OH − ions. The H 3 O + and OH − ions neutralize each other, so only those of the two that were in excess remain, and their concentration determines the pH. Thus, the solution is initially acidic (pH < 7), but eventually all the hydronium ions present from the original acid are neutralized, and the solution becomes neutral. As more base is added, the solution turns basic.
The total initial amount of the hydronium ions is:
\[\mathrm{n(H^+)_0=[H_3O^+]_0×0.02500\: L=0.002500\: mol} \nonumber \]
Once X mL of the 0.100- M base solution is added, the number of moles of the OH − ions introduced is:
The total volume becomes:
\[V=\mathrm{(25.00\: mL+X\: mL)\left(\dfrac{1\: L}{1000\: mL}\right)} \nonumber \]
The number of moles of H 3 O + becomes:
\[\mathrm{n(H^+)=n(H^+)_0-n(OH^-)_0=0.002500\: mol-0.100\:\mathit{M}×X\: mL×\left(\dfrac{1\: L}{1000\: mL}\right)} \nonumber \]
The concentration of H 3 O + is:
\[\mathrm{[H_3O^+]=\dfrac{n(H^+)}{V}=\dfrac{0.002500\: mol-0.100\:\mathit{M}×X\: mL×\left(\dfrac{1\: L}{1000\: mL}\right)}{(25.00\: mL+X\: mL)\left(\dfrac{1\: L}{1000\: mL}\right)}} \nonumber \]
\[\mathrm{=\dfrac{0.002500\: mol×\left(\dfrac{1000\: mL}{1\: L}\right)-0.100\:\mathit{M}×X\: mL}{25.00\: mL+X\: mL}} \nonumber \]
with the definition of \(\mathrm{pH}\):
\[\mathrm{pH=−\log([H_3O^+])} \label{phdef} \]
The preceding calculations work if \(\mathrm{n(H^+)_0-n(OH^-)_0>0}\) and so n(H + ) > 0. When \(\mathrm{n(H^+)_0=n(OH^-)_0}\), the H 3 O + ions from the acid and the OH − ions from the base mutually neutralize. At this point, the only hydronium ions left are those from the autoionization of water, and there are no OH − particles to neutralize them. Therefore, in this case:
\[\ce{[H3O+]}=\ce{[OH- ]},\:\ce{[H3O+]}=K_\ce{w}=1.0\times 10^{-14};\:\ce{[H3O+]}=1.0\times 10^{-7} \nonumber \]
\[\mathrm{pH=-log(1.0\times 10^{-7})=7.00} \nonumber \]
Finally, when \(\mathrm{n(OH^-)_0>n(H^+)_0}\), there are not enough H 3 O + ions to neutralize all the OH − ions, and instead of \(\mathrm{n(H^+)=n(H^+)_0-n(OH^-)_0}\), we calculate: \(\mathrm{n(OH^-)=n(OH^-)_0-n(H^+)_0}\)
In this case:
\[\mathrm{[OH^-]=\dfrac{n(OH^-)}{\mathit{V}}=\dfrac{0.100\:\mathit{M}×X\: mL×\left(\dfrac{1\: L}{1000\: mL}\right)-0.002500\: mol}{(25.00\: mL+X\: mL)\left(\dfrac{1\: L}{1000\: mL}\right)}} \nonumber \]
\[\mathrm{=\dfrac{0.100\:\mathit{M}×X\: mL-0.002500\: mol×\left(\dfrac{1000\: mL}{1\: L}\right)}{25.00\: mL+X\: mL}} \nonumber \]
then using the definition of \(pOH\) and its relationship to \(pH\) in room temperature aqueous solutios (Equation \ref{phdef}):
\[\begin{align} pH &=14-pOH \nonumber \\&=14+\log([OH^-]) \nonumber\end{align} \nonumber \]
Let us now consider the four specific cases presented in this problem:
(a) X = 0 mL
\[\mathrm{[H_3O^+]=\dfrac{n(H^+)}{\mathit{V}}=\dfrac{0.002500\: mol×\left(\dfrac{1000\: mL}{1\: L}\right)}{25.00\: mL}=0.1\:\mathit{M}} \nonumber \]
then using the definition of \(pH\) (Equation \ref{phdef}):
\[\begin{align} pH &= −\log(0.100) \nonumber \\ &= 1.000 \nonumber\end{align} \nonumber \]
(b) X = 12.50 mL
\[\mathrm{[H_3O^+]=\dfrac{n(H^+)}{\mathit{V}}=\dfrac{0.002500\: mol×\left(\dfrac{1000\: mL}{1\: L}\right)-0.100\:\mathit{M}×12.50\: mL}{25.00\: mL+12.50\: mL}=0.0333\:\mathit{M}} \nonumber \]
then using the definition of \(pH\) (Equation \ref{phdef}):
\[ \begin{align} pH &= −\log(0.0333) \nonumber \\ &= 1.477 \nonumber\end{align} \nonumber \]
(c) X = 25.00 mL
Since the volumes and concentrations of the acid and base solutions are the same:
\[\mathrm{n(H^+)_0=n(OH^-)_0} \nonumber \]
and
\[pH = 7.000 \nonumber \]
as described earlier.
(d) X = 37.50 mL
In this case:
\[\mathrm{n(OH^-)_0>n(H^+)_0} \nonumber \]
\[\mathrm{[OH^-]=\dfrac{n(OH^-)}{\mathit{V}}=\dfrac{0.100\:\mathit{M}×35.70\: mL-0.002500\: mol×\left(\dfrac{1000\: mL}{1\: L}\right)}{25.00\: mL+37.50\: mL}=0.0200\:\mathit{M}} \nonumber \]
then using the definition of \(pH\) (Equation \ref{phdef}):
\[ \begin{align}[pH = 14 − pOH \nonumber\\ &= 14 + \log([OH^{−}]) \nonumber \\ &= 14 + \log(0.0200) \nonumber \\ &= 12.30 \nonumber \end{align} \nonumber \]
Calculate the pH for the strong acid/strong base titration between 50.0 mL of 0.100 M HNO 3 ( aq ) and 0.200 M NaOH (titrant) at the listed volumes of added base:
- 0.00 mL,
- 15.0 mL,
- 25.0 mL, and
- 40.0 mL.
- Answer a
-
0.00: 1.000
- Answer b
-
15.0: 1.5111
- Answer c
-
25.0: 7e. Do not delete this text first.
- Answer d
-
40.0: 12.523
In Example \(\PageIndex{1}\), we calculated pH at four points during a titration. Table \(\PageIndex{1}\) shows a detailed sequence of changes in the pH of a strong acid and a weak acid in a titration with NaOH.
| Volume of 0.100 M NaOH Added (mL) | Moles of NaOH Added | pH Values 0.100 M HCl 1 | pH Values 0.100 M \(CH_3CO_2H\) 2 |
|---|---|---|---|
| 0.0 | 0.0 | 1.00 | 2.87 |
| 5.0 | 0.00050 | 1.18 | 4.14 |
| 10.0 | 0.00100 | 1.37 | 4.57 |
| 15.0 | 0.00150 | 1.60 | 4.92 |
| 20.0 | 0.00200 | 1.95 | 5.35 |
| 22.0 | 0.00220 | 2.20 | 5.61 |
| 24.0 | 0.00240 | 2.69 | 6.13 |
| 24.5 | 0.00245 | 3.00 | 6.44 |
| 24.9 | 0.00249 | 3.70 | 7.14 |
| 25.0 | 0.00250 | 7.00 | 8.72 |
| 25.1 | 0.00251 | 10.30 | 10.30 |
| 25.5 | 0.00255 | 11.00 | 11.00 |
| 26.0 | 0.00260 | 11.29 | 11.29 |
| 28.0 | 0.00280 | 11.75 | 11.75 |
| 30.0 | 0.00300 | 11.96 | 11.96 |
| 35.0 | 0.00350 | 12.22 | 12.22 |
| 40.0 | 0.00400 | 12.36 | 12.36 |
| 45.0 | 0.00450 | 12.46 | 12.46 |
| 50.0 | 0.00500 | 12.52 | 12.52 |
|
The simplest acid-base reactions are those of a strong acid with a strong base. Table \(\PageIndex{1}\) shows data for the titration of a 25.0-mL sample of 0.100 M hydrochloric acid with 0.100 M sodium hydroxide. The values of the pH measured after successive additions of small amounts of NaOH are listed in the first column of this table, and are graphed in Figure \(\PageIndex{1}\), in a form that is called a titration curve . The pH increases slowly at first, increases rapidly in the middle portion of the curve, and then increases slowly again. The point of inflection (located at the midpoint of the vertical part of the curve) is the equivalence point for the titration. It indicates when equivalent quantities of acid and base are present. For the titration of a strong acid with a strong base, the equivalence point occurs at a pH of 7.00 and the points on the titration curve can be calculated using solution stoichiometry (Table \(\PageIndex{1}\) and Figure \(\PageIndex{1}\)).
The titration of a weak acid with a strong base (or of a weak base with a strong acid) is somewhat more complicated than that just discussed, but it follows the same general principles. Let us consider the titration of 25.0 mL of 0.100 M acetic acid (a weak acid) with 0.100 M sodium hydroxide and compare the titration curve with that of the strong acid. Table \(\PageIndex{1}\) gives the pH values during the titration, Figure \(\PageIndex{1b}\) shows the titration curve.
Although the initial volume and molarity of the acids are the same, there are important differences between the two titration curves. The titration curve for the weak acid begins at a higher value (less acidic) and maintains higher pH values up to the equivalence point. This is because acetic acid is a weak acid, which is only partially ionized. The pH at the equivalence point is also higher (8.72 rather than 7.00) due to the hydrolysis of acetate, a weak base that raises the pH:
\[\ce{CH3CO2-}(aq)+\ce{H2O}(l)⇌\ce{CH3CO2H}(l)+\ce{OH-}(aq) \nonumber \]
After the equivalence point, the two curves are identical because the pH is dependent on the excess of hydroxide ion in both cases.
The titration curve shown in Figure \(\PageIndex{1b}\) is for the titration of 25.00 mL of 0.100 M CH 3 CO 2 H with 0.100 M NaOH. The reaction can be represented as:
\[\ce{CH3CO2H + OH- ⟶ CH3CO2- + H2O} \nonumber \]
- What is the initial pH before any amount of the NaOH solution has been added? K a = 1.8 × 10 −5 for CH 3 CO 2 H.
- Find the pH after 25.00 mL of the NaOH solution have been added.
- Find the pH after 12.50 mL of the NaOH solution has been added.
- Find the pH after 37.50 mL of the NaOH solution has been added.
Solution
(a) Assuming that the dissociated amount is small compared to 0.100 M , we find that:
\[K_\ce{a}=\ce{\dfrac{[H3O+][CH3CO2- ]}{[CH3CO2H]}}≈\ce{\dfrac{[H3O+]^2}{[CH3CO2H]_0}} \nonumber \]
and
\[\ce{[H3O+]}=\sqrt{K_\ce{a}×\ce{[CH3CO2H]}}=\sqrt{1.8\times 10^{-5}×0.100}=1.3\times 10^{-3} \nonumber \]
\[\mathrm{pH=-\log(1.3\times 10^{-3})=2.87} \nonumber \]
(b) After 25.00 mL of NaOH are added, the number of moles of NaOH and CH 3 CO 2 H are equal because the amounts of the solutions and their concentrations are the same. All of the CH 3 CO 2 H has been converted to \(\ce{CH3CO2-}\). The concentration of the \(\ce{CH3CO2-}\) ion is:
\[\mathrm{\dfrac{0.00250\: mol}{0.0500\: L}=0.0500\: \ce{MCH3CO2-}} \nonumber \]
The equilibrium that must be focused on now is the basicity equilibrium for \(\ce{CH3CO2-}\):
\[\ce{CH3CO2-}(aq)+\ce{H2O}(l)⇌\ce{CH3CO2H}(aq)+\ce{OH-}(aq) \nonumber \]
so we must determine K b for the base by using the ion product constant for water:
\[K_\ce{b}=\ce{\dfrac{[CH3CO2H][OH- ]}{[CH3CO2- ]}} \nonumber \]
\[K_\ce{a}=\ce{\dfrac{[CH3CO2- ][H+]}{[CH3CO2H]}},\textrm{ so }\ce{\dfrac{[CH3CO2H]}{[CH3CO2- ]}}=\dfrac{\ce{[H+]}}{K_\ce{a}}. \nonumber \]
Since K w = [H + ][OH − ]:
\[\begin{align} K_\ce{b} &=\dfrac{\ce{[H+][OH- ]}}{K_\ce{a}} \\ &=\dfrac{K_\ce{w}}{K_\ce{a}} \\ &=\dfrac{1.0\times 10^{-14}}{1.8\times 10^{-5}} \\ &=5.6\times 10^{-10} \end{align} \nonumber \]
Let us denote the concentration of each of the products of this reaction, CH 3 CO 2 H and OH − , as x . Using the assumption that x is small compared to 0.0500 M , \(K_\ce{b}=\dfrac{x^2}{0.0500\:M}\), and then:
\[\ce{pOH}=-\log(5.3\times 10^{-6})=5.28 \nonumber \]
\[\ce{pH}=14.00−5.28=8.72 \nonumber \]
Note that the pH at the equivalence point of this titration is significantly greater than 7.
(c) In (a), 25.00 mL of the NaOH solution was added, and so practically all the CH 3 CO 2 H was converted into \(\ce{CH3CO2-}\). In this case, only 12.50 mL of the base solution has been introduced, and so only half of all the CH 3 CO 2 H is converted into \(\ce{CH3CO2-}\). The total initial number of moles of CH 3 CO 2 H is 0.02500L × 0.100 M = 0.00250 mol, and so after adding the NaOH, the numbers of moles of CH 3 CO 2 H and \(\ce{CH3CO2-}\) are both approximately equal to \(\mathrm{\dfrac{0.00250\: mol}{2}=0.00125\: mol}\), and their concentrations are the same.
Since the amount of the added base is smaller than the original amount of the acid, the equivalence point has not been reached, the solution remains a buffer, and we can use the Henderson-Hasselbalch equation:
(as the concentrations of \(\ce{CH3CO2-}\) and CH 3 CO 2 H are the same)
Thus:
\[\ce{pH}=−\log(1.8\times 10^{−5})=4.74 \nonumber \]
(the pH = the p K a at the halfway point in a titration of a weak acid)
(d) After 37.50 mL of NaOH is added, the amount of NaOH is 0.03750 L × 0.100 M = 0.003750 mol NaOH. Since this is past the equivalence point, the excess hydroxide ions will make the solution basic, and we can again use stoichiometric calculations to determine the pH:
So:
Note that this result is the same as for the strong acid-strong base titration example provided, since the amount of the strong base added moves the solution past the equivalence point.
Calculate the pH for the weak acid/strong base titration between 50.0 mL of 0.100 M HCOOH( aq ) (formic acid) and 0.200 M NaOH (titrant) at the listed volumes of added base:
- 0.00 mL,
- 15.0 mL,
- 25.0 mL, and
- 30.0 mL.
- Answer a
-
0.00 mL: 2.37
- Answer b
-
15.0 mL: 3.92
- Answer c
-
25.00 mL: 8.29
- Answer d
-
30.0 mL: 12.097
Acid-Base Indicators
Certain organic substances change color in dilute solution when the hydronium ion concentration reaches a particular value. For example, phenolphthalein is a colorless substance in any aqueous solution with a hydronium ion concentration greater than 5.0 × 10 −9 M (pH < 8.3). In more basic solutions where the hydronium ion concentration is less than 5.0 × 10 −9 M (pH > 8.3), it is red or pink. Substances such as phenolphthalein, which can be used to determine the pH of a solution, are called acid-base indicators . Acid-base indicators are either weak organic acids or weak organic bases.
The equilibrium in a solution of the acid-base indicator methyl orange, a weak acid, can be represented by an equation in which we use \(\ce{HIn}\) as a simple representation for the complex methyl orange molecule:
\[\underbrace{\ce{HIn (aq)}}_{\ce{red}}+\ce{H2O (l)}⇌\ce{H3O^{+} (aq)}+\underbrace{\ce{In^{-} (aq)}}_{\ce{yellow}} \nonumber \]
\[K_\ce{a}=\ce{\dfrac{[H3O+][In- ]}{[HIn]}}=4.0\times 10^{−4} \nonumber \]
The anion of methyl orange, \(\ce{In^{-}}\), is yellow, and the nonionized form, \(\ce{HIn}\), is red. When we add acid to a solution of methyl orange, the increased hydronium ion concentration shifts the equilibrium toward the nonionized red form, in accordance with Le Chatelier’s principle. If we add base, we shift the equilibrium towards the yellow form. This behavior is completely analogous to the action of buffers.
An indicator’s color is the visible result of the ratio of the concentrations of the two species In − and \(\ce{HIn}\). If most of the indicator (typically about 60−90% or more) is present as \(\ce{In^{-}}\), then we see the color of the \(\ce{In^{-}}\) ion, which would be yellow for methyl orange. If most is present as \(\ce{HIn}\), then we see the color of the \(\ce{HIn}\) molecule: red for methyl orange. For methyl orange, we can rearrange the equation for K a and write:
\[\mathrm{\dfrac{[In^-]}{[HIn]}=\dfrac{[substance\: with\: yellow\: color]}{[substance\: with\: red\: color]}=\dfrac{\mathit{K}_a}{[H_3O^+]}} \label{ABeq2} \]
Equation \ref{ABeq2} shows us how the ratio of \(\ce{\dfrac{[In- ]}{[HIn]}}\) varies with the concentration of hydronium ion. The above expression describing the indicator equilibrium can be rearranged:
\[ \begin{align} \dfrac{[H_3O^+]}{\mathit{K}_a} &=\dfrac{[HIn]}{[In^- ]} \\[8pt] \log\left(\dfrac{[H_3O^+]}{\mathit{K}_a}\right) &= \log\left(\dfrac{[HIn]}{[In^- ]}\right) \\[8pt] \log([H_3O^+])-\log(\mathit{K}_a) &=-\log\left(\dfrac{[In^-]}{[HIn]}\right) \\[8pt] -pH+p\mathit{K}_a & =-\log\left(\dfrac{[In^-]}{[HIn]}\right) \\[8pt] pH &=p\mathit{K}_a+\log\left(\dfrac{[In^-]}{[HIn]}\right) \end {align} \nonumber \]
or in general terms
\[pH=p\mathit{K}_a+\log\left(\dfrac{[base]}{[acid]}\right) \label{HHeq} \]
Equation \ref{HHeq} is the same as the Henderson-Hasselbalch equation , which can be used to describe the equilibrium of indicators.
When [H 3 O + ] has the same numerical value as K a , the ratio of [In − ] to [HIn] is equal to 1, meaning that 50% of the indicator is present in the red form (HIn) and 50% is in the yellow ionic form (In − ), and the solution appears orange in color. When the hydronium ion concentration increases to 8 × 10 −4 M (a pH of 3.1), the solution turns red. No change in color is visible for any further increase in the hydronium ion concentration (decrease in pH). At a hydronium ion concentration of 4 × 10 −5 M (a pH of 4.4), most of the indicator is in the yellow ionic form, and a further decrease in the hydronium ion concentration (increase in pH) does not produce a visible color change. The pH range between 3.1 (red) and 4.4 (yellow) is the color-change interval of methyl orange; the pronounced color change takes place between these pH values.
There are many different acid-base indicators that cover a wide range of pH values and can be used to determine the approximate pH of an unknown solution by a process of elimination. Universal indicators and pH paper contain a mixture of indicators and exhibit different colors at different pHs. Figure \(\PageIndex{2}\) presents several indicators, their colors, and their color-change intervals.
Titration curves help us pick an indicator that will provide a sharp color change at the equivalence point. The best selection would be an indicator that has a color change interval that brackets the pH at the equivalence point of the titration.
The color change intervals of three indicators are shown in Figure \(\PageIndex{3}\). The equivalence points of both the titration of the strong acid and of the weak acid are located in the color-change interval of phenolphthalein. We can use it for titrations of either strong acid with strong base or weak acid with strong base.
Litmus is a suitable indicator for the HCl titration because its color change brackets the equivalence point. However, we should not use litmus for the CH 3 CO 2 H titration because the pH is within the color-change interval of litmus when only about 12 mL of NaOH has been added, and it does not leave the range until 25 mL has been added. The color change would be very gradual, taking place during the addition of 13 mL of NaOH, making litmus useless as an indicator of the equivalence point.
We could use methyl orange for the HCl titration, but it would not give very accurate results: (1) It completes its color change slightly before the equivalence point is reached (but very close to it, so this is not too serious); (2) it changes color, as Figure \(\PageIndex{2}\) demonstrates, during the addition of nearly 0.5 mL of NaOH, which is not so sharp a color change as that of litmus or phenolphthalein; and (3) it goes from yellow to orange to red, making detection of a precise endpoint much more challenging than the colorless to pink change of phenolphthalein. Figure \(\PageIndex{2}\) shows us that methyl orange would be completely useless as an indicator for the CH 3 CO 2 H titration. Its color change begins after about 1 mL of NaOH has been added and ends when about 8 mL has been added. The color change is completed long before the equivalence point (which occurs when 25.0 mL of NaOH has been added) is reached and hence provides no indication of the equivalence point.
We base our choice of indicator on a calculated pH, the pH at the equivalence point. At the equivalence point, equimolar amounts of acid and base have been mixed, and the calculation becomes that of the pH of a solution of the salt resulting from the titration.
Summary
A titration curve is a graph that relates the change in pH of an acidic or basic solution to the volume of added titrant. The characteristics of the titration curve are dependent on the specific solutions being titrated. The pH of the solution at the equivalence point may be greater than, equal to, or less than 7.00. The choice of an indicator for a given titration depends on the expected pH at the equivalence point of the titration, and the range of the color change of the indicator.
Glossary
- acid-base indicator
- organic acid or base whose color changes depending on the pH of the solution it is in
- color-change interval
- range in pH over which the color change of an indicator takes place
- titration curve
- plot of the pH of a solution of acid or base versus the volume of base or acid added during a titration


5.2.4: Representing Subtraction
Lesson
Let's subtract signed numbers.
Exercise \(\PageIndex{1}\): Equivalent Equations
Consider the equation \(2+3=5\). Here are some more equations, using the same numbers, that express the same relationship in a different way:
\(3+2=5\qquad\qquad 5-3=2\qquad\qquad 5-2=3\)
For each equation, write two more equations, using the same numbers, that express the same relationship in a different way.
- \(9+(-1)=8\)
- \(-11+x=7\)
Exercise \(\PageIndex{2}\): Subtraction with Number Lines
1. Here is an unfinished number line diagram that represents a sum of 8.
- How long should the arrow be?
-
For an equation that goes with this diagram, Mai writes \(3+?=8\).
Tyler writes \(8-3=?\). Do you agree with either of them? - What is the unknown number? How do you know?
2. Here are two more unfinished diagrams that represent sums.
For each diagram:
- What equation would Mai write if she used the same reasoning as before?
- What equation would Tyler write if he used the same reasoning as before?
- How long should the other arrow be?
- What number would complete each equation? Be prepared to explain your reasoning.
3. Draw a number line diagram for \((-8)-(3)=?\) What is the unknown number? How do you know?
Exercise \(\PageIndex{3}\): We can Add Instead
1. Match each diagram to one of these expressions:
\(3+7\qquad\qquad 3-7\qquad\qquad 3+(-7)\qquad\qquad 3-(-7)\)
a.
b.
c.
d.
2. Which expressions in the first question have the same value? What do you notice?
3. Complete each of these tables. What do you notice?
| expression | value |
|---|---|
| \(8+(-8)\) | |
| \(8-8\) | |
| \(8+(-5)\) | |
| \(8-5\) | |
| \(8+(-12)\) | |
| \(8-12\) |
| expression | value |
|---|---|
| \(-5+5\) | |
| \(-5-(-5)\) | |
| \(-5+9\) | |
| \(-5-(-9)\) | |
| \(-5+2\) | |
| \(-5-(-2)\) |
Are you ready for more?
It is possible to make a new number system using only the numbers 0, 1, 2, and 3. We will write the symbols for adding and subtracting in this system like this: \(2\oplus 1=3\) and \(2\ominus 1=1\). The table shows some of the sums.
| \(\oplus\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 |
- In this system, \(1\oplus 2=3\) and \(2\oplus 3=1\). How can you see that in the table?
- What do you think \(3\oplus 1\) should be?
- What about \(3\oplus 3\)?
- What do you think \(3\ominus 1\) should be?
- What about \(2\ominus 3\)?
- Can you think of any uses for this number system?
Summary
The equation \(7-5=?\) is equivalent to \(?+5=7\). The diagram illustrates the second equation.
Notice that the value of \(7+(-5)\) is 2.
We can solve the equation \(?+5=7\) by adding -5 to both sides. This shows that \(7-5=7+(-5)\)
Likewise, \(3-5=?\) is equivalent to \(?+5=3\).
Notice that the value of \(3+(-5)\) is \(-2\).
We can solve the equation \(?+5=3\) by adding -5 to both sides. This shows that \(3-5=3+(-5)\)
In general:
\(a-b=a+(-b)\)
If \(a-b=x\), then \(x+b=a\). We can add \(-b\) to both sides of this second equation to get that \(x=a+(-b)\)
Glossary Entries
Definition: Deposit
When you put money into an account, it is called a deposit .
For example, a person added $60 to their bank account. Before the deposit, they had $435. After the deposit, they had $495, because \(435+60=495\).
Definition: Withdrawal
When you take money out of an account, it is called a withdrawal.
For example, a person removed $25 from their bank account. Before the withdrawal, they had $350. After the withdrawal, they had $325, because \(350-25=325\).
Practice
Exercise \(\PageIndex{4}\)
Write each subtraction equation as an addition equation.
- \(a-9=6\)
- \(p-20=-30\)
- \(z-(-12)=15\)
- \(x-(-7)=-10\)
Exercise \(\PageIndex{5}\)
Find each difference. If you get stuck, consider drawing a number line diagram.
- \(9-4\)
- \(4-9\)
- \(9-(-4)\)
- \(-9-(-4)\)
- \(-9-4\)
- \(4-(-9)\)
- \(-4-(-9)\)
- \(-4-9\)
Exercise \(\PageIndex{6}\)
A restaurant bill is $59 and you pay $72. What percentage gratuity did you pay?
(From Unit 4.3.1)
Exercise \(\PageIndex{7}\)
Find the solution to each equation mentally.
- \(30+a=40\)
- \(500+b=200\)
- \(-1+c=-2\)
- \(d+3,567=0\)
Exercise \(\PageIndex{8}\)
One kilogram is 2.2 pounds. Complete the tables. What is the interpretation of the constant of proportionality in each case?
| pounds | kilograms |
|---|---|
| \(2.2\) | \(1\) |
| \(11\) | |
| \(5.5\) | |
| \(1\) |
______ kilogram per pound
| kilograms | pounds |
|---|---|
| \(1\) | \(2.2\) |
| \(7\) | |
| \(30\) | |
| \(0.5\) |
______ pounds per kilogram
(From Unit 2.1.3)


2.1.6: Mendelian Genetics
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In the previous section, we explored the mechanisms of biological variation and evolution at the most fundamental level - cells and DNA. In this section, we are going to look at how genetics functions for individuals. This field of genetics is often called Classical or Mendelian Genetics. It focuses on how various traits are passed from one individual to the next.
Some working vocabulary
- Locus (sing.) and Loci (pl.): the specific position of a gene on a chromosome
- Gene : a discrete unit of a DNA sequence that codes for a protein (or polypeptide) or for a characteristic, process, or trait
- Allele : alternate forms of a gene at a given locus (ex. T and t)
- Genotype : the genetic makeup of an individual
- Phenotype : the outward expression of the genotype, as affected by the environment and culture of an individual
- Dominant allele : an allele that masks the effect of other alleles in the phenotype (ex. in the case of Tt, T masks the effect of t)
- Recessive allele : an allele that is masked in the phenotype unless present in a double dose (ex. in the case of tt, there is no T to mask the effect of t)
- Codominant allele : both alleles are expressed in the phenotype (ex. in the ABO blood types, there are three alleles (A,B,O); A and B are codominant alleles)
- Homozygous : two of the same alleles at a given locus on homologous chromosomes (ex. TT or tt)
- Homozygous dominant : both alleles are the dominant allele (ex. TT)
- Homozygous recessive : both alleles are the recessive allele (ex. tt)
- Heterozygous : two different alleles (ex. Tt)
Note : Capital letters are usually shorthand for the dominant allele and lower-case is usually shorthand for the recessive allele when discussing Mendelian genetics.
Traits
Traits come in two types: Simple (also called Mendelian or discrete traits) and complex (sometimes called continuous or quantitative traits).
Simple traits
These are either/or traits (they usually have 2 phenotypes):
- The trait is either present or absent (e.g., hairy pinna).
- or the trait is one state or the other (e.g., attached or unattached earlobes: see image).
- It's controlled by one gene, which has one effect.
They are biologically (genetically) determined:
- They cannot be changed.
- There is little to no influence from the environment.
- They have a finite number of genotypes, usually very small. For example, 1 trait with 2 options (T and t) = 3 genotypes (ex. TT, tt, Tt).
There are approximately 4000 traits identified as discrete or simple traits of inheritance. Some additional examples include:
- Soft vs. flaky earwax
- Widow's peak
- Blood type
- Dimples
- Chin cleft (Figure \(\PageIndex{1}\))
Figure \(\PageIndex{1}\) - Cleft chin
Complex traits
- These traits have a continuous distibution (e.g., height: there are people ranging from short to tall, not just short and tall).
- The phenotype is the result of two factors; the number of loci involved and environmental influence (this includes culture).
- The trait is influenced by more than one gene; this trait is referred to as polygenic ).
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Depending on the number of genes involved, there can be many genotypes. The more genes that are involved, the finer the genotype categories until they converge into a continuous distribution.
- 1 gene = 3 genotypes
- 2 genes = 9 genotypes
- 3 genes = 27 genotypes
- 4 genes = 81 genotypes
- It's often difficult to try to divide them into categories due to an infinite number of phenotypes
The opposite of this, where one gene has multiple effects, is called pleiotropy .
Due to the large role that the environment plays in complex traits, we have a statistical concept in genetics called heritability. Heritability is the proportion of the total variation observed in the population that can be attributed to genetics rather than to the environment.
Gregor Mendel and his Peas
For simple traits, it is easy to trace patterns of inheritance and use them to show how traits are passed on from one generation to the next. Charles Darwin thought that traits were passed down by " blending ": each parent contributes equally to the offspring, and these contributions are mixed together and then halved at each successive generation (Figure \(\PageIndex{2}\)).
Figure \(\PageIndex{2}\): Blending inheritance
However, there was a contemporary of his that Darwin didn't know about, a monk named Gregor Mendel (pictured below), who figured out something important about simple traits and genes. Mendel is famous for a series of experiments that he completed with pea plants (between about 1856 and 1868) and the paper that he wrote about it ( Experiments with Plant Hybrids ). Unfortunately, despite the fact that Mendel published this work in 1865, it was relatively unknown until 1900. What he concluded was that genetic information is inherited in discrete units (he called them "factors", we call them "genes"). His work demonstrated that inheritance is non-blending or " particulate ".
Figure \(\PageIndex{3}\) - Gregor Mendel
Mendel's pea experiments
Mendel first noticed that pea plants have seven simple traits:
- Seed shape and color
- Cotyledon color
- Flower color
- Pod shape
- Pod color
- Location of pods and flowers on the stem
- Plant size
He also noted that when one selectively bred the plants with the various traits together, he could predict the ratios of offspring characteristics. For example, he noticed that when he bred pure-bred plants with violet flowers with those with white flowers, the first generation (F1) all had violet flowers. If blending inheritance was functioning, one would expect all of the flowers to be a pale violet, halfway between the original violet and white.
Figure \(\PageIndex{4}\) - Mendel's seven traits
Then when he took those F1 violet-flowered plants and bred them together, the second generation (F2) had a ratio of 3:1 violet to white flowers (there were still no pale-violet hybrids). This phenomenon was consistent across the pea plants' other six traits.
He determined that this is what was going on:
- The pure-bred violet plant was homozygous for its violet allele (BB).
- The pure-bred white plant was homozygous for its white allele (bb).
- So, when the two plants were crossed -- all of the offspring (F1) were heterozygous (Bb).
- And, when the heterozygous (Bb) offspring were crossed, this was the result: the genotype is 1 BB: 2 Bb: 1 bb and the phenotype is 3 violet: 1 white.
Figure \(\PageIndex{5}\) - A Punnett square of pea flowers
Punnett squares
This diagram in Figure \(\PageIndex{5}\) is called a Punnett square (after an early 20th c. English geneticist named Reginald Punnett). They're really pretty simple to construct. One just puts the alleles of each parent (one parent on the top and the other down the side) and then you fill in the boxes of what the offspring will be. In order to devise the ratios of the various genotypes and phenotypes, just count the number of boxes of each.
For further information on how to construct Punnett squares, go here or watch this video from Bozeman Science.
Mendel's very thorough experiments (some estimates say that he bred approximately 29,000 pea plants) lead him to write these four postulates.
Mendels' postulates
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Genetic characteristics are controlled by genes that exist in pairs in individual organisms.
- The pairs of genes that he's referring to are those on each pair of homologous chromosomes.
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When an individual has two different genes for a particular trait, only one is expressed and is dominant to the other, which is recessive.
- In other words, when an individual has a T allele and a t allele, the T allele will mask the expression of the t allele. Therefore, the phenotype expressed will be that of T.
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During the formation of gametes (AKA sex cells, sperm and eggs), the genes separate, or segregate, randomly so that each sex cell receives one or the other with equal likelihood.
- In other words, the offspring have an equal likelihood of getting a particular gene from one parent or the other.
- This is also known as "Mendel's Law of Segregation".
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During the formation of gametes, segregating pairs of genes assort independently of each other.
- In other words, the segregation of any particular pair of genes does not affect the segregation of any other pair.
- This is also known as " Mendel's Law of Independent Assortment "
Some examples of inheritance patterns in humans
Autosomal dominant traits
Figure \(\PageIndex{6}\) - Autosomal dominant inheritance
- An individual only has to have one copy of the allele in order to exhibit these traits.
- If an individual has the trait, then at least one of their parents has the trait.
- Both males and females are affected equally and should be capable of transmitting the trait.
- There is no skipping of generations.
- If one of the parents has it, then on average, 50% of the children will show the trait.
Examples of this include:
Huntington's disease
Neurofibromatosis-1
Autosomal recessive traits
- An individual needs both recessive alleles to express the trait
- Heterozygotes are considered to be "carriers"
- Both males and females should be affected equally and should be capable of transmitting the trait.
Figure \(\PageIndex{7}\) - Autosomal recessive inheritance
Examples of this include:
Cystic fibrosis, sickle cell anemia, and Tay Sachs disease
Sex-linked traits
- Remember that all females have two X sex chromosomes (XX) and males have one X and one Y sex chromosome (XY)
- These traits are inherited on the sex chromosomes (X and Y)
- There have different rates for males and females
There are three types:
- X-linked dominant
- X-linked recessive
- Y-linked
X-linked dominant
Figure \(\PageIndex{8}\) - X linked dominant inheritance
- Both males and females will express the trait if they possess at least one copy of the allele.
An example of this is:
Incontinentia pigmenti (IP)
X-linked recessive
- For females, they need two copies of the allele to express the trait, but only one to be a "carrier".
- For males, they need only one copy of the allele to express the trait. They will pass the allele down to all of their daughters.
Figure \(\PageIndex{9}\) - X linked recessive inheritance
Examples of this include:
Red-green colorblindness and Hemophilia A
Y-linked
- No female will carry or express the allele (because they don't have a y chromosome)
- All sons will carry and express the allele (because they must inherit a y chromosome from their father)
An example of this is:
Hairy pinna
References
- Klug, WS and Cummings, MR. 1991. Concepts of Genetics, 3rd. edition. Macmillan Publishing Company: New York.


9.2: Arnold Schoenberg
Arnold Schoenberg is one of the most influential composers of the twentieth century. He championed atonality in music composition, first through freely composed, expressionist works such as Pierrot Lunaire (one song from that cycle, “Madonna,” is on our playlist), and later through his own system of composition commonly referred to as as twelve-tone music (the Piano Suite, a portion of which is on our list, was composed using this method). This system of atonal composition became the dominant musical idiom at music conservatories in America and Europe during the latter half of the twentieth century. Though the influence of twelve-tone composition appears to be waning, its impact on the music of the last century is enormous. Love it or hate it, the music of Schoenberg walks large on the stage of history.
Biography
Early Life
Arnold Schoenberg was born into a lower middle-class Jewish family in the Leopoldstadt district (in earlier times a Jewish ghetto) of Vienna, at “Obere Donaustraße 5.” His father Samuel, a native of Bratislava, was a shopkeeper, and his mother Pauline was native of Prague. Arnold was largely self-taught. He took only counterpoint lessons with the composer Alexander von Zemlinsky, who was to become his first brother-in-law.
In his twenties, Schoenberg earned a living by orchestrating operettas, while composing his own works, such as the string sextet Verklärte Nacht (“Transfigured Night”). He later made an orchestral version of this, which became one of his most popular pieces. Both Richard Strauss and Gustav Mahler recognized Schoenberg’s significance as a composer; Strauss when he encountered Schoenberg’s Gurre-Lieder , and Mahler after hearing several of Schoenberg’s early works.
Strauss turned to a more conservative idiom in his own work after 1909, and at that point dismissed Schoenberg. Mahler adopted him as a protégé and continued to support him, even after Schoenberg’s style reached a point Mahler could no longer understand. Mahler worried about who would look after him after his death. Schoenberg, who had initially despised and mocked Mahler’s music, was converted by the “thunderbolt” of Mahler’s Third Symphony , which he considered a work of genius. Afterward he “spoke of Mahler as a saint.”
In 1898 Schoenberg converted to Christianity in the Lutheran church. According to MacDonald this was partly to strengthen his attachment to Western European cultural traditions, and partly as a means of self-defence “in a time of resurgent anti-Semitism.” In 1933, after long meditation, he returned to Judaism, because he realised that “his racial and religious heritage was inescapable,” and to take up an unmistakable position on the side opposing Nazism. He would self-identify as a member of the Jewish religion later in life.
In October 1901, he married Mathilde Zemlinsky, the sister of the conductor and composer Alexander von Zemlinsky, with whom Schoenberg had been studying since about 1894. Mathilde bore him two children, Gertrud (1902–1947) and Georg (1906–1974). Gertrud would marry Schoenberg’s pupil Felix Greissle in 1921. During the summer of 1908, his wife Mathilde left him for several months for a young Austrian painter, Richard Gerstl. This period marked a distinct change in Schoenberg’s work. It was during the absence of his wife that he composed “You lean against a silver-willow” (German: Du lehnest wider eine Silberweide ), the thirteenth song in the cycle Das Buch der Hängenden Gärten , Op. 15, based on the collection of the same name by the German mystical poet Stefan George. This was the first composition without any reference at all to a key. Also in this year, he completed one of his most revolutionary compositions, the String Quartet No. 2, whose first two movements, though chromatic in color, use traditional key signatures, yet whose final two movements, also settings of George, daringly weaken the links with traditional tonality. Both movements end on tonic chords, and the work is not fully non-tonal. Breaking with previous string-quartet practice, it incorporates a soprano vocal line.
During the summer of 1910, Schoenberg wrote his Harmonielehre ( Theory of Harmony , Schoenberg 1922), which remains one of the most influential music-theory books. From about 1911, Schoenberg belonged to a circle of artists and intellectuals who included Lene Schneider-Kainer, Franz Werfel, Herwarth Walden and the latter’s wife, Else Lasker-Schüler.
In 1910 he met Edward Clark, an English music journalist then working in Germany. Clark became his sole English student, and in his later capacity as a producer for the BBC he was responsible for introducing many of Schoenberg’s works, and Schoenberg himself, to Britain (as well as Webern, Berg and others).
Another of his most important works from this atonal or pantonal period is the highly influential Pierrot Lunaire , Op. 21, of 1912, a novel cycle of expressionist songs set to a German translation of poems by the Belgian-French poet Albert Giraud. Utilizing the technique of Sprechstimme , or melodramatically spoken recitation, the work pairs a female vocalist with a small ensemble of five musicians. The ensemble, which is now commonly referred to as the Pierrot ensemble, consists of flute (doubling on piccolo), clarinet (doubling on bass clarinet), violin (doubling on viola), violoncello, speaker, and piano.
Wilhelm Bopp, director of the Vienna Conservatory from 1907, wanted a break from the stale environment personified for him by Robert Fuchs and Hermann Graedener. Having considered many candidates, he offered teaching positions to Schoenberg and Franz Schreker in 1912. At the time Schoenberg lived in Berlin. He was not completely cut off from the Vienna Conservatory, having taught a private theory course a year earlier. He seriously considered the offer, but he declined. Writing afterward to Alban Berg, he cited his “aversion to Vienna” as the main reason for his decision, while contemplating that it might have been the wrong one financially, but having made it he felt content. A couple of months later he wrote to Schreker suggesting that it might have been a bad idea for him as well to accept the teaching position.
World War I
World War I brought a crisis in his development. Military service disrupted his life when at the age of 42 he was in the army. He was never able to work uninterrupted or over a period of time, and as a result he left many unfinished works and undeveloped “beginnings”. On one occasion, a superior officer demanded to know if he was “this notorious Schoenberg, then”; Schoenberg replied: “Beg to report, sir, yes. Nobody wanted to be, someone had to be, so I let it be me” (according to Norman Lebrecht (2001), this is a reference to Schoenberg’s apparent “destiny” as the “Emancipator of Dissonance”).
In what Ross calls an “act of war psychosis,” Schoenberg drew comparisons between Germany’s assault on France and his assault on decadent bourgeois artistic values. In August 1914, while denouncing the music of Bizet, Stravinsky andRavel, he wrote: “Now comes the reckoning! Now we will throw these mediocre kitschmongers into slavery, and teach them to venerate the German spirit and to worship the German God.”
The deteriorating relation between contemporary composers and the public led him to found the Society for Private Musical Performances ( Verein für musikalische Privataufführungen in German) in Vienna in 1918. He sought to provide a forum in which modern musical compositions could be carefully prepared and rehearsed, and properly performed under conditions protected from the dictates of fashion and pressures of commerce. From its inception through 1921, when it ended because of economic reasons, the Society presented 353 performances to paid members, sometimes at the rate of one per week. During the first year and a half, Schoenberg did not let any of his own works be performed. Instead, audiences at the Society’s concerts heard difficult contemporary compositions by Scriabin, Debussy, Mahler, Webern, Berg, Reger, and other leading figures of early 20th-century music.
Development of the Twelve-Tone Method
Later, Schoenberg was to develop the most influential version of the dodecaphonic (also known as twelve-tone) method of composition, which in French and English was given the alternative name serialism by René Leibowitzand Humphrey Searle in 1947. This technique was taken up by many of his students, who constituted the so-called Second Viennese School. They included Anton Webern, Alban Berg and Hanns Eisler, all of whom were profoundly influenced by Schoenberg. He published a number of books, ranging from his famous Harmonielehre (Theory of Harmony) to Fundamentals of Musical Composition , many of which are still in print and used by musicians and developing composers.
Schoenberg viewed his development as a natural progression, and he did not deprecate his earlier works when he ventured into serialism. In 1923 he wrote to the Swiss philanthropist Werner Reinhart:
For the present, it matters more to me if people understand my older works. . . . They are the natural forerunners of my later works, and only those who understand and comprehend these will be able to gain an understanding of the later works that goes beyond a fashionable bare minimum. I do not attach so much importance to being a musical bogey-man as to being a natural continuer of properly-understood good old tradition!
His first wife died in October 1923, and in August of the next year Schoenberg married Gertrud Kolisch (1898–1967), sister of his pupil, the violinist Rudolf Kolisch. She wrote the libretto for Schoenberg’s one-act opera Von heute auf morgen under the pseudonym Max Blonda. At her request Schoenberg’s (ultimately unfinished) piece, Die Jakobsleiter was prepared for performance by Schoenberg’s student Winfried Zillig. After her husband’s death in 1951 she founded Belmont Music Publishers devoted to the publication of his works. Arnold used the notes G and E ♭ (German: Es, i.e., “S”) for “Gertrud Schoenberg”, in the Suite , for septet, Op. 29 (1925).
Following the 1924 death of composer Ferruccio Busoni, who had served as Director of a Master Class in Composition at the Prussian Academy of Arts in Berlin, Schoenberg was appointed to this post the next year, but because of health problems was unable to take up his post until 1926. Among his notable students during this period were the composers Roberto Gerhard, Nikos Skalkottas, and Josef Rufer.
Along with his twelve-tone works, 1930 marks Schoenberg’s return to tonality, with numbers 4 and 6 of the Six Pieces for Male Chorus Op.35, the other pieces being dodecaphonic.
Third Reich and Move to America
Schoenberg continued in his post until the Nazis came to power under Adolf Hitler in 1933. While vacationing in France, he was warned that returning to Germany would be dangerous. Schoenberg formally reclaimed membership in the Jewish religion at a Paris synagogue, then traveled with his family to the United States. However, this happened only after his attempts to move to Britain came to nothing. He enlisted the aid of his former student and great champion Edward Clark, now a senior producer with the BBC, in helping him gain a British teaching post or even a British publisher, but to no avail.
His first teaching position in the United States was at the Malkin Conservatory in Boston. He moved to Los Angeles, where he taught at the University of Southern California and the University of California, Los Angeles, both of which later named a music building on their respective campuses Schoenberg Hall. He was appointed visiting professor at UCLA in 1935 on the recommendation of Otto Klemperer, music director and conductor of the Los Angeles Philharmonic Orchestra; and the next year was promoted to professor at a salary of $5,100 per year, which enabled him in either May 1936 or 1937 to buy a Spanish Revival house at 116 North Rockingham in Brentwood Park, near the UCLA campus, for $18,000. This address was directly across the street from Shirley Temple’s house, and there he befriended fellow composer (and tennis partner) George Gershwin. The Schoenbergs were able to employ domestic help and began holding Sunday afternoon gatherings that were known for excellent coffee and Viennese pastries. Frequent guests included Otto Klemperer (who studied composition privately with Schoenberg beginning in April 1936), Edgard Varèse, Joseph Achron, Louis Gruenberg, Ernst Toch, and, on occasion, well-known actors such as Harpo Marx and Peter Lorre. Composers Leonard Rosenman and George Tremblay studied with Schoenberg at this time.
After his move to the United States in 1934, the composer used the alternative spelling of his surname Schoenberg , rather than Schönberg , in what he called “deference to American practice,” though according to one writer he first made the change a year earlier.
He lived there the rest of his life, but at first he was not settled. In around 1934, he applied for a position of teacher of harmony and theory at the New South Wales State Conservatorium in Sydney. The Director, Edgar Bainton, rejected him for being Jewish and for having “modernist ideas and dangerous tendencies.” Schoenberg also at one time explored the idea of emigrating to New Zealand. His secretary and student (and nephew of Schoenberg’s mother-in-law Henriette Kolisch), was Richard (Dick) Hoffmann Jr, Viennese-born but who lived in New Zealand 1935–47, and Schoenberg had since childhood been fascinated with islands, and with New Zealand in particular, possibly because of the beauty of the postage stamps issued by that country.
During this final period, he composed several notable works, including the difficult Violin Concerto, Op. 36 (1934/36), the Kol Nidre , Op. 39, for chorus and orchestra (1938), the Ode to Napoleon Buonaparte , Op. 41 (1942), the haunting Piano Concerto, Op. 42 (1942), and his memorial to the victims of the Holocaust, A Survivor from Warsaw , Op. 46 (1947). He was unable to complete his opera Moses und Aron (1932/33), which was one of the first works of its genre written completely using dodecaphonic composition. Along with twelve-tone music, Schoenberg also returned to tonality with works during his last period, like the Suite for Strings in G major (1935), the Chamber Symphony No. 2 in E ♭ minor, Op. 38 (begun in 1906, completed in 1939), the Variations on a Recitative in D minor, Op. 40 (1941). During this period his notable students included John Cage and Lou Harrison.
In 1941 he became a citizen of the United States.
Later Years and Death
Schoenberg’s superstitious nature may have triggered his death. The composer had triskaidekaphobia (the fear of the number 13), and according to friend Katia Mann, he feared he would die during a year that was a multiple of 13. He dreaded his sixty-fifth birthday in 1939 so much that a friend asked the composer and astrologer Dane Rudhyar to prepare Schoenberg’s horoscope. Rudhyar did this and told Schoenberg that the year was dangerous, but not fatal.
But in 1950, on his seventy-sixth birthday, an astrologer wrote Schoenberg a note warning him that the year was a critical one: 7 + 6 = 13. This stunned and depressed the composer, for up to that point he had only been wary of multiples of 13 and never considered adding the digits of his age. He died on Friday, 13 July 1951, shortly before midnight. Schoenberg had stayed in bed all day, sick, anxious and depressed. His wife Gertrud reported in a telegram to her sister-in-law Ottilie the next day that Arnold died at 11:45 pm, 15 minutes before midnight. In a letter to Ottilie dated 4 August 1951, Gertrud explained, “About a quarter to twelve I looked at the clock and said to myself: another quarter of an hour and then the worst is over. Then the doctor called me. Arnold’s throat rattled twice, his heart gave a powerful beat and that was the end.”
Schoenberg’s ashes were later interred at the Zentralfriedhof in Vienna on 6 June 1974.
Music
Listen: Second String Quartet
Please listen to the following audio file to hear the fourth movement played by the Carmel Quartet with soprano Rona Israel-Kolatt, in 2007
An audio element has been excluded from this version of the text. You can listen to it online here: http://pb.libretexts.org/map/?p=366
Schoenberg’s significant compositions in the repertory of modern art music extend over a period of more than 50 years. Traditionally they are divided into three periods though this division is arguably arbitrary as the music in each of these periods is considerably varied. The idea that his twelve-tone period “represents a stylistically unified body of works is simply not supported by the musical evidence,” and important musical characteristics—especially those related to motivic development—transcend these boundaries completely. The first of these periods, 1894–1907, is identified in the legacy of the high-Romantic composers of the late nineteenth century, as well as with “expressionist” movements in poetry and art. The second, 1908–1922, is typified by the abandonment of key centers, a move often described (though not by Schoenberg) as “free atonality.” The third, from 1923 onward, commences with Schoenberg’s invention of dodecaphonic, or “twelve-tone” compositional method. Schoenberg’s best-known students, Hanns Eisler, Alban Berg, andAnton Webern, followed Schoenberg faithfully through each of these intellectual and aesthetic transitions, though not without considerable experimentation and variety of approach.
First Period: Late Romanticism
Beginning with songs and string quartets written around the turn of the century, Schoenberg’s concerns as a composer positioned him uniquely among his peers, in that his procedures exhibited characteristics of both Brahms and Wagner, who for most contemporary listeners, were considered polar opposites, representing mutually exclusive directions in the legacy of German music. Schoenberg’s Six Songs, Op. 3 (1899–1903), for example, exhibit a conservative clarity of tonal organization typical of Brahms and Mahler, reflecting an interest in balanced phrases and an undisturbed hierarchy of key relationships. However, the songs also explore unusually bold incidentalchromaticism, and seem to aspire to a Wagnerian “representational” approach to motivic identity. The synthesis of these approaches reaches an apex in his Verklärte Nacht , Op. 4 (1899), a programmatic work for string sextet that develops several distinctive “leitmotif”-like themes, each one eclipsing and subordinating the last. The only motivic elements that persist throughout the work are those that are perpetually dissolved, varied, and re-combined, in a technique, identified primarily in Brahms’s music, that Schoenberg called “developing variation”. Schoenberg’s procedures in the work are organized in two ways simultaneously; at once suggesting a Wagnerian narrative of motivic ideas, as well as a Brahmsian approach to motivic development and tonal cohesion.
Second Period: Free Atonality
Schoenberg’s music from 1908 onward experiments in a variety of ways with the absence of traditional keys or tonal centers. His first explicitly atonal piece was the second string quartet, Op. 10, with soprano. The last movement of this piece has no key signature, marking Schoenberg’s formal divorce from diatonic harmonies. Other important works of the era include his song cycle Das Buch der Hängenden Gärten , Op. 15 (1908–1909), his Five Orchestral Pieces, Op. 16 (1909), the influential Pierrot Lunaire , Op. 21 (1912), as well as his dramatic Erwartung , Op. 17 (1909). The urgency of musical constructions lacking in tonal centers, or traditional dissonance-consonance relationships, however, can be traced as far back as his Chamber Symphony No. 1, Op. 9 (1906), a work remarkable for its tonal development of whole-tone and quartal harmony, and its initiation of dynamic and unusual ensemble relationships, involving dramatic interruption and unpredictable instrumental allegiances; many of these features would typify the timbre-oriented chamber music aesthetic of the coming century.
Third Period: Twelve-Tone and Tonal Works
In the early 1920s, he worked at evolving a means of order that would make his musical texture simpler and clearer. This resulted in the “method of composing with twelve tones which are related only with one another,” in which the twelve pitches of the octave (unrealized compositionally) are regarded as equal, and no one note or tonality is given the emphasis it occupied in classical harmony. He regarded it as the equivalent in music of Albert Einstein’s discoveries in physics. Schoenberg announced it characteristically, during a walk with his friend Josef Rufer, when he said, “I have made a discovery which will ensure the supremacy of German music for the next hundred years.” This period included the Variations for Orchestra , Op. 31 (1928); Piano Pieces , Opp. 33a & b (1931), and the Piano Concerto, Op. 42 (1942). Contrary to his reputation for strictness, Schoenberg’s use of the technique varied widely according to the demands of each individual composition. Thus the structure of his unfinished opera Moses und Aron is unlike that of his Fantasy for Violin and Piano , Op. 47 (1949).
Ten features of Schoenberg’s mature twelve-tone practice are characteristic, interdependent, and interactive:
- Hexachordal inversional combinatoriality
- Aggregates
- Linear set presentation
- Partitioning
- Isomorphic partitioning
- Invariants
- Hexachordal levels
- Harmony, “consistent with and derived from the properties of the referential set”
- Metre, established through “pitch-relational characteristics”
- Multidimensional set presentations
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Arnold Schoenberg. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Arnold_Schoenberg . License : CC BY-SA: Attribution-ShareAlike


35.17: Raiding, Capturing, and Slaughtering in the Name of God
What caused the beheading of five colonists? Throughout the period that surrounded the colonial encounter, there were many reasons for such cruel acts. Protection from colonists or retributions for past wrongdoings seem to be the most common rationales for violence and war from the indigenous people. In the case regarding the kidnapping and executing of five plantation workers in the year of 1618, the reasoning was based on these same notions, but was driven extensively by another ideal entirely. Prior to the systematic execution of the five Europeans, the gentleman who the narrator refers to as the indigenous peoples’ “king” gives an explanation for his future actions by stating that “[y]ou designed to kill us, but we hurt no man who has not first offended us; our God has given you into our hands, and you must die” (Winkfield 48). Here there are notes of the aforementioned retribution, however there is also a strong religious tie in the quote as well. This is not the only place it is mentioned during this encounter, as the entirety of the speech the “king” delivers is based around their worshipping of the Sun. Here, an act of extreme violence is justified, in the eyes of the accusers, by basing the motivations in a religious understanding. This justification can, historically, be seen to drive many of the worst areas of the colonial encounter, from both sides. Religion acts as a justification, but it is not the motivation. In this circumstance, the group of indigenous peoples wish to seek retribution for the past wrongdoings of white, European colonists, and are able to find a way to connect their actions to the wants of their God. The motivation behind the attack is revenge, but the justification is that the captors are delivered from God and are sent to die. Similarly to this, white colonists referenced God, Christianity, and purification to justify genocidal acts against indigenous peoples. These are the well-meaning words that are seen in journals, manuscripts, and memoirs from the time, used to justify the slaughter, pillage, and rape of an entire species of people. What they fail to mention is the monetary gains, political control, and power that they also receive from their actions–a healthy little byproduct of doing the Lord’s work. Acting violently without cause makes you a savage. Acting violently for God makes you a disciple. Does killing due to a will of God justify the beheading of five men? Does it justify it more so than revenge for the killing of a large number of their people? Whatever the answers to these questions may be, these people justify their actions as being right in the eyes of God.


14.6: Light Emitting Diode
Let's talk about the recombining electrons for a minute. When the electron falls down from the conduction band and fills in a hole in the valence band, there is an obvious loss of energy. The question is; where does that energy go? In silicon, the answer is not very interesting. Silicon is what is known as an indirect band-gap material . What this means is that as an electron goes from the bottom of the conduction band to the top of the valence band, it must also undergo a significant change in momentum. This all comes about from the details of the band structure for the material, which we will not concern ourselves with here. As we all know, whenever something changes state, we must still conserve not only energy, but also momentum. In the case of an electron going from the conduction band to the valence band in silicon, both of these things can only be conserved if the transition also creates a quantized set of lattice vibrations, called phonons , or "heat". Phonons posses both energy and momentum, and their creation upon the recombination of an electron and hole allows for complete conservation of both energy and momentum. All of the energy which the electron gives up in going from the conduction band to the valence band (1.1 eV) ends up in phonons, which is another way of saying that the electron heats up the crystal.
In some other semiconductors, something else occurs. In a class of materials called direct band-gap semiconductors , the transition from conduction band to valence band involves essentially no change in momentum. Photons, it turns out, possess a fair amount of energy (several eV per photon in some cases) but they have very little momentum associated with them. Thus, for a direct band gap material, the excess energy of the electron-hole recombination can either be taken away as heat, or more likely, as a photon of light. This radiative transition then conserves energy and momentum by giving off light whenever an electron and hole recombine. This gives rise to (for us) a new type of device, the light emitting diode (LED). Emission of a photon in an LED is shown schematically in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\): Radiative recombination in a direct band-gap semiconductorIt was Planck who postulated that the energy of a photon was related to its frequency by a constant, which was later named after him. If the frequency of oscillation is given by the Greek letter "nu" \((\nu)\), then the energy of the photon is just \(h \nu\) where \(h\) is Planck's constant, which has a value of \(4.14 \times 10^{-15} \mathrm{~eV} \cdot \mathrm{seconds}\). \[E = h \nu \nonumber \]
When we talk about light it is conventional to specify its wavelength, \(\lambda\), instead of its frequency. Visible light has a wavelength on the order of nanometers (red is about 600 nm, green about 500 nm and blue is in the 450 nm region.) A handy "rule of thumb" can be derived from the fact that \(\lambda = \frac{c}{\nu}\), where \(c\) is the speed of light. Since \(c = 3 \times 10^{8} \ \frac{\mathrm{m}}{\mathrm{sec}}\) or \(c = 3 \times 10^{17} \ \frac{\mathrm{nm}}{\mathrm{sec}}\), \[\begin{array}{l} \lambda (\mathrm{nm}) &= \frac{hc}{E (\mathrm{eV})} \\ &= \frac{1242}{E (\mathrm{eV})} \end{array} \nonumber \]
Thus, a semiconductor with a \(2 \mathrm{~eV}\) band-gap should give off light at about \(620 \mathrm{~nm}\) (in the range of red light). A \(3 \mathrm{~eV}\) band-gap material would emit at \(414 \mathrm{~nm}\), in the violet. The human eye, of course, is not equally responsive to all colors. We show this in Figure \(\PageIndex{2}\), where we have also included the materials which are used for important light emitting diodes (LEDs) for each of the different spectral regions.
Figure \(\PageIndex{2}\): Relative response of the human eye to various colorsAs you no doubt notice, a number of the important LEDs are based on the \(\mathrm{GaAsP}\) system. \(\mathrm{GaAs}\) is a direct band-gap semiconductor with a band gap of \(1.42 \mathrm{~eV}\) (in the infrared). \(\mathrm{GaP}\) is an indirect band-gap material with a band gap of \(2.26 \mathrm{~eV}\) (550 nm, or green). Both \(\mathrm{As}\) and \(\mathrm{P}\) are group V elements. (Hence the nomenclature of the materials as III-V compound semiconductors .) We can replace some of the \(\mathrm{As}\) with \(\mathrm{P}\) in \mathrm{GaAs}\) and make a mixed compound semiconductor \(\mathrm{Ga As}_{1-x} \mathrm{P}_{x}\). When the mole fraction of phosphorous is less than about \(0.45\) the band gap is direct, and so we can "engineer" the desired color of LED that we want by simply growing a crystal with the proper phosphorus concentration!
The properties of the GaAsP system are shown in Figure \(\PageIndex{3}\). It turns out that for this system, there are actually two different band gaps, as shown in the inset of Figure \(\PageIndex{3}\). One is a direct gap (no change in momentum) and the other is indirect. In \(\mathrm{GaAs}\), the direct gap has lower energy than the indirect one (like in the inset) and so the transition is a radiative one. As we start adding phosphorous to the system, both the direct and indirect band gaps increase in energy. However, the direct gap energy increases faster with phosphorous fraction than does the indirect one. At a mole fraction \(x\) of about \(0.45\), the gap energies cross over and the material goes from being a direct gap semiconductor to an indirect gap semiconductor. At \(x=0.35\) the band gap is about \(1.97 \mathrm{~eV}\) (630 nm), and so we would only expect to get light up to the red using the \(\mathrm{GaAsP}\) system for making LEDs. Fortunately, people discovered that you could add an impurity (nitrogen) to the \(\mathrm{GaAsP}\) system, which introduced a new level in the system. An electron could go from the indirect conduction band (for a mixture with a mole fraction greater than \(0.45\)) to the nitrogen site, changing its momentum, but not its energy. It could then make a direct transition to the valence band, and light with colors all the way to the green became possible. The use of a nitrogen recombination center is depicted in Figure \(\PageIndex{4}\).
Figure \(\PageIndex{3}\): Band gap for the \(\mathrm{GaAsP}\) system Figure \(\PageIndex{4}\): Addition of a nitrogen recombination center to indirect \(\mathrm{GaAsP}\)If we want colors with wavelengths shorter than the green, we must abandon the \(\mathrm{GaAsP}\) system and look for more suitable materials. A compound semiconductor made from the II-VI elements \(\mathrm{Zn}\) and \(\mathrm{Se}\) make up one promising system, and several research groups have successfully made blue and blue-green LEDs from \(\mathrm{ZnSe}\). \(\mathrm{SiC}\) is another (weak) blue emitter which is commercially available on the market. Recently, workers at a tiny, unknown chemical company stunned the "display world" by announcing that they had successfully fabricated a blue LED using the II-V material \(\mathrm{GaN}\). A good blue LED has been the "holy grail" of the display and CD-ROM research community for a number of years now. Obviously, adding blue to the already working green and red LEDs completes the set of 3 primary colors necessary for a full-color flat panel display (hang a TV screen on your wall like a picture?). Using a blue LED or laser in a CD-ROM would more than quadruple its data capacity, as bit diameter scales as \(\lambda\), and hence the area as \(\lambda^{2}\).


1.3: Critical Thinking and Clinical Reasoning
Before learning how to use the nursing process, it is important to understand some basic concepts related to critical thinking and nursing practice. Let’s take a deeper look at how nurses think.
Critical Thinking and Clinical Reasoning
Nurses make decisions while providing patient care by using critical thinking and clinical reasoning. Critical thinking is a broad term used in nursing that includes “reasoning about clinical issues such as teamwork, collaboration, and streamlining workflow.” [1] Using critical thinking means that nurses take extra steps to maintain patient safety and don’t just “follow orders.” It also means the accuracy of patient information is validated and plans for caring for patients are based on their needs, current clinical practice, and research.
“Critical thinkers” possess certain attitudes that foster rational thinking. These attitudes are as follows:
- Independence of thought: Thinking on your own
- Fair-mindedness: Treating every viewpoint in an unbiased, unprejudiced way
- Insight into egocentricity and sociocentricity: Thinking of the greater good and not just thinking of yourself. Knowing when you are thinking of yourself (egocentricity) and when you are thinking or acting for the greater good (sociocentricity)
- Intellectual humility: Recognizing your intellectual limitations and abilities
- Nonjudgmental: Using professional ethical standards and not basing your judgments on your own personal or moral standards
- Integrity: Being honest and demonstrating strong moral principles
- Perseverance: Persisting in doing something despite it being difficult
- Confidence: Believing in yourself to complete a task or activity
- Interest in exploring thoughts and feelings: Wanting to explore different ways of knowing
- Curiosity: Asking “why” and wanting to know more
Clinical reasoning is defined as, “A complex cognitive process that uses formal and informal thinking strategies to gather and analyze patient information, evaluate the significance of this information, and weigh alternative actions.” To make sound judgments about patient care, nurses must generate alternatives, weigh them against the evidence, and choose the best course of action. The ability to clinically reason develops over time and is based on knowledge and experience. [3]
The ANA’s Standards of Professional Nursing Practice associated with each component of the nursing process are described below.
Assessment
Assessment is the first step of the nursing process. The American Nurses Association (ANA) “Assessment” Standard of Practice is defined as, “The registered nurse collects pertinent data and information relative to the health care consumer’s health or the situation.” This includes collecting “pertinent data related to the health and quality of life in a systematic, ongoing manner, with compassion and respect for the wholeness, inherent dignity, worth, and unique attributes of every person, including but not limited to, demographics, environmental and occupational exposures, social determinants of health, health disparities, physical, functional, psychosocial, emotional, cognitive, spiritual/transpersonal, sexual, sociocultural, age-related, environmental, and lifestyle/economic assessments.” [1]
A registered nurse uses a systematic method to collect and analyze patient data. Assessment includes physiological data, as well as psychological, sociocultural, spiritual, economic, and lifestyle data. For example, a nurse’s assessment of a hospitalized patient in pain includes the patient’s response to pain, such as the inability to get out of bed, refusal to eat, withdrawal from family members, or anger directed at hospital staff. Nurses assess patients to gather clues, make generalizations, and diagnose human responses to health conditions and life processes. Patient data is considered either subjective or objective, and it can be collected from multiple sources.
Diagnosis
The “Diagnosis” Standard of Practice is defined as, “The registered nurse analyzes the assessment data to determine actual or potential diagnoses, problems, and issues.” [13] A nursing diagnosis is the nurse’s clinical judgment about the client's response to actual or potential health conditions or needs. Nursing diagnoses are the bases for the nurse’s care plan and are different than medical diagnoses.
Outcomes Identification
The “Outcomes Identification” Standard of Practice is defined as, “The registered nurse identifies expected outcomes for a plan individualized to the health care consumer or the situation.” The nurse sets measurable and achievable short- and long-term goals and specific outcomes in collaboration with the patient based on their assessment data and nursing diagnoses.
Planning
The “Planning” Standard of Practice is defined as, “The registered nurse develops a collaborative plan encompassing strategies to achieve expected outcomes.” [16] Assessment data, diagnoses, and goals are used to select evidence-based nursing interventions customized to each patient’s needs and concerns. Goals, expected outcomes, and nursing interventions are documented in the patient’s nursing care plan so that nurses, as well as other health professionals, have access to it for continuity of care. [17]
Nursing Care Plans
Creating nursing care plans is a part of the “Planning” step of the nursing process. A nursing care plan is a type of documentation that demonstrates the individualized planning and delivery of nursing care for each specific patient using the nursing process. Registered nurses (RNs) create nursing care plans so that the care provided to the patient across shifts is consistent among health care personnel.
Implementation
The “Implementation” Standard of Practice is defined as, “The nurse implements the identified plan.” Nursing interventions are implemented or delegated with supervision according to the care plan to assure continuity of care across multiple nurses and health professionals caring for the patient. Interventions are also documented in the patient’s electronic medical record as they are completed.
Evaluation
The “Evaluation” Standard of Practice is defined as, “The registered nurse evaluates progress toward attainment of goals and outcomes.” During evaluation, nurses assess the patient and compare the findings against the initial assessment to determine the effectiveness of the interventions and overall nursing care plan. Both the patient’s status and the effectiveness of the nursing care must be continuously evaluated and modified as needed.
Benefits of Using the Nursing Process
Using the nursing process has many benefits for nurses, patients, and other members of the health care team. The benefits of using the nursing process include the following:
- Promotes quality patient care
- Decreases omissions and duplications
- Provides a guide for all staff involved to provide consistent and responsive care
- Encourages collaborative management of a patient’s health care problems
- Improves patient safety
- Improves patient satisfaction
- Identifies a patient’s goals and strategies to attain them
- Increases the likelihood of achieving positive patient outcomes
- Saves time, energy, and frustration by creating a care plan or path to follow
By using these components of the nursing process as a critical thinking model, nurses plan interventions customized to the patient’s needs, plan outcomes and interventions, and determine whether those actions are effective in meeting the patient’s needs. In the remaining sections of this chapter, we will take an in-depth look at each of these components of the nursing process. Using the nursing process and implementing evidence-based practices are referred to as the “science of nursing.” Let’s review concepts related to the “art of nursing” while providing holistic care in a caring manner using the nursing process.
Holistic Nursing Care
The American Nurses Association (ANA) recently updated the definition of nursing as, “Nursing integrates the art and science of caring and focuses on the protection, promotion, and optimization of health and human functioning; prevention of illness and injury; facilitation of healing; and alleviation of suffering through compassionate presence. Nursing is the diagnosis and treatment of human responses and advocacy in the care of individuals, families, groups, communities, and populations in the recognition of the connection of all humanity.”
The ANA further describes nursing is a learned profession built on a core body of knowledge that integrates both the art and science of nursing. The art of nursing is defined as, “Unconditionally accepting the humanity of others, respecting their need for dignity and worth, while providing compassionate, comforting care.”
Nurses care for individuals holistically, including their emotional, spiritual, psychosocial, cultural, and physical needs. They consider problems, issues, and needs that the person experiences as a part of a family and a community as they use the nursing process.
Caring and the Nursing Process
The American Nurses Association (ANA) states, “The act of caring is foundational to the practice of nursing.” Successful use of the nursing process requires the development of a care relationship with the patient. A care relationship is a mutual relationship that requires the development of trust between both parties. This trust is often referred to as the development of rapport and underlies the art of nursing. While establishing a caring relationship, the whole person is assessed, including the individual’s beliefs, values, and attitudes, while also acknowledging the vulnerability and dignity of the patient and family. Assessing and caring for the whole person takes into account the physical, mental, emotional, and spiritual aspects of being a human being. Caring interventions can be demonstrated in simple gestures such as active listening, making eye contact, touching, and verbal reassurances while also respecting and being sensitive to the care recipient’s cultural beliefs and meanings associated with caring behaviors.
- Klenke-Borgmann, L., Cantrell, M. A., & Mariani, B. (2020). Nurse educator’s guide to clinical judgment: A review of conceptualization, measurement, and development. Nursing Education Perspectives, 41 (4), 215-221. ↵
- Klenke-Borgmann, L., Cantrell, M. A., & Mariani, B. (2020). Nurse educator’s guide to clinical judgment: A review of conceptualization, measurement, and development. Nursing Education Perspectives, 41 (4), 215-221. ↵
- Powers, L., Pagel, J., & Herron, E. (2020). Nurse preceptors and new graduate success. American Nurse Journal, 15 (7), 37-39. ↵
- “ The Detective ” by paurian is licensed under CC BY 2.0 ↵
- “ In the Quiet Zone… ” by C.O.D. Library is licensed under CC BY-NC-SA 2.0 ↵
- NCSBN. (n.d.). NCSBN clinical judgment model . https://www.ncsbn.org/14798.htm ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- “ The Nursing Process ” by Kim Ernstmeyer at Chippewa Valley Technical College is licensed under CC BY 4.0 ↵
- “Patient Image in LTC.JPG” by ARISE project is licensed under CC BY 4.0 ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (n.d.). The nursing process. https://www.nursingworld.org/practice-policy/workforce/what-is-nursing/the-nursing-process/ ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (n.d.). The nursing process . https://www.nursingworld.org/practice-policy/workforce/what-is-nursing/the-nursing-process/ ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (n.d.). The nursing process. https://www.nursingworld.org/practice-policy/workforce/what-is-nursing/the-nursing-process/ ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (3rd ed.). American Nurses Association. ↵
- American Nurses Association. (n.d.) The nursing process. https://www.nursingworld.org/practice-policy/workforce/what-is-nursing/the-nursing-process / ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (n.d.). The nursing process. https://www.nursingworld.org/practice-policy/workforce/what-is-nursing/the-nursing-process / ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- Walivaara, B., Savenstedt, S., & Axelsson, K. (2013). Caring relationships in home-based nursing care - registered nurses’ experiences. The Open Journal of Nursing, 7 , 89-95. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3722540/pdf/TONURSJ-7-89.pdf ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- “ hospice-1793998_1280.jpg ” by truthseeker08 is licensed under CC0 ↵
- Watson Caring Science Institute. (n.d.). Watson Caring Science Institute. Jean Watson, PHD, RN, AHN-BC, FAAN, (LL-AAN) . https://www.watsoncaringscience.org/jean-bio/ ↵


7: Iterative Methods
Iterative methods are often used for solving a system of nonlinear equations. Even for linear systems, iterative methods have some advantages. They may require less memory and may be computationally faster. They are also easier to code. Here, without detailing the theoretical numerical analysis, we will simply explain the related iterative methods that go by the names of the Jacobi method, the Gauss-Seidel method, and the Successive Over Relaxation method (or SOR). We illustrate these three methods by showing how to solve the two-dimensional Poisson equation, an equation that we will later need to solve to determine the flow field past an obstacle
The Poisson equation is given by
\[-\nabla^{2} \Phi=f, \nonumber \]
where
\[\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} \nonumber \]
is the usual two-dimensional Laplacian. This could be a linear equation with \(f\) independent of \(\Phi\) , or a nonlinear equation where \(f\) may be some nonlinear function of \(\Phi\) .
After discretizing the Poisson equation using the centered finite difference approximation for the second derivatives, we obtain on a grid with uniform grid spacing \(h\) ,
\[4 \Phi_{i, j}-\Phi_{i+1, j}-\Phi_{i-1, j}-\Phi_{i, j+1}-\Phi_{i, j-1}=h^{2} f_{i, j} . \nonumber \]
The Jacobi method simply solves the discretized equation (7.1) for \(\Phi_{i, j}\) iteratively. With superscripts indicating iteration steps, we have
\[\Phi_{i, j}^{(n+1)}=\frac{1}{4}\left(\Phi_{i+1, j}^{(n)}+\Phi_{i-1, j}^{(n)}+\Phi_{i, j+1}^{(n)}+\Phi_{i, j-1}^{(n)}+h^{2} f_{i, j}^{(n)}\right) . \nonumber \]
In the old FORTRAN-77 scientific programming language, implementing the Jacobi method required the saving of \(\Phi\) in two different arrays, one corresponding to the \(n\) -th iteration, and one corresponding to the \((n+1)\) -st iteration. When the iteration was done with a single array, the method was called Gauss-Seidel. In the standard Gauss-Seidel method, the array was updated row-by-row and had the form
\[\Phi_{i, j}^{(n+1)}=\frac{1}{4}\left(\Phi_{i+1, j}^{(n)}+\Phi_{i-1, j}^{(n+1)}+\Phi_{i, j+1}^{(n)}+\Phi_{i, j-1}^{(n+1)}+h^{2} f_{i, j}^{(n)}\right) . \nonumber \]
The Gauss-Seidel method had the double advantage of requiring less memory and converging more rapidly than the Jacobi method.
A variant of Gauss-Seidel that can also be found in textbooks is called red-black Gauss-Seidel. In this algorithm, the grid is viewed as a checkerboard with red and black squares. An updating of \(\Phi_{i, j}\) is done in two passes: in the first pass, \(\Phi_{i, j}\) is updated only on the red squares; in the second pass, only on the black squares. Because of the structure of the Laplacian finite difference operator, when solving \(\nabla^{2} \Phi=0\) the updated values of \(\Phi\) on the red squares depend only on the values of \(\Phi\) on the black squares, and the updated values on the black squares depend only on the red squares.
Translating FORTRAN-77 directly into MATLAB, the Gauss-Seidel method in the standard form could be implemented on a uniform grid with \(N-1\) and \(M-1\) internal grid points in the \(x\) -and \(y\) -directions, respectively, as
Nowadays, however, programming languages such as MATLAB operate on vectors, which are optimized for modern computers, and the explicit loops that used to be the workhorse of FORTRAN77 are now vectorized.
So to properly implement the Jacobi method, say, in MATLAB, one can predefine the vectors index 1 and index 2 that contain the index numbers of the first and second indices of the matrix variable phi. These definitions would be
where indexl and index 2 reference the internal grid points of the domain. The variable phi is assumed to be known on the boundaries of the domain corresponding to the indices \((1,1)\) , \((N+1,1),(1, M+1)\) , and \((N+1, M+1)\) . The Jacobi method in MATLAB can then be coded in one line as
The red-black Gauss-Seidel method could be implemented in a somewhat similar fashion. One must now predefine the vectors even 1, odd1, even 2, and odd2, with definitions
The red-black Gauss-Seidel method then requires the following four coding lines to implement:
Each iteration of the red-black Gauss-Seidel method will run slower than that of the Jacobi method, and the red-black Gauss-Seidel method will only be useful if the slower iterations are compensated by a faster convergence. In practice, however, the Jacobi method or the red-black Gauss Seidel method is replaced by the corresponding Successive Over Relaxation method (SOR method). We will illustrate this method using the Jacobi method, though the better approach is to use red-black Gauss-Seidel.
The Jacobi method is first rewritten by adding and subtracting the diagonal term \(\Phi_{i, j}^{(n)}\) :
\[\Phi_{i, j}^{(n+1)}=\Phi_{i, j}^{(n)}+\frac{1}{4}\left(\Phi_{i+1, j}^{(n)}+\Phi_{i-1, j}^{(n)}+\Phi_{i, j+1}^{(n)}+\Phi_{i, j-1}^{(n)}-4 \Phi_{i, j}^{(n)}+h^{2} f_{i, j}^{(n)}\right) \nonumber \]
In this form, we see that the Jacobi method updates the value of \(\Phi_{i, j}\) at each iteration. We can either magnify or diminish this update by introducing a relaxation parameter \(\lambda\) . We have
\[\Phi_{i, j}^{(n+1)}=\Phi_{i, j}^{(n)}+\frac{\lambda}{4}\left(\Phi_{i+1, j}^{(n)}+\Phi_{i-1, j}^{(n)}+\Phi_{i, j+1}^{(n)}+\Phi_{i, j-1}^{(n)}-4 \Phi_{i, j}^{(n)}+h^{2} f_{i, j}^{(n)}\right) \nonumber \]
which can be written more efficiently as
\[\Phi_{i, j}^{(n+1)}=(1-\lambda) \Phi_{i, j}^{(n)}+\frac{\lambda}{4}\left(\Phi_{i+1, j}^{(n)}+\Phi_{i-1, j}^{(n)}+\Phi_{i, j+1}^{(n)}+\Phi_{i, j-1}^{(n)}+h^{2} f_{i, j}^{(n)}\right) \nonumber \]
When used with Gauss-Seidel, a value of \(\lambda\) in the range \(1<\lambda<2\) can often be used to accelerate convergence of the iteration. When \(\lambda>1\) , the modifier over in Successive Over Relaxation is apt. When the right-hand-side of the Poisson equation is a nonlinear function of \(\Phi\) , however, the \(\lambda=1\) Gauss-Seidel method may fail to converge. In this case, it may be reasonable to choose a value of \(\lambda\) less than one, and perhaps a better name for the method would be Successive Under Relaxation. When under relaxing, the convergence of the iteration will of course be slowed. But this is the cost that must sometimes be paid for stability.
Newton’s method for a system of nonlinear equations
A system of nonlinear equations can be solved using a version of the iterative Newton’s Method for root finding. Although in practice, Newton’s method is often applied to a large nonlinear system, we will illustrate the method here for the simple system of two equations and two unknowns.
Suppose that we want to solve
\[f(x, y)=0, \quad g(x, y)=0, \nonumber \]
for the unknowns \(x\) and \(y\) . We want to construct two simultaneous sequences \(x_{0}, x_{1}, x_{2}, \ldots\) and \(y_{0}, y_{1}, y_{2}, \ldots\) that converge to the roots. To construct these sequences, we Taylor series expand \(f\left(x_{n+1}, y_{n+1}\right)\) and \(g\left(x_{n+1}, y_{n+1}\right)\) about the point \(\left(x_{n}, y_{n}\right) .\) Using the partial derivatives \(f_{x}=\) \(\partial f / \partial x, f_{y}=\partial f / \partial y\) , etc., the two-dimensional Taylor series expansions, displaying only the linear terms, are given by
\[\begin{array}{ll} f\left(x_{n+1}, y_{n+1}\right)=f\left(x_{n}, y_{n}\right)+\left(x_{n+1}-x_{n}\right) f_{x}\left(x_{n}, y_{n}\right) & +\left(y_{n+1}-y_{n}\right) f_{y}\left(x_{n}, y_{n}\right)+\ldots \\ g\left(x_{n+1}, y_{n+1}\right)=g\left(x_{n}, y_{n}\right)+\left(x_{n+1}-x_{n}\right) g_{x}\left(x_{n}, y_{n}\right) & +\left(y_{n+1}-y_{n}\right) g_{y}\left(x_{n}, y_{n}\right)+\ldots . \end{array} \nonumber \]
To obtain Newton’s method, we take \(f\left(x_{n+1}, y_{n+1}\right)=0, g\left(x_{n+1}, y_{n+1}\right)=0\) , and drop higherorder terms above linear. Although one can then find a system of linear equations for \(x_{n+1}\) and \(y_{n+1}\) , it is more convenient to define the variables
\[\Delta x_{n}=x_{n+1}-x_{n}, \quad \Delta y_{n}=y_{n+1}-y_{n} \nonumber \]
The iteration equations will then be given by
\[x_{n+1}=x_{n}+\Delta x_{n}, \quad y_{n+1}=y_{n}+\Delta y_{n} \nonumber \]
and the linear equations to be solved for \(\Delta x_{n}\) and \(\Delta y_{n}\) are given by
\[\left(\begin{array}{ll} f_{x} & f_{y} \\ g_{x} & g_{y} \end{array}\right)\left(\begin{array}{l} \Delta x_{n} \\ \Delta y_{n} \end{array}\right)=\left(\begin{array}{l} -f \\ -g \end{array}\right), \nonumber \]
where \(f, g, f_{x}, f_{y}, g_{x}\) , and \(g_{y}\) are all evaluated at the point \(\left(x_{n}, y_{n}\right) .\) The two-dimensional case is easily generalized to \(n\) dimensions. The matrix of partial derivatives is called the Jacobian Matrix.
We illustrate Newton’s Method by finding numerically the steady state solution of the Lorenz equations, given by
\[\begin{aligned} \sigma(y-x) &=0, \\ r x-y-x z &=0, \\ x y-b z &=0, \end{aligned} \nonumber \]
where \(x, y\) , and \(z\) are the unknown variables and \(\sigma, r\) , and \(b\) are the known parameters. Here, we have a three-dimensional homogeneous system \(f=0, g=0\) , and \(h=0\) , with
\[\begin{aligned} &f(x, y, z)=\sigma(y-x), \\ &g(x, y, z)=r x-y-x z, \\ &h(x, y, z)=x y-b z . \end{aligned} \nonumber \]
The partial derivatives can be computed to be
\[\begin{array}{lll} f_{x}=-\sigma, & f_{y}=\sigma, & f_{z}=0, \\ g_{x}=r-z, & g_{y}=-1, & g_{z}=-x, \\ h_{x}=y, & h_{y}=x, & h_{z}=-b . \end{array} \nonumber \]
The iteration equation is therefore
\[\left(\begin{array}{ccc} -\sigma & \sigma & 0 \\ r-z_{n} & -1 & -x_{n} \\ y_{n} & x_{n} & -b \end{array}\right)\left(\begin{array}{c} \Delta x_{n} \\ \Delta y_{n} \\ \Delta z_{n} \end{array}\right)=-\left(\begin{array}{c} \sigma\left(y_{n}-x_{n}\right) \\ r x_{n}-y_{n}-x_{n} z_{n} \\ x_{n} y_{n}-b z_{n} \end{array}\right) \nonumber \]
with
\[\begin{aligned} &x_{n+1}=x_{n}+\Delta x_{n}, \\ &y_{n+1}=y_{n}+\Delta y_{n}, \\ &z_{n+1}=z_{n}+\Delta z_{n} . \end{aligned} \nonumber \]
The sample MATLAB program that solves this system is in the m-file newton_system.m.