6.1: Directional and Partial Derivatives
In Chapter 5 we considered functions \(f : E^{1} \rightarrow E\) of one real variable.
Now we take up functions \(f : E^{\prime} \rightarrow E\) where both \(E^{\prime}\) and \(E\) are normed spaces.
The scalar field of both is always assumed the same: \(E^{1}\) or \(C\) (the complex field). The case \(E=E^{*}\) is excluded here; thus all is assumed finite.
We mostly use arrowed letters \(\vec{p}, \vec{q}, \ldots, \vec{x}, \vec{y}, \vec{z}\) for vectors in the domain space \(E^{\prime},\) and nonarrowed letters for those in \(E\) and for scalars.
As before, we adopt the convention that \(f\) is defined on all of \(E^{\prime},\) with \(f(\vec{x})=0\) if not defined otherwise.
Note that, if \(\vec{p} \in E^{\prime},\) one can express any point \(\vec{x} \in E^{\prime}\) as
\[\vec{x}=\vec{p}+t \vec{u},\]
with \(t \in E^{1}\) and \(\vec{u}\) a unit vector. For if \(\vec{x} \neq \vec{p},\) set
\[t=|\vec{x}-\vec{p}| \text { and } \vec{u}=\frac{1}{t}(\vec{x}-\vec{p});\]
and if \(\vec{x}=\vec{p},\) set \(t=0,\) and any\(\vec{u}\) will do. We often use the notation
\[\vec{t}=\Delta \vec{x}=\vec{x}-\vec{p}=t \vec{u} \quad\left(t \in E^{1}, \vec{t}, \vec{u} \in E^{\prime}\right).\]
First of all, we generalize Definition 1 in Chapter 5, §1.
Given \(f : E^{\prime} \rightarrow E\) and \(\vec{p}, \vec{u} \in E^{\prime}(\vec{u} \neq \overrightarrow{0}),\) we define the directional derivative of \(f\) along \(\vec{u}\) (or \(\vec{u}\) -directed derivative of \(f )\) at \(\vec{p}\) by
\[D_{\vec{u}} f(\vec{p})=\lim _{t \rightarrow 0} \frac{1}{t}[f(\vec{p}+t \vec{u})-f(\vec{p})],\]
if this limit exists in \(E\) (finite).
We also define the \(\vec{u}\) -directed derived function,
\[D_{\vec{u}} f : E^{\prime} \rightarrow E,\]
as follows. For any \(\vec{p} \in E^{\prime}\),
\[D_{\vec{u} f(\vec{p})}=\left\{\begin{array}{ll}{\lim _{t \rightarrow 0} \frac{1}{t}[f(\vec{p}+t \vec{u})-f(\vec{p})]} & {\text { if this limit exists, }} \\ {0} & {\text { otherwise. }}\end{array}\right.\]
Thus \(D_{\vec{u}} f\) is always defined, but the name derivative is used only if the limit (1) exists (finite). If it exists for each \(\vec{p}\) in a set \(B \subseteq E^{\prime},\) we call \(D_{\vec{u}} f\) (in classical notation \(\partial f / \partial \vec{u} )\) the \(\vec{u}\) -directed derivative of \(f\) on \(B\).
Note that, as \(t \rightarrow 0, \vec{x}\) tends to \(\vec{p}\) over the line \(\vec{x}=\vec{p}+t \vec{u}.\) Thus \(D_{\vec{u}} f(\vec{p})\) can be treated as a relative limit over that line. Observe that it depends on both the direction and the length of \(\vec{u}.\) Indeed, we have the following result.
Given \(f : E^{\prime} \rightarrow E, \vec{u} \neq \overrightarrow{0},\) and a scalar \(s \neq 0,\) we have
\[D_{s \vec{u}} f=s D_{\vec{u}} f.\]
Moreover, \(D_{s \vec{u}} f(\vec{p})\) is a genuine derivative iff \(D_{\vec{u}} f(\vec{p})\) is.
- Proof
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Set \(t=s \theta\) in (1) to get
\[s D_{\vec{u}} f(\vec{p})=\lim _{\theta \rightarrow 0} \frac{1}{\theta}[f(\vec{p}+\theta s \vec{u})-f(\vec{p})]=D_{s \vec{u}} f(\vec{p}). \quad \square\]
In particular, taking \(s=1 /|\vec{u}|,\) we have
\[|s \vec{u}|=\frac{|\vec{u}|}{|\vec{u}|}=1 \text { and } D_{\vec{u}} f=\frac{1}{s} D_{s \vec{u}} f.\]
Thus all reduces to the case \(D_{\vec{v}} f,\) where \(\vec{v}=s \vec{u}\) is a unit vector. This device, called normalization, is often used, but actually it does not simplify matters.
If \(E^{\prime}=E^{n}\left(C^{n}\right),\) then \(f\) is a function of \(n\) scalar variables \(x_{k}(k=1, \ldots, n)\) and \(E^{\prime}\) has the \(n\) basic unit vectors \(\vec{e}_{k}.\) This example leads us to the following definition.
If in formula (1), \(E^{\prime}=E^{n}\left(C^{n}\right)\) and \(\vec{u}=\vec{e}_{k}\) for a fixed \(k \leq n,\) we call \(D_{\vec{u}} f\) the partially derived function for \(f,\) with respect to \(x_{k},\) denoted
\[D_{k} f \text { or } \frac{\partial f}{\partial x_{k}},\]
and the limit (1) is called the partial derivative of \(f\) at \(\vec{p},\) with respect to \(x_{k},\) denoted
\[D_{k} f(\vec{p}), \text { or } \frac{\partial}{\partial x_{k}} f(\vec{p}), \text { or }\left.\frac{\partial f}{\partial x_{k}}\right|_{\vec{x}=\vec{p}}.\]
If it exists for all \(\vec{p} \in B,\) we call \(D_{k} f\) the partial derivative (briefly, partial) of \(f\) on \(B,\) with respect to \(x_{k}\).
In any case, the derived functions \(D_{k} f(k=1, \ldots, n)\) are always defined on all of \(E^{n}\left(C^{n}\right).\)
If \(E^{\prime}=E^{3}\left(C^{3}\right),\) we often write \(x, y, z\) for \(x_{1}, x_{2}, x_{3},\) and
\[\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \text { for } D_{k} f \quad(k=1,2,3).\]
Note 1. If \(E^{\prime}=E^{1},\) scalars are also "vectors," and \(D_{1} f\) coincides with \(f^{\prime}\) as defined in Chapter 5, §1 (except where \(f^{\prime}=\pm \infty\)). Explain!
Note 2. As we have observed, the \(\vec{u}\) -directed derivative (1) is obtained by keeping \(\vec{x}\) on the line \(\vec{x}=\vec{p}+t \vec{u}.\)
If \(\vec{u}=\vec{e}_{k},\) the line is parallel to the \(k\)th axis; so all coordinates of \(\vec{x},\) except \(x_{k},\) remain fixed \(\left(x_{i}=p_{i}, i \neq k\right),\) and \(f\) behaves like a function of one variable, \(x_{k}.\) Thus we can compute \(D_{k} f\) by the usual rules of differentiation, treating all \(x_{i}(i \neq k)\) as constants and \(x_{k}\) as the only variable.
For example, let \(f(x, y)=x^{2} y.\) Then
\[\frac{\partial f}{\partial x}=D_{1} f(x, y)=2 x y \text { and } \frac{\partial f}{\partial y}=D_{2} f(x, y)=x^{2}.\]
Note 3. More generally, given \(\vec{p}\) and \(\vec{u} \neq \overrightarrow{0},\) set
\[h(t)=f(\vec{p}+t \vec{u}), \quad t \in E^{1}.\]
Then \(h(0)=f(\vec{p});\) so
\[\begin{aligned} D_{\vec{u}} f(\vec{p}) &=\lim _{t \rightarrow 0} \frac{1}{t}[f(\vec{p}+t \vec{u})-f(\vec{p})] \\ &=\lim _{t \rightarrow 0} \frac{h(t)-h(0)}{t-0} \\ &=h^{\prime}(0) \end{aligned}\]
if the limit exists. Thus all reduces to a function \(h\) of one real variable.
For functions \(f : E^{1} \rightarrow E,\) the existence of a finite derivative ("differentiability") at \(p\) implies continuity at \(p\) (Theorem 1 of Chapter 5, §1). But in the general case, \(f : E^{\prime} \rightarrow E,\) this may fail even if \(D_{\vec{u}} f(\vec{p})\) exists for all \(\vec{u} \neq \overrightarrow{0}\).
(a) Define \(f : E^{2} \rightarrow E^{1}\) by
\[f(x, y)=\frac{x^{2} y}{x^{4}+y^{2}}, \quad f(0,0)=0.\]
Fix a unit vector \(\vec{u}=\left(u_{1}, u_{2}\right)\) in \(E^{2}.\) Let \(\vec{p}=(0,0).\) To find \(D_{\vec{u}} f(p),\) use the \(h\) of Note 3 :
\[h(t)=f(\vec{p}+t \vec{u})=f(t \vec{u})=f\left(t u_{1}, t u_{2}\right)=\frac{t u_{1}^{2} u_{2}}{t^{2} u_{1}^{4}+u_{2}^{2}} \text { if } u_{2} \neq 0,\]
and \(h=0\) if \(u_{2}=0.\) Hence
\[D_{\vec{u}} f(\vec{p})=h^{\prime}(0)=\frac{u_{1}^{2}}{u_{2}} \text { if } u_{2} \neq 0,\]
and \(h^{\prime}(0)=0\) if \(u_{2}=0.\) Thus \(D_{\vec{u}}(\overrightarrow{0})\) exists for all \(\vec{u}.\) Yet \(f\) is discontinuous at \(\overrightarrow{0}\) (see Problem 9 in Chapter 4, §3).
(b) Let
\[f(x, y)=\left\{\begin{array}{ll}{x+y} & {\text { if } x y=0,} \\ {1} & {\text { otherwise.}}\end{array}\right.\]
Then \(f(x, y)=x\) on the \(x\)-axis; so \(D_{1} f(0,0)=1\).
Yet \(f\) is discontinuous at \(\overrightarrow{0}\) (even relatively so) over any line \(y=a x\) \((a \neq 0).\) For on that line, \(f(x, y)=1\) if \((x, y) \neq(0,0);\) so \(f(x, y) \rightarrow 1\) but \(f(0,0)=0+0=0\).
Thus continuity at \(\overrightarrow{0}\) fails. (But see Theorem 1 below!)
Hence, if differentiability is to imply continuity, it must be defined in a stronger manner. We do it in §3. For now, we prove only some theorems on partial and directional derivatives, based on those of Chapter 5.
If \(f : E^{\prime} \rightarrow E\) has a \(\vec{u}\)-directed derivative at \(\vec{p} \in E^{\prime},\) then \(f\) is relatively continuous at \(\vec{p}\) over the line
\[\vec{x}=\vec{p}+t \vec{u} \quad\left(\overrightarrow{0} \neq \vec{u} \in E^{\prime}\right).\]
- Proof
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Set \(h(t)=f(\vec{p}+t \vec{u}), t \in E^{1}\).
By Note 3, our assumption implies that \(h\) (a function on \(E^{1}\)) is differentiable at \(0.\)
By Theorem 1 in Chapter 5, §1, then, \(h\) is continuous at \(0;\) so
\[\lim _{t \rightarrow 0} h(t)=h(0)=f(\vec{p}),\]
i.e.,
\[\lim _{t \rightarrow 0} f(\vec{p}+t \vec{u})=f(\vec{p}).\]
But this means that \(f(\vec{x}) \rightarrow f(\vec{p})\) as \(\vec{x} \rightarrow \vec{p}\) over the line \(\vec{x}=\vec{p}+t \vec{u},\) for, on that line, \(\vec{x}=\vec{p}+t \vec{u}.\)
Thus, indeed, \(f\) is relatively continuous at \(\vec{p},\) as stated. \(\quad \square\)
Note that we actually used the substitution \(\vec{x}=\vec{p}+t \vec{u}.\) This is admissible since the dependence between \(x\) and \(t\) is one-to-tone (Corollary 2(iii) of Chapter 4, §2). Why?
Let \(E^{\prime} \ni \vec{u}=\vec{q}-\vec{p}, \vec{u} \neq \overrightarrow{0}\).
If \(f : E^{\prime} \rightarrow E\) is relatively continuous on the segment \(I=L[\vec{p}, \vec{q}]\) and has a \(\vec{u}\)-directed derivative on \(I-Q\) (\(Q\) countable), then
\[|f(\vec{q})-f(\vec{p})| \leq \sup \left|D_{\vec{u}} f(\vec{x})\right|, \quad \vec{x} \in I-Q.\]
- Proof
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Set again \(h(t)=f(\vec{p}+t \vec{u})\) and \(g(t)=\vec{p}+t \vec{u}\).
Then \(h=f \circ g,\) and \(g\) is continuous on \(E^{1}.\) (Why?)
As \(f\) is relatively continuous on \(I=L[\vec{p}, \vec{q}],\) so is \(h=f \circ g\) on the interval \(J=[0,1] \subset E^{1}\) (cf. Chapter 4, §8, Example (1)).
Now fix \(t_{0} \in J.\) If \(\vec{x}_{0}=\vec{p}+t_{0} \vec{u} \in I-Q,\) our assumptions imply the existence of
\[\begin{aligned} D_{\vec{u}} f\left(\vec{x}_{0}\right) &=\lim _{t \rightarrow 0} \frac{1}{t}\left[f\left(\vec{x}_{0}+t \vec{u}\right)-f\left(\vec{x}_{0}\right)\right] \\ &=\lim _{t \rightarrow 0} \frac{1}{t}\left[f\left(\vec{p}+t_{0} \vec{u}+t \vec{u}\right)-f\left(\vec{p}+t_{0} \vec{u}\right)\right] \\ &=\lim _{t \rightarrow 0} \frac{1}{t}\left[h\left(t_{0}+t\right)-h\left(t_{0}\right)\right] \\ &=h^{\prime}\left(t_{0}\right) . \quad \text {(Explain!)} \end{aligned}\]
This can fail for at most a countable set \(Q^{\prime}\) of points \(t_{0} \in J\) (those for which \(\vec{x}_{0} \in Q).\)
Thus \(h\) is differentiable on \(J-Q^{\prime};\) and so, by Corollary 1 in Chapter 5, §4,
\[|h(1)-h(0)| \leq \sup _{t \in J-Q^{\prime}}\left|h^{\prime}(t)\right|=\sup _{\vec{x} \in I-Q}\left|D_{\vec{u}} f(\vec{x})\right|.\]
Now as \(h(1)=f(\vec{p}+\vec{u})=f(\vec{q})\) and \(h(\overrightarrow{0})=f(\vec{p}),\) formula (2) follows. \(\quad \square\)
If in Theorem 2, \(E=E^{1}\) and if \(f\) has a \(\vec{u}\)-directed derivative at least on the open line segment \(L(\vec{p}, \vec{q}),\) then
\[f(\vec{q})-f(\vec{p})=D_{\vec{u}} f\left(\vec{x}_{0}\right)\]
for some \(\vec{x}_{0} \in L(\vec{p}, \vec{q})\).
- Proof
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The proof is as in Theorem 2, based on Corollary 3 in Chapter 5, §2 (instead of Corollary 1 in Chapter 5, §4).
Theorems 2 and 3 are often used in "normalized" form, as follows.
If in Theorems 2 and 3, we set
\[r=|\vec{u}|=|\vec{q}-\vec{p}| \neq 0 \text { and } \vec{v}=\frac{1}{r} \vec{u},\]
then formulas (2) and (3) can be written as
\[|f(\vec{q})-f(\vec{p})| \leq|\vec{q}-\vec{p}| \sup \left|D_{\vec{v}} f(\vec{x})\right|, \quad \vec{x} \in I-Q,\]
and
\[f(\vec{q})-f(\vec{p})=|\vec{q}-\vec{p}| D_{\vec{v}} f\left(\vec{x}_{0}\right)\]
for some \(\vec{x}_{0} \in L(\vec{p}, \vec{q})\).
For by Corollary 1,
\[D_{\vec{u}} f=r D_{\vec{v}} f=|\vec{q}-\vec{p}| D_{\vec{v}} f;\]
so (2') and (3') follow.


2.S: Polar Covalent Bonds; Acids and Bases (Summary)
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Concepts & Vocabulary
2.1 Polar Covalent Bonds: Electronegativity
- The difference in electronegativity values of two atoms determines whether the bond between those atoms is classified as either ionic , polar covalent , or non-polar covalent .
- Ionic bonds result from large differences in electronegativity values, such as that between a metal and non-metal atom (Na and Cl).
- Covalent bonding generally results when both atoms are non-metals, like C, H, O, N and the halides.
- When both atoms the same and/or have the same electronegativity value, then the bonding electrons are shared equally and the bond is classified as non-polar covalent .
- Polar covalent bonds occur when the difference in electronegativity values is small, and the bonding electrons are not shared equally.
2.2 Polar Covalent Bonds: Dipole Moments
- The molecular dipole moment is the sum of all the bond dipoles within a molecule and depends on both the molecular geometry and the bond polarity.
- Molecules that contain no polar bonds, like CH 4 , and/or completely symmetrical molecules, like CO 2 , generally have no net dipole moment.
- Asymmetrical molecules that contain bonds of different polarities or non-bonding lone pairs typically have a molecular dipole moment.
- Formal Charge compares how many valence electrons surround a free atom versus how many surround that same type of atom bonded with a molecule or ion.
- Formal Charge = (# of valence electrons in free atom) - (# of lone-pair electrons) - (1/2 # of bond pair electrons) Eqn. 2.3.1
- Formal charges of zero generally represent the most stable structures.
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These bonding patterns for the atoms commonly found in organic molecules result in a formal charge of zero
- Carbon - 4 bonds, no lone pairs
- Hydrogen - 1 bond, no lone pairs
- Nitrogen - 3 bonds, 1 lone pair
- Oxygen - 2 bonds, 2 lone pairs
- Halogens - 1 bond, 3 lone pairs.
- Resonance Theory is often used when the observed chemical and physical properties of a molecule or ion cannot be adequately described by a single Lewis Structure. A classic example is the benzene molecule, C 6 H 6 . The Lewis Structure of benzene could be drawn in two different ways. Both structures have alternating double bond and single bonds between the carbons. The only difference is the location of the pi bonds.
If these structures are correct, then the benzene molecule should have two different C-C bond lengths and bond energies, corresponding to a C-C single bond and to a C=C double bond. However, analysis shows that benzene contains only one type of carbon-carbon bond and it's bond length and energy are half between those of a single bond and double bond. Resonance theory states that benzene exists as the "average" of the two structures called a resonance hybrid , in which the six pi electrons delocalized over all six carbon atoms. Each C-C bond in benzene would be the average of a single bond and double bond or a "bond and a half". Dashed lines are often used to show type of "partial" bonding in a resonance hybrid of benzene
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The rules for estimating stability of resonance structures are
- The resonance form in which all atoms have complete valence shells is more stable.
- The greater the number of covalent bonds , the greater the stability since more atoms will have complete octets
- The structure with the least number of formal charges is more stable
- The structure with the least separation of formal charges is more stable
- A structure with a negative charge on the more electronegative atom will be more stable
- Positive charges on the least electronegative atom (most electropositive) is more stable
- Resonance forms that are equivalent have no difference in stability and contribute equally. (eg. benzene)
- If these rules are applied to the two Lewis Structures of benzene, the result would be that both structures will have the same relative stability and will both contribute equally to the character of the resonance hybrid.
- In resonance structures, the electrons are able to move to help stabilize the molecule. This movement of the electrons is called delocalization .
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The rules for drawing resonance structures are:
- Resonance structures should have the same number of electrons, do not add or subtract any electrons. (You can check the number of electrons by counting them)
- All resonance structures must follow the rules of writing Lewis Structures .
- The hybridization of the structure must stay the same.
- The skeleton of the structure can not be changed (only the electrons move).
- Resonance structures must also have the same amount of lone pairs.
2.7 Acids and Bases - The Brønsted-Lowry Definition
- A Brønsted-Lowry acid is a proton (H + ) donor and a Brønsted-Lowry base is a proton acceptor.
- The strength of Brønsted-Lowry acids is measured indicated by its pKa value. The lower the pKa - the stronger the acid.
- A strong acid will have a weak conjugate base. A strong base will have a weak conjugate acid.
2.9 Predicting Acid-Base Reactions from pKa Values
- The equilibrium of an acid-base reaction favors the formation of weaker acids from stronger acids. To predict the direction of the equilibrium, identify Brønsted-Lowry acid on each side of the reaction. Assign/look up pKa values for each acid. The equilibrium will favor the side that has the weakest acid (the highest pKa).
2.10 Organic Acids and Organic Bases
- Organic acids are stronger when the conjugate base that is formed upon loss of a proton is more stable.
- Some factors that effect the stability of the conjugate base (often an anion) are the anionic atom's size and electronegativity, resonance effects, inductive effects, and solvation.
2.11: Acids and Bases - The Lewis Definition
- A Lewis acid is a lone pair acceptor and a Lewis base is a lone pair donor.
2.12: Non-covalent Interactions between Molecules
- Non-covalent Interactions, also known as Intermolecular Forces, significantly effect the physical properties of organic molecules. Hydrogen bonding is the most important of these interactions, but others include ion-dipole, dipole-dipole, and London Dispersion Forces.
Summary Problems
Exercise \(\PageIndex{1}\)
Draw all possible resonance structures to demonstrate delocalization of the positive charge in the following molecule. Circle the most stable resonance structure and explain your answer. Also, draw the resonance hybrid.
- Answer
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The circled structure is the most stable because it is the only structure with a full octet on all atoms. (Remember that carbocations must have an incomplete octet, and an oxygen with a positive charge and a full octet is more stable than a carbocation.)
As a reminder, the resonance hybrid structure is a combination of all of your resonance structure showing partial pi bonds and partial formal charges.
Exercise \(\PageIndex{2}\)
Draw all possible resonance structures to demonstrate delocalization of the negative charge in the following molecule. Also, draw the resonance hybrid and circle the most stable resonance contributor among all of your resonance structures.
- Answer
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The circled structure is the most stable because the negative charge is on oxygen, the most electronegative element sharing the negative charge. As a reminder, in resonance structures with negative charges, all of the atoms have octets, so you only need to focus on electronegativity differences to find the most stable structure.
Exercise \(\PageIndex{3}\)
For the equilibrium shown below, answer the following questions: a) Draw curved arrows to illustrate bond breakage and formation in the reaction. b) At equilibrium, are the products or reactants favored? Explain. c) What percent reactants and percent products are present at equilibrium? d) Use resonance structures to explain why N is the most basic atom in the conjugate base. e) Use resonance structures to explain why the hydrogen that is removed from the conjugate acid is the most acidic proton.
- Answer
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a)
b) The lower pKa is the stronger acid, and the equilibrium will always favor the weak acid (higher pKa), so this reaction favors the reactants. Note: It's normal to assume that the neutral products would be favored, but this is a good example that shows our chemical intuition isn't always right. To make sure we understand the equilibrium, we need to have pKa values to compare.
c) Keq = 10 12.5-17.5 = 10 -5 = 0.00001 So, the ratio of reactants to products is 1:0.00001. The percent reactants is 99.999% and the percent products is 0.001%.
d) There are two atoms with lone pairs in the conjugate base, O and N, so we should evaluate both to show that N is the most basic atom. Protonating O yields a positive charge localized on O. Protonating N yields a charge delocalized over 3 Cs, 1 N, and 1 O. The N is most basic because the charge in the acid is delocalized and there are two structures with full octets (positive charge on N and O).
e) To find the most acidic proton, we should focus on Hs on atoms next to (not on) double bonds. This will yield a delocalized negative charge when the H is removed. In the conjugate acid, we have two options. Option 1 yields a compound with a negative charge delocalized over 2 atoms, O and C. Option 2 yields the base from the original reaction because the charge is delocalized over 4 atoms, C and 3 Os. This is very stable because it is spread over 3 electronegative oxygen atoms.
Exercise \(\PageIndex{4}\)
This question focuses on the reaction of the two molecules shown below. a) Draw an acid-base reaction of these molecules. Clearly label the acid, the base, the conjugate acid, and the conjugate base. Draw curved arrows to clearly indicate electron movement involved in bond formation and bond cleavage. For the base and the conjugate base, label the hybridization of the charged atoms. b) If you drew the reaction correctly, the pKa of the conjugate acid is 9. Use the table in Section 2.8 to determine the pKa of the acid. Are the reactants or products favored at equilibrium? What is the approximate ratio of reactants to products? What is the approximate percentage of reactants and products? c) Draw the structures of the two charged molecules in the reaction. Draw resonance structures to illustrate charge delocalization in these molecules. Based on your analysis in part b, which charged molecule is more stable? Briefly explain this result.
- Answer
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a)
Don't forget that the charged atoms in the base and conjugate base are sp 2 hybridized so that the lone pair can be in a p orbital and then delocalized by resonance. For negative charges to be delocalized, the lone pair must be in a p orbital so that electrons can flow to adjacent pi bonds by resonance.
b)
The acid has the larger pKa, so it is the weaker acid and the reactants are favored. The approximate difference in pKa values is 1. So, Keq is 10 -1 and for every 1 product molecule there are 10 reactant molecules. (Remember, since the reactants are favored, there are more reactants and the Keq must be less than 1.) The approximate percentage of reactants is 90% and products is 10%. (10/(10+1) is approximately 90% and 1/(10+1) is approximately 10%)
c)
We know that the equilibrium favors the reactants. We also know that in acid base reactions, the stronger acid reacts with the stronger base to yield the weaker base and the weaker acid. In this reaction, the products are the stronger acid-base pair. We also know that "stronger" means more reactive and less stable, while "weaker" means less reactive and more stable. So, since the reactants are favored, that means the base (as labeled above) is more stable than the conjugate base. This seems strange since the charge is more delocalized in the conjugate base. This comparison highlights the importance of quality of resonance structures versus quantity of resonance structures. The charge is more stable in the base since in two of the resonance structures, the negative charge is on O. In the conjugate base, only one of the structures has the negative charge on O.
Exercise \(\PageIndex{5}\)
Determine the position of the most acidic proton in the following molecule. You should draw resonance structures and a resonance hybrid to justify your answer.
- Answer
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Donation of the most acidic proton (H + ) yields the most stable conjugate base; the one that has the charge on the most electronegative element and/or the most delocalized charge. You should focus on protons attached to electronegative elements (e.g., O or N) and protons on atoms next to pi bonds, but not attached to atoms with pi bonds. In this molecule, all Hs are attached to carbon. So, we should focus on the Hs next to pi bonds. These are Hs on sp 3 hybridized carbons next to sp 2 hybridized carbons. Loss of these protons will yield a negative charge that can be delocalized by resonance.
Option #1 yields a conjugate base with the charge delocalized over 1 carbon and 1 oxygen. Option #2 yields a conjugate base with a charge delocalized over 3 carbons and 2 oxygens. This is the most stable conjugate base, so the most acidic proton is the red proton (removed in option #2).
Exercise \(\PageIndex{6}\)
Determine the position of the most basic atom in the following molecule. You should draw resonance structures and a resonance hybrid to justify your answer.
- Answer
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There are four basic atoms (atoms with lone pair electrons) in this molecule. The strongest base will yield the most stable conjugate acid (the one with the most delocalized positive charge and/or the charge on the least electronegative atom).
Options #1 and #2 yield conjugate acids with localized charges. Options #3 and #4 yield conjugate acids with delocalized charges.This illustrates a key point: For atoms that can't have an expanded octet, atoms with a lone pair and no pi bond will accept a proton and yield a localized charge in the conjugate acid while atoms with both a lone pair and a pi bond will accept a proton and yield a delocalized charge in the conjugate acid. So, we should focus our energy on the latter atoms. In this problem, that means Options #3 and #4. In option #3, the charge is delocalized over 3 atoms, 2 that have full octets on all atoms (positive charge on N and O). In option #4, the charge is delocalized over 4 atoms, 3 that have full octets on all atoms (positive charge on N and 2 Os). So, Option #4 has more total resonance structures and more structures that have a full octet. This is the most stable conjugate acid which means the circled O is the most basic atom.
Skills to Master
- Skill 2.1 Predict whether a bond is ionic, polar covalent, or non-polar covalent based on the position of the atoms in the periodic table.
- Skill 2.2 Identify the partial positive and partial negative atoms of a polar covalent bond based on relative electronegativity.
- Skill 2.3 Determine the dipole moment of a molecule based on molecular geometry and bond polarity.
- Skill 2.4 Identify the chemicals in a reaction as Brønsted-Lowry acids or bases, and conjugate acids and bases.
- Skill 2.5 Predict the products of an acid-base reaction.
- Skill 2.6 Use pKa values to predict the equilibrium direction of an acid-base reaction.
- Skill 2.7 Predict the relative strength of an organic acid by examining the stability of the conjugate base.
- Skill 2.8 Use molecular structure and analysis of intermolecular forces to rank a series of organic molecules with respect to physical properties like melting point and boiling point.
- Skill 2.9 Identify the chemicals in a reaction as Lewis acids or bases.
Memorization Tasks (MT)
- MT 2.1 Memorize that the C-H bond is considered to be non-polar.
- MT 2.2 Memorize the common bonding patterns for C, H, N, O and the halogens that have a zero formal charge.
- MT 2.3 Memorize the factors that affect the relativity stability of conjugate bases.


12.2L: In Vivo Testing
- Describe how animals can be used for diagnostic antibody production
In Vivo Testing
In vivo methods refer to the use of animals as a conduit to generate purified polyclonal antibody solutions ( antiserum ) for research purposes. Polyclonal antibodies are applied in immunological assays to diagnose disease.
In vivo testing follows strict guidelines and humane animal use ethics. The protocol for diagnostic antibody production in animals follows multiple steps. Animals are injected with microbes or antigenic fragments that elicit an immune response; the immune response is allowed to develop for 1-2 weeks, after which blood is harvested. This blood now contains antibodies created from the antigens that were introduced into the animals. Antibodies are purified from the serum to make antiserum or a purified antibody solution for one particular antigen.
These preparations will produce multiple antibody types that recognize different epitopes on the antigen, hence the term polyclonal. Polyclonal antibodies have various applications in the clinic and in research laboratories. Animals are also used to model human diseases in the research field. They are useful vehicles to understand how our bodies work, find cures and treatments for diseases, test new drugs for safety, and evaluate medical procedures before they are used on patients.
Mice, and other rodents such as rats and hamsters, make up over 90% of the animals used in biomedical research. In addition to having bodies that work similar to humans and other animals, rodents are small in size, easy to handle, relatively inexpensive to buy and keep, and produce many offspring in a short period of time. In vivo testing remains a crucial step for the evaluation of in vitro experimental findings and the production of immunological solutions needed for the diagnosis of human diseases.
Key Points
- In vivo testing is necessary for medical and research purposes. The medical field benefits from animal models to test the safety of drugs before they are used on patients. The research field benefits from in vivo testing by validating in vitro findings in vertebrates closest to humans.
- The most used animal models are mice, rats, and other rodents.
- In vivo testing is useful for the production of polyclonal antibodies applied in immunoassays and diagnostic immunology.
Key Terms
- in vitro : In an artificial environment outside the living organism.
- antiserum : a serum prepared from human or animal sources containing antigens specific for combatting an infectious disease
- in vivo : Within a living organism.


16.1: Assignment- Create Capstone Presentation
You are now ready for your final capstone assignment. For this assignment you will need to download four files: one each for Word, Excel, PowerPoint, and Access. In this scenario, you have been asked to give a presentation on the new RowanGo app. You will create the presentation and a document, integrating Excel and Access in each and using tools covered throughout the course.
Follow the directions, then submit your assignment. If you get stuck on a step, review the modules in this course and ask your classmates for help in the discussion forum.
Part 1: PowerPoint presentation
- Download the PowerPoint presentation and Word document as well as the Access database and Excel table that you will integrate into both documents. Save all four files to the Rowan folder on your desktop.
- Open the PowerPoint presentation. Save it to the Rowan folder on your desktop as BA132_LastName_Capstone .pptx , replacing “LastName” with your own last name. (Example: BA132_Hywater_Capstone) It is a good idea to save your work periodically.
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Here is what the final presentation should contain:
- Five total slides
- New theme from original one
- Excel chart from file (the formatting may or may not change between Excel and PowerPoint and it is completely fine, just make it legible in the slide.)
- Access abbreviated contact table from file.
- Word document file first paragraph on left side, and any image of a hand holding a smartphone from internet on right side in slide.
- Embed video from https://www.youtube.com/watch?v=N_ZlMNCl9UQ
- Add transitions to slides
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Here is an example of what the final version may look like. Use your own theme, style, colors and make sure to remember to make it readable from a distance for a presentation. This is an example of what yours may look like:
- Once you are done save your work. Congratulations, you are now done with Part 1!
Part 2: Word document
- Open the Word document you downloaded in Part 1. Save it to the Rowan folder on your desktop as BA132_LastName_Capstone .docx , replacing “LastName” with your own last name. (Example: BA132_Hywater_Memo) It is a good idea to save your work periodically.
-
Here is what the final presentation should contain:
- No watermark.
- Easy to tell document has a new theme.
- Excel table and chart centered.
- Below the chart the words: You can review the presentation here and contact information examples here. Each ‘here’ links to the PowerPoint presentation file and the Access customer contact file respectively.
-
Here is an example of what the final version may look like. Use your own theme, style, colors and make sure to remember to make it readable. This is an example of what yours may look like:
- Once you are done save your work. Congratulations, you are now done with Part 2!
- Save your work and submit both your PowerPoint presentation and Word document in your course online.
Contributors and Attributions
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- Assignment: Create Capstone Presentation. Authored by : Lumen Learning. License : CC BY: Attribution


6.4: Types of Inventory
The Role of Inventory in the Supply Chain
Managing inventory is one of the most important activities in a supply chain. Materials/goods are needed to provide manufacturers with the exact items that they need, in the right order, the right quality, delivered to the right location, and at the right time. Without all of this happening, it will be impossible to produce high quality goods and meet commitments to our customers. In addition, when goods are ready for shipment, the outbound supply chain needs to be organized in such a way that customers receive their requested orders in a cost-efficient manner.
Types of Inventory in the Supply Chain:
- Finished goods
- Raw materials
- Purchased components and operating supplies
- Work-in-process
Watch the following 2:30 video, which illustrates the four types of inventory:
Reasons for holding inventories:
Many reasons exist for keeping stocks of inventory. Some of the most common include:
-
Manufacturers often build up inventories throughout the year because of
seasonal demand
.
- An example is a Chocolate manufacturer who does not have the capacity to produce all the product that is needed for Christmas. They may begin building inventory in late spring in order to have enough on hand for orders in November and December.
- At the same time, a manufacturer may carry large amounts of inventory if they have some uncertainty or risk in their supply base . If suppliers have some risk of shortages, work stoppages, poor quality or late deliveries then more stock may be carried.
- Firms may be tempted by extra discounts often provided by purchasing large order sizes. Perhaps they may want to minimize transportation costs. There may also be some worry about future price increases that can cause organizations to build up their inventories.
- Retailers carry inventory to ensure that they do not run out of what they anticipate their customers may want. Distributors and retailers may try and balance the cost of keeping large inventories on hand with providing excellent customer service with few or no disappointed customers. However, it is often a challenge to anticipate exact customer behaviour.
- It is a challenge to synchronize incoming flow of materials and goods in order to meet production schedules and ship to customers as promised. As a result, inventory may be stored at many locations along the supply chain. This causes extra cost and inefficiencies for each organization.
"Supply Chain" Introduction to Operations Management Copyright © by Hamid Faramarzi and Mary Drane is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.


2.7: Glossary
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- Chippewa Valley Technical College via OpenRN
American Nurses Association (ANA): The professional organization that represents the interests of the nation’s 4 million registered nurses.
Beneficence: To “do good.”
Black Box Warnings: The strongest warnings issued by the Federal Drug Association (FDA) that signify a drug carries a significant risk of serious or life-threatening adverse effects.
Code of Ethics for Nurses: Developed by the American Nurses Association as a guide for carrying out nursing responsibilities in a manner consistent with quality in nursing care and the ethical obligations of the profession.
Cultural Competence: The process by which nurses demonstrate culturally congruent practice.
Culturally Congruent Practice: The application of evidence-based nursing that is in agreement with the preferred cultural values, beliefs, worldview, and practices of the healthcare consumer and other stakeholders.
Do Not Crush List: A list of medications that should not be crushed, often due to a sustained-release formulation.
Drug Diversion: The transfer of any legally prescribed controlled substance from the individual for whom it was prescribed to another person for any illicit use.
Error-Prone Abbreviations: Abbreviations, symbols, and dose designations that are frequently misinterpreted and involved in harmful medication errors.
Five Rights of Medication Administration: Standards of practice that require the following information is confirmed prior to each administration of medication: right patient, right drug, right dose, right time, and right route.
Health Literacy: The degree to which individuals have the capacity to obtain, process, and understand basic health information needed to make appropriate health decisions.
High-Risk Medications: Drugs that bear a heightened risk of causing significant patient harm when they are used in error.
Inappropriate Polypharmacy: Present when one or more medicines are prescribed that are not or no longer needed.
Joint Commission: A national organization that accredits and certifies health care organizations in the United States.
Look-Alike and Sound-Alike (LASA) Drugs: Medications that require special safeguards to reduce the risk of errors and minimize harm.
Maleficence: Causing harm to patients.
National Patient Safety Goals (NPSGs): Goals established by the Joint Commission to help accredited organizations address specific areas of concern related to patient safety.
Nursing: The protection, promotion, and optimization of health and abilities, prevention of illness and injury, facilitation of healing, alleviation of suffering through the diagnosis and treatment of human response, and advocacy in the care of individuals, families, groups, communities, and populations, as defined by the American Nurses Association.
Nursing Process: Standards of Practice that include Assessment; Diagnosis; Outcome Identification; Planning; Implementation; and Evaluation components of providing patient care.
Nursing Scope and Standards of Practice: A document created by the American Nurses Association that outlines professional nursing performance according to national standards.
Polypharmacy: The concurrent use of multiple medications.
Prescription Drug Monitoring Programs (PDMP): A statewide electronic database that collects designated data on substances dispensed in a state to address prescription drug abuse, addiction, and diversion.
Professional Assistance Procedure: A voluntary non-disciplinary program to provide support for credentialed professionals in Wisconsin with substance abuse disorder who are committed to their own recovery.
Registered Nurse (RN): An individual who is educationally prepared and licensed by a state to practice as a registered nurse.
Root Cause Analysis: An analysis after an error occurs to help identify not only what and how an event occurred, but also why it happened. When investigators are able to determine why an event or failure occurred, they can create workable corrective measures that prevent future errors from occurring.
Safety Culture: The culture of a health care agency that empowers staff to speak up about risks to patients and to report errors and near misses, all of which drive improvement in patient care and reduce the incident of patient harm.
Scheduled Medications: The Controlled Substances Act (CSA) places all substances that are regulated under existing federal law into one of five schedules, ranging from Schedule I drugs with a high potential for abuse and the potential to create severe psychological and/or physical dependence, to Schedule V drugs with the least potential for abuse.
Social Determinants of Health: Poverty, education, safe medication, and other healthcare disparities that affect a patient’s health.
Standards of Practice: Authoritative statements of duties by the American Nursing Association that all registered nurses, regardless of role, population, or specialty, are expected to perform competently. Standards of Practice include Assessment, Diagnosis, Outcome Identification, Planning, Implementation, and Evaluation components of providing patient care.
Standards of Professional Performance: Describe a competent level of behavior in the professional role, including activities related to ethics, culturally congruent practice, communication, collaboration, leadership, education, evidence-based practice, and quality of practice as defined by the American Nursing Association.
State Nurse Practice Act: Laws enacted by state legislatures setting professional standards of nursing care to which nurses are held accountable by the State Board of Nursing.
State Board of Nursing: A group of officials who enforce the State Nurse Practice Act.
Substance Use Disorder: A pattern of behaviors that ranges from misuse to dependency or addiction, whether it is alcohol, legal drugs, or illegal drugs. Addiction is a complex disease with serious physical, emotional, financial, and legal consequences.


8.1: Why It Matters- Early Adulthood
Why learn about development changes during early adulthood?
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When we are children and teens, we eagerly anticipate each and every birthday, waiting for the next big one…when we’ll finally be grown up and have all the freedoms and rights enjoyed by those who are older than us. Indeed, there are opportunities to drive, buy a car, vote, go to college, join the military, drink, move out on our own, date, live together, get married, work, have children, buy a house, and more. This can be an awesome time in our lives, as we tend to be physically and cognitively strong and healthy, we dream and make plans for the future, find people to share our experiences, and try out new roles. It can also be challenging, stressful, and scary as we realize that a lot of responsibility comes with such freedom. We have probably all seen the coffee mugs that proclaim, “Adulting is hard,” or the t-shirts that announce, “I can’t adult today” (typically worn by young adults!).
Development is a process, and we aren’t suddenly adults at a certain age. In fact, we may even take longer to grow up these days. In this module, we’ll learn about norms, trends, and theories about why certain patterns are forming. It’s even been proposed that there is a new stage of development between adolescence and early adulthood, called “emerging adulthood,” when young people don’t quite feel like they are adults yet and wait longer to join the workforce, move out on their own, get married, and have children. Yet by the end of early adulthood, most of us will have accomplished the important developmental tasks of becoming more autonomous, taking care of ourselves and even others, committing to relationships and jobs/careers, getting married, raising families, and becoming part of our communities. There are, of course, many individual and cultural differences.
 When will you feel like an adult? Or do you already feel like an adult? Why or why not? Did your parents become adults earlier or later in their lives, compared to you?


 Mental illness is not the same as experiencing normal stress and sadness. One main difference is that mental illness causes significant distress and impairs functioning, making it difficult to cope with the demands of everyday life (Canadian Mental Health Association CMHA, n.d; Mental Health Foundation, n.d; Mental Health Foundation of Australia, n.d; National Alliance on Mental Health, n.d; Public Health Agency of Canada PHAC, n.d.). In other words, when people are mentally ill, they can’t manage activities of daily living, work effectively, or maintain relationships. Their overwhelming distress is not just a reaction to daily events, and they are not behaving that way on purpose. Just like a physical illness, when a mental illness is not recognized and treated, it can worsen and will last for an unnecessarily long time.
It is not easy to recognize when the distress experienced by a person with an intellectual disability is actually a symptom of mental illness. The stigma associated with mental illness makes most people reluctant to talk about their experiences of having strange thoughts or deep sadness. For individuals with intellectual disabilities who already struggle to find the right words to express themselves, talking about their unusual experiences can be overwhelming.
When health professionals diagnose psychiatric disorders, they rely on information obtained during interviews with clients. However, clients with intellectual disabilities may not have the language or memory skills needed to explain what has been happening. When these clients have had only limited opportunities to socialize with others, they may be unsure about whether their experiences are “normal” or not. People in their lives may have ignored their symptoms of mental illness because they believed these symptoms were simply part of the disability.
In this chapter, we describe indicators of mental illness in three key areas: disorders of thinking, disorders of mood, and disorders of behaviour. Although only qualified clinicians can diagnose psychiatric disorders in individuals with disabilities, all those who care for them can help report the kinds of indicators that could be symptoms of mental illness. The chapter begins with a brief background of mental health care; identifies key indicators associated with disorders of thinking, mood, and behaviour; and emphasizes relevant observations that caregivers must report.
Background of Mental Health Care
History
Until the 17th century, most people who were considered to be insane were viewed as sinful, weak, or possessed by demons. Cast out of most societies, they were incarcerated in jails or poorhouses. Although asylums or mental hospitals were eventually created to house people with mental illness, early treatment approaches were crude.
Individuals with intellectual disabilities, traditionally referred to as mentally retarded , were often housed in these asylums as well. Distinctions between insanity and retardation were not always clear.  Those with the label severely retarded were believed to be unable to express feelings and therefore not able to experience emotional distress (Werges, 2007). Inmates in the asylums were not viewed as people with an illness; rather, they were considered untreatable and were punished for any unusual behaviour.
By the early 20th century, mental health care became viewed as a public responsibility, and insane asylums were replaced by hospitals providing medical care. During the 1950s and 1960s, research led to an increased understanding of mental illness, and specific psychiatric disorders and associated treatments were identified. As it became clear that psychosocial factors can contribute to mental illness, psychiatric disorders became more normalized, or viewed as conditions that could happen to anyone. However, advances in understanding mental illness in individuals with intellectual disabilities occurred at a much slower pace. Separate institutions were established for these individuals and unusual behaviour was still not always considered a symptom of mental illness.
 Admissions to hospital beds were decreased and community agencies were expected to provide care. Unfortunately, for many individuals the expected help from community agencies still remains elusive. Today numerous individuals with mental illness find refuge only in jails and homeless shelters. Those with severe and persistent mental illness present in general hospital emergency rooms and yet, if admitted at all, may be discharged within days without having received the help they need. These shorter hospital stays and limited community supports have made it even more difficult to understand the unique needs of those diagnosed with both intellectual disability and mental illness (dually diagnosed).
Reversing the Trend
In an effort to reverse the international trend of neglecting health services and care for people who are mentally ill, the World Health Organization (2013) presented their Mental Health Action Plan 2013 –2020 , calling for governments to find better ways to recognize mental illness and improve treatment. Similarly, the National Institutes of Mental Health (2008) in the United States, the Mental Health Commission of Canada (2012), the Mental Health Council of Australia (n.d.), and the government of the United Kingdom (Her Majesty’s Government, 2011) all developed strategic plans to improve mental health. It is hoped that these strategic plans will be the start of changing society’s perceptions of mental illness.
Diagnosis
Throughout history, the process of classifying different types of mental illness and finding ways to consistently explain people’s unusual experiences has taken many forms. Today, psychiatrists and clinicians use two manuals to classify and diagnose psychiatric disorders. The first is the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5, American Psychiatric Association, 2013). The DSM was first published in 1952 and has been revised several times. The DSM-5 is a reference manual that outlines a set of criteria that health professionals can use to provide consistent treatment of mental illness, including symptoms, descriptions, markers, and treatments of mental disorders.
The second is the International Statistical Classification of Diseases and Related Health Problems, 10th Revision (ICD-10, World Health Organization, 2010). This manual uses a system of coding created by the World Health Organization to track health interventions. More than 40% of people with intellectual disabilities are dually diagnosed with a psychiatric disorder (Cooper et al., 2007; Lunsky, Klein-Geltink, & Yates, 2013).
For clinicians to use diagnostic manuals and other assessment tools to accurately diagnose psychiatric disorders, they need in-depth descriptions of how clients are thinking, what their mood is like, and how they are behaving. Knowing that people with intellectual disabilities may not be able to provide all the information needed, their caregivers can contribute by sharing relevant observations. In the following sections, we highlight indicators known to be associated with common psychiatric disorders.
Indicators Associated with Disorders of Thinking
Schizophrenia
According to the World Fellowship of Schizophrenia and Allied Disorders (n.d.), schizophrenia strikes 1% of the world’s population. Individuals with intellectual disabilities are affected by schizophrenia more frequently than the general population, and they are more likely to be hospitalized for their symptoms (Balogh et al., 2010). Schizophrenia affects people from all walks of life and usually first appears between the ages of 15 and 30. Not everyone will experience the same symptoms, but some symptoms are common to many, such as withdrawing, hearing voices, talking to oneself, seeing things that are not there, neglecting personal hygiene, and showing low energy.
Schizophrenia refers to a group of severe, disabling psychiatric disorders marked by withdrawal from reality, illogical thinking, delusions (fixed false beliefs that cannot be changed through reasoning), hallucinations (hearing, seeing, smelling, tasting, or feeling touched by things that are not there), and flat affect (lack of observable expressions of emotions, monotone voice, expressionless face, immobile body).
Distinguishing between positive or hard, and negative or soft, symptoms is particularly important with schizophrenia. Notice that in this context, the word positive is not the same as good . Rather, positive symptoms are psychotic and demonstrate how the individual has lost touch with reality. Positive symptoms are those that do exist but should not exist, such as delusions, hallucinations, and disorganized thinking and behaviour. Delusions fall into several categories. An individual with a persecutory delusion may believe he is being tormented, followed, tricked, or spied on. An individual with a grandiose delusion may believe she has special powers. An individual with a reference delusion may believe that passages in books, newspapers, television shows, song lyrics, or other environmental cues are directed to him. In delusions of thought withdrawal or thought insertion, the individual believes others are reading her mind, her thoughts are being transmitted to others, or outside forces are imposing their thoughts or impulses on her.
In contrast, negative symptoms are those characteristics that should be there but are lacking. For example, negative symptoms include apathy (lack of interest in people, things, and activities), lack of motivation, blunted affect (reduced displays of emotion), poverty of speech (brief terse replies to questions that lack content), anhedonia (lack of interest in activities once enjoyed), and asociality (avoidance of relationships). Keep in mind that the inability to show emotion associated with a blunted affect does not reflect an inability to feel emotion. Similarly, it is helpful to understand that withdrawing from others is a coping mechanism for an individual with schizophrenia and not a rejection of those who initiate contact.
Indicators Associated with Disorders of Thinking: Schizophrenia
Key Points for Caregivers
1. With schizophrenia, all those who care for verbal clients must try to know the content of any hallucination or delusion their client is experiencing or has experienced. A typical hallucination might involve a deep, loud commanding male voice repeatedly telling a client that he or she is “worthless and must die.” Caregivers are expected to pose a clear question, such as “Are you hearing voices right now?” when clients are talking to themselves.
2. Caregivers of non-verbal clients must try to determine if hallucinations or delusions are occurring . Indicators of hallucinations include:
a. Staring to the side and nodding and making hand motions as though listening to a conversation others do not hear
b. Physically attacking an unseen other person
c. Staring with an angry or frightened expression at strangers or well-liked companions in a situation that does not require those emotions
d. Covering the eyes or ears as if to protect oneself, when threats are not present
e. Frowning as though something smelled or tasted foul
f. Dressing in protective ways such as wrapping on extra scarves or blankets when the weather does not require it
3. The nature of schizophrenia prevents the client from understanding that the voices or other hallucinations or delusions are not real. The only treatment that is effective in decreasing the thought changes in schizophrenia is antipsychotic medication. Without medication, trying to orient clients to reality can bring on profound anxiety and even aggressive actions. Caregivers must know what psychiatric medications have been prescribed .
4. The major side effects of psychiatric medication are a further challenge to treating schizophrenia. Hallucinations and delusions often include the belief that antipsychotic medication is poison . In response to this belief, psychotic clients frequently pretend to swallow pills, or cheek . Given this unchangeable reluctance to swallowing pills, clients often receive long-acting antipsychotic medications intramuscularly during monthly clinic visits. However, caregivers must always watch clients swallow their psychiatric medications .
Psychosis
Traditionally, explanations of psychiatric disorders were introduced as conditions considered either psychotic (not in touch with reality, such as schizophrenia) or neurotic (in touch with reality but deeply troubled, such as anxiety). However, as research extends knowledge within the field, the experience of psychosis is now understood to be associated with a number of non-psychiatric conditions. Both children and adults with intellectual disabilities may engage in self-talk and conversations with imaginary friends. These fantasy conversations can appear as though they are indicators of psychosis such as delusions or hallucinations. When clients are able to indicate some awareness that these conversations are imaginary, and when clients seem to be directing or controlling them, they are not considered indicators of psychosis. Psychotic symptoms may be present with clients who are dehydrated, experiencing seizure disorders, on new medications, or taking street drugs.
Indicators Associated with Disorders of Thinking: Psychosis
Key Points for Caregivers
- With psychosis, determine whether clients know that fantasy conversations are not real.
- Rule out dehydration , particularly in clients who are elderly.
- Ask whether the client has been having seizures or has a seizure disorder .
- Note whether a new medication or new medical condition is present.
- Document any street drug use, as many drugs induce psychotic-like behaviour.
Dementia
Dementia is not a specific disease; rather, it is a group of symptoms that affect thinking and social abilities enough to seriously interfere with daily functioning (Alzheimer’s Association, n.d.). Dementia occurs at a higher rate in people with intellectual disabilities than it does in the general population (Janicki & Dalton, 2000; Jokien et al., 2013; National Task Group on Intellectual Disabilities and Dementia Practices, 2013; Shooshtari et al., 2011). Adults with Down syndrome are particularly susceptible to Alzheimer’s disease as they approach middle age (Wilkinson & Janicki, n.d.).
Progressive memory loss that occurs over a period of time is often associated with Alzheimer’s disease and other dementias. However, in clients with intellectual disabilities, indicators may be seen first in symptoms of physical deterioration. Previously mastered skills with daily living may become a problem. Clients may have difficulty with their sight, hearing, and speech. They may remain inactive for long periods of time. Their gait (the way they walk) may change. They may lose their sense of balance and fall frequently. Seizures may occur in those previously unaffected. As their condition worsens, clients find it very hard to learn new information. They become disoriented and no longer recognize caregivers and family. During all stages of dementia, feelings of intense anxiety and fearfulness are common.
When memory loss occurs, most people feel frightened, frustrated, and angry. When they can’t answer a question they once could, they may confabulate (unintentionally make up) a reply. Keep in mind that confabulating is not lying and people are not deliberately being deceitful.
Sundowning, or sundown syndrome, is expected in clients with dementia. In this state, clients show increased agitation, restlessness, and confusion in the late afternoon, evening, and night (Khachiyants et al., 2011). At sundown, clients may become more anxious and disoriented. They may pace, wander, and even become aggressive and yell or scream as the day comes to a close. Clients can be more difficult to redirect at this time. It is important to note that dark, quiet rooms may be frightening to sundowning clients.
Indicators Associated with Disorders of Thinking: Dementia
Key Points for Caregivers
- Keep records of changes in behaviour and loss of previous abilities. This is especially important to health care workers diagnosing dementia. Therefore, maintaining clear records of how clients function throughout their adult life is useful. These records can be formal medical charts obtained from physicians, or they can be informal records obtained from family members or clients themselves. Scrapbooks, photo albums, video home movies, and audio recordings will illustrate how clients spoke, moved, and responded when they were well and healthy. Comparing a previous picture, audio, or video with a current cell phone capture of a client can clarify changes that are happening.
- Record specific examples of physical weakening, particularly in speaking and walking. For example, identify how a client’s gait (or how they walk) is changing. Keep a record of any falls, stumbles, or times when clients seem to lose their balance.
- Prepare for and seek medical help immediately if seizures occur. A seizure in someone previously unaffected and that is not part of a medical condition can indicate dementia.
- Understand and identify how often confabulation answers are occurring. Confabulation means making up answers without intending to lie. Knowing that clients with dementia are not lying or making things up on purpose can help caregivers understand that this behaviour is part of a psychiatric disorder and not acting-out behaviour.
- Understand and identify how often sundowning is occurring. Sundowning means increased agitation in the evening. Clients may seem to be expressing anger by behaviours such as screaming and yelling, but it is important to understand that this behaviour is likely a response to an intense and constant inner fear that clients are unable to escape.
- Create and communicate supports that help memory lapses. People with dementia are forgetful, get lost, and can have difficulty recognizing friends, family, and staff. Whenever possible, create cues to help with memory loss, such as calendars of events, photo collections, or scrapbooks. Ensure that these cues are consistent and that as many people as possible in the client’s life know about them.
Differentiating between Dementia, Delirium, Dehydration, and Depression
Dementia should not be confused with delirium, dehydration, or depression, even though all these conditions can leave clients feeling confused. Dementia, as explained above, is a progressive disorder where people become confused over a long period of time. Delirium is a disorder where people become confused quite suddenly and it can be caused by a medical condition such as an infection. Dehydration is also a condition where people become confused quite suddenly; it is caused by not taking in enough fluids. Depression, as explained in the following section, is a mood disorder where deep sadness can make people seem as though they are confused.
Indicators Associated with Disorders of Mood
Depression
Depression is a mood disorder where people feel very sad even when things are going well. Mood disorders are disturbances in the regulation of mood, behaviour, and affect that go beyond the normal ups and downs that most people experience. Mood refers to a pervading feeling (a feeling that is there all the time) that lasts for more than two weeks. With depression, a person’s mood becomes so intense that others just cannot help to cheer them up. Medications, such as antidespressant drugs, are needed to treat depression when it is serious enough to be diagnosed as a psychiatric disorder.
One out of every five people experiences depression during his or her life, and the rate of depression among people with intellectual disabilities is significantly higher (Shooshtari et al., 2011).  For example, about half of those with a depressive disorder also suffer from an anxiety disorder.
Health care providers rely heavily on descriptions of clients’ affect to diagnose depression. Affect refers to the outward expression of emotion on people’s faces, their bodies, and the way they speak. People cannot usually control their affect. Even when people who are depressed try to smile, their affect may seem “flat “or without any real emotion. People who are depressed sometimes have a restricted affect where they do not seem to show any emotion at all. They may also have an inappropriate affect where the emotions that they show do not match the content of what they are saying; for example, laughing when describing a loved one’s recent death. Affect can also be labile (change quickly) with no apparent connection to events in the environment.
Types of depression include major or unipolar depression, dysthymia, and bipolar affective disorder.
Major or unipolar depression is a mood disturbance in which the major symptoms—depressed mood and loss of interest or pleasure in all or almost all activities—occur daily for at least two weeks and severely incapacitate a person’s usual ability to function. In extreme cases, psychosis can be present in depressive disorders. When people with intellectual disabilities experience major depression, they may try to stay in bed all day, become mute, eat too little or too much, and refuse to wash or even use the toilet.
Major or unipolar depression can be further described as agitated or retarded . An agitated depression is accompanied by psychomotor agitation such as restlessness, sobbing, and excessive talking.
Depressive disorders can be exogenous or endogenous. Exogenous means that the illness follows a clearly defined stress such as a death or unexpected change in living arrangements. Endogenous means that no obvious stressors exist. It is important to emphasize that an endogenous major depression is NOT a reaction to loss. A lively approach by caregivers or others seeking to cheer up clients is ineffective.
Dysthymia is a mood disorder where people do feel sad for most of the day but they can still function, particularly toward the end of the day. Dysthymia can last for years and often goes unrecognized.
Bipolar affective disorder , formerly known as manic depressive illness, is marked by serious mood swings. Typically, clients experience extreme highs (mania or hypomania) alternating with extreme lows (depression). People feel normal only in the periods between the highs and lows. For some people, the cycles occur so rapidly that they hardly ever feel a sense of control over their mood swings.
In the manic phase of their bipolar illness, clients demonstrate grandiosity (feeling grand and better than others), jocularity (joking more than is appropriate), flight of ideas (rapidly skipping from one idea to the next in conversation), and decreased impulse control. They may go several days without sleeping at all. They may demonstrate problematic social behaviours such as aggression, property destruction, fecal smearing, stripping off clothing, and urinating in inappropriate places (McKee et al., 2004). People with intellectual disabilities may demonstrate grandiosity by believing they can drive a car when they cannot, for example. They may demonstrate decreased impulse control by spending all their money instead of saving in a planned way. They may engage in risky sexual activity that results in pregnancy or sexually transmitted diseases. They can be expected to neglect their needs to eat well, to sleep adequately, and to care for themselves.
Suicide
Thinking about suicide is common in people who are depressed. Caregivers must always ask if clients have ever thought about hurting or killing themselves, or otherwise try to assess clients who are or may be depressed. People who are thinking about killing themselves may engage in risky behaviour, such as walking into traffic with their eyes closed. They may seem preoccupied with the topics of death, of others who have died, and of funerals. It is important to ask verbal clients three questions:
- Have you considered taking your own life?

- What stops you?
In clients with little or no ability to express their thoughts of suicide, unusual behaviours may be associated with a wish to stop unrelenting mental distress by dying. A behaviour may not actually seem potentially lethal, such as manual self-strangulation. However, when people intend to die and believe what they are doing will cause death, their attempts to kill themselves must be taken seriously (Byrne, Hurley, and James, 2007)
It is important to emphasize that suicide gestures and attempts are sometimes made when a client appears to be getting better and caregivers believe that the depression has lifted. However, a brighter and more cheerful affect may signify the client’s feelings of relief. The client’s feelings about wanting to die may not have changed. In fact, when people begin to feel their energy returning, they are more able to follow through on their decision to end their life. Similarly, given that antidepressant medications can be expected to take 7 to 21 days to become fully effective, they may actually provide clients with the increase in energy needed to act on their suicidal ideas.
A further key consideration is to ensure that any items that might be used for self-harm (razors, knives, pills, belts, or cords) on a client’s person or in their environment are taken away. Keeping clients who are depressed and suicidal safe requires consistent monitoring. Increased caregiver time, attention, and presence are needed.
 A vegetative shift occurs when vegetative signs shift or change. Critical vegetative signs include affect, energy, pleasure, appetite, sleep, and thoughts of suicide. A memory aid for indicators of vegetative shift is “A E P A S S.”
Indicators Associated with Disorders of Mood: Vegetative Shift
Key Points for Caregivers
Vegetative shift is a term used by mental health professionals to describe problems associated with depressive disorders. A vegetative shift occurs when vegetative signs shift or change. Critical vegetative signs include affect, energy, pleasure, appetite, sleep, and thoughts of suicide. A memory aid for indicators of vegetative shift is “A E P A S S”: Affect, Energy, Pleasure, Appetite, Sleep, Suicide.
Use the “A E P A S S” memory aid to document a shift in vegetative signs.
- A–an Affect of sadness and poor concentration. Identify sad facial, postural, and emotional responses that last for weeks at a time and do not seem to fit the context. Sadness in clients may or may not be a response to something that has happened. Clients may show their depression through crying, becoming withdrawn and non-responsive, or becoming agitated and verbally and physically aggressive.
- E–Energy loss and fatigue. Write down the number of times clients are refusing to leave their beds, take care of activities of daily living such as bathing and dressing, and attend work or leisure activities. Include times when either constipation or incontinence occurs in relation to lacking the energy to go to the bathroom. Include the number of hours clients spend sitting or lying down for long periods of time just watching TV or doing nothing.
- P–Pleasure or interest. Notice whether clients are no longer taking pleasure in activities they once enjoyed. Notice also when clients with bipolar disorder are cycling into a manic or hyperactive state, when they may carry out pleasurable activities impulsively. In particular, monitor reckless behaviour in spending money and sexual activity.
- A–Appetite changes. Appetite can increase or decrease in depressive disorders. Carefully record client weight and identify any gains or losses. Tantrums at meals, and stealing or hoarding food for constant nibbling, can reflect a need to soothe the emotional distress of depression. In contrast, refusing to take the time to eat can reflect the hyperactivity associated with a manic state.
- S–Sleep disturbances. Keep a record of the number of hours clients sleep each night, including the time they go to bed, the number of times they get up in the night, and the time they wake up. Difficulty falling asleep can be an indicator of anxiety. Waking up early can be an indicator of depression. Sleeping in late can be a response to dysthymia, where people feel sad in the morning but then feel better as the day progresses. Remaining in bed and sleeping or dozing for long periods of time during both night and day can be a way of withdrawing that is often associated with depression. In contrast, going to bed late, getting up frequently in the night, and getting out of bed early can indicate a pattern of hyperactivity associated with a manic state.
- S–Suicidal thinking. Ask verbal clients if they have thought about taking their own life, if they have a plan to do so, and what (if anything) stops them. Assess if clients seem to speak often about death when such discussions are out of context.  Important: Remove any objects that could be used for self-harm.
Anxiety
Anxiety disorders are a group of conditions marked by pathological or extreme anxiety or dread. People with anxiety experience disturbances of mood, behaviour, and most systems in the body, making them unable to continue with everyday activities. Many feel anxious most of the time for no apparent reason.
Anxiety is different from fear. Fear is a response to an event or object that a person is aware of. The psychiatric disorder of anxiety occurs when the intensity and duration of anxiety does not match the potential for harm or threat to the affected person. Symptoms of anxiety disorders often include aches and pains throughout the body, and treating these can distract caregivers from recognizing anxiety as a cause of the physical pain.
People with intellectual disabilities struggle with anxiety more often than the general population. In part, this may be due to slightly different brain structures, limited expressive language abilities, and difficulty understanding social situations, which seem unpredictable and scary (Cooray & Bakala, 2005). Anxiety can be expressed with physical symptoms such as stomach aches, headaches, increased heart rate or pounding heart, feeling startled easily, trembling, sweating, and difficulty swallowing. It can be expressed behaviourally through increased agitation, crying, repeating comments about feeling afraid, withdrawing, regressing toward more childlike behaviour, clinging to caregivers, and freezing (not being able to move). As intellectually disabled people often depend on or feel they are dependent on others for their survival, their feelings of constant vulnerability can lead to frequent expressions of anxiety.
Treatment for verbal clients can include cognitive behavioural therapy. Anti-anxiety medications can help both verbal and non-verbal clients feel a much-needed sense of peace. However, repeatedly probing to find out what clients fear may not uncover a threat that others would recognize or believe is harmful. Therefore, finding ways to offer comfort to clients rather than simply seeking to understand a cause for their anxiety disorder is helpful.
Two disorders commonly suffered by people with intellectual disabilities are post-traumatic stress disorder and obsessive-compulsive disorder.
Post-traumatic stress disorder (PTSD) is a disorder in which an overwhelming traumatic event is re-experienced, causing intense fear, helplessness, horror, and avoidance of stimuli associated with the trauma. Nightmares, insomnia, flashbacks, mistrust, intense psychological distress, and self-medication with substance abuse are common in people with PTSD (Mayo Clinic, n.d.).
People with intellectual disabilities experience PTSD somewhat differently as they may not be able to describe the event that caused the trauma. Caregivers may not be aware of nightmares that are occurring. The flashbacks and memories may be vague and distorted and the memories confused. The flashbacks may seem so real that clients feel they are being traumatized again and again (Byrne, Hurley, & James, 2007). Conversely, flashbacks may be reported as additional traumatic events, with the result that records become inaccurate and make treatment difficult.
In one study, 75% of participants with mild to moderate intellectual disabilities had experienced at least one traumatic event during their lifespan, putting them at significant risk for PTSD (Martorell et al., 2009). Examples of traumas that intellectually disabled people face include physical and sexual abuse, loss of a parent, removal of children, involvement in vehicle or other collisions, and natural catastrophes. Reliving the trauma can result in acting-out behaviour (aggression or fighting back, inappropriate displays of sexuality, withdrawing in fear) that cannot be redirected.
Indicators Associated with Disorders of Mood: Anxiety
Post-traumatic Stress Disorder (PTSD)
Key Points for Caregivers
- Monitor sleeping and identify whether nightmares are occurring . When possible, identify what the nightmares are about.
- Record situations when clients seem fearful , particularly when there does not seem to be anything or anyone around who could hurt them. Notice if clients avoid certain situations or individuals by verbal and non-verbal resistance and behaviours.
- Provide records of known traumatic events to caregivers and family members. Exchange information about these events to determine whether they are new traumas or flashbacks .
Obsessive-compulsive disorder (OCD) , once thought to be simply a response to anxiety, is now considered a biological condition. Obsessions (the thinking component) are uncontrollable, persistent, and repeated thoughts or impulses. They frequently revolve around themes of contamination, aggression, orderliness, sex, and religion.
Compulsions (the behaving component) are the actions people take to try and relieve the pressure of the unrelenting obsessions. Compulsions are ritualistic patterns of behaviour often involving excessive cleaning, washing, counting, repeating, ordering, and arranging, or hoarding and checking. Repeated eye blinking, hair pulling, head banging, unusual behaviours when urinating or defecating, and self-mutilation are common.
People with OCD do not want to have obsessive thoughts, as those thoughts cause overpowering feelings of pressure and anxiety. They do not want to act on the thoughts either. However, because of the biological nature of the disorder, people are driven to believe that the only way to relieve the pressure of the obsessions is to keep acting on the compulsions. When people are in the process of acting on their compulsions, they can become very angry when their actions are interrupted. In essence, interruptions make the pressure and anxiety feel much worse.
People with intellectual disabilities often respond to interruptions of their ritualistic behaviour with aggressive retaliation such as hitting, spitting, and biting. They may not know why they must act this way, but they feel better when they do and desperately distressed when they do not.
In spite of a preoccupation with cleaning-centred behaviours, clients with this disorder may not achieve personal cleanliness and will present with extreme self-care deficits. Experts suggest that interrupting the behaviour simply increases the anxiety. With this in mind, caregivers can help by avoiding interrupting. For example, rather than interrupting a ritual such as repeated hand washing, a caregiver can ensure that hand lotion is available. With this disorder, the goal is not to prevent ritualistic behaviour, but to delay it.
The tricyclic antidepressant clomipramine (Anafranil) can have an anti-obsession effect. Side effects of the drug include a potential for seizures and withdrawal reactions.
Indicators Associated with Disorders of Mood: Anxiety
Obsessive-compulsive Disorder (OCD)
Key Points for Caregivers
- Delay rather than interrupt ritualistic behaviours. Expect that clients may respond with aggression when redirected away from their repetitive behaviours. When caregivers view the behaviours through the eyes of clients, caregivers can acknowledge a client’s belief that these behaviours are the only possible relief from unbearable anxiety.
- Prepare for and seek medical help immediately if seizures occur , particularly if clients have, or have the potential for, seizure disorders and are taking the antidepressant medication clomipramine (also called Anafranil).
- Make comfort measures available when ritualistic behaviours result in physical symptoms. For example, keep available hand lotion for excessive hand washing, and padding or helmets for head banging.
Indicators Associated with Disorders of Behaviour
Personality Disorders
In the preceding sections, we have discussed psychiatric disorders that include or could include elements of psychosis (not being in touch with reality). Clients diagnosed with a thought disorder such as schizophrenia can be expected to have psychotic symptoms before they are treated with antipsychotic medications. We now turn our attention to disorders of behaviour, in which psychotic breaks from reality are not expected.
A personality disorder occurs when personality traits—behaviour patterns that reflect how a person perceives and relates to others and self—become rigid, maladaptive, and fixed. The disorder primarily affects the way an individual gets along with others. In the general population, people with personality disorders have a great deal of difficulty getting along with others and functioning well. Their behaviour can be disturbing to those around them. Often a pattern of repeating the behaviour occurs in different situations, despite efforts to make changes. Personality disorders are believed to be present from a young age, and while medications may be helpful, there are no antipersonality disorder drugs available as there are antipsychotic and antidepressive drugs.
In the intellectually disabled population, clinicians continue to question whether the same diagnostic criteria for personality disorders should be used (Alexander & Cooray, 2003; Chester, 2010). In part, this is because people with intellectual disabilities (especially those who are moderately and severely disabled) can have difficulty getting along with others for a variety of reasons, many of which can be explained in context. Often, caregivers and family members can support clients toward changing their behaviours, but this is not always possible for people diagnosed with a personality disorder. Without diagnostic criteria that have been adapted to the unique needs of the intellectually disabled, clinicians are often reluctant to impose on clients the stigma that can come with this diagnosis.
Borderline Personality Disorder
Borderline personality disorder is characterized by displays of hostility, emotional dysfunction, mood swings, and aggression (Wilson, 2001). Clients who have this disorder can overreact to typical requests and events. They can be verbally aggressive in ways that are very disturbing to victims. This verbal aggression can quickly lead to physical aggression. They frequently engage in self-injuring and suicidal behaviour.
Clients with borderline personality disorder do not seem able to see the connection between their behaviour and consequences. They may have encounters with the justice system. They tend to view people, circumstances, and events in their lives as either all good or all bad—a tendency that is referred to as splitting . This tendency can be destructive in that clients often polarize the people working with them into groups of either “good” or “bad.” Clients may become overattached to some people and make serious accusations against others. It is critical for caregivers to avoid siding with clients against other people or caregivers. Let clients know their accusations will be recorded, discussed, and acted on professionally during team meetings. It is important to ensure that communication with these clients does not suggest an alignment of the caregiver and client against any other member of the team.
Indicators Associated with Disorders of Behaviour: Borderline Personality Disorder
Key Points for Caregivers
- Gather as much information as possible about clients’ patterns of overreacting and behaviour that is harmful to self or others. Identify if the patterns began occurring at a young age and whether any elements of the behaviour have changed and improved over time and with support.
- With borderline personality disorder, consistent communication among caregivers is critically important. Divisions among caregivers can occur if clients split or view individuals around them only as all good or all bad.
Conclusion
In this chapter we have discussed the psychiatric disorders of thinking, mood, and behaviour. These disorders cause so much distress that clients cannot function and enjoy their lives. We have described key indicators of these disorders and emphasized relevant observations that caregivers must watch for and attend to.
Disorders of thinking include schizophrenia, dementia, and psychosis. Disorders of mood include depression (for example, major depression or bipolar affective disorder) and anxiety (for example, post-traumatic stress disorder and obsessive-compulsive disorder). Disorders of behaviour include borderline personality disorder. None of the indicators that are present with these disorders are displayed on purpose. Just as the nature of physical illnesses causes people’s minds and bodies to act in certain ways, mental illness does as well. Understanding the links between difficult behaviours and diagnosed psychiatric disorders can make all the difference in supporting dually diagnosed clients toward success.
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