13.7: Exact Calculations and Approximations
- Understand the exact equations that are involves in complex acid-base equilibria in aqueous solutions
The methods for dealing with acid-base equilibria that we developed in the earlier units of this series are widely used in ordinary practice. Although many of these involve approximations of various kinds, the results are usually good enough for most purposes. Sometimes, however — for example, in problems involving very dilute solutions, the approximations break down, often because they ignore the small quantities of H + and OH – ions always present in pure water. In this unit, we look at exact, or "comprehensive" treatment of some of the more common kinds of acid-base equilibria problems.
Strong acid-base systems
The usual definition of a “strong” acid or base is one that is completely dissociated in aqueous solution. Hydrochloric acid is a common example of a strong acid. When HCl gas is dissolved in water, the resulting solution contains the ions H 3 O + , OH – , and Cl – , but except in very concentrated solutions, the concentration of HCl is negligible; for all practical purposes, molecules of “hydrochloric acid”, HCl, do not exist in dilute aqueous solutions. To specify the concentrations of the three species present in an aqueous solution of HCl, we need three independent relations between them.
Limiting Conditions
These relations are obtained by observing that certain conditions must always hold for aqueous solutions:
- The dissociation equilibrium of water must always be satisfied : \[[H^+][OH^–] = K_w \label{1-1}\]
- The undissociated acid and its conjugate base must be in mass balance . The actual concentrations of the acid and its conjugate base can depend on a number of factors, but their sum must be constant, and equal to the “nominal concentration”, which we designate here as \(C_a\). For a solution of HCl, this equation would be \[[HCl] + [Cl^–] = C_a \label{1-2}\] Because a strong acid is by definition completely dissociated, we can neglect the first term and write the mass balance condition as \[[Cl^–] = C_a \label{1-3}\]
- In any ionic solution, the sum of the positive and negative electric charges must be zero ; in other words, all solutions are electrically neutral. This is known as the electroneutrality principle . \[[H^+] = [OH^–] + [Cl^–] \label{1-4}\]
The next step is to combine these three limiting conditions into a single expression that relates the hydronium ion concentration to \(C_a\). This is best done by starting with an equation that relates several quantities and substituting the terms that we want to eliminate. Thus we can get rid of the \([Cl^–]\) term by substituting Equation \(\ref{1-3}\) into Equation \(\ref{1-4}\) :
\[[H^+] = [OH^–] + C_a \label{1-5}\]
The \([OH^–]\) term can be eliminated by the use of Equation \(\ref{1-1}\):
\[[H^+] = C_a + \dfrac{K_w}{[H^+]} \label{1-6}\]
This equation tells us that the hydronium ion concentration will be the same as the nominal concentration of a strong acid as long as the solution is not very dilute. As the acid concentration falls below about 10 –6 M, however, the second term predominates; \([H^+]\) approaches \(\sqrt{K_w}\) or \(10^{–7} M\) at 25 °C. The hydronium ion concentration can of course never fall below this value; no amount of dilution can make the solution alkaline!
No amount of dilution can make the solution of a strong acid alkaline!
Notice that Equation \(\ref{1-6}\) is a quadratic equation; in regular polynomial form it would be rewritten as
\[[H^+]^2 – C_a[H^+] – K_w = 0 \label{1-7}\]
Most practical problems involving strong acids are concerned with more concentrated solutions in which the second term of Equation \(\ref{1-7}\) can be dropped, yielding the simple relation
\[[H^+] \approx [A^–]\]
Activities and Concentrated Solutions of Strong Acids
In more concentrated solutions, interactions between ions cause their “effective” concentrations, known as their activities , to deviate from their “analytical” concentrations. Thus in a solution prepared by adding 0.5 mole of the very strong acid HClO 4 to sufficient water to make the volume 1 liter, freezing-point depression measurements indicate that the concentrations of hydronium and perchlorate ions are only about 0.4 M . This does not mean that the acid is only 80% dissociated; there is no evidence of HClO 4 molecules in the solution. What has happened is that about 20% of the H 3 O + and ClO 4 – ions have formed ion-pair complexes in which the oppositely-charged species are loosely bound by electrostatic forces. Similarly, in a 0.10 M solution of hydrochloric acid, the activity of H + is 0.81, or only 81% of its concentration. (See the green box below for more on this.)
Activities are important because only these work properly in equilibrium calculations. The relation between the concentration of a species and its activity is expressed by the activity coefficient \(\gamma\):
\[a = \gamma C \label{1-8}\]
As a solution becomes more dilute, \(\gamma\) approaches unity. At ionic concentrations below about 0.001 M , concentrations can generally be used in place of activities with negligible error. Recall that pH is defined as the negative logarithm of the hydrogen ion activity , not its concentration.
Activities of single ions cannot be determined, so activity coefficients in ionic solutions are always the average, or mean , of those for all ionic species present. This quantity is denoted as \(\gamma_{\pm}\).
At very high concentrations, activities can depart wildly from concentrations. This is a practical consideration when dealing with strong mineral acids which are available at concentrations of 10 M or greater. In a 12 M solution of hydrochloric acid, for example, the mean ionic activity coefficient* is 207. This means that under these conditions with [H + ] = 12, the activity {H + } = 2500, corresponding to a pH of about –3.4, instead of –1.1 as might be predicted if concentrations were being used. These very high activity coefficients also explain another phenomenon: why you can detect the odor of HCl over a concentrated hydrochloric acid solution even though this acid is supposedly "100% dissociated".
At these high concentrations, a pair of "dissociated" ions \(H^+\) and \(Cl^–\) will occasionally find themselves so close together that they may momentarily act as an HCl unit; some of these may escape as \(HC l (g)\) before thermal motions break them up again. Under these conditions, “dissociation” begins to lose its meaning so that in effect, dissociation is no longer complete. Although the concentration of \(HCl (aq)\) will always be very small, its own activity coefficient can be as great as 2000, which means that its escaping tendency from the solution is extremely high, so that the presence of even a tiny amount is very noticeable.
Weak monoprotic acids and bases
Most acids are weak; there are hundreds of thousands of them, whereas there are no more than a few dozen strong acids. We can treat weak acid solutions in exactly the same general way as we did for strong acids. The only difference is that we must now include the equilibrium expression for the acid. We will start with the simple case of the pure acid in water, and then go from there to the more general one in which strong cations are present. In this exposition, we will refer to “hydrogen ions” and \([H^+]\) for brevity, and will assume that the acid \(HA\) dissociates into \(H^+\) and its conjugate base \(A^-\).
Pure acid in water
In addition to the species H + , OH – , and A− which we had in the strong-acid case, we now have the undissociated acid HA; four variables, requiring four equations.
- Equilibria
\[ [H^+][OH^–] = K_w \label{2-1}\]
\[ K_a = \dfrac{[H^+][A^–]}{[HA]} \label{2-2}\]
- Mass balance
\[ C_a = [HA] + [A^–] \label{2-3}\]
- Charge balance
\[[H^+] = [OH^–] + [HA^–] \label{2-4}\]
To eliminate [HA] from Equation \(\ref{2-2}\), we solve Equation \(\ref{2-4}\) for this term, and substitute the resulting expression into the numerator:
\[ K_a =\dfrac{[H^+]([H^+] - [OH^-])}{C_a-([H^+] - [OH^-]) } \label{2-5}\]
The latter equation is simplified by multiplying out and replacing [H + ][OH – ] with K w . We then get rid of the [OH – ] term by replacing it with K w /[H + ]
\[[H^+] C_b + [H^+]^2 – [H^+][OH^–] = K_a C_a – K_a [H^+] + K_a [OH^–]\]
\[[H^+]^2 C_b + [H^+]^3 – [H^+] K_w = K_a C_a – K_a [H^+] + \dfrac{K_a K_w}{[H^+]}\]
Rearranged into standard polynomial form, this becomes
\[[H^+]^3 + K_a[H^+]^2 – (K_w + C_aK_a) [H^+] – K_a K_w = 0 \label{2-5a}\]
For most practical applications, we can make approximations that eliminate the need to solve a cubic equation.
Approximation 1: Neglecting Hydroxide Population
Unless the acid is extremely weak or the solution is very dilute, the concentration of OH – can be neglected in comparison to that of [H + ]. If we assume that [OH – ] ≪ [H + ], then Equation \(\ref{2-5a}\) can be simplified to
\[K_a \approx \dfrac{[H^+]^2}{C_a-[H^+]} \label{2-6}\]
which is a quadratic equation:
\[[H^+]^2 +K_a[H^+]– K_aC_a \approx 0 \label{2-7}\]
and thus, from the quadratic formula,
\[ [H^+] \approx \dfrac{K_a + \sqrt{K_a + 4K_aC_a}}{2} \label{2-8}\]
Calculate the pH of a 0.0010 M solution of acetic acid, \(K_a = 1.74 \times 10^{–5}\).
Solution
First approximation:
\[[H^+] = \sqrt{(1.0 \times 10^{–3}) × (1.74 \times 10^{–5}} = \sqrt{1.74 \times 10^{–8}} = 1.3 \times 10^{–4}\; M. \nonumber \]
Applying the "5% test",
\[\dfrac{1.3 \times 10^{–4}}{1.0 \times 10^{–3}} = 0.13\nonumber \]
This exceeds 0.05, so we must explicitly solve the quadratic Equation \(\ref{2-7}\) to obtain two roots: \(+1.2 \times 10^{–4}\) and \(–1.4 \times 10^{-4}\). Taking the positive root, we have
\[pH = –\log (1.2 \times 10^{–4}) = 3.9 \nonumber \]
Approximation 2: Very Concentrated Acids
If the acid is fairly concentrated (usually more than 10 –3 M), a further simplification can frequently be achieved by making the assumption that \([H^+] \ll C_a\). This is justified when most of the acid remains in its protonated form [HA], so that relatively little H + is produced. In this event, Equation \(\ref{2-6}\) reduces to
\[ K_a \approx \dfrac{[H^+]^2}{C_a} \label{2-9}\]
or
\[[H^+] \approx \sqrt{K_aC_a} \label{2-10}\]
Calculate the pH and percent ionization of 0.10 M acetic acid "HAc" (CH 3 COOH), \(K_a = 1.74 \times 10^{–5}\).
Solution:
It is usually best to start by using Equation \(\ref{2-9}\) as a first approximation:
\[[H^+] = \sqrt{(0.10)(1.74 \times 10^{–5})} = \sqrt{1.74 \times 10^{–6}} = 1.3 \times 10^{–3}\; M\nonumber \]
This approximation is generally considered valid if [H + ] is less than 5% of C a ; in this case, [H + ]/ C a = 0.013, which is smaller than 0.05 and thus within the limit. This same quantity also corresponds to the ionization fraction, so the percent ionization is 1.3%. The pH of the solution is
\[pH = –\log 1.2 \times 10^{-3} = 2.9\nonumber \]
Approximation 3: Very Weak and Acidic
If the acid is very weak or its concentration is very low, the \(H^+\) produced by its dissociation may be little greater than that due to the ionization of water. However, if the solution is still acidic, it may still be possible to avoid solving the cubic equation \(\ref{2-5a}\) by assuming that the term \(([H^+] - [OH^–]) \ll C_a\) in Equation \(\ref{2-5}\):
\[ K_a = \dfrac{[H^+]^2}{C_a - [H^+]} \label{2-11}\]
This can be rearranged into standard quadratic form
\[[H^+]^2 + K_a [H^+] – K_a C_a = 0 \label{2-12}\]
For dilute solutions of weak acids, an exact treatment may be required. With the aid of a computer or graphic calculator, solving a cubic polynomial is now far less formidable than it used to be. However, round-off errors can cause these computerized cubic solvers to blow up; it is generally safer to use a quadratic approximation.
Boric acid, B(OH) 3 ("H 3 BO 3 ") is a weak acid found in the ocean and in some natural waters. As with many boron compounds, there is some question about its true nature, but for most practical purposes it can be considered to be monoprotic with \(K_a = 7.3 \times 10^{–10}\):
\[Bi(OH)_3 + 2 H_2O \rightleftharpoons Bi(OH)_4^– + H_3O^+\nonumber \]
Find the [H + ] and pH of a 0.00050 M solution of boric acid in pure water.
Solution
Because this acid is quite weak and its concentration low, we will use the quadratic form Equation \(\ref{2-7}\), which yields the positive root \(6.12 \times 10^{–7}\), corresponding to pH = 6.21. Notice that this is only six times the concentration of \(H^+\) present in pure water!
It is instructive to compare this result with what the quadratic approximation would yield, which yield \([H^+] = 6.04 \times 10^{–7}\) so \(pH = 6.22\).
Weak bases
The weak bases most commonly encountered are:
- anions A – of weak acids:
\[A^– + H_2O \rightleftharpoons HA + OH^–\]
\[CO_3^{2–} + H_2O \rightleftharpoons HCO_3^– + OH^–\]
- ammonia
\[NH_3 + H_2O \rightleftharpoons NH_4^+ + OH^–\]
- amines, e.g. methylamine
\[CH_3NH_2 + H_2O \rightleftharpoons CH_3NH_3^++ H_2O\]
Solution of an anion of a weak acid
Note that, in order to maintain electroneutrality, anions must be accompanied by sufficient cations to balance their charges. Thus for a C b M solution of the salt NaA in water, we have the following conditions:
- Species
Na + , A – , HA, H 2 O, H + , OH –
- Equilibria
\[[H^+][OH^–] = K_w \label{2-13}\]
\[ K_b =\dfrac{[HA][OH^-]}{[A^-]} \label{2-14}\]
- Mass balance
\[C_b = [Na^+] = [HA] + [A^–] \label{2-15}\]
- Charge balance
\[[Na^+] + [H^+] = [OH^–] + [A^–] \label{2-16}\]
Replacing the [Na + ] term in Equation \(\ref{2-15}\) by \(C_b\) and combining with \(K_w\) and the mass balance, a relation is obtained that is analogous to that of Equation \(\ref{2-5}\) for weak acids:
\[K_b =\dfrac{[OH^-] ([OH^-] - [H^+])}{C_b - ([OH^-] - [H^+])} \label{2-17}\]
The approximations
\[ K_b \approx \dfrac{[OH^-]^2}{C_b - [OH^-]} \label{2-18}\]
and
\[[OH^–] \approx \sqrt{K_b C_b} \label{2-19}\]
can be derived in a similar manner.
Calculate the pH and the concentrations of all species in a 0.01 M solution of methylamine, CH 3 NH 2 (\(K_b = 4.2 \times 10^{–4}\)).
Solution
We begin by using the simplest approximation Equation \(\ref{2-14}\):
\[[OH^–] = \sqrt{(K_b C_b}- = \sqrt{(4.2 \times 10^{-4})(10^{–2})} = 2.1 \times 10^{–3}\nonumber \]
To see if this approximation is justified, we apply a criterion similar to what we used for a weak acid: [OH – ] must not exceed 5% of C b . In this case,
\[ \dfrac{[OH^–]}{ C_b} = \dfrac{(2.1 \times 10^{-3}} { 10^{–2}} = 0.21\nonumber \]
so we must use the quadratic form Equation \(\ref{2-12}\) that yields the positive root \(1.9 \times 10^{–3}\) which corresponds to \([OH^–]\)
\[[H^+] = \dfrac{K_w}{[OH^–} = \dfrac{1 \times 10^{-14}}{1.9 \times 10^{–3}} = 5.3 \times 10^{-12}\nonumber \]
and
\[pH = –\log 5.3 \times 10^{–12} = 11.3.\nonumber \]
From the charge balance equation, solve for
\[[CH_3NH_2] = [OH^–] – [H^+] \approx [OH^–] = 5.3 \times 10^{–12}\; M. \nonumber \]
For the concentration of the acid form (methylaminium ion CH 3 NH 3 + ), use the mass balance equation:
\[[CH_3NH_3^+] = C_b – [CH_3NH_2] = 0.01 – 0.0019 =0.0081\; M.\nonumber \]
Mixtures of Acids
Many practical problems relating to environmental and physiological chemistry involve solutions containing more than one acid. In this section, we will restrict ourselves to a much simpler case of two acids, with a view toward showing the general method of approaching such problems by starting with charge- and mass-balance equations and making simplifying assumptions when justified. In general, the hydrogen ions produced by the stronger acid will tend to suppress dissociation of the weaker one, and both will tend to suppress the dissociation of water, thus reducing the sources of H + that must be dealt with.
Consider a mixture of two weak acids HX and HY; their respective nominal concentrations and equilibrium constants are denoted by C x , C y , K x and K y ,
Starting with the charge balance expression
\[ [H^+] = [X^–] + [Y^–] + [OH^–] \label{3-1}\]
We use the equilibrium constants to replace the conjugate base concentrations with expressions of the form
\[ [X^-] = K_x \dfrac{[HX]}{[H^+]} \label{3-2}\]
to yield
\[ [H^+] = \dfrac{[HX]}{K_x} + \dfrac{[HY]}{K_y} + K_w \label{3-3}\]
If neither acid is very strong or very dilute, we can replace equilibrium concentrations with nominal concentrations:
\[ [H^+] \approx \sqrt{C_cK_x + C_yK_y K_w} \label{3-4}\]
Estimate the pH of a solution that is 0.10M in acetic acid (\(K_a = 1.8 \times 10^{–5}\)) and 0.01M in formic acid (\(K_a = 1.7 \times 10^{–4}\)).
Solution
Because K w is negligible compared to the C a K a products, we can simplify \Equation \(ref{3-4}\):
\[[H^+] = \sqrt{1.8 \times 10^{–6} + 1.7 \times 10^{-6}} = 0.0019\nonumber \]
Which corresponds to a pH of \(–\log 0.0019 = 2.7\)
Note that the pH of each acid separately at its specified concentration would be around 2.8. However, if 0.001 M chloroacetic acid ( K a = 0.0014) is used in place of formic acid, the above expression becomes
\[ [H^+] \approx \sqrt{ 1.4 \times 10^{-6} + 1.75 \times 10^{-14}} = 0.00188 \label{3-5}\]
which exceeds the concentration of the stronger acid; because the acetic acid makes a negligible contribution to [H + ] here, the simple approximation given above \Equation \(\ref{3-3}\) is clearly invalid. We now use the mass balance expression for the stronger acid
\[[HX] + [X^–] = C_x \label{3-6}\]
to solve for [X – ] which is combined with the equilibrium constant K x to yield
\[[X^-] = C_x - \dfrac{[H^+][X^]}{K_x} \label{3-7}\]
Solving this for [X – ] gives
\[ [X^-] = \dfrac{C_xK_x}{K_x + [H^+]} \label{3-8}\]
The approximation for the weaker acetic acid (HY) is still valid, so we retain it in the substituted electronegativity expression:
\[ [H^+] \dfrac{C_xK_x}{K_x+[H^+]} + \dfrac{C_yK_y}{[H^+]} \label{3-9}\]
which is a cubic equation that can be solved by approximation.
Several methods have been published for calculating the hydrogen ion concentration in solutions containing an arbitrary number of acids and bases. These generally involve iterative calculations carried out by a computer. See, for example, J. Chem. Education 67(6) 501-503 (1990) and 67(12) 1036-1037 (1990).
Equilibria of Polyprotic Acids
Owing to the large number of species involved, exact solutions of problems involving polyprotic acids can become very complicated. Thus for phosphoric acid H 3 PO 4 , the three "dissociation" steps yield three conjugate bases:
Fortunately, it is usually possible to make simplifying assumptions in most practical applications. In the section that follows, we will show how this is done for the less-complicated case of a diprotic acid. A diprotic acid HA can donate its protons in two steps, yielding first a monoprotonated species HA – and then the completely deprotonated form A 2– .
Since there are five unknowns (the concentrations of the acid, of the two conjugate bases and of H + and OH – ), we need five equations to define the relations between these quantities. These are
- Equilibria
\[[H^+][OH^–] = K_w \label{4-1}\]
\[ K_1 = \dfrac{[H^+][HA^-]}{[H_2A]} \label{4-2}\]
\[ K_1 = \dfrac{[H^+][HA^{2-}]}{[HA^-]} \label{4-3}\]
- Mass balance
\[C_a = [H_2A] + [HA^–] + [A^{2–}] \label{4-4}\]
- Charge balance
\[[H^+] = [OH^–] + [HA^–] + 2 [A^{2–}] \label{4-5}\]
(It takes 2 moles of \(H^+\) to balance the charge of 1 mole of \(A^{2–}\))
Solving these five equations simultaneously for \(K_1\) yields the rather intimidating expression
\[ K_1 = \dfrac{[H^+] \left( [H^+] - [OH^-] \dfrac{2K_2[H^+] - [OH^-]}{[H^+ + 2K_2} \right)}{C_a - \left( [H^+] - [OH^-] \dfrac{K_2 [H^+] -[OH^-]}{[H^+] + 2K_2} \right)} \label{4-6}\]
which is of little practical use except insofar as it provides the starting point for various simplifying approximations.
If the solution is even slightly acidic, then ([H + ] – [OH – ]) ≈ [H + ] and
\[ K_1 = \dfrac{[H^+] \left( [H^+] \dfrac{2K_2[H^+]}{[H^+ + 2K_2} \right)}{C_a - \left( [H^+] \dfrac{K_2 [H^+]}{[H^+] + 2K_2} \right)} \label{4-7}\]
For any of the common diprotic acids, \(K_2\) is much smaller than \(K_1\). If the solution is sufficiently acidic that \(K_2 \ll [H^+]\), then a further simplification can be made that removes \(K_2\) from Equation \(\ref{4-7}\); this is the starting point for most practical calculations.
\[ K_1 \approx \dfrac{[H^+]^2}{C_a-[H^+]} \label{4-8}\]
Finally, if the solution is sufficiently concentrated and \(K_1\) sufficiently small so that \([H^+] \ll C_a\), then Equation \(\ref{4-8}\) reduces to:
\[ [H^+] \approx \sqrt{K_a C_a}\]
Acid with conjugate base: Buffer solutions
Solutions containing a weak acid together with a salt of the acid are collectively known as buffers . When they are employed to control the pH of a solution (such as in a microbial growth medium), a sodium or potassium salt is commonly used and the concentrations are usually high enough for the Henderson-Hasselbalch equation to yield adequate results.
Exact solution
In this section, we will develop an exact analytical treatment of weak acid-salt solutions, and show how the H – H equation arises as an approximation. A typical buffer system is formed by adding a quantity of strong base such as sodium hydroxide to a solution of a weak acid HA. Alternatively, the same system can be made by combining appropriate amounts of a weak acid and its salt NaA. A system of this kind can be treated in much the same way as a weak acid, but now with the parameter C b in addition to C a .
- Species
Na + , A – , HA, H 2 O, H + , OH –
- Equilibria
\[[H^+][OH^–] = K_w \label{5-1}\]
\[K_a = \dfrac{[H^+][A^-]}{[HA]} \label{5-2}\]
- Mass balance
\[C_a + C-b = [HA] + [A^–] \label{5-3}\]
\[C_b = [Na^+] \label{5-4}\]
- Charge balance
\[[Na^+] + [H^+] = [OH^–] + [A^–] \label{5-5}\]
Substituting Equation \(\ref{5-4}\) into Equation \(\ref{5-5}\) yields an expression for [A – ]:
\[[A^–] = C_b + [H^+] – [OH^–] \label{5-6}\]
Inserting this into Equation \(\ref{5-3}\) and solving for [HA] yields
\[[HA] = C_b + [H^+] – [OH^–] \label{5-7)}\]
Finally, we substitute these last two expressions into the equilibrium constant (Equation \(\ref{5-2}\)):
\[ [H^+] = K_a \dfrac{C_a - [H^+] + [OH^-]}{C_b + [H^+] - [OH^-]} \label{5-8}\]
which becomes cubic in [H + ] when [OH – ] is replaced by ( K w / [H + ]).
\[[H^+]^3 +(C_b +K_a)[H^+]^2 – (K_w + C_aK_a) [H^+] – K_aK_w = 0 \label{5-8a}\]
Approximations
In almost all practical cases it is possible to make simplifying assumptions. Thus if the solution is known to be acidic or alkaline, then the [OH – ] or [H + ] terms in Equation \(\ref{5-8}\) can be neglected. In acidic solutions, for example, Equation \(\ref{5-8}\) becomes
\[ [H^+] = K_a \dfrac{C_a - [H^+]}{C_b + [H^+]} \label{5-9}\]
which can be rearranged into a quadratic in standard polynomial form:
\[ [H^+]^2 + (C_b + C_a) [H^+] – K_a C_a = 0 \label{5-10}\]
If the concentrations C a and C b are sufficiently large, it may be possible to neglect the [H + ] terms entirely, leading to the commonly-seen Henderson-Hasselbalch Approximation .
\[ \color{red} [H^+] \approx K_a \dfrac{C_a}{C_b} \label{5-11}\]
It's important to bear in mind that the Henderson-Hasselbalch Approximation is an "approximation of an approximation" that is generally valid only for combinations of K a and concentrations that fall within the colored portion of this plot.
Most buffer solutions tend to be fairly concentrated, with C a and C b typically around 0.01 - 0.1 M . For more dilute buffers and larger K a 's that bring you near the boundary of the colored area, it is safer to start with Equation \(\ref{5-9}\).
Chlorous acid HClO 2 has a pK a of 1.94. Calculate the pH of a solution made by adding 0.01 M/L of sodium hydroxide to a -.02 M/L solution of chloric acid.
Solution
In the resulting solution, C a = C b = 0.01 M . On the plots shown above, the intersection of the log C a = –2 line with the plot for pK a = 2 falls near the left boundary of the colored area, so we will use the quadratic form \(\ref{5-10}\).
Substitution in Equation \(\ref{5-10}\) yields
\[H^+ + 0.02 H^+ – (10^{–1.9} x 10^{–2}) = 0 \nonumber\]
which yields a positive root 0.0047 = [H + ] that corresponds to pH = 2.3 .
Note: Using the Henderson-Hassalbach Approximateion (Equation \(\ref{5-11}\)) would give pH = pK a = 1.9.


3.5: Addition of Velocities
Learning Objectives
By the end of this section, you will be able to:
- Apply principles of vector addition to determine relative velocity.
- Explain the significance of the observer in the measurement of velocity.
Relative Velocity
If a person rows a boat across a rapidly flowing river and tries to head directly for the other shore, the boat instead moves diagonallyrelative to the shore, as in Figure \(\displaystyle \PageIndex{1}\). The boat does not move in the direction in which it is pointed. The reason, of course, is that the river carries the boat downstream. Similarly, if a small airplane flies overhead in a strong crosswind, you can sometimes see that the plane is not moving in the direction in which it is pointed, as illustrated in Figure \(\displaystyle \PageIndex{2}\). The plane is moving straight ahead relative to the air, but the movement of the air mass relative to the ground carries it sideways.
In each of these situations, an object has a velocity relative to a medium (such as a river) and that medium has a velocity relative to an observer on solid ground. The velocity of the object relative to the observer is the sum of these velocity vectors, as indicated in Figures \(\displaystyle \PageIndex{1}\) and \(\displaystyle \PageIndex{2}\). These situations are only two of many in which it is useful to add velocities. In this module, we first re-examine how to add velocities and then consider certain aspects of what relative velocity means.
How do we add velocities? Velocity is a vector (it has both magnitude and direction); the rules of vector addition discussed inVector Addition and Subtraction: Graphical Methods and Vector Addition and Subtraction: Analytical Methods apply to the addition of velocities, just as they do for any other vectors. In one-dimensional motion, the addition of velocities is simple—they add like ordinary numbers. For example, if a field hockey player is moving at 5 m/s straight toward the goal and drives the ball in the same direction with a velocity of 30 m/s relative to her body, then the velocity of the ball is 35 m/s relative to the stationary, profusely sweating goalkeeper standing in front of the goal.
In two-dimensional motion, either graphical or analytical techniques can be used to add velocities. We will concentrate on analytical techniques. The following equations give the relationships between the magnitude and direction of velocity (\(\displaystyle v\) and \(\displaystyle θ\) ) and its components (\(\displaystyle v_x\) and \(\displaystyle v_y\) ) along the x- and y-axes of an appropriately chosen coordinate system:
\[v_x=v\cos θ\]
\[v_y=v\sin θ\]
\[ v=\sqrt{v^2x+v^2} \]
\[ θ=\tan^{−1}\left(\dfrac{v_y}{v_x}\right).\]
These equations are valid for any vectors and are adapted specifically for velocity. The first two equations are used to find the components of a velocity when its magnitude and direction are known. The last two are used to find the magnitude and direction of velocity when its components are known.
TAKE-HOME EXPERIMENT: RELATIVE VELOCITY OF A BOAT
Fill a bathtub half-full of water. Take a toy boat or some other object that floats in water. Unplug the drain so water starts to drain. Try pushing the boat from one side of the tub to the other and perpendicular to the flow of water. Which way do you need to push the boat so that it ends up immediately opposite? Compare the directions of the flow of water, heading of the boat, and actual velocity of the boat.
Example \(\displaystyle \PageIndex{1}\): Adding Velocities - A Boat on a River
Refer to Figure \(\displaystyle \PageIndex{4}\), which shows a boat trying to go straight across the river. Let us calculate the magnitude and direction of the boat’s velocity relative to an observer on the shore, \(\displaystyle v_{tot}\). The velocity of the boat, vboat , is 0.75 m/s in the y -direction relative to the river and the velocity of the river, \(\displaystyle v_{river}\), is 1.20 m/s to the right.
Strategy
We start by choosing a coordinate system with its \(\displaystyle x\)-axis parallel to the velocity of the river, as shown in Figure. Because the boat is directed straight toward the other shore, its velocity relative to the water is parallel to the y-axis and perpendicular to the velocity of the river. Thus, we can add the two velocities by using the equations \(\displaystyle v_{tot}=\sqrt{v^2_x+v^2_y}\) and \(\displaystyle θ=tan^{−1}(v_y/v_x)\) directly.
Solution
The magnitude of the total velocity is
\(\displaystyle v_{tot}=\sqrt{v^2_x+v^2_y}\),
where
\(\displaystyle v_x=v_{river}=1.20 m/s\)
and
\(\displaystyle v_y=v_{boat}=0.750 m/s.\)
Thus,
\(\displaystyle v_{tot}=\sqrt{(1.20 m/s)^2+(0.750 m/s)^2}\)
yielding
\(\displaystyle v_{tot}=1.42 m/s.\)
The direction of the total velocity \(\displaystyle θ\) is given by:
\(\displaystyle θ=tan^{−1}(v_y/v_x)=tan^{−1}(0.750/1.20).\)
This equation gives
\(\displaystyle θ=32.0º.\)
Discussion
Both the magnitude v and the direction \(\displaystyle θ\) of the total velocity are consistent with Figure. Note that because the velocity of the river is large compared with the velocity of the boat, it is swept rapidly downstream. This result is evidenced by the small angle (only 32.0º ) the total velocity has relative to the riverbank.
Example \(\displaystyle \PageIndex{2}\): Calculating Velocity - Wind Velocity Causes an Airplane to Drift
Calculate the wind velocity for the situation shown in Figure \(\displaystyle \PageIndex{5}\). The plane is known to be moving at 45.0 m/s due north relative to the air mass, while its velocity relative to the ground (its total velocity) is 38.0 m/s in a direction 20.0º west of north.
Strategy
In this problem, somewhat different from the previous example, we know the total velocity \(\displaystyle v_{to}\)t and that it is the sum of two other velocities, \(\displaystyle v_w\) (the wind) and \(\displaystyle v_p\) (the plane relative to the air mass). The quantity \(\displaystyle v_p\) is known, and we are asked to find vw. None of the velocities are perpendicular, but it is possible to find their components along a common set of perpendicular axes. If we can find the components of \(\displaystyle v_w\), then we can combine them to solve for its magnitude and direction. As shown in Figure, we choose a coordinate system with its x- axis due east and its y- axis due north (parallel to \(\displaystyle v_p\)). (You may wish to look back at the discussion of the addition of vectors using perpendicular components in Vector Addition and Subtraction: Analytical Methods.)
Solution
Because \(\displaystyle v_{tot}\) is the vector sum of the \(\displaystyle v_w\) and \(\displaystyle v_p\), its x- and y- components are the sums of the x- and y- components of the wind and plane velocities. Note that the plane only has vertical component of velocity so \(\displaystyle v_{px}=0\) and \(\displaystyle v_{py}=v_p\). That is,
\(\displaystyle v_{tot}x=v_{wx}\)
and
\(\displaystyle v_{toty}=v_{wy}+v_p\).
We can use the first of these two equations to find \(\displaystyle v_{wx}\):
\(\displaystyle v_{wx}=v_{totx}=v_{tot}cos 110º.\)
Because \(\displaystyle v_{tot}=38.0 m/s\) and \(\displaystyle cos 110º=–0.342\) we have
\(\displaystyle v_{wx}=(38.0 m/s)(–0.342)=–13 m/s.\)
The minus sign indicates motion west which is consistent with the diagram.
Now, to find \(\displaystyle v_{wy}\) we note that
\(\displaystyle v_{toty}=v_{wy}+v_p\)
Here \(\displaystyle v_{toty}=v_{tot}sin 110º\); thus,
\(\displaystyle v_{wy}=(38.0 m/s)(0.940)−45.0 m/s=−9.29 m/s.\)
This minus sign indicates motion south which is consistent with the diagram.
Now that the perpendicular components of the wind velocity \(\displaystyle v_{wx}\) and \(\displaystyle v_{wy}\) are known, we can find the magnitude and direction of \(\displaystyle v_w\). First, the magnitude is
\(\displaystyle v_w=\sqrt{v^2_{wx}+v^2_{wy}}=\sqrt{(−13.0 m/s)^2+(−9.29 m/s)^2}\)
so that
\(\displaystyle v_w=16.0 m/s.\)
The direction is:
\(\displaystyle θ=tan^{−1}(v_{wy}/v_{wx})=tan^{−1}(−9.29/−13.0)\)
giving
\(\displaystyle θ=35.6º.\)
Discussion
The wind’s speed and direction are consistent with the significant effect the wind has on the total velocity of the plane, as seen in Figure. Because the plane is fighting a strong combination of crosswind and head-wind, it ends up with a total velocity significantly less than its velocity relative to the air mass as well as heading in a different direction.
Note that in both of the last two examples, we were able to make the mathematics easier by choosing a coordinate system with one axis parallel to one of the velocities. We will repeatedly find that choosing an appropriate coordinate system makes problem solving easier. For example, in projectile motion we always use a coordinate system with one axis parallel to gravity.
Relative Velocities and Classical Relativity
When adding velocities, we have been careful to specify that the velocity is relative to some reference frame . These velocities are called relative velocities . For example, the velocity of an airplane relative to an air mass is different from its velocity relative to the ground. Both are quite different from the velocity of an airplane relative to its passengers (which should be close to zero). Relative velocities are one aspect of relativity , which is defined to be the study of how different observers moving relative to each other measure the same phenomenon.
Nearly everyone has heard of relativity and immediately associates it with Albert Einstein (1879–1955), the greatest physicist of the 20th century. Einstein revolutionized our view of nature with his modern theory of relativity, which we shall study in later chapters. The relative velocities in this section are actually aspects of classical relativity, first discussed correctly by Galileo and Isaac Newton. Classical relativity is limited to situations where speeds are less than about 1% of the speed of light—that is, less than . Most things we encounter in daily life move slower than this speed.
Let us consider an example of what two different observers see in a situation analyzed long ago by Galileo. Suppose a sailor at the top of a mast on a moving ship drops his binoculars. Where will it hit the deck? Will it hit at the base of the mast, or will it hit behind the mast because the ship is moving forward? The answer is that if air resistance is negligible, the binoculars will hit at the base of the mast at a point directly below its point of release. Now let us consider what two different observers see when the binoculars drop. One observer is on the ship and the other on shore. The binoculars have no horizontal velocity relative to the observer on the ship, and so he sees them fall straight down the mast. (Figure \(\displaystyle \PageIndex{1}\); blue curve) To the observer on shore, the binoculars and the ship have the same horizontal velocity, so both move the same distance forward while the binoculars are falling. This observer sees the red curved path shown in Figure \(\displaystyle \PageIndex{6}\). Although the paths look different to the different observers, each sees the same result—the binoculars hit at the base of the mast and not behind it. To get the correct description, it is crucial to correctly specify the velocities relative to the observer.
Example \(\displaystyle \PageIndex{3}\): Calculating Relative Velocity: An Airline Passenger Drops a Coin
An airline passenger drops a coin while the plane is moving at 260 m/s. What is the velocity of the coin when it strikes the floor 1.50 m below its point of release: (a) Measured relative to the plane? (b) Measured relative to the Earth?
Strategy
Both problems can be solved with the techniques for falling objects and projectiles. In part (a), the initial velocity of the coin is zero relative to the plane, so the motion is that of a falling object (one-dimensional). In part (b), the initial velocity is 260 m/s horizontal relative to the Earth and gravity is vertical, so this motion is a projectile motion. In both parts, it is best to use a coordinate system with vertical and horizontal axes.
Solution for (a)
Using the given information, we note that the initial velocity and position are zero, and the final position is 1.50 m. The final velocity can be found using the equation:
\(\displaystyle v_y^2=v_{0y}^2−2g(y−y_0).\)
Substituting known values into the equation, we get
\(\displaystyle v_y^2=0^2−2(9.80m/s^2)(−1.50m−0 m)=29.4m^2/s^2\)
yielding
\(\displaystyle v_y=−5.42 m/s.\)
We know that the square root of 29.4 has two roots: 5.42 and -5.42. We choose the negative root because we know that the velocity is directed downwards, and we have defined the positive direction to be upwards. There is no initial horizontal velocity relative to the plane and no horizontal acceleration, and so the motion is straight down relative to the plane.
Solution for (b)
Because the initial vertical velocity is zero relative to the ground and vertical motion is independent of horizontal motion, the final vertical velocity for the coin relative to the ground is \(\displaystyle v_y=−5.42m/s\), the same as found in part (a). In contrast to part (a), there now is a horizontal component of the velocity. However, since there is no horizontal acceleration, the initial and final horizontal velocities are the same and \(\displaystyle v_x=260 m/s\). The x- and y- components of velocity can be combined to find the magnitude of the final velocity:
\(\displaystyle v=\sqrt{v_x^2+v_y^2}\).
Thus,
\(\displaystyle v=\sqrt{(260 m/s)^2+(−5.42 m/s)^2}\)
yielding
\(\displaystyle v=260.06 m/s.\)
The direction is given by:
\(\displaystyle θ=tan^{−1}(v_y/v_x)=tan^{−1}(−5.42/260)\)
so that
\(\displaystyle θ=tan^{−1}(−0.0208)=−1.19º.\)
Discussion
In part (a), the final velocity relative to the plane is the same as it would be if the coin were dropped from rest on the Earth and fell 1.50 m. This result fits our experience; objects in a plane fall the same way when the plane is flying horizontally as when it is at rest on the ground. This result is also true in moving cars. In part (b), an observer on the ground sees a much different motion for the coin. The plane is moving so fast horizontally to begin with that its final velocity is barely greater than the initial velocity. Once again, we see that in two dimensions, vectors do not add like ordinary numbers—the final velocity v in part (b) is not \(\displaystyle (260 – 5.42) m/s\); rather, it is \(\displaystyle 260.06 m/s.\) The velocity’s magnitude had to be calculated to five digits to see any difference from that of the airplane. The motions as seen by different observers (one in the plane and one on the ground) in this example are analogous to those discussed for the binoculars dropped from the mast of a moving ship, except that the velocity of the plane is much larger, so that the two observers see very different paths. (See Figure.) In addition, both observers see the coin fall 1.50 m vertically, but the one on the ground also sees it move forward 144 m (this calculation is left for the reader). Thus, one observer sees a vertical path, the other a nearly horizontal path.
MAKING CONNECTIONS: RELATIVITY AND EINSTEIN
Because Einstein was able to clearly define how measurements are made (some involve light) and because the speed of light is the same for all observers, the outcomes are spectacularly unexpected. Time varies with observer, energy is stored as increased mass, and more surprises await.
PHET EXPLORATIONS: MOTION IN 2D
Try the new "Ladybug Motion 2D" simulation for the latest updated version. Learn about position, velocity, and acceleration vectors. Move the ball with the mouse or let the simulation move the ball in four types of motion (2 types of linear, simple harmonic, circle).
Summary
- Velocities in two dimensions are added using the same analytical vector techniques, which are rewritten as
\(\displaystyle v_x=vcosθ\)
\(\displaystyle v_y=vsinθ\)
\(\displaystyle v=\sqrt{v^2_x+v^2+y}\)
\(\displaystyle θ=tan^{−1}(v_y/v_x)\).
- Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies dramatically with reference frame.
- Relativity is the study of how different observers measure the same phenomenon, particularly when the observers move relative to one another. Classical relativity is limited to situations where speed is less than about 1% of the speed of light (3000 km/s).
Glossary
- classical relativity
- the study of relative velocities in situations where speeds are less than about 1% of the speed of light—that is, less than 3000 km/s
- relative velocity
- the velocity of an object as observed from a particular reference frame
- relativity
- the study of how different observers moving relative to each other measure the same phenomenon
- velocity
- speed in a given direction
- vector addition
- the rules that apply to adding vectors together


7.3E: Vitamin B₆ (Pyridoxine)
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Vitamin B 6 refers to a group of chemically similar compounds which can be interconverted in biological systems. Vitamin B 6 is part of the vitamin B group of essential nutrients. Its active form, pyridoxal 5′-phosphate, serves as a coenzyme in some 100 enzyme reactions in amino acid, glucose, and lipid metabolism. Several forms
- Pyridoxine 5′-phosphate (P5P)
- Pyridoxal (PL)
- Pyridoxal 5′-phosphate (PLP), the metabolically active form (sold as P-5-P vitamin supplement)
- Pyridoxamine (PM)
- Pyridoxamine 5′-phosphate (PMP)
- 4-Pyridoxic acid (PA), the catabolite which is excreted in urine
- Pyritinol, a semi-synthetic derivative of pyridoxine, where two pyridoxine moieties are bound by a disulfide bridge.
All forms except pyridoxic acid and pyritinol can be interconverted. Absorbed pyridoxamine is converted to PMP by pyridoxal kinase, which is further converted to PLP by pyridoxamine-phosphate transaminase or pyridoxine 5′-phosphate oxidase which also catalyzes the conversion of PNP to PLP. Pyridoxine 5′-phosphate oxidase is dependent on flavin mononucleotide (FMN) as a cofactor which is produced from riboflavin (vitamin B 2 ) i.e. in this biochemical pathway, dietary vitamin B 6 cannot be used without vitamin B 2 .
Pyridoxine and Pyridoxamine
PLP, the metabolically active form of vitamin B6, is involved in many aspects of mcronutrient metabolism, neurotransmitter synthesis, histamine synthesis, hemoglobin synthesis and function, and gene expression. PLP generally serves as a coenzyme (cofactor) for many reactions including decarboxylation, transamination, racemization, elimination, replacement, and beta-group interconversion. The liver is the site for vitamin B6 metabolism.Amino acid metabolism PLP is a cofactor in the biosynthesis of five important neurotransmitters: serotonin, dopamine, epinephrine, norepinephrine, and gamma-aminobutyric acid (GABA). PLP is also involved in the synthesis of histamine.
Dietary Sources
Vitamin B 6 is widely distributed in foods in both its free and bound forms. Cooking, storage, and processing losses of vitamin B 6 vary and in some foods may be more than 50%, depending on the form of vitamin present in the food. Plant foods lose the least during processing, as they contain mostly pyridoxine, which is far more stable than the pyridoxal or pyridoxamine found in animal foods. For example, milk can lose 30–70% of its vitamin B 6 content when dried. Vitamin B 6 is found in the germ and aleurone layer of grains, and milling results in the reduction of this vitamin in white flour. The heating that occurs before most freezing and canning processes are other methods that may result in the loss of vitamin B 6 in foods.
Foods that contain large amounts of vitamin B6 include:
- fortified breakfast cereals
- pork
- turkey
- beef
- bananas
- chickpeas
- potatoes
- pistachios
Dietary Reference Intake
The Food and Nutrition Board (FNB) of the U.S. Institute of Medicine updated Estimated Average Requirements (EARs) and Recommended Dietary Allowances (RDAs) for vitamin B 6 in 1998. The current EARs for vitamin B 6 for women and men ages 14 and up increase with age from 1.0 to 1.3 mg/day and from 1.1 to 1.4 mg/day, respectively; the RDAs increase with age from 1.2 to 1.5 and from 1.3 to 1.7 mg/day, respectively. RDAs are higher than EARs so as to identify amounts that will cover people with higher than average requirements. RDA for pregnancy equals 1.9 mg/day. RDA for lactation equals 2.0 mg/day. For infants up to 12 months the Adequate Intake (AI) is 0.1–0.3 mg/day. and for children ages 1–13 years the RDA increases with age from 0.5 to 1.0 mg/day. As for safety, the FNB sets Tolerable Upper Intake Levels (known as ULs) for vitamins and minerals when evidence is sufficient. In the case of vitamin B 6 the UL is set at 100 mg/day. The European Food Safety Authority reviewed the same safety question and set its UL at 25 mg/day. Safety issues are presented at length in the Toxicity section.
For U.S. food and dietary supplement labeling purposes the amount in a serving is expressed as a percent of daily value (%DV). For vitamin B 6 labeling purposes 100% of the Daily Value was 2.0 mg, but as of May 2016 it has been revised to 1.7 mg. Food and supplement companies have until July 28, 2018 to comply with the change.
Absorption and excretion
Vitamin B 6 is absorbed in the jejunum and ileum by passive diffusion. With the capacity for absorption being so great, animals are able to absorb quantities much greater than necessary for physiological demands. The absorption of pyridoxal phosphate and pyridoxamine phosphate involves their dephosphorylation catalyzed by a membrane-bound alkaline phosphatase. Those products and nonphosphorylated forms in the digestive tract are absorbed by diffusion, which is driven by trapping of the vitamin as 5′-phosphates through the action of phosphorylation (by a pyridoxal kinase) in the jejunal mucosa. The trapped pyridoxine and pyridoxamine are oxidized to pyridoxal phosphate in the tissue.
The products of vitamin B 6 metabolism are excreted in the urine, the major product of which is 4-pyridoxic acid. An estimated 40–60% of ingested vitamin B 6 is oxidized to 4-pyridoxic acid. Several studies have shown that 4-pyridoxic acid is undetectable in the urine of vitamin B 6 -deficient subjects, making it a useful clinical marker to assess the vitamin B 6 status of an individual. Other products of vitamin B 6 metabolism excreted in the urine when high doses of the vitamin have been given include pyridoxal, pyridoxamine, and pyridoxine and their phosphates. A small amount of vitamin B 6 is also excreted in the feces.
Deficiencies
The classic clinical syndrome for vitamin B6 deficiency is a seborrhoeic dermatitis-like eruption, atrophic glossitis with ulceration, angular cheilitis, conjunctivitis, intertrigo, and neurologic symptoms of somnolence, confusion, and neuropathy (due to impaired sphingosine synthesis) and sideroblastic anemia (due to impaired heme synthesis).
Less severe cases present with metabolic disease associated with insufficient activities of the coenzyme PLP. The most prominent of the lesions is due to impaired tryptophan–niacin conversion. This can be detected based on urinary excretion of xanthurenic acid after an oral tryptophan load. Vitamin B6 deficiency can also result in impaired transsulfuration of methionine to cysteine. The PLP-dependent transaminases and glycogen phosphorylase provide the vitamin with its role in gluconeogenesis, so deprivation of vitamin B6 results in impaired glucose tolerance.
A deficiency of vitamin B6 alone is relatively uncommon and often occurs in association with other vitamins of the B complex. The elderly and alcoholics have an increased risk of vitamin B6 deficiency, as well as other micronutrient deficiencies. Evidence exists for decreased levels of vitamin B6 in women with type 1 diabetes and in patients with systemic inflammation, liver disease, rheumatoid arthritis, and those infected with HIV.
Contributors and Attributions
- Wikipedia. Content is copyrighted under a CC-BY-SA 4.0 license.


12.1: Introduction to Contraception, Abortion, and Sexually Transmitted Infections
After completing this module, students should have a working knowledge of the following:
- The various methods of birth control
- Reproductive rights and Abortion
- Sexually Transmitted Disease and Prevention
Introduction
From the dawn of time, people have sought sexual gratification without the desire to procreate. One of the first forms of birth control on record is a condom made of pig intestine which dates back to 1640. In this chapter we will explore the various methods of birth control also known as Contraception. We will understand abortion and the legal battle that continues to challenge the right to choose almost 50 years since Roe v. Wade. We will look at some common Sexually Transmitted Infections (STIs aka STDs) and how to prevent and treat them. We will also explore how stigma plays a role in how we treat issues surrounding contraception, abortion and STIs.


9.13: The ADDIE Model
- Describe the ADDIE Model of instructional design
ADDIE is the classic model of instructional design that is used for developing educational and training programs and instructional materials. ADDIE stands for Analysis, Design, Development, Implementation and Evaluation, the five steps in the design process, detailed below: [1]
Analysis: Identify the performance gap
The analysis phase involves identifying and clarifying the instructional problem or, from a training standpoint, identifying the performance gap and desired outcomes. This phase includes identifying participant characteristics (for example, current knowledge and skills, level of experience, language proficiency and motivation), learning resources and budget and time constraints, defining the learning environment and establishing instructional goals and objectives.
Design: Identify the learning objectives
The design phase involves subject matter design broadly, including defining specific learning objectives and instructional strategies, structuring content and assessments. The design should reflect a logical flow. Assessments should provide feedback on the learner’s progress in achieving the learning objectives.
Development: Develop a performance solution
As the title indicates, the development phase involves creating/curating and assembling the content specified in the design phase. This phase also involves stakeholder review and validation and any required revisions. This phase may involve integration of technology and related testing.
Implementation: Deliver the performance solution
Implementation involves development of the training framework, including course curriculum, learning outcomes and the learning space. The process should also include confirming the availability of required materials and associated applications or websites and preparing learners to use any required tools or technology. The final step, of course, is participant engagement.
Evaluation: Evaluate the results relative to the performance objectives.
Although evaluation is listed last, in practice it is included in every aspect of the process. That is, the overall design process is meant to be iterative, with elements fine-tuned along the way. Interim evaluations, referred to as formative evaluations, are those that are conducted prior to implementation to confirm that the learning resources meet the specifications established in the design phase. A summative evaluation would be conducted after implementation to determine training effectiveness on three bases: participant satisfaction, participant learning and participant performance.
-
Branch, R. M. "
Instructional Design: The ADDIE approach
." Springer. 2009. Accessed July 18, 2019. ↵
Contributors and Attributions
- Image: The ADDIE Model. Provided by : Lumen Learning. License : CC BY: Attribution


10.2: Stability and Classiﬁcation of Isolated Critical Points
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Isolated Critical Points and Almost Linear Systems
A critical point is isolated if it is the only critical point in some small "neighborhood" of the point. That is, if we zoom in far enough it is the only critical point we see. In the above example, the critical point was isolated. If on the other hand there would be a whole curve of critical points, then it would not be isolated.
A system is called almost linear (at a critical point \((x_0,y_0)\)) if the critical point is isolated and the Jacobian at the point is invertible, or equivalently if the linearized system has an isolated critical point. In such a case, the nonlinear terms will be very small and the system will behave like its linearization, at least if we are close to the critical point.
In particular the system we have just seen in Examples 8.1.1 and 8.1.2 has two isolated critical points \((0,0)\) and \((0,1)\), and is almost linear at both critical points as both of the Jacobian matrices \( \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\) and \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \) are invertible .
On the other hand a system such as \(x' = x^2\),\(y' = y^2\) has an isolated critical point at \((0,0)\), however the Jacobian matrix
\[\begin{bmatrix} 2x & 0 \\ 0 & 2y \end{bmatrix} \nonumber \]
is zero when \((x,y) = (0,0)\). Therefore the system is not almost linear. Even a worse example is the system \(x' = x\),\(y' = x^2\),which does not have an isolated critical point, as \(x'\) and \(y'\) are both zero whenever \(x=0\), that is, the entire \(y\) axis.
Fortunately, most often critical points are isolated, and the system is almost linear at the critical points. So if we learn what happens here, we have figured out the majority of situations that arise in applications.
Stability and Classification of Isolated Critical Points
Once we have an isolated critical point, the system is almost linear at that critical point, and we computed the associated linearized system, we can classify what happens to the solutions. We more or less use the classification for linear two-variable systems from Section 3.5 , with one minor caveat. Let us list the behaviors depending on the eigenvalues of the Jacobian matrix at the critical point in Table \(\PageIndex{1}\). This table is very similar to Table 3.5.1, with the exception of missing “center” points. We will discuss centers later, as they are more complicated.
| Eigenvalues of the Jacobian matrix | Behavior | Stability |
|---|---|---|
| real and both positive | source / unstable node | unstable |
| real and both negative | sink / stable node | asymptotically stable |
| real and opposite signs | saddle | unstable |
| complex with positive real part | spiral source | unstable |
| complex with negative real part | spiral sink | asymptotically stable |
In the new third column, we have marked points as asymptotically stable or unstable . Formally, a stable critical point \((x_0,y_0)\) is one where given any small distance \(\epsilon\) to \((x_0,y_0)\),and any initial condition within a perhaps smaller radius around \((x_0,y_0)\),the trajectory of the system will never go further away from \((x_0,y_0)\) than \(\epsilon\). An unstable critical point is one that is not stable. Informally, a point is stable if we start close to a critical point and follow a trajectory we will either go towards, or at least not get away from, this critical point.
A stable critical point \((x_0,y_0)\) is called asymptotically stable if given any initial condition sufficiently close to \((x_0,y_0)\) and any solution \(\bigl( x(t), y(t) \bigr)\) given that condition, then
\[\lim_{t \to \infty} \bigl( x(t), y(t) \bigr) = (x_0,y_0) . \nonumber \]
That is, the critical point is asymptotically stable if any trajectory for a sufficiently close initial condition goes towards the critical point \((x_0,y_0)\).
Example \(\PageIndex{1}\)
Consider \(x'=-y-x^2\), \(y'=-x+y^2\). See Figure \(\PageIndex{1}\) for the phase diagram. Let us find the critical points. These are the points where \(-y-x^2 = 0\) and \(-x+y^2=0\). The first equation means \(y = -x^2\), and so \(y^2 = x^4\). Plugging into the second equation we obtain \(-x+x^4 = 0\). Factoring we obtain \(x(1-x^3)=0\). Since we are looking only for real solutions we get either \(x=0\) or \(x=1\). Solving for the corresponding \(y\) using \(y = -x^2\),we get two critical points, one being \((0,0)\) and the other being \((1,-1)\). Clearly the critical points are isolated. Let us compute the Jacobian matrix:
\[\begin{bmatrix}-2x & -1 \\-1 & 2y\end{bmatrix} . \nonumber \]
At the point \((0,0)\) we get the matrix \(\left[ \begin{smallmatrix} 0 & -1 \\ -1 & 0 \end{smallmatrix} \right]\) and so the two eigenvalues are \(1\) and \(-1\). As the matrix is invertible, the system is almost linear at \((0,0)\). As the eigenvalues are real and of opposite signs, we get a saddle point, which is an unstable equilibrium point.
At the point \((1,-1)\) we get the matrix \(\left[ \begin{smallmatrix} -2 & -1 \\ -1 & -2 \end{smallmatrix} \right]\) and computing the eigenvalues we get \(-1\),\(-3\).The matrix is invertible, and so the system is almost linear at \((1,-1)\). As we have real eigenvalues both negative, the critical point is a sink, and therefore an asymptotically stable equilibrium point. That is, if we start with any point \((x_i,y_i)\) close to \((1,-1)\) as an initial condition and plot a trajectory, it will approach \((1,-1)\). In other words,
\[\lim_{t \to \infty} \bigl( x(t), y(t) \bigr) = (1,-1) . \nonumber \]
As you can see from the diagram, this behavior is true even for some initial points quite far from \((1,-1)\),but it is definitely not true for all initial points.
Example \(\PageIndex{2}\)
Let us look at \(x'=y+y^2e^x\), \(y'=x\). First let us find the critical points. These are the points where \(y+y^2e^x = 0\) and \(x=0\). Simplifying we get \(0=y+y^2 = y(y+1)\). So the critical points are \((0,0)\) and \((0,-1)\),and hence are isolated. Let us compute the Jacobian matrix:
\[\begin{bmatrix}y^2e^x & 1+2ye^x \\1 & 0\end{bmatrix}. \nonumber \]
At the point \((0,0)\) we get the matrix \(\left[ \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right]\) and so the two eigenvalues are \(1\) and \(-1\). As the matrix is invertible, the system is almost linear at \((0,0)\). And, as the eigenvalues are real and of opposite signs, we get a saddle point, which is an unstable equilibrium point.
At the point \((0,-1)\) we get the matrix \(\left[ \begin{smallmatrix} 1 & -1 \\ 1 & 0 \end{smallmatrix} \right]\) whose eigenvalues are \(\frac{1}{2} \pm i \frac{\sqrt{3}}{2}\). The matrix is invertible, and so the system is almost linear at \((0,-1)\). As we have complex eigenvalues with positive real part, the critical point is a spiral source, and therefore an unstable equilibrium point.
See Figure \(\PageIndex{2}\) for the phase diagram. Notice the two critical points, and the behavior of the arrows in the vector field around these points.
Trouble with Centers
Recall, a linear system with a center meant that trajectories traveled in closed elliptical orbits in some direction around the critical point. Such a critical point we would call a center or a stable center . It would not be an asymptotically stable critical point, as the trajectories would never approach the critical point, but at least if you start sufficiently close to the critical point, you will stay close to the critical point. The simplest example of such behavior is the linear system with a center. Another example is the critical point \((0,0)\) in Example 8.1.1.
The trouble with a center in a nonlinear system is that whether the trajectory goes towards or away from the critical point is governed by the sign of the real part of the eigenvalues of the Jacobian. Since this real part is zero at the critical point itself, it can have either sign nearby, meaning the trajectory could be pulled towards or away from the critical point.
Example \(\PageIndex{3}\)
An easy example where such a problematic behavior is exhibited is the system \(x'=y, y' = -x+y^3\). The only critical point is the origin \((0,0)\). The Jacobian matrix is
\[\begin{bmatrix}0 & 1 \\-1 & 3 y^2 \\ \end{bmatrix} . \nonumber \]
At \((0,0)\) the Jacobian matrix is \(\left[ \begin{smallmatrix}0 & 1 \\-1 & 0 \\\end{smallmatrix} \right]\),which has eigenvalues \(\pm i\). Therefore, the linearization has a center.
Using the quadratic equation, the eigenvalues of the Jacobian matrix at any point \((x,y)\) are
\[\lambda = \frac{3}{2}y^2 \pm i \frac{\sqrt{4-9y^4}}{2} . \nonumber \]
At any point where \(y \not= 0\) (so at most points near the origin), the eigenvalues have a positive real part (\(y^2\) can never be negative). This positive real part will pull the trajectory away from the origin. A sample trajectory for an initial condition near the origin is given in Figure \(\PageIndex{3}\).
The moral of the example is that further analysis is needed when the linearization has a center. The analysis will in general be more complicated than in the above example, and is more likely to involve case-by-case consideration. Such a complication should not be surprising to you. By now in your mathematical career, you have seen many places where a simple test is inconclusive, perhaps starting with the second derivative test for maxima or minima, and requires more careful, and perhaps ad hoc analysis of the situation.
Conservative Equations
An equation of the form
\[x'' + f(x) = 0 \nonumber \]
for an arbitrary function \(f(x)\) is called a conservative equation . For example the pendulum equation is a conservative equation. The equations are conservative as there is no friction in the system so the energy in the system is "conserved." Let us write this equation as a system of nonlinear ODE.
\[x' = y, \qquad y' = -f(x) . \nonumber \]
These types of equations have the advantage that we can solve for their trajectories easily. The trick is to first think of \(y\) as a function of \(x\) for a moment. Then use the chain rule
\[x'' = y' = y \frac{dy}{dx} , \nonumber \]
where the prime indicates a derivative with respect to \(t\). We obtain \(y \frac{dy}{dx} + f(x) = 0\). We integrate with respect to \(x\) to get \(\int y \frac{dy}{dx} \,dx + \int f(x)\, dx = C\). In other words
\[\frac{1}{2} y^2 + \int f(x)\, dx = C . \nonumber \]
We obtained an implicit equation for the trajectories, with different \(C\) giving different trajectories. The value of \(C\) is conserved on any trajectory. This expression is sometimes called the Hamiltonian or the energy of the system. If you look back to Section 1.8 , you will notice that \(y\frac{dy}{dx} + f(x) = 0\) is an exact equation, and we just found a potential function.
Example \(\PageIndex{4}\)
Let us find the trajectories for the equation \(x'' + x-x^2 = 0\), which is the equation from Example 8.1.1. The corresponding first order system is
\[x' = y , \qquad y' = -x+x^2 . \nonumber \]
Trajectories satisfy
\[\frac{1}{2} y^2 + \frac{1}{2} x^2 - \frac{1}{3} x^3 = C . \nonumber \]
We solve for \(y\)
\[y = \pm \sqrt{-x^2 + \frac{2}{3} x^3 + 2C} . \nonumber \]
Plotting these graphs we get exactly the trajectories in Figure 8.1.2. In particular we notice that near the origin the trajectories are closed curves : they keep going around the origin, never spiraling in or out. Therefore we discovered a way to verify that the critical point at \((0,0)\) is a stable center. The critical point at \((0,1)\) is a saddle as we already noticed. This example is typical for conservative equations.
Consider an arbitrary conservative equation. The trajectories are given by
\[y = \pm \sqrt{ - 2 \int f(x)\, dx + 2C} . \nonumber \]
So all trajectories are mirrored across the \(x\)-axis. In particular, there can be no spiral sources nor sinks. All critical points occur when \(y=0\) (the \(x\)-axis), that is when \(x' = 0\). The critical points are simply those points on the \(x\)-axis where \(f(x) = 0\). The Jacobian matrix is
\[\begin{bmatrix}0 & 1 \\-f'(x) & 0\end{bmatrix} . \nonumber \]
So the critical point is almost linear if \(f'(x) \not= 0\) at the critical point. Let \(J\) denote the Jacobian matrix, then the eigenvalues of \(J\) are solutions to
\[0 = \det(J - \lambda I) = \lambda^2 + f'(x) . \nonumber \]
Therefore \(\lambda = \pm \sqrt{-f'(x)}\). In other words, either we get real eigenvalues of opposite signs, or we get purely imaginary eigenvalues. There are only two possibilities for critical points, either an unstable saddle point , or a stable center . There are never any asymptotically stable points, sinks, or sources.


4.3: The Medieval Period (400 CE-1400 CE)
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In contrast to the Classical Period, which saw tremendous growth and innovation in the study of communication, the Medieval Period might be considered the dark ages of academic study in our field. During this era, the Greco-Roman culture was dominated by Christian influence after the fall of the Roman Empire. The church felt threatened by secular rhetorical works they considered full of pagan thought. While the church preserved many of the classical teachings of rhetoric, it made them scarce to those not in direct service to the church. A secular education was extremely hard to obtain during the Medieval Period for almost everyone.
Even though Christianity condemned communication study as pagan and corrupt, it embraced several aspects of the Classical Period to serve its specific purposes. The ideas from the Classical Period were too valuable for the church to completely ignore. Thus, they focused on communication study to help them develop better preaching and letter writing skills to persuade people to Christianity. Emphasis was placed on persuasion and developing public presentations, both oral and written. Like the Classical Period, those in power continued to stifle women’s participation in communication study, keeping them largely illiterate while men served as the overseers of the church and the direction of academic inquiry.
One of the most recognizable people from this era was Augustine (354 CE-430 CE), a Christian clergyman and renowned rhetorician who actually argued for the continued development of ideas that had originated during the Classical Period. He thought that the study of persuasion, in particular, was a particularly worthwhile pursuit for the church. Augustine was a teacher by trade and used his teaching skills as well as knowledge of communication to move “men” toward truth, which for him was the word of God (Baldwin).
With the exception of Augustine, the formal study of communication took a back seat to a focus on theological issues during the Medieval Period. Fortunately, the study of communication managed to survive as one of the seven branches of a liberal education during this period, but it remained focused on developing presentational styles apt for preaching. Boethius and the Archbishop Isidore of Seville made small efforts to preserve classical learning by reviving the works of Cicero and Quintilian to persuade people to be just and good. Nevertheless, aside from Augustine’s work, little progress was made during the remaining Medieval years; the formal study of communication literally plunged into the “dark ages” before reemerging during the Renaissance.


18.1: The Four-Minute Mile
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To understand why it matters to me, I will begin with a story of a crossing – a very recent one – between my colonial boyhood and my late-middle-aged self. This particular one took place not in South Africa, the country in which I was born and of which I am a citizen, but in England.
To explain why the crossing between past and present matters to my own understanding of IR, some personal background is required. Growing up in colonial South Africa, my home was littered with the culture of England – a country that my South African-born mother never visited until she was fifty. In addition, the boarding school that I attended was loosely modelled on the English public school tradition. So, we were encouraged to participate in the forms of organised sport that were England’s ‘gift’ to the world. Understandably then, my earliest thinking about what made the international was set by the cultural authority of England and the political sweep of the British Empire. Given this, the story of Roger Bannister’s sub-four-minute mile had a particular appeal for my young self. To explain: the measured mile became an important test in competitive athletics in the early 1950s. It was long believed that no person could run a mile in under four minutes. But, in the aftermath of the Second World War, when physical training and nutrition techniques improved along with the instruments for timing, the four-minute mile came closer and closer to being conquered. Indeed, breaking the barrier became a sort of milestone competitive goal for both individual athletes and the countries they represented.
My initial fascination with the four-minute mile was ignited by an edition of the Eagle Sports Album , which had been sent to the school library from London. In its pages, much was made of the importance of Bannister’s feat for Britain and Britons like my family, who were located in distant parts of the world. The drama of the event whetted a life-long interest in athletics. Finally, while on a trip to Oxford in October 2015, I visited the field on the Iffley Road where Bannister ran the famous measured mile. Like many a pilgrimage, the visit was exciting, elating and enlightening. As I stood on the ‘Roger Bannister running track’ – as the field is now called – I looked for the church flagpole that Bannister had spotted seconds before his famous run. When a young man carrying spiked running shoes walked by, I remembered, if only for a fleeting moment, the thrill of competitive running. But more important than the rush was the slow realisation that what had happened on that famous day offered lessons in how I had first come to know and understand the world of IR.
Until the visit, it never occurred to me that what had taken place on the day of the event was a quintessential moment of modernity – the conquering of space by time. In IR, of course, the control of territory through the instruments and techniques of administration and the control that follows is the very essence of the discipline. So, the idea of the international has no meaning unless territory is under the control of sovereignty. As a result, bringing ungoverned places into the idea of the international is the very first order of business in international relations. The notion of sovereignty, which is the enabling force of IR, follows upon this demarcation of space opening towards the exercise of control along a boundary-line between ‘the international’ and ‘the domestic’. Technology, in the form of maps and their making, helped to make ‘permanent’ such boundaries in the minds of rulers – especially colonial ones (see Branch 2014).
Strictly speaking, without boundaries there can be no IR. But, the divide between the boundaries drawn by the instruments of modernity are not the tightly patrolled frontier with its technology of control – passports, visa, immigration documents and the like. It is a liminal space where inclusion and exclusion is negotiated continuously. So, there were – as there remain – forms of interaction between groups who have resisted incorporation into the command and control that orthodox IR insists is the gift of statehood. This betwixt-and-between space has been a site of great tragedy, as the migrant crisis in Europe that began in 2015 shows. In many places, outside of the authoritative gaze of modern media, frontiers were killing fields. European colonisation, which drew the furthest corners on the planet into a single political whole under the banner of civilisation and Christianity, was extremely violent. If killing was one dimension of this, another was the disruption to the ways of living of millions upon millions. This violent disruption in the lives of people continued into the 1960s as the idea of the international spread across the world.
One example was a 1965 agreement in which Britain gave an archipelago of islands in the Indian Ocean to the United States. The residents of these islands, known collectively as the Chagos islands, were forcibly moved. In the past fifty years, the islanders themselves and their descendants have made numerous unsuccessful legal attempts to overturn this decision. Generally speaking, tragedies like these – which occur at the margin of the world – have been ignored in IR – although anthropologists, historians and international lawyers have explored them.
The second issue that occurred to me was the power of who pronounces on these matters. In literature – and increasingly in social science – this is the issue of ‘voice’: who gets to speak, how they get to speak and why this happens. At the policy end of IR, alas, the issue of voice is seldom considered a priority issue, notwithstanding the path-breaking insights that feminists have brought to the discipline. They have exposed the multiple ways in which women experience the international differently to men and how they are silenced in the story of the international despite the significant roles they have and continue to play in its creation.
Two signs on Iffley Road declare Bannister’s triumph. The first, which is mounted on a stone gatepost, is informational. It reads, ‘Here at the Iffley Road track the first sub-four-minute mile was run on 6th May 1954 by ROGER BANNISTER. Oxford University.’ The second is positioned above a wooden fence facing Iffley Road. Under the crest of Oxford University, it reads, ‘Here, on 6 May 1954, Roger Bannister set a new World Mile record of 3 minutes 59.4 seconds. The first Mile ever run under 4 minutes.’ If the first sign informs, the second proclaims Bannister’s achievement as truth. Here, in the historical conquest of space by time, there is no room for ambiguity.
Let us be clear about several things. Of course, the Iffley Road field was the site of the first ‘timed’, ‘authenticated’ or ‘measured’ mile run under four minutes. But – and this is why critical questioning is important in IR, as it is in all forms of knowledge – it seems unlikely that nobody else, anywhere else, across human history had ever run this distance in under four minutes. Indeed, medical science today suggests that humans with particular kinds of physiological traits are able to run faster over distance than those without them. For our purposes of understanding my appreciation of IR, this signage – its declaration and its claim to authority – is rooted in a white, Western, male-dominated world. This is the world into which I was born and raised. Outside of this, nothing is worthy of recognition. It confirms that the lateimperial gaze of the early 1950s, when Bannister ran his famous mile, had little understanding of, or interest in, the non-West.
It seems obvious that prejudices like these need to be challenged, but this is difficult because mainstream IR has elevated its denial of the non-Western world to an art form. For many, the business of IR remains mortgaged to the commonsense understandings of race, class and gender that marked the early decades of the twentieth century when IR emerged as a formal academic discipline. As a result, in many corners of the world, IR is called a ‘mutant’ discipline (Vale 2016a). This is because IR seems to have no conceptual capacity – no grammar or vocabulary, as social theorists might say – to explain the everyday lives of people who live beyond or beneath sovereign borders. And, because it has no adequate category to include them, IR fails to understand them.


Section 1.1: Prelude to Community Building
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- Geraldine L. Palmer, Todd L. Rogers, Judah Viola, & Maronica Engel
- Rebus Community
The concept of community building has no universal definition. However, this could be more positive than negative because having no universal definition also keeps the conception of community fluid, which we believe speaks more accurately to its diverse complexities. To have some idea of what we mean by community building we (the editors) start this overview of the first three chapters by first giving you a look at how we conceptualize or think about community. Community can be geographical, virtual, groups of people with similarities such as a community of community practitioners. Importantly, we see community overall as a way of being and doing or navigating life with others. Therefore community building means actively seeking ways and resources to bring forward co-liberation. This might mean we find ourselves building community by holding space for difference. For example, as editors of this textbook, we are community psychologists working in academia and the community, an industrial/organizational psychologist, and an instructional designer. Our commonality is our passion for social and racial justice. We believe that this passion also shows up in the case stories you will find in this section.
The first case story, Cultural Development in Underrepresented Communities: Using an Empowerment and Citizen Participation Framework is offered by Dr. Jacqueline Samuel. Dr. Samuel’s work illustrates community building with those who seek to use art and culture as viable avenues for building community. Students interested in arts and culture, and the health and well-being of communities of all types will find a plethora of information that might guide your journey, as using the arts to inform and reclaim power is not at all a far-reaching idea.
The second case story in this section is Dr. Amber Kelly’s and Dr. Kathleen McAuliff’s (2021) work alongside members of the disability community, seeking to advance inclusivity in all facets of community life. Dare2Dialogue brings awareness around negative stigma associated with a disability and ways to address these challenges through dialogue and centering and raising the voices of people with disabilities.
The third case story in this section is the work of Dr. August Hoffman. Dr. Hoffman’s passion is visible as he comes alongside of a student using green space to promote communal experiences for the student completing an Individualized Studies program with an emphasis on Gardening Development at Metropolitan State University. The case study moves the reader through the student’s experiences with growing healthy foods for the community as part of education, with the ultimate goal of helping children understand the benefits of healthier eating and the actual origins of the foods they consume.


5.7: References
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Evans, D. (2001). Emotion: The science of sentiment. Oxford University Press.
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4.S: Organic Compounds- Cycloalkanes and their Stereochemistry (Summary)
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Concepts & Vocabulary
- Cycloalkanes are saturated hydrocarbons that have the generic formula C n H 2n , where n is the number of carbons in the ring.
- The IUPAC rules for naming cycloalkanes is very similar to the rules used for naming alkanes.
4.2: Cis-Trans Isomerism in Cycloalkanes
- Stereoisomers are molecules that have the same molecular formula, the same atom connectivity, but they differ in the relative spatial orientation of the atoms.
- Di-substituted cycloalkanes exhibit cis - / trans - stereoisomerism. The cis- isomer has both substituents on the same face of the ring, while the trans- isomer has groups on opposite faces of the ring.
4.3: Stability of Cycloalkanes - Ring Strain
- Ring strain is the total strain in a ring due to torsional strain , steric strain and angle strain .
- Angle strain is when the C-C-C bond angles in rings are different than 109.5 o , the optimal bond angle for sp 3 hybridized carbons.
- Ring strain causes small cycloalkanes, like cyclopropane and cyclobutane, to be much less stable than other cycloalkanes.
4.4: Conformations of Cycloalkanes
- Cyclopentane has less ring strain than cyclopropane and cyclobutane, because its ring carbons have more flexibility to rotate away from planarity, resulting in lower angle and torsional strains.
4.5: Conformations of Cyclohexane
- Cyclohexane has significantly lower ring strain than smaller cycloalkanes, because cyclohexane can adopt non-planar structures, which minimize angle strain and torsional strain.
- The common non-planar structures of cyclohexane are the boat, twist-boat, and chair conformations. The most stable, and hence, the most common, is the chair conformation.
4.6: Axial and Equatorial Bonds in Cyclohexane
- The two chair conformations of cyclohexane interconvert rapidly at room temperature in a process called chair flip or ring flip .
- In the chair conformation of cyclohexane, of the two groups attached to each ring carbon, one of the groups occupies the axial position, while the other group occupies the equatorial position.
- A group that was axial will switch to the equatorial position during a ring flip, and vice versa.
4.7: Conformations of Monosubstituted Cyclohexanes
- To minimize the steric effects of 1,3-diaxial interactions , the single group on a monosubstituted cyclohexane ring will prefer to be in the equatorial position over the axial position. The larger the group, the greater is the preference shifts.
4.8: Conformations of Disubstituted Cyclohexanes
- The preference for large groups to be in the equatorial position effects the relative stability of the cis and trans isomers of disubstituted cyclohexanes. Conformational analysis is the process used to determine which isomer, cis or trans , is most stable.
Summary Problems
Exercise \(\PageIndex{1}\)
The following molecule, quinic acid, is a natural product that can be obtained from a variety of sources including the coffee bean. Draw both chair conformations for this molecule, identify each substituent in both structures as axial or equatorial, and clearly indicate which chair conformation is the most stable.
- Answer
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The circled conformation is more stable because it has more equatorial substituents (3 versus 2) and the largest group (the carboxylic acid) is equatorial.
Exercise \(\PageIndex{2}\)
Convert the following name to a skeletal structure: cis -1- t -butyl-2-ethylcyclohexane.  Circle the most stable of the two conformations.
- Answer
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Remember that, due to its large size, the t -butyl substituent locks the cyclohexane ring into one conformation with the t -butyl in the equatorial position. Thus, this isn't an equilibrium. It exists only as the circled conformation.
Exercise \(\PageIndex{3}\)
Convert the following name to a skeletal structure: trans -3-isobutylcyclohexanol. Then, draw the two chair conformations, label substituents as axial or equatorial, and circle the more stable conformation.
- Answer
-
The circled molecule is most stable because the larger substituent is equatorial.
Skills to Master
- Skill 4.1 Be able to name and draw cycloalkanes
- Skill 4.2 Identify and draw the cis- and trans- stereoisomers of disubstituted cycloalkanes.
- Skill 4.3 Determine the effects of torsional strain, steric strain, and angle strain on the overall ring strain of a cycloalkane.
- Skill 4.4 Draw the chair conformers of cyclohexane.
- Skill 4.5 Draw and identify the axial and equatorial positions in a chair conformer of cyclohexane and its ring-flip conformer.
- Skill 4.6 Use conformational analysis to determine the most stable stereoisomer in disubstituted and polysubstituted cyclohexanes.


3.9: Chapter 3 References and Resources for Further Reading
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Bandura, A. (1997). Self-efficacy: The exercise of control . New York: Freeman.
Cameron, J. & Pierce, W. (1994). Reinforcement, reward, and intrinsic motivation: A meta-analysis. Review of Educational Research, 64 , 363–423.
Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology , 53 (1), 109–132. https://doi-org.ezproxy.hacc.edu/10.....100901.135153 [doi-org.ezproxy.hacc.edu]
National Center for Education Statistics. (2020). The condition of education . NCES. National Center for Education Statistics The Condition of Education [nces.ed.gov]
Pajares, F., & Schunk, D. H. (2002). Self and self-belief in psychology and education: A historical perspective. In J. Aronson (Ed.), Improving academic achievement: Impact of psychological factors on education (p. 3–21). Academic Press. https://doi.org/10.1016/B978-012064455-1/50004-X .
Wigfield, A., Eccles, J. S., Fredricks, J. A., Simpkins, S., Roeser, R. W., & Schiefele, U. (2015). Development of achievement motivation and engagement. In M. E. Lamb & R. M. Lerner (Eds.), Handbook of child psychology and developmental science: Socioemotional processes (p. 657–700). John Wiley & Sons, Inc. https://doi.org/10.1002/9781118963418.childpsy316 .
Wigfield, A., Tonk, S., & Eccles, J. (2004). Expectancy-value theory in cross-cultural perspective. In D. McInerney & S. van Etten (Eds.), Research on Sociocultural Influences on Motivation and Learning . Greenwich, CT: Information Age Publishers.
Wigfield, A., & Eccles, J. S. (2000). Expectancy--value theory of achievement motivation. Contemporary Educational Psychology , 25 (1), 68–81. https://doi-org.ezproxy.hacc.edu/10....ceps.1999.1015 .