7.1: Mathematical Continuum Descriptions of Membranes
Membranes play an important role in cellular functions. They act as barriers to the outside environment, but also must allow the exchange of nutrients and waste. They must also be physically flexible to allow for cellular growth and movement, such as during cell division and migrations [1]. Membranes are composed of phospholipid bilayers.  The membrane is composed of two adjacent layers of lipids, with the tails facing each other, and the hydrophilic heads facing the outside aqueous environment (Figure \(\PageIndex{1}\)A). These membranes have a high aspect ratio with a low thickness of about 5 nm [1]. This is since the membrane is only two lipid molecules thick. However, the hydrophobic interior of the bilayer is a strong barrier against the transport of hydrophilic molecules across the membrane. Diffusion across the membrane is very slow for hydrophilic molecules and ions (transport requires transporter proteins) (Figure \(\PageIndex{1}\)B). In addition, lipids can flip between leaflets, but this is very slow, as it requires the hydrophilic head pass through the hydrophobic region (Figure \(\PageIndex{1}\)B). Lateral diffusion can be quite quick for lipids (~ 1 µm/s 2 and even transmembrane proteins (0.1 µm/s 2 ) (Figure \(\PageIndex{1}\)B). Due to the described properties of the cellular membranes, they are commonly modeled as a 2-dimensional planar, elastic sheet in a 3-dimensional space (Figure \(\PageIndex{2}\)). Biological structures like vesicles and even entire cells can be described as a closed form made by these sheets. These descriptions are known as continuum descriptions of the membrane. This is in contrast to molecular- and particle-based descriptions of the membrane, which focus on smaller time scales and length scales of membranes.
To understand the shape and dynamics of membranes, we will need to describe the geometry of the membrane surface itself. In addition, we will describe how the energy changes for different membrane configurations. In particular, we will look at how the energy changes when a membrane is stretched, bent, sheared and changed in the thickness (Figure \(\PageIndex{2}\)).
Area Changes
The area deformation can easily be parameterized by a single variable \(\Delta a\) or a function \(\Delta a(x, y) \) if the area deformation is dependent on position. We can treat the lipids in the membrane as a system of connected elastic springs, with increasing the area equivalent to stretching the springs (Figure \(\PageIndex{3}\)).
As such, the energy of stretching a piece of membrane is:
\[E_{a}=\frac{1}{2} \frac{K_{a} \Delta a^{2}}{a_{0}}\]
where \(K_a\) is the area modulus, which is intrinsic to the membrane, and \(a_0\) is some reference area from which change in area is calculated (i.e. \(\Delta a=a-a_{0}\)). This equation is equivalent to a Hookean spring. Of course, when looking at a whole system, this term has to be integrated over the total area:
\[E_{a}=\frac{K_{a}}{2} \int (\frac{ \Delta a}{a_{0}})^2 da \]
If the area change is constant over the surface of the membrane, then the integral reduces to Equation (1).
The area modulus has values of about 55-70 k B T or 230-290 mN/m [1].
Thickness Changes
Membrane compression can simply be described by a function \(w(x,y)\) that outputs the thickness at point \((x,y)\). Again, we can model the thickness change as a system of springs (Figure \(\PageIndex{4}\)). If the half-width of the membrane (essentially the lipid length in the membrane at equilibrium) is defined as \(w_0\), and the thickness is \(2w_0\) (Figure \(\PageIndex{4}\)), then we get a similar equation for the energy penalty of changing the thickness:
\[E_{\text { thickness }}=\frac{K_{t}}{2} \int\left(\frac{w(x, y)-w_{0}}{w_{0}}\right)^{2} d a\]
where \(K_t\) is the stiffness for thickness variation, which is intrinsic to the membrane. It has units of energy/area and is approximately 60 k B T/nm 2 [1].
Shear Changes:
Pure phospholipid membranes are essentially two-dimensional fluids. Due to the fluid nature of phospholipid bilayer, the membrane cannot support a shear strain and does not have a shear modulus. Therefore, as seen in Figure \(\PageIndex{2}\), static shear change does not deform a fluid membrane and does not cost any energy . However, most biological systems are not pure fluid membranes. For example, red blood cells have an underlying cytoskeletal network, which can support shear. If the shear elasticity is an important effect that needs to be modeled, it must be accounted for to model the equilibrium shapes and the dynamics of such cells. An example of this is described later.
An Interlude on Membrane Geometry
To understand how the energy costs of bending are determined, it is important to describe the geometry of the membrane.
If we take a reference two-dimensional plane, we can define a function \(h(x,y)\) that characterizes the height of the membrane above the reference plane at point \((x,y)\). In most situations, this can accurately represent the shape of the membrane. This representation is known as the Monge parameterization (Figure \(\PageIndex{5}\)). From the Monge parameterization, we can now calculate the curvature of a membrane. This is important as the free energy cost of bending the membrane is dependent on the curvature.
In a single dimension, if we have a smooth function \(y=h(x)\), then at a point \(x\) we can define the curvature \(\kappa (x) \) to be the inverse of the radius of a circle that closely approximates the curve at point \(x\) (Figure \(\PageIndex{6}\)A):
\[\kappa=\frac{1}{R}\]
It might be expected that a straight line has a curvature of 0. Indeed, since the circle that approximates the straight line is that of an infinite radius, the curvature is \( \kappa=\frac{1}{\infty}=0 \) (Figure \(\PageIndex{7}\)).
To generalize this to the two-dimensional case, we could cut through the two-dimensional surface with a plane, obtaining a curve, from which we can calculate the curvature. However, depending on the orientation of the plane used to cut the surface, the value of the curvature for that orientation is different. It can be shown that the curvatures in two directions describe a single point on a surface. Specifically, there is one particular choice of orthogonal planes that the curvatures take extreme values. These values \(c_1 , c_2 \) are known as the principal curvatures (Figure \(\PageIndex{6}\)B). In fact, one can calculate the curvature in any other direction using the formula:
\[c_{X}=c_{1} \cos ^{2} \alpha_{X}+c_{2} \sin ^{2} \alpha_{X}\]
where \(c_{x}\) is the curvature in the direction X, and \(\alpha_{X}\) is the angle between the direction X and the direction belonging to curvature \(c_{1}\). These statements are known as the Euler Curvature Formula.
As an example of principal curvatures, the principal curvatures of a cylinder would be \(1/R\) around the circumference of the cylinder (as the radius of curvature is \( R \), but \(0\) along the length of the cylinder.
Other curvature quantities calculated from the principal curvatures are also commonly defined:
\[ \begin{array}{c}{K=c_{1}+c_{2}} \\ {K_{G}=c_{1} c_{2}} \\ {H=\frac{K}{2}=\frac{c_{1}+c_{2}}{2}}\end{array} \]
where \( K \) is the is the extrinsic curvature, \(G\) is the Gaussian curvature, and \(H\) is the mean curvature.
With the Monge parameterization, the curvatures can be expressed in terms of the height function. It can be shown that the extrinsic curvature is:
\[K=\nabla \cdot\left(\frac{\nabla h(\boldsymbol{r})}{\sqrt{1+(\nabla h(\boldsymbol{r}))^{2}}}\right)\]
For a small gradient, the extrinsic curvature can be approximated as \( K \approx \Delta h(\boldsymbol{r}) \).
Why is the Monge parameterization and expressing the curvature using the parameterization important? These are important as the bending energy is dependent on the curvature, and to determine the equilibrium shapes of a membrane system, the height function that minimizes the total free energy of the system will be determined to represent the final membrane shape.
It is important to note that Monge parameterization is not the only way of describing a surface. It is not the most general parameterization or the most convenient parameterization in many cases. It will not work if there are folds or overlaps of the surface.
This is only an introduction to membrane/surface geometry, which itself is only a small part of the field of differential geometry. For more information on differential geometry in relation to membranes, the reader is referred to [3]. For a more general and advanced introduction to differential geometry, the reader is referred to [4].
Bending Changes – Helfrich theory:
The bending energy of a membrane is described by the well-known Helfrich theory. The Helfrich bending energy is defined in terms of the extrinsic and Gaussian curvature:
\[ E_{\text {bend}}=\int\left(\frac{1}{2} \kappa_{b}\left(c_{1}+c_{2}-c_{0}\right)^{2}+\overline{\kappa}_{b} c_{1} c_{2}\right) d A=\int\left(\frac{1}{2} \kappa_{b}\left(K-c_{0}\right)+\overline{\kappa}_{b} G\right) d A \]
where \(\kappa_b\) is the bending rigidity and \(\overline{\kappa}_{b}\) is the Gaussian bending rigidity. This equation also contains the spontaneous curvature \(c_0\). It is nonzero if there is some curvature in the membrane at an equilibrium state, which occurs if there is an asymmetric bilayer (different lipid composition for each monolayer). Typical values of \(\kappa_b\) is 10-20 k B T [1].
It is important to note that that Gaussian bending term can be ignored in most cases due to the Gauss-Bonnet theorem. The Gauss-Bonnet theorem says that if the topology of the system remains unchanged, the integral over the Gaussian curvature will be a constant. The topology (genus) can be determined by the number of holes/handles the surface has. For example, a sphere has no holes and therefore has a genus of 0, but a donut has one hole and a genus of 1. Therefore, unless the genus of the system changes, the Gaussian term vanishes. This is the case for many biological membranes. However, for complicated morphologies of the endoplasmic reticulum and mitochondria, Gaussian curvature could have a significant contribution as membrane holes, tubes, and tori are created and destroyed.
We can simplify the above equation for Monge parameterization, with no spontaneous curvature. It can be shown for the Monge parameterization that the area element is \( d A=d x d y \sqrt{1+(\nabla h)^{2}} \). For the small gradient approximation, this is equivalent to \( d A=\left(1+\frac{1}{2}(\nabla h)^{2}\right) d x d y \) (based on a simple Taylor expansion). We can also penalize area stretching with a term \( \frac{1}{2} \sum(\nabla h)^{2} \) where \(\Sigma \) is membrane tension. Therefore, a Helfrich energy equation for a Monge parameterization, under the small gradient approximation, and includes bending and tension is:
\[ E_{b e n d}=\frac{1}{2} \int d x d y\left[\kappa_{b}(\Delta h)^{2}+\Sigma(\nabla h)^{2}\right] \]
Vesicles and Free Energy Cost
Vesicles are spherical lipid bilayer structures. Vesicles are used for many cellular processes for transporting large amounts of cargo within cells or even between cells. An example of the latter is the transport of neurotransmitters between neurons at the synaptic cleft.
We can calculate the free energy cost of the vesicles. The principal curvatures are the same: \( 1/R \) Therefore, the free energy cost is:
\[ E_{v e s i c l e}=\frac{\kappa_{b}}{2} \int\left(\frac{2}{R}\right)^{2} d A=\frac{\kappa_{b}}{2}\left(\frac{2}{R}\right)^{2} 4 \pi R^{2}=8 \pi \kappa_{b} \]
Interestingly, this means there is a fixed energy cost to create a vesicle regardless of the size of the vesicle. If \(\kappa_{b} \approx 10 k_{B} T \) then \( E_{\text {vesicle}} \approx 250 \mathrm{k}_{B} T \), which means there must be an active mechanism for the formation of vesicles in cells. Indeed vesicle-coating proteins like clathrin decrease this energy barrier such that vesicle formation is favorable.
Applications
Membrane tethers
When a pulling force is applied to a single point on a vesicle, it will generate an extruded membrane tubule known as membrane tethers. Membrane tethers have many biological implications. For example, neutrophils can use membrane tethers to attach to the endothelial wall when rolling. Membrane tethers also have an important role in the endomembrane system. In addition, the energetics and dynamics of membrane tethers are similar to that of other biological processes, such as endocytosis, and actin projections during cell motility.
Experiments use optical tweezers or molecular motors to apply a force at a single point on the membrane. We will use a simplified model (Figure \(\PageIndex{8}\)) of the process where we have a sphere of radius \(R\) attached to a cylinder of radius \(r\) and length \(L\). At the end of the cylinder is a hemispherical cap.
We can write the free energy of the vesicle+tether system by calculating the contributions of bending, area stretching, and compression. The bending energy of the system is:
\[ \begin{array}{c}{E_{\text {bending}}=8 \pi \kappa_{b}+\pi \kappa_{b} \frac{L}{r}+4 \pi \kappa_{b}} \\ {=12 \pi \kappa_{b}+\pi \kappa_{b} \frac{L}{r}}\end{array} \]
The first term corresponds to the spherical vesicle (as demonstrated previously). The second term corresponds to the cylinder, which is calculated by squaring the sum of the principal curvatures \( \left(0+\frac{1}{r}\right)^{2} \) and multiplying by the area. The last term corresponds to the hemispherical cap.
Area stretching also contributes to the free energy of the system:
\[ E_{\text {area}}=\frac{K_{a}}{2} \frac{\left(a-a_{0}\right)^{2}}{a_{0}} \]
where \( a_0 \) is the reference area and \( a=4 \pi R^{2}+2 \pi r L \) is the area of the system. The first term corresponds to a sphere of radius \( R\) and the second term corresponds to a cylinder of raidus \( r \). Note that the area of the hemispherical cap \( 4 \pi r^2 \) is ignored as it is assumed \( r \ll R, L \).
Note that this system has no thickness change of the membrane and therefore does not contribute to the free energy of the system.
In the case of a closed bilayer like a vesicle, the pressure difference across the membrane \( \Delta p \) also contributes to the energy of the system:
\[ E_{p V}=-\Delta p \int d V=-\Delta p\left(\frac{4}{3} \pi R^{3}+r^{2} \pi L\right) \]
Again, the contribution of the hemispherical cap is ignored. The minus sign is present because excess internal pressure increases the volume.
Finally, the force pulling the membrane must also contribute to the free energy of the system:
\[ E_{l o a d}=-f L \]
where \( f \) is the force applied. The minus sign is present because increased force leads to a high tether length.
Summing equations 11, 12, 13, and 14 together, we get:
\[ E_{t o t}=12 \pi \kappa_{b}+\pi \kappa_{b} \frac{L}{r}+\frac{K_{a}}{2} \frac{\left(a-a_{0}\right)^{2}}{a_{0}}-\Delta p\left(\frac{4}{3} \pi R^{3}+r^{2} \pi L\right)-f L \]
At equilibrium, the total energy is minimized. Therefore, the equilibrium shape can be determined by minimizing the free energy.
In this case, we can take the derivatives of Equation 15 with respect to each of the variables to minimize the energy.
Note that
\[ \frac{\partial E_{\text {area}}}{\partial R}=K_{a} \frac{a-a_{0}}{a_{0}} \frac{\partial a}{\partial R}=\tau 8 \pi R \]
where \( \tau \) is the membrane surface tension, which can be related to area stretching by the formula \( \tau = \frac{K_{a}\left(a-a_{0}\right)}{a_{0}} \). Similarly, it can be shown that \( \partial E_{\text {area}} / \partial r=\tau 2 \pi L \) and \( \partial E_{\text {area}} / \partial L=\tau 2 \pi r \). Therefore:
\[ \begin{array}{l}{\frac{\partial E_{t o t}}{\partial R}=0 \rightarrow 8 \pi \tau R-4 \pi \Delta p R^{2}=0 \rightarrow \Delta p=\frac{2 \tau}{R}} \\ {\frac{\partial E_{t o t}}{\partial r}=0 \rightarrow-\pi \kappa_{b} \frac{L}{r^{2}}+2 \pi \tau L-2 \pi \Delta p r L=0} \\ {\frac{\partial E_{t o t}}{\partial L}=0 \rightarrow \pi \kappa_{b} \frac{1}{r}+2 \pi \tau r-\pi \Delta p r^{2}-f=0}\end{array} \]
The first minimization leads to an equation known as the Laplace-Young relation, which is a well-known equation that relates the pressure difference \( \Delta p \) across a fluid interface (here the membrane) and the surface tension \( \tau \). A more general form takes into account both radii of curvature:
\[ \Delta p=\tau\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right) \]
We can plug in the Laplace-Young relation into the other minimized equations:
\[ \begin{array}{l}{-\pi \kappa_{b} \frac{L}{r^{2}}+2 \pi \tau L-\frac{4 \pi \tau r L}{R}=0} \\ {\pi \kappa_{b} \frac{1}{r}+2 \pi \tau r-f-\frac{2 \pi \tau r^{2}}{R}=0}\end{array} \]
Since \( r \ll R \) (Figure \(\PageIndex{8}\)) the last terms can be ignored:
\[ 2 \pi \tau L=\pi \kappa_{b} \frac{L}{r^{2}} \]
\[ r=\sqrt{\frac{\kappa_{b}}{2 \tau}} \]
\[ \pi \kappa_{b} \frac{1}{r}+2 \pi \tau r-f=0 \rightarrow \pi \kappa_{b} \sqrt{\frac{2 \tau}{\kappa_{b}}}+2 \pi \tau \sqrt{\frac{\kappa_{b}}{2 \tau}}=f \]
\[ f=2 \pi \sqrt{2 K_{b} \tau} \]
We now get a relation between the radius of tether, bending energy, and membrane tension. In addition, we get a relation between the force applied, bending energy and tension. Similar relations can actually be used to determining the bending rigidity \( \kappa_{b} \) using a micropipette aspiration and optical tweezers used to make membrane tethers [5].
A general framework of determining equilibrium shapes (variational calculus):
The previous section looked at how relationships between meaningful physical quantities can be derived if the shape of the system is known. This was done by calculating the free energy cost for the membrane shape and minimizing it with respect to the variables of the system. However, a different approach can also be taken. What if the physical quantities of the system are known, but the equilibrium shapes need to be determined? This can be done by minimizing an energy functional of the system, which is the basis of variational calculus.
We can define a functional as a function that assigns a scalar value for a given function. In this case, we can think of the free energy of the system \( E_{tot} \) as a functional that is dependent on the height function \( h(x,y) \). Variational calculus provides tools to find functions that minimize the functionals. This is exactly what we need to determine the equilibrium shapes! Variational calculus can be used to find the shape function \( h(x,y) \) that minimizes the energy \( E_{tot} \)
In variational calculus, in a one-dimensional case, the minimization of the functional:
\[ J[y]=\int L\left(x, y, y^{\prime}\right) d x \]
is solved by the solution of the differential equation:
\[ \frac{\partial L}{\partial f}-\frac{d}{d x} \frac{\partial L}{\partial f^{\prime}}=0 \]
where \( y=f(x) \) minimizes the functional. This equation is known as the Euler-Lagrange equation. Indeed, the equation for \( E_{tot} \) has this form.
As a side note, many equations in physics have this form and the Euler-Lagrange equation is used. In fact, the dynamics of a physical system can be represented as a functional known as the Lagrangian, and solving the Euler-Lagrange equations for this Lagrangian lead to the equations of motion. This approach to classical mechanics is known as Lagrangian mechanics.
Going back to membranes, if we use Equation (9), we can get the following Euler-Lagrange equation:
\[ \kappa \Delta \Delta h-\Sigma \Delta h=0 \]
This partial differential equation can be solved for the height function that describes the shape of the membrane at equilibrium. This is called the shape equation.
Minimization of red blood cells:
While solving the shape equations are non-trivial and won’t be shown here, some results in the case of red blood cells will instead be shown.
It is important to note that there are often two additional energy terms added to the Helfrich bending energy to obtain the total free energy for red blood cells.
When an initially flat membrane is bent, the outer monolayer is stretched, and the inner monolayer is compressed and the number of lipids per unit area is higher for the inner surface and lower for the outer surface. The energy cost of this is added to the model as the following model:
\[ E_{A D E}=\frac{\kappa_{A D E}}{2} \frac{\pi}{A D^{2}}\left(\Delta A-\Delta A_{0}\right)^{2} \]
where \(\kappa_{ADE} \) is a separate bending energy, which a material parameter, \(A\) is the membrane area, and \(D\) is the membrane thickness. This is known as the area-difference-elasticity (ADE) model.
In addition, an elastic term is added of the form:
\[ E_{e}=\frac{K_{A}}{2} \oint \alpha^{2} d A+\mu \oint \beta d A \]
where \( \alpha^{2} \) and \(\beta^{2} \) are the local area and strain invariants, \( K_{A} \) and \( \mu \) are elastic moduli for stretching and strain, respectively.
When the total free energy \( E_{b e n d}+E_{A D E}+E_{e} \) is minimized, the equilibrium shapes of the red blood cells can be determined. The below Figure shows a comparison of the theoretically determined shapes and those that have been found experimentally.
References:
- Phillips, R.; Kondev, J.; Theriot, J.; Garcia, H. G. Physical Biology of the Cell, 2013 , Garland Science, Taylor & Francis Group, LLC: Abingdon, UK.
- Boal, D. Mechanics of the Cell. Cambridge University Press : Cambridge, UK. 2002
- Deserno, Markus. Fluid Lipid Membranes – A Primer. Max-Planck-Institut fu ̈r Polymerforschung. Mainz, Germany. 2006
- Deserno, Markus. Notes on Differential Geometry. Max-Planck-Institut fu ̈r Polymerforschung. Mainz, Germany. 2004
- Bo, Lin, and Richard E. Waugh. “Determination of Bilayer Membrane Bending Stiffness by Tether Formation from Giant, Thin-Walled Vesicles.” Biophysical Journal , vol. 55, doi:10.1016/S0006-3495(89)82844-9. Accessed 13 June 2019.
- Li, Xuejin, et al. Continuum- and Particle-Based Modeling of Shapes and Dynamics of Red Blood Cells in Health and Disease. Soft Matter , vol. 9, no. 1, 2013, pp. 28–37., doi:10.1039/c2sm26891d.
- W., G. Lim H., et al. “Stomatocyte-Discocyte-Echinocyte Sequence of the Human Red Blood Cell: Evidence for the Bilayer- Couple Hypothesis from Membrane Mechanics.” Proceedings of the National Academy of Sciences , vol. 99, no. 26, 2002, pp. 16766–16769., doi:10.1073/pnas.202617299.


7: Vitamins
A vitamin is an organic compound and a vital nutrient that an organism requires in limited amounts. An organic chemical compound (or related set of compounds) is called a vitamin when the organism cannot synthesize the compound in sufficient quantities, and it must be obtained through the diet; thus, the term "vitamin" is conditional upon the circumstances and the particular organism.
-
- 7.1: Vitamins: Basic Concepts
- Vitamins are organic compounds (meaning they contain Carbon) and they are vital nutrients that organisms require in limited amounts. Just like other essential nutrients, vitamins must be obtained from the diet since the organism can not make them (or can not make enough of them) for survival. Interestingly, not all organisms have the same vitamin requirements.  What makes them "fat soluble" is the fact that they dissolve in fats and oils but they won't dissolve in water. If you've ever seen a vitamin E capsule, you already know that inside the capsule is an oily substance containing vitamin E. The Fat soluble vitamins are present in many foods but they are especially high in foods like nuts and seeds that contain a high concentration of fats and oils.
-
- 7.3: Water Soluble Vitamins
- Nine vitamins are considered water soluble, including: vitamin C and all of the B vitamins (Riboflavin, Niacin, Thiamin, B₆ , Folate, B₁₂, Pantothenic Acid and Biotin). In contrast to the four fat-soluble vitamins, water-soluble vitamins dissolve easily in water and, in general, are readily excreted from the body, to the degree that urinary output is a strong predictor of vitamin consumption. Because they are not as readily stored, more consistent intake is important.


Skip to main content
-
-
9.1: Introduction to Muscle Tissue
-
When most people think of muscles, they think of the muscles that are visible just under the skin, particularly of the limbs. These are skeletal muscles, so-named because most of them move the skeleton. But there are two other types of muscle in the body, with distinctly different jobs.
-
-
9.2: Overview of Muscle Tissues
-
Muscle is one of the four primary tissue types of the body, and the body contains three types of muscle tissue: skeletal muscle, cardiac muscle, and smooth muscle (Figure 10.1.1). All three muscle tissues have some properties in common; they all exhibit a quality called excitability as their plasma membranes can change their electrical states (from polarized to depolarized) and send an electrical wave called an action potential along the entire length of the membrane.
-
-
9.3: Skeletal Muscle
-
Skeletal muscles act not only to produce movement but also to stop movement, such as resisting gravity to maintain posture. Small, constant adjustments of the skeletal muscles are needed to hold a body upright or balanced in any position. Muscles also prevent excess movement of the bones and joints, maintaining skeletal stability and preventing skeletal structure damage or deformation. Joints can become misaligned or dislocated entirely; muscles work to keep joints stable.
-
-
9.4: Muscle Fiber Contraction and Relaxation
-
The sequence of events that result in the contraction of an individual muscle fiber begins with a signal—the neurotransmitter, ACh—from the motor neuron innervating that fiber. The local membrane of the fiber will depolarize as positively charged sodium ions (Na+) enter, triggering an action potential that spreads to the rest of the membrane will depolarize, including the T-tubules.
-
-
9.5: Types of Muscle Fibers
-
There are three main types of skeletal muscle fibers. Slow oxidative fibers contract relatively slowly and use aerobic respiration (oxygen and glucose) to produce ATP. Fast oxidative fibers have fast contractions and primarily use aerobic respiration, but because they may switch to anaerobic respiration (glycolysis), can fatigue more quickly than SO fibers. Lastly, fast glycolytic fibers have fast contractions and primarily use anaerobic glycolysis - these fatigue more quickly than the others.
-
-
9.6: Development and Regeneration of Muscle Tissue
-
Most muscle tissue of the body arises from embryonic mesoderm. Paraxial mesodermal cells adjacent to the neural tube form blocks of cells called somites. Skeletal muscles, excluding those of the head and limbs, develop from mesodermal somites, whereas skeletal muscle in the head and limbs develop from general mesoderm. Somites give rise to myoblasts. A myoblast is a muscle-forming stem cell that migrates to different regions in the body and then fuse(s) to form a syncytium, or myotube.


11.29: Reading- Pricing Objectives
Introduction
Companies set the prices of their products in order to achieve specific objectives. Consider the following examples:
In 2014 Nike initiated a new pricing strategy. The company determined from a market analysis that its customers appreciated the value that the brand provided, which meant that it could charge a higher price for its products. Nike began to raise its prices 4–5 percent a year. Footwear News reported on the impact of their strategy:
“The ability to raise prices is a key long-term advantage in the branded apparel and footwear industry—we are particularly encouraged that Nike is able to drive pricing while most U.S. apparel names are calling for elevated promotional [and] markdown levels in the near-term,” said UBS analyst Michael Binetti. Binetti said Nike’s new strategy is an emerging competitive advantage. [1]
Nike’s understanding of customer value enabled it to raise prices and achieve company growth objectives, increasing U.S. athletic footwear sales by $168 million in one year.
In 2015 the U.S. airline industry lost $12 billion in value in one day because of concerns about potential price wars. When Southwest Airlines announced that it was increasing its capacity by 1 percent, the CEO of American Airlines—the world’s largest airline—responded that American would not lose customers to price competition and would match lower fares. Forbes magazine reported on the consequences:
This induced panic among investors, as they feared that this would trigger a price war among the airlines. The investors believe that competing on prices would undermine the airline’s ability to charge profitable fares, pull down their profits, and push them back into the shackles of heavy losses. Thus, the worried investors sold off stocks of major airlines, wiping out nearly $12 billion of market value of the airline industry in a single trading day. [2]
Common Pricing Objectives
Not surprising, product pricing has a big effect on company objectives. (You’ll recall that objectives are essentially a company’s business goals.) Pricing can be used strategically to adjust performance to meet revenue or profit objectives, as in the Nike example above. Or, as the airline-industry example shows, pricing can also have unintended or adverse effects on a company’s objectives. Product pricing will impact each of the objectives below:
- Profit objective: For example, “Increase net profit in 2016 by 5 percent”
- Competitive objective: For example, “Capture 30 percent market share in the product category”
- Customer objective: For example, “Increase customer retention”
Of course, over the long run, no company can really say, “We don’t care about profits. We are pricing to beat competitors.” Nor can the company focus only on profits and ignore how it delivers customer value. For this reason, marketers talk about a company’s “orientation” in pricing. Orientation describes the relative importance of one factor compared to the others. All companies must consider customer value in pricing, but some have an orientation toward profit. We would call this profit-oriented pricing.
Profit-Oriented Pricing
Profit-oriented pricing places an emphasis on the finances of the product and business. A business’s profit is the money left after all costs are covered. In other words, profit = revenue – costs. In profit-oriented pricing, the price per product is set higher than the total cost of producing and selling each product to ensure that the company makes a profit on each sale.
The benefit of profit-oriented pricing is obvious: the company is guaranteed a profit on every sale. There are real risks to this strategy, though. If a competitor has lower costs, then it can easily undercut the pricing and steal market share. Even if a competitor does not have lower costs, it might choose a more aggressive pricing strategy to gain momentum in the market.
Also, customers don’t really care about the company’s costs. Price is a component of the value equation, but if the product fails to deliver value, it will be difficult to generate sales.
Finally, profit-oriented pricing is often a difficult strategy for marketers to succeed with, because it limits flexibility. If the price is too high, then the marketer has to adjust other aspects of the marketing mix to create more value. If the marketer invests in the other three Ps—by, say, making improvements to the product, increasing promotion, or adding distribution channels—that investment will probably require additional budget, which will further raise the price.
It’s fairly standard for retailers to use some profit-oriented pricing—applying a standard mark-up over wholesale prices for products, for instance—but that’s rarely their only strategy. Successful retailers will also adjust pricing for some or all products in order to increase the value they provide to customers.
Competitor-Oriented Pricing
Sometimes prices are set almost completely according to competitor prices. A company simply copies the competitor’s pricing strategy or seeks to use price as one of the features that differentiates the product. That could mean either pricing the product higher than competitive products, to indicate that the firm believes it to provide greater value, or lower than competitive products in order to be a low-price solution.
This is a fairly simple way to price, especially with products whose pricing information is easily collected and compared. Like profit-oriented pricing, it carries some risks, though. Competitor-oriented pricing doesn’t fully take into account the value of the product to the customer vis-à-vis the value of competitive products. As a result, the product might be priced too low for the value it provides, or too high.
As the airline example illustrates, competitor-oriented pricing can contribute to a difficult market dynamic. If players in a market compete exclusively on price, they will erode their profits and, over time, limit their ability to add value to products.
Customer-Oriented Pricing
Customer-oriented pricing is also referred to as value-oriented pricing. Given the centrality of the customer in a marketing orientation (and this marketing course!), it will come as no surprise that customer-oriented pricing is the recommended pricing approach because its focus is on providing value to the customer. Customer-oriented pricing looks at the full price-value equation (Figure 1, above; discussed earlier in the module in “Demonstrating Customer Value”) and establishes the price that balances the value. The company seeks to charge the highest price that supports the value received by the customer.
Customer-oriented pricing requires an analysis of the customer and the market. The company must understand the buyer persona, the value that the buyer is seeking, and the degree to which the product meets the customer need. The market analysis shows competitive pricing but also pricing for substitutes.
In an attempt to bring the customer voice into pricing decisions, many companies conduct primary market research with target customers. Crafting questions to get at the value perceptions of the customer is difficult, though, so marketers often turn to something called the Van Westerndorp price-sensitivity meter. This method uses the following four questions to understand customer perceptions of pricing:
- At what price would you consider the product to be so expensive that you would not consider buying it? (Too expensive)
- At what price would you consider the product to be priced so low that you would feel the quality couldn’t be very good? (Too cheap)
- At what price would you consider the product starting to get expensive, such that it’s not out of the question, but you would have to give some thought to buying it? (Expensive/High Side)
- At what price would you consider the product to be a bargain—a great buy for the money? (Cheap/Good Value)
Each of these questions asks about the customer’s perspective on the product value, with price as one component of the value equation.
The responses of many potential buyers can be plotted on a graph (see Figure 2, below). Each line shows the different customer responses to each of the questions at different price points. For example, 100 percent of those interviewed think the product is too cheap at $0, and 40 percent think that it’s still too cheap at a price of $500. The graph shows an acceptable price range in which the customers’ responses cross one another. They become torn between whether the prices are cheap or expensive but are not clearly landing on one side or the other. The results of this graph suggest a price band between $500 and $1,200.
For the purposes of this course, we won’t be getting into a full analysis of these data or the price-sensitivity meter; the important point is that marketers need to balance the customer’s perception of the value provided with the customer’s perception of the right price (“perceived costs” in Figure 1, above) in the value equation.
- footwearnews.com/2014/business/news/nike-price-hikes-drive-u-s-sneaker-growth-144128/ ↵
- http://www.forbes.com/sites/greatspeculations/2015/06/11/airlines-stocks-drop-as-fear-of-price-war-clouds-the-industry/#2715e4857a0b103622d442d5 ↵
Contributors and Attributions
- Pricing Objectives. Provided by : Lumen Learning. License : CC BY: Attribution
- Van Westendorp Price Sensitivity Meter. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Van_Westendorp%27s_Price_Sensitivity_Meter . License : CC BY-SA: Attribution-ShareAlike
- Best Price. Located at : pixabay.com/en/award-price-tag-note-board-offer-73084/. License : CC0: No Rights Reserved


2.4.4: Two Graphs for Each Relationship
Lesson
Let's use tables, equations, and graphs to answer questions about proportional relationships.
Exercise \(\PageIndex{1}\): True or False: Fractions and Decimals
Decice whether each equation is true or false. Be prepared to explain your reasoning.
- \(\frac{3}{2}\cdot 16=3\cdot 8\)
- \(\frac{3}{4}\div\frac{1}{2}=\frac{6}{4}\div\frac{1}{4}\)
- \((2.8)\cdot (13)=(0.7)\cdot (52)\)
Exercise \(\PageIndex{2}\): Tables, Graphs, and Equations
Explore the graph. Start by dragging the gray bar on the left across the screen until you can see both the table and the graph. Notice the values in the table and the coordinates of the labeled point. Grab the point and move it around.
- What stays the same and what changes in the table? in the equation? on the graph?
- Choose one row in the table and write it here. To what does this row correspond on the graph?
| \(x\) | \(y\) |
|---|---|
Grab and drag the point until you see the equation \(y=\frac{3}{2}x\).
- Do not move the point. Choose three rows from the table, other than the origin. Record \(x\) and \(y\), and compute \(\frac{y}{x}\).
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|---|---|---|
- What do you notice? What does this have to do with the equation of the line?
- Do not move the point. Check the box to view the coordinates \((1,?)\). What are the coordinates of this point? What does this correspond to in the table? What does this correspond to in the equation?
- Drag the point to a different location. Record the equation of the line, the coordinates of three points, and the value of \(\frac{y}{x}\).
Equation of the line: _______________________________
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|---|---|---|
- Based on your observations, summarize any connections you see between the table, characteristics of the graph, and the equation.
Are you ready for more?
The graph of an equation of the form \(y=kx\), where \(k\) is a positive number, is a line through \((0,0)\) and the point \((1,k)\).
- Name at least one line through \((0,0)\) that cannot be represented by an equation like this.
- If you could draw the graphs of all of the equations of this form in the same coordinate plane, what would it look like?
Exercise \(\PageIndex{3}\): Hot Dog Eating Contest
Andre and Jada were in a hot dog eating contest. Andre ate 10 hot dogs in 3 minutes. Jada ate 12 hot dogs in 5 minutes.
- The points shown on the first set of axes display information about Andre’s and Jada’s consumption. Which point indicates Andre’s consumption? Which indicates Jada’s consumption? Label them.
- Draw two lines: one through the origin and Andre’s point, and one through the origin and Jada’s point. Write an equation for each line. Use to represent time in minutes, and to represent number of hot dogs.
- For each equation, what does the constant of proportionality tell you?
- The points shown on the second set of axes display information about Andre’s and Jada’s consumption. Which point indicates Andre’s consumption? Which indicates Jada’s consumption? Label them.
- Draw lines from the origin through each of the two points. Write an equation for each line. What does the constant of proportionality tell you in each case?
Summary
Imagine that a faucet is leaking at a constant rate and that every 2 minutes, 10 milliliters of water leaks from the faucet. There is a proportional relationship between the volume of water and elapsed time.
- We could say that the elapsed time is proportional to the volume of water. The corresponding constant of proportionality tells us that the faucet is leaking at a rate of \(\frac{1}{5}\) of a minute per milliliter.
- We could say that the volume of water is proportional to the elapsed time. The corresponding constant of proportionality tells us that the faucet is leaking at a rate of 5 milliliters per minute.
Let’s use \(v\) to represent volume in milliliters and \(t\) to represent time in minutes. Here are graphs and equations that represent both ways of thinking about this relationship:
Even though the relationship between time and volume is the same, we are making a different choice in each case about which variable to view as the independent variable. The graph on the left has \(v\) as the independent variable, and the graph on the right has \(t\) as the independent variable.
Glossary Entries
Definition: Coordinate Plane
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3,2)\) on the coordinate plane, because it is three units to the right and two units up.
Definition: Origin
The origin is the point \((0,0)\) in the coordinate plane. This is where the horizontal axis and the vertical axis cross.
Practice
Exercise \(\PageIndex{4}\)
At the supermarket you can fill your own honey bear container. A customer buys 12 oz of honey for $5.40.
- How much does honey cost per ounce?
- How much honey can you buy per dollar?
- Write two different equations that represent this situation. Use \(h\) for ounces of honey and \(c\) for cost in dollars.
- Choose one of your equations, and sketch its graph. Be sure to label the axes.
Exercise \(\PageIndex{5}\)
The point \((3,\frac{6}{5})\) lies on the graph representing a proportional relationship. Which of the following points also lie on the same graph? Select all that apply.
- \((1,0.4)\)
- \(1.5, \frac{6}{10})\)
- \(\frac{6}{5},3)\)
- \((4,\frac{11}{5})\)
- \((15,6)\)
Exercise \(\PageIndex{6}\)
A trail mix recipe asks for 4 cups of raisins for every 6 cups of peanuts. There is proportional relationship between the amount of raisins, \(r\) (cups), and the amount of peanuts, \(p\) (cups), in this recipe.
- Write the equation for the relationship that has constant of proportionality greater than 1. Graph the relationship.
- Write the equation for the relationship that has constant of proportionality less than 1. Graph the relationship.
Exercise \(\PageIndex{7}\)
Here is a graph that represents a proportional relationship.
- Come up with a situation that could be represented by this graph.
- Label the axes with the quantities in your situation.
- Give the graph a title.
- Choose a point on the graph. What do the coordinates represent in your situation?
(From Unit 2.4.2)


7.6: Applications of Percents
- be able to distinguish between base, percent, and percentage
- be able to find the percentage, the percent, and the base
Base, Percent, and Percentage
There are three basic types of percent problems. Each type involves a base, a percent, and a percentage, and when they are translated from words to mathematical symbols each becomes a multiplication statement . Examples of these types of problems are the following:
- What number is 30% of 50? (Missing product statement.)
- 15 is what percent of 50? (Missing factor statement.)
- 15 is 30% of what number? (Missing factor statement.)
In problem 1, the product is missing. To solve the problem, we represent the missing product with \(P\).
\(P = 30\% \cdot 50\)
The missing product \(P\) is called the percentage . Percentage means part , or portion . In \(P = 30\% \cdot 50\). \(P\) represents a particular part of 50.
In problem 2, one of the factors is missing. Here we represent the missing factor with \(Q\).
\(15 = Q \cdot 50\)
Percent
The missing factor is the
percent
. Percent, we know, means
per
100, or
part
of 100. In \(15 = Q \cdot 50\). \(Q\) indicates what part of 50 is being taken or considered. Specifically, \(15 = Q \cdot 50\) means that if 50 was to be divided into 100 equal parts, then \(Q\) indicates 15 are being considered.
In problem 3, one of the factors is missing. Represent the missing factor with \(B\).
\(15 = 30\% \cdot B\)
Base
The missing factor is the
base
. Some meanings of base are a
source of supply
, or a
starting place
. In \(15 = 30\% \cdot B\), \(B\) indicates the amount of supply. Specifically, \(15 = 30\% \cdot B\) indicates that 15 represents 30% of the total supply.
Each of these three types of problems is of the form
\(\text{(percentage)} = \text{(percent)} \cdot \text{(base)}\)
We can determine any one of the three values given the other two using the methods discussed in [link] .

A salesperson, who gets a commission of 12% of each sale she makes, makes a sale of $8,400.00. How much is her commission?
Solution
We need to determine what part of $8,400.00 is to be taken. What part indicates percentage .
\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {12\%} & {\text{of}} & {8,400.00?} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {12\%} & {\cdot} & {8,400.00} & {\text{Convert to decimals.}} \\ {P} & {=} & {.12} & {\cdot} & {8,400.00} & {\text{Multiply.}} \\ {P} & {=} & {1008.00} & {} & {} & {} \end{array}\)
Thus, the salesperson's commission is $1,008.00.
A girl, by practicing typing on her home computer, has been able to increase her typing speed by 110%. If she originally typed 16 words per minute, by how many words per minute was she able to increase her speed?
Solution
We need to determine what part of 16 has been taken. What part indicates percentage .
\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {110\%} & {\text{of}} & {16?} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {110\%} & {\cdot} & {16} & {\text{Convert to decimals.}} \\ {P} & {=} & {1.10} & {\cdot} & {16} & {\text{Multiply.}} \\ {P} & {=} & {17.6} & {} & {} & {} \end{array}\)
Thus, the girl has increased her typing speed by 17.6 words per minute. Her new speed is \(16 + 17.6 = 33.6\) words per minute.
A student who makes $125 a month working part-time receives a 4% salary raise. What is the student's new monthly salary?
Solution
With a 4% raise, this student will make 100% of the original salary + 4% of the original salary. This means the new salary will be 104% of the original salary. We need to determine what part of $125 is to be taken. What part indicates percentage .
\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {104\%} & {\text{of}} & {125} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {104\%} & {\cdot} & {125} & {\text{Convert to decimals.}} \\ {P} & {=} & {1.04} & {\cdot} & {125} & {\text{Multiply.}} \\ {P} & {=} & {130} & {} & {} & {} \end{array}\)
Thus, this student's new monthly salary is $130.
An article of clothing is on sale at 15% off the marked price. If the marked price is $24.95, what is the sale price?
Solution
Since the item is discounted 15%, the new price will be \(100\% - 15\% = 85\%\) of the marked price. We need to determine what part of 24.95 is to be taken. What part indicates percentage .
\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {85\%} & {\text{of}} & {$24.95} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {85\%} & {\cdot} & {24.95} & {\text{Convert to decimals.}} \\ {P} & {=} & {.85} & {\cdot} & {24.95} & {\text{Multiply.}} \\ {P} & {=} & {21.2075} & {} & {} & {\text{Since this number represents money,}} \\ {} & {} & {} & {} & {} & {\text{we'll round to 2 decimal places}} \\ {P} & {=} & {21.21} & {} & {} & {} \end{array}\)
Thus, the sale price of the item is $21.21.
Practice Set A
What number is 42% of 85?
- Answer
-
35.7
Practice Set A
A sales person makes a commission of 16% on each sale he makes. How much is his commission if he makes a sale of $8,500?
- Answer
-
$1,360
Practice Set A
An assembly line worker can assemble 14 parts of a product in one hour. If he can increase his assembly speed by 35%, by how many parts per hour would he increase his assembly of products?
- Answer
-
4.9
Practice Set A
A computer scientist in the Silicon Valley makes $42,000 annually. What would this scientist's new annual salary be if she were to receive an 8% raise?
- Answer
-
$45,360
Finding the Percent
\(\begin{array} {cccccl} {15} & {\text{is}} & {\text{What number}} & {\text{of}} & {50?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {15} & {=} & {Q} & {\cdot} & {50} & {} \end{array}\)
Recall that (missing factor) = (product) \(\div\) (known factor).
\(\begin{array} {rccl} {Q} & = & {15 \div 50} & {\text{Divide.}} \\ {Q} & = & {0.3} & {\text{Convert to a percent}} \\ {Q} & = & {30\%} & {} \end{array}\)
Thus, 15 is 30% of 50.
\(\begin{array} {cccccl} {4.32} & {\text{is}} & {\text{What percent}} & {\text{of}} & {72?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {4.32} & {=} & {Q} & {\cdot} & {72} & {} \end{array}\)
\(\begin{array} {rccl} {Q} & = & {4.32 \div 72} & {\text{Divide.}} \\ {Q} & = & {0.06} & {\text{Convert to a percent}} \\ {Q} & = & {6\%} & {} \end{array}\)
Thus, 4.32 is 6% of 72.
On a 160 question exam, a student got 125 correct answers. What percent is this? Round the result to two decimal places.
Solution
We need to determine the percent.
\(\begin{array} {cccccl} {125} & {\text{is}} & {\text{What percent}} & {\text{of}} & {160?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {125} & {=} & {Q} & {\cdot} & {160} & {} \end{array}\)
\(\begin{array} {rccl} {Q} & = & {125 \div 160} & {\text{Divide.}} \\ {Q} & = & {0.78125} & {\text{Round to two decimal places}} \\ {Q} & = & {.78} & {} \end{array}\)
Thus, this student received a 78% on the exam.
A bottle contains 80 milliliters of hydrochloric acid (HCl) and 30 milliliters of water. What percent of HCl does the bottle contain? Round the result to two decimal places.
Solution
We need to determine the percent. The total amount of liquid in the bottle is
\(\text{80 milliliters + 30 milliliters = 110 milliliters}\)
\(\begin{array} {cccccl} {80} & {\text{is}} & {\text{What percent}} & {\text{of}} & {110?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {80} & {=} & {Q} & {\cdot} & {110} & {} \end{array}\)
\(\begin{array} {rccl} {Q} & = & {80 \div 110} & {\text{Divide.}} \\ {Q} & = & {0.727272...} & {\text{Round to two decimal places}} \\ {Q} & \approx & {73\%} & {\text{The symbol "} \approx \text{" is read as "approximately."}} \end{array}\)
Thus, this bottle contains approximately 73% HCl.
Five years ago a woman had an annual income of $19,200. She presently earns $42,000 annually. By what percent has her salary increased? Round the result to two decimal places.
Solution
We need to determine the percent.
\(\begin{array} {cccccl} {42,000} & {\text{is}} & {\text{What percent}} & {\text{of}} & {19,200?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {42,000} & {=} & {Q} & {\cdot} & {19,200} & {} \end{array}\)
\(\begin{array} {rccl} {Q} & = & {42,000 \div 19,200} & {\text{Divide.}} \\ {Q} & = & {2.1875} & {\text{Round to two decimal places}} \\ {Q} & = & {2.19} & {\text{Convert to a percent.}} \\ {Q} & = & {219\%} & {\text{Convert to a percent.}} \end{array}\)
Thus, this woman's annual salary has increased 219%.
Practice Set B
99.13 is what percent of 431?
- Answer
-
23%
Practice Set B
On an 80 question exam, a student got 72 correct answers. What percent did the student get on the exam?
- Answer
-
90%
Practice Set B
A bottle contains 45 milliliters of sugar and 67 milliliters of water. What fraction of sugar does the bottle contain? Round the result to two decimal places (then express as a percent).
- Answer
-
40%
Finding the Base
\(\begin{array} {cccccl} {15} & {\text{is}} & {30\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {15} & {=} & {30\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {15} & {=} & {.30} & {\cdot} & {B} & {\text{[\text{(missing factor)} = \text{(product)} \div \text{(known factor)}]}} \end{array}\)
\(\begin{array} {rcl} {B} & = & {15 \div .30} \\ {B} & = & {50} \end{array}\)
Thus, 15 is 30% of 50.
\(\begin{array} {cccccl} {56.43} & {\text{is}} & {33\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {56.43} & {=} & {33\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {56.43} & {=} & {.33} & {\cdot} & {B} & {\text{Divide.}} \end{array}\)
\(\begin{array} {rcl} {B} & = & {56.43 \div .33} \\ {B} & = & {171} \end{array}\)
Thus, 56.43 is 33% of 171.
Fifteen milliliters of water represents 2% of a hydrochloric acid (HCl) solution. How many milliliters of solution are there?
Solution
We need to determine the total supply. The word supply indicates base .
\(\begin{array} {cccccl} {15} & {\text{is}} & {2\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {15} & {=} & {2\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {15} & {=} & {.02} & {\cdot} & {B} & {\text{Divide.}} \end{array}\)
\(\begin{array} {rcl} {B} & = & {15 \div .02} \\ {B} & = & {750} \end{array}\)
Thus, there are 750 milliliters of solution in the bottle.
In a particular city, a sales tax of \(6 \dfrac{1}{2}\) % is charged on items purchased in local stores. If the tax on an item is $2.99, what is the price of the item?
Solution
We need to determine the price of the item. We can think of price as the starting place . Starting place indicates base . We need to determine the base.
\(\begin{array} {cccccl} {2.99} & {\text{is}} & {6 \dfrac{1}{2}\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {2.99} & {=} & {6 \dfrac{1}{2}\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {2.99} & {=} & {6.5\%} & {\cdot} & {B} & {} \\ {2.99} & {=} & {0.065} & {\cdot} & {B} & {[\text{(missing factor)} = \text{(product)} \div \text{(known factor)}]} \end{array}\)
\(\begin{array} {rcll} {B} & = & {2.99 \div .065} & {\text{Divide.}} \\ {B} & = & {46} & {} \end{array}\)
Thus, the price of the item is $46.00.
A clothing item is priced at $20.40. This marked price includes a 15% discount. What is the original price?
Solution
We need to determine the original price. We can think of the original price as the starting place . Starting place indicates base . We need to determine the base. The new price, $20.40, represents \(100\% - 15\% = 85\%\) of the original price.
\(\begin{array} {cccccl} {20.40} & {\text{is}} & {85\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {20.40} & {=} & {85\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {20.40} & {=} & {0.85} & {\cdot} & {B} & {[\text{(missing factor)} = \text{(product)} \div \text{(known factor)}]} \end{array}\)
\(\begin{array} {rcll} {B} & = & {20.40 \div .85} & {\text{Divide.}} \\ {B} & = & {24} & {} \end{array}\)
Thus, the original price of the item is $24.00.
Practice Set C
1.98 is 2% of what number?
- Answer
-
99
Practice Set C
3.3 milliliters of HCl represents 25% of an HCl solution. How many milliliters of solution are there?
- Answer
-
13.2ml
Practice Set C
A salesman, who makes a commission of \(18 \dfrac{1}{4}\)% on each sale, makes a commission of $152.39 on a particular sale. Rounded to the nearest dollar, what is the amount of the sale?
- Answer
-
$835
Practice Set C
At "super-long play," \(2\dfrac{1}{2}\) hours of play of a video cassette recorder represents 31.25% of the total playing time. What is the total playing time?
- Answer
-
8 hours
Exercises
For the following 25 problems, find each indicated quantity.
Exercise \(\PageIndex{1}\)
What is 21% of 104?
- Answer
-
21.84
Exercise \(\PageIndex{2}\)
What is 8% of 36?
Exercise \(\PageIndex{3}\)
What is 98% of 545?
- Answer
-
534.1
Exercise \(\PageIndex{4}\)
What is 143% of 33?
Exercise \(\PageIndex{5}\)
What is \(10 \dfrac{1}{2}\)% of 20?
- Answer
-
2.1
Exercise \(\PageIndex{6}\)
3.25 is what percent of 88?
Exercise \(\PageIndex{7}\)
22.44 is what percent of 44?
- Answer
-
51
Exercise \(\PageIndex{8}\)
0.0036 is what percent of 0.03?
Exercise \(\PageIndex{9}\)
31.2 is what percent of 26?
- Answer
-
120
Exercise \(\PageIndex{10}\)
266.4 is what percent of 74?
Exercise \(\PageIndex{11}\)
0.0101 is what percent of 0.0505?
- Answer
-
20
Exercise \(\PageIndex{12}\)
2.4 is 24% of what number?
Exercise \(\PageIndex{13}\)
24.19 is 41% of what number?
- Answer
-
59
Exercise \(\PageIndex{14}\)
61.12 is 16% of what number?
Exercise \(\PageIndex{15}\)
82.81 is 91% of what number?
- Answer
-
91
Exercise \(\PageIndex{16}\)
115.5 is 20% of what number?
Exercise \(\PageIndex{17}\)
43.92 is 480% of what number?
- Answer
-
9.15
Exercise \(\PageIndex{18}\)
What is 85% of 62?
Exercise \(\PageIndex{19}\)
29.14 is what percent of 5.13?
- Answer
-
568
Exercise \(\PageIndex{20}\)
0.6156 is what percent of 5.13?
Exercise \(\PageIndex{21}\)
What is 0.41% of 291.1?
- Answer
-
1.19351
Exercise \(\PageIndex{22}\)
26.136 is 121% of what number?
Exercise \(\PageIndex{23}\)
1,937.5 is what percent of 775?
- Answer
-
250
Exercise \(\PageIndex{24}\)
1 is what percent of 2,000?
Exercise \(\PageIndex{25}\)
0 is what percent of 59?
- Answer
-
0
Exercise \(\PageIndex{26}\)
An item of clothing is on sale for 10% off the marked price. If the marked price is $14.95, what is the sale price? (Round to two decimal places.)
Exercise \(\PageIndex{27}\)
A grocery clerk, who makes $365 per month, receives a 7% raise. How much is her new monthly salary?
- Answer
-
390.55
Exercise \(\PageIndex{28}\)
An item of clothing which originally sells for $55.00 is marked down to $46.75. What percent has it been marked down?
Exercise \(\PageIndex{29}\)
On a 25 question exam, a student gets 21 correct. What percent is this?
- Answer
-
84
Exercise \(\PageIndex{30}\)
On a 45 question exam, a student gets 40%. How many questions did this student get correct?
Exercise \(\PageIndex{31}\)
A vitamin tablet, which weighs 250 milligrams, contains 35 milligrams of vitamin C. What percent of the weight of this tablet is vitamin C?
- Answer
-
14
Exercise \(\PageIndex{32}\)
Five years ago a secretary made $11,200 annually. The secretary now makes $17,920 annually. By what percent has this secretary's salary been increased?
Exercise \(\PageIndex{33}\)
A baseball team wins \(48 \dfrac{3}{4}\)% of all their games. If they won 78 games, how many games did they play?
- Answer
-
160
Exercise \(\PageIndex{34}\)
A typist was able to increase his speed by 120% to 42 words per minute. What was his original typing speed?
Exercise \(\PageIndex{35}\)
A salesperson makes a commission of 12% on the total amount of each sale. If, in one month, she makes a total of $8,520 in sales, how much has she made in commission?
- Answer
-
$1,022.40
Exercise \(\PageIndex{36}\)
A salesperson receives a salary of $850 per month plus a commission of \(8\dfrac{1}{2}\) % of her sales. If, in a particular month, she sells $22,800 worth of merchandise, what will be her monthly earnings?
Exercise \(\PageIndex{37}\)
A man borrows $1150.00 from a loan company. If he makes 12 equal monthly payments of $130.60, what percent of the loan is he paying in interest?
- Answer
-
36.28%
Exercise \(\PageIndex{38}\)
The distance from the sun to the earth is approximately 93,000,000 miles. The distance from the sun to Pluto is approximately 860.2% of the distance from the sun to the Earth. Approximately, how many miles is Pluto from the sun?
Exercise \(\PageIndex{39}\)
The number of people on food stamps in Maine in 1975 was 151,000. By 1980, the number had decreased to 59,200. By what percent did the number of people on food stamps decrease? (Round the result to the nearest percent.)
- Answer
-
61
Exercise \(\PageIndex{40}\)
In Nebraska, in 1960, there were 734,000 motor-vehicle registrations. By 1979, the total had increased by about 165.6%. About how many motor-vehicle registrations were there in Nebraska in 1979?
Exercise \(\PageIndex{41}\)
From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scientists to 52,000. How many were there in 1973?
- Answer
-
19,500
Exercise \(\PageIndex{42}\)
In 1950, in the United States, there were 1,894 daily newspapers. That number decreased to 1,747 by 1981. What percent did the number of daily newspapers decrease?
Exercise \(\PageIndex{43}\)
A particular alloy is 27% copper. How many pounds of copper are there in 55 pounds of the alloy?
- Answer
-
14.85
Exercise \(\PageIndex{44}\)
A bottle containing a solution of hydrochloric acid (HCl) is marked 15% (meaning that 15% of the HCl solution is acid). If a bottle contains 65 milliliters of solution, how many milliliters of water does it contain?
Exercise \(\PageIndex{45}\)
A bottle containing a solution of HCl is marked 45%. A test shows that 36 of the 80 milliliters contained in the bottle are hydrochloric acid. Is the bottle marked correctly? If not, how should it be remarked?
- Answer
-
Marked correctly
Exercises For Review
Exercise \(\PageIndex{46}\)
Use the numbers 4 and 7 to illustrate the commutative property of multiplication.
Exercise \(\PageIndex{47}\)
Convert \(\dfrac{14}{5}\) to a mixed number.
- Answer
-
\(2\dfrac{4}{5}\)
Exercise \(\PageIndex{48}\)
Arrange the numbers \(\dfrac{7}{12}\), \(\dfrac{5}{9}\) and \(\dfrac{4}{7}\) in increasing order.
Exercise \(\PageIndex{49}\)
Convert 4.006 to a mixed number.
- Answer
-
\(4 \dfrac{3}{500}\)
Exercise \(\PageIndex{50}\)
Convert \(\dfrac{7}{8}\)% to a fraction.


17.9: Protection of Alcohols
-
- Last updated
- Save as PDF
Objectives
After completing this section, you should be able to
- explain what is meant by “protecting” a functional group during an organic synthesis.
- describe one common method for protecting the hydroxy group of an alcohol, and give an example of its use (e.g., in the preparation of a Grignard reagent).
Often during the synthesis of complex molecules one functional group in a molecule interferes with an intended reaction on a second functional group on the same molecule. An excellent example is the fact that a Grignard reagent can't be prepared from halo alcohol because the C-Mg bond is not compatible with the acidic -OH group.
When situations like this occurs chemists circumvent the problem by changing the interfering functional group into one that does not interfere with the intended reaction. Chemists call this process protection of a functional group. Functional group protection involves three steps:
- Blocking the interfering functionality by introducing a protecting group.
- Performing the intended reaction.
- Removing the protecting group and reforming the original functional group.
Protecting Alcohols Through the Formation of Trialkylsilyl Ethers
There are several methods for protecting an alcohol, however, the most common is the reaction with a chlorotrialkylsilane, Cl-SiR 3 This reactions forms a trialkylsilyl ether, R'O-SiR 3 . One of the most common reagents is chlorotrimethylsilane [(CH 3 ) 3 SiCl] which is often used in conjunction with a base, such as triethylamine. The base helps to form the alkoxide anion and remove the HCl produced by the reaction. The produced trimethylsilyl ether is commonly abbreviated -OTMS. This silyl ether no longer has the fairly acidic OH proton as the alcohol group has been changed (protected).
General Reaction
Example \(\PageIndex{1}\)
Mechanism
Notice this reaction occurs through an S N 2-like mechanism with the alcohol attacking a trialkyl-substitued silicon atom. Normally, an S N 2 reaction would not be viable for the analogous tert -butyl chloride due to the steric hindrance of the tertiary electrophilic carbon. Being a third-row atom, silicon is larger and forms longer bonds that carbon. The Si-C bonds of chlorotrimethylsilane are 195 pm compared to the 154 pm C-C bonds or tert -butylchloride. The three methyl groups attached to silicon are more spread out and offer less steric hindrance so the S N 2-like reaction can occur.
Deprotection
Trimethylsilyl ethers, like most other ethers, are relatively unreactive towards many reagents such as oxidizing agents, reducing agents, or Grignard reagents. However, the TMS ether protecting group can be removed by reaction with an aqueous acid or the fluoride ion (F - ) to regenerate the alcohol. Common sources of the fluoride ion are lithium fluoride (LiF) and tetrabutylammoniumfluoride (TBAF) [(CH 3 CH 2 CH 2 CH 2 ) 4 NF]. This step is commonly referred to as a deprotection .
Utilizing a Protecting Group for a Grignard Reaction
The problem posed at the beginning of this section can be solved through the use of a TMS protecting group. Once the alcohol is converted into a TMS ether the acidic hydrogen will no longer be present and a Grignard reagent can be formed.
1) Protection of the Alcohol
2) Form the Grignard Reagent
3) Perform the Grignard Reaction
4) Deprotection
Exercises
Exercise \(\PageIndex{1}\)
Propose a multiple-step synthesis to transform 4-bromo-1-butanol into 2-methylhexane-2,6-diol.
- Answer
-
Contributors and Attributions
- Layne Morsch (University of Illinois Springfield)
- James Kabrhel (University of Wisconsin - Green Bay, Sheboygan Campus)


13: Employment Law
13.1 Introduction
LEARNING OBJECTIVES
- Understand the principle of employment at will and the exceptions to the doctrine.
- Learn important employment laws that affect businesses across industries.
- Examine the laws that govern the relationship between employers and employees who belong to a union.
Until the early Twentieth Century, there were not many laws that regulated the employer-employee relationship. The belief was that the free market system would ensure that employers treat employees fairly or else they would not be able to attract and keep good workers. However, the reality of the Industrial Revolution proved that the traditional employment relationship favored the interests of the employers at the expense of workers, including children. As a result, Congress and state legislatures started passing employment and labor laws to protect the interests of employees. Today, employment law is a very robust area of the law that impacts businesses across industries.
13.2 Employment At Will
In general, employees are free to quit a job at any time for any reason, with or without notice. Similarly, employers are free to end a worker’s employment at any time for any reason, with or without notice. This principle is called employment at will . This doctrine is based on the concept that employment is a form of an implied contractual relationship. Therefore, as long as both parties want to continue their contract to work together, the law presumes they will. When one party does not want to continue, then they may end their working relationship.
There are five major exceptions to the employment at will doctrine:
- Contract;
- Good cause;
- Discrimination against an employee based on membership in a protected class;
- Violation of public policy; and
- Whistleblowing.
Figure 13.1 Exceptions to the Employment at Will Doctrine
Contract
Employers and employees may modify the employment at will doctrine through contract. In addition to formal, written employment contracts, courts have also enforced oral promises made during the hiring process. Promises made to job applicants are generally enforceable, even when the promises are not approved by the employer’s executives or upper management. Therefore, it is important when hiring employees that companies do not make promises that can be reasonably interpreted to be guaranteed employment or employment for a certain period of time.
Employee handbooks also may create implied contracts that modify the employment at will doctrine. Often handbooks state that the company follows a progressive discipline policy and that employees may only be fired for “just cause” or after receiving warnings, notice, hearing, or other procedures. Policies such as this create an implied contract that require businesses to follow the progressive discipline policy before terminating a worker’s employment, in absence of good cause.
Good Cause
The definition of “good cause” for terminating the employment of a worker varies from state to state. And businesses often define “good cause” in their policies. However, most states recognize the following as good cause to fire an employee without going through progressive discipline first:
- Theft;
- Fraud;
- Damage to company property;
- Being under the influence of illegal drugs or alcohol at work;
- Fighting;
- Threats against other employees or customers;
- Domestic violence;
- Having weapons on premises;
- Unethical behavior; and
- Willful or malicious misbehavior.
Poor performance also constitutes good cause for firing an employee. Employers need to be careful, however, to document the performance issues and to engage in progressive discipline when appropriate. If an employee has not been counseled that their performance is unsatisfactory, then the employee is more likely to bring a charge of wrongful discharge against the employer. Courts are more likely to rule that poor performance constitutes good cause when an employee has notice of the performance issue(s) and has a reasonable opportunity to fix them.
Discrimination Against a Protected Class Member
Anti-discrimination laws make it illegal to take adverse actions against a member of a protected class based on their membership in the class. A protected class is a group of people who are protected by laws that prohibit discrimination based on a personal characteristic, such as race, color, religion, gender, national origin, age or disability.
As discussed in Chapter 14, adverse actions include failure to hire, failure to promote, demotion, and termination of employment. That is not to say that someone who is a member of a protected class may never be fired. Rather, it is illegal to fire them because of their race, color, religion, gender, national origin, age or disability.
Violation of Public Policy
The public policy exception occurs when an employee is fired for:
- Refusing to perform an action that violates a law or public policy; or
- For exercising a legal right or advancing a public policy.
In other words, an employee can’t be fired for refusing to do something illegal or doing something legal that the employer does not want done. For example, an employee cannot be fired for not falsifying reports, for refusing to testify falsely in court, for filing a workers’ compensation claim, or for serving on a jury.
Whistleblowing
A whistleblower is an employee who reports the employer’s illegal behavior to a governmental or law enforcement agency. Many different laws have whistleblower provisions that encourage people who have knowledge of illegal activity to report it without fear of retribution or losing their jobs. Whistleblower protections apply to good faith reports of wrongdoing, even if it turns out that the activity is not illegal. However, whistleblower protection does not usually protect employees who make reports that they either know, or should have known, do not include illegal activity.
13.3 Common Employment Law Torts
Employees may assert various tort claims against their employers. Tort claims are often decided on the basis of generalized duties of care rather than specific types of conduct prohibited by law. Tort claims vary state to state but most states recognize the following claims between employers and employees:
- Negligent hiring, retention, and supervision;
- Negligent investigation;
- Negligent infliction of emotional distress;
- Intentional infliction of emotional distress (also called outrageous conduct);
- Tortious interference with contract and/or prospective business advantage;
- Defamation (libel and slander);
- Invasion of privacy; and
- Fraud.
Torts are discussed in more detail in Chapter 9. However, it is important for businesses to understand that they owe their employees and managers a duty of care. If they violate that duty, then they may be subject to legal liability.
13.4 Wage and Hour Laws
The Fair Labor Standards Act (FLSA) is a federal law that was passed in 1938 that nationalized standards for pay, record keeping and child labor for businesses with two or more employees that engage in interstate commerce. The FLSA prohibits “oppressive child labor,” which means that children under fourteen cannot work unless it is a family business, babysitting, newspaper delivery, entertainment, or agriculture. Fourteen and fifteen year olds are permitted to work limited hours after school in nonhazardous jobs, such as retail and restaurants. Sixteen and seventeen year olds may work unlimited hours in nonhazardous jobs.
Figure 13.2 FLSA Facts
The FLSA also provides for a national minimum wage and a standard forty hour work week. Under the FLSA, an employee who works more than forty hours is entitled to overtime pay. However, the Act provides exemptions to the overtime requirement. Employees who earn more than a certain amount and perform specific types of work are not entitled to overtime pay.
| Exempt Categories to Overtime Pay under the FLSA | ||||||||||||
|


1.3.5: Exothermic and Endothermic Reactions
Learning Objectives
- Describe three ways to speed up a reaction.
- Differentiate between reversible exothermic, irreversible exothermic, and endothermic reactions and draw appropriate graphs to represent each which include axis labels, activation energy, and the effect a catalyst would have on the graph.
Energy
Energy can be defined as the capacity to supply heat or do work. One type of work ( w ) is the process of causing matter to move against an opposing force. For example, we do work when we inflate a bicycle tire—we move matter (the air in the pump) against the opposing force of the air surrounding the tire.
Like matter, energy comes in different types. One scheme classifies energy into two types: potential energy , the energy an object has because of its relative position, composition, or condition, and kinetic energy , the energy that an object possesses because of its motion. Water at the top of a waterfall or dam has potential energy because of its position; when it flows downward through generators, it has kinetic energy that can be used to do work and produce electricity in a hydroelectric plant (Figure \(\PageIndex{2}\)). A battery has potential energy because the chemicals within it can produce electricity that can do work.
Energy can be converted from one form into another, but all of the energy present before a change occurs always exists in some form after the change is completed. This observation is expressed in the law of conservation of energy: during a chemical or physical change, energy can be neither created nor destroyed, although it can be changed in form. (This is also one version of the first law of thermodynamics, as you will learn later.)
When one substance is converted into another, there is always an associated conversion of one form of energy into another. Heat is usually released or absorbed, but sometimes the conversion involves light, electrical energy, or some other form of energy. For example, chemical energy (a type of potential energy) is stored in the molecules that compose gasoline. When gasoline is combusted within the cylinders of a car’s engine, the rapidly expanding gaseous products of this chemical reaction generate mechanical energy (a type of kinetic energy) when they move the cylinders’ pistons.
According to the law of conservation of matter (seen in an earlier chapter), there is no detectable change in the total amount of matter during a chemical change. When chemical reactions occur, the energy changes are relatively modest and the mass changes are too small to measure, so the laws of conservation of matter and energy hold well. However, in nuclear reactions, the energy changes are much larger (by factors of a million or so), the mass changes are measurable, and matter-energy conversions are significant. This will be examined in more detail in a later chapter on nuclear chemistry. To encompass both chemical and nuclear changes, we combine these laws into one statement: The total quantity of matter and energy in the universe is fixed.
Thermal Energy, Temperature, and Heat
Thermal energy is kinetic energy associated with the random motion of atoms and molecules. Temperature is a quantitative measure of “hot” or “cold.” When the atoms and molecules in an object are moving or vibrating quickly, they have a higher average kinetic energy (KE), and we say that the object is “hot.” When the atoms and molecules are moving slowly, they have lower KE, and we say that the object is “cold” (Figure \(\PageIndex{3}\)). Assuming that no chemical reaction or phase change (such as melting or vaporizing) occurs, increasing the amount of thermal energy in a sample of matter will cause its temperature to increase. And, assuming that no chemical reaction or phase change (such as condensation or freezing) occurs, decreasing the amount of thermal energy in a sample of matter will cause its temperature to decrease.
'
Figure \(\PageIndex{3}\): (a) The molecules in a sample of hot water move more rapidly than (b) those in a sample of cold water.
Most substances expand as their temperature increases and contract as their temperature decreases. This property can be used to measure temperature changes, as shown in Figure \(\PageIndex{4}\). The operation of many thermometers depends on the expansion and contraction of substances in response to temperature changes.
Heat ( q ) is the transfer of thermal energy between two bodies at different temperatures. Heat flow (a redundant term, but one commonly used) increases the thermal energy of one body and decreases the thermal energy of the other. Suppose we initially have a high temperature (and high thermal energy) substance (H) and a low temperature (and low thermal energy) substance (L). The atoms and molecules in H have a higher average KE than those in L. If we place substance H in contact with substance L, the thermal energy will flow spontaneously from substance H to substance L. The temperature of substance H will decrease, as will the average KE of its molecules; the temperature of substance L will increase, along with the average KE of its molecules. Heat flow will continue until the two substances are at the same temperature (Figure \(\PageIndex{5}\)).
Matter undergoing chemical reactions and physical changes can release or absorb heat. A change that releases heat is called an exothermic process . For example, the combustion reaction that occurs when using an oxyacetylene torch is an exothermic process—this process also releases energy in the form of light as evidenced by the torch’s flame (Figure \(\PageIndex{6a}\)). A reaction or change that absorbs heat is an endothermic process . A cold pack used to treat muscle strains provides an example of an endothermic process. When the substances in the cold pack (water and a salt like ammonium nitrate) are brought together, the resulting process absorbs heat, leading to the sensation of cold.
Figure \(\PageIndex{6}\): (a) An oxyacetylene torch produces heat by the combustion of acetylene in oxygen. The energy released by this exothermic reaction heats and then melts the metal being cut. The sparks are tiny bits of the molten metal flying away. (b) A cold pack uses an endothermic process to create the sensation of cold. (credit a: modification of work by “Skatebiker”/Wikimedia commons).
Each chemical bond is stored energy. For a reaction to occur, the bonds of the reactants must be broken. This process always requires energy. At the same time, the bonds between the products must form, this process always releases energy. If the energy required to break the bonds of the reactants is less than that given off by the formation of bonds in the products, the excess energy will be released from the reaction. If the energy required to break the bonds of the reactants is greater than that given off by the formation of bonds in the products, the reaction will need an external supply of energy and the reaction will be endothermic.
Activation energy : Also called energy of activation. It is the amount of energy required to get the reaction started. Some reactions require very little energy, in others a great deal of energy is required.
Catalyst: Is a substance that when added to the reaction reduces the amount of energy required to start the reaction, but itself does not become part of the product and thus can be re-used in the reaction. If a catalyst is involved in a reaction, it is written above the arrow, as it does not become a part of the product.
HCl
Sucrose + H 2 O → Glucose + fructose
Reversible reaction : The energy hill may be approached from either side.
Irreversible reaction: The change in energy is so great, that the energy hill cannot be approached from the opposite side.
Glossary
- endothermic process
- chemical reaction or physical change that absorbs heat
- exothermic process
- chemical reaction or physical change that releases heat
- heat ( q )
- transfer of thermal energy between two bodies
- kinetic energy
- energy of a moving body, in joules, equal to \(\dfrac{1}{2}mv^2\) (where m = mass and v = velocity)
- potential energy
- energy of a particle or system of particles derived from relative position, composition, or condition
- thermal energy
- kinetic energy associated with the random motion of atoms and molecules
- thermochemistry
- study of measuring the amount of heat absorbed or released during a chemical reaction or a physical change
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


20.7: Review Questions
The nurse admits a client with an acute pulmonary embolism and is instructed to start heparin. Which of the following tests can the nurse best use to adjust the dose of heparin?
- Answer
-
Partial thromboplastin time (PTT)
A client who received heparin developed uncontrollable bleeding. What medication is the provider most likely to use as an antidote to the heparin?
- Answer
-
Protamine
The nurse is answering the client’s question about the risk for heparin-induced thrombocytopenia (HIT). Which of the following information should be included in the discussion?
- Answer
-
The condition is immune mediated.
A client is admitted to the emergency department with an acute myocardial infarction. Which medication does the nurse anticipate being ordered to dissolve the clot in the coronary artery?
- Answer
-
Alteplase
The nurse is teaching a client with a pulmonary embolism about their new medication, heparin. What will the nurse tell the client about the purpose of heparin?
- Answer
-
It will prevent further clot formation.
A nurse is teaching a client about warfarin dietary considerations. Which of the following statements by the client indicates understanding of the restrictions?
- Answer
-
“I should eat a consistent amount of vitamin K–containing foods in my diet.”
A nurse admits a client who has been started on warfarin for a pulmonary embolism. Which test will the nurse monitor to assess warfarin’s effect?
- Answer
-
International normalized ratio (INR)
The nurse is monitoring for medication-related adverse effects in a client receiving heparin. Which of the following would indicate that the client should be assessed for heparin-induced thrombocytopenia (HIT)?
- Answer
-
A decrease in platelet count and concomitant deep vein thrombosis
Which of the following direct-acting oral anticoagulant drugs must be taken with food for effective absorption?
- Answer
-
Rivaroxaban
An emergency department nurse is assessing a client with an acute ischemic stroke prior to administering alteplase. Which finding will the nurse need to discuss with the provider prior to administering alteplase therapy?
- Answer
-
Blood pressure of 197/110 mm Hg


3.11: Audience Analysis
- Describe strategies for analyzing the audience
Writing in college is all about your audience, which means you’ll need to spend some time getting to know your audience. This is particularly true for argumentative writing, where you are writing to persuade someone about a potentially controversial topic on which they might already have a strong opinion. Knowing the ideas and potential biases of your audience is key to making your writing effective.
This doesn’t mean you have to hang out with the people for whom you are writing. It does mean you need to do some thinking and possibly some researching about who your audience is, what that audience knows about your topic or issue, and what biases or opinions that audience may already have.
Let’s look at an example. Let’s say you have decided to write about reducing the costs of textbooks at your college.  Since most of your fellow college students would agree with you on an issue like this, if you’re going to make a difference, you would need to target a different audience—perhaps college administrators. A good first step would be to head to the web to see what college administrators think and have written about this issue. Then, as a second step, if you could interview a college administrator on your campus, you would have even more information.
Clementa Carlos Pinckney was a Democratic senator in the South Carolina legislature. He represented the 45th District from 2000 until his death. From 1997 through 2000, he was a member of the South Carolina House of Representatives.
In the video below, President Obama’s addresses mourners gathered in Charleston, South Carolina, to honor Reverend Pinckney following his death. While Obama’s address was to those in attendance, he is clearly aware of and speaking to a national audience.
Skim through the following video (or read the transcript) to begin thinking about intended and unintended audiences. It’s easier to think about audiences for a public speech than it is for an essay, but the same rhetorical principles of audience analysis apply.
You can view the transcript for “The President Honors the Life of Reverend Clementa Pinckney” here (opens in new window).
Think about the video or the transcript as you answer the practice questions below.
https://assessments.lumenlearning.co...sessments/5176
Now you know a little bit about the people to whom President Obama was speaking and others whom he did not address or consider directly but who may have heard or read his speech nonetheless. The president had an immediate purpose to eulogize Reverend Pinckney for an audience of mourners, but he was also speaking to a more general audience of U.S. citizens concerned about racism, gun violence, civil rights, and inequality. The claims and evidence from his speech may, therefore, be relevant to other audiences who are concerned about similar issues and share some of the same ideas and values.
This presentation explains important components to consider when creating an argument for your audience.
https://lumenlearning.h5p.com/content/1290929781859018238/embed
Audience Awareness
Even if you aren’t planning to become a politician, you have to think about the audience for everything you say and do. This is even more important with social media. Some of the best examples of not thinking about unintended audiences or failing to build common ground come from social media. Think about the following fake social media post in terms of audience and common ground:
What’s wrong here? Louise works at Generic Store and makes a social media post speaking poorly of her employer and admitting to criminal behavior. To add to that, she tags the store’s social media account in the post. Obviously, Louise is not considering unintended audiences. She may have meant to share her post only with “Badrhetoricgirl” and “nocommonground1,” but at least sixteen others have seen and responded to her post. It’s very likely that her employer would come across this post and take disciplinary action against Louise.
This example may seem ridiculous: maybe you are already wise to social media privacy settings, but this same kind of mistake takes place frequently in student writing and speaking.
Check your understanding of audience awareness in the following interactive.
https://lumenlearning.h5p.com/content/1290929788524115158/embed
Contributors and Attributions
- Audience Analysis. Provided by : University of Mississippi. License : CC BY: Attribution
- Modification, adaptation, and original content. Provided by : Lumen Learning. License : CC BY-SA: Attribution-ShareAlike
- Argument and Analysis. Provided by : Excelsior College Online Writing Lab. Located at : https://owl.excelsior.edu/argument-and-critical-thinking/argument-and-audience/ . License : CC BY: Attribution
- Knowing Your Audience. Provided by : Georgia State University Dept. of English. Project : APLU-PLC Adaptive Courseware for English Comp Project . License : CC BY-SA: Attribution-ShareAlike
- Using what you know about audience. Provided by : University of Mississippi. Project : PLATO. License : CC BY-SA: Attribution-ShareAlike
- The President Honors the Life of Reverend Clementa Pinckney. Authored by : The Obama White House. Located at : https://www.youtube.com/watch?v=rRvBzzR5tdA&feature=youtu.be . License : All Rights Reserved . License Terms : Standard YouTube License


1.3: Dimensional Analysis
-
- Last updated
- Save as PDF
- Howard DeVoe
- University of Maryland
Sometimes you can catch an error in the form of an equation or expression, or in the dimensions of a quantity used for a calculation, by checking for dimensional consistency. Here are some rules that must be satisfied:
-
In this e-book the
differential
of a function, such as \(\df\), refers to an
infinitesimal
quantity. If one side of an equation is an infinitesimal quantity, the other side must also be. Thus, the equation \(\df = a\dx + b\dif y\) (where \(ax\) and \(by\) have the same dimensions as \(f\)) makes mathematical sense, but \(\df = ax+b\dif y\) does not.
Derivatives, partial derivatives, and integrals have dimensions that we must take into account when determining the overall dimensions of an expression that includes them. For instance:
-
Some examples of applying these principles are given here using symbols described in Sec. 1.2.
Example 1. Since the gas constant \(R\) may be expressed in units of J K\(^{-1}\) mol\(^{-1}\), it has dimensions of energy divided by thermodynamic temperature and amount. Thus, \(RT\) has dimensions of energy divided by amount, and \(nRT\) has dimensions of energy. The products \(RT\) and \(nRT\) appear frequently in thermodynamic expressions.
Example 3. Find the dimensions of the constants \(a\) and \(b\) in the van der Waals equation \[ p = \frac {nRT}{V-nb} - \frac {n^{2}a} {V^2} \] Dimensional analysis tells us that, because \(nb\) is subtracted from \(V\), \(nb\) has dimensions of volume and therefore \(b\) has dimensions of volume/amount. Furthermore, since the right side of the equation is a difference of two terms, these terms have the same dimensions as the left side, which is pressure. Therefore, the second term \(n^{2}a/V^{2}\) has dimensions of pressure, and \(a\) has dimensions of pressure \(\times\) volume\(^{2}\) \(\times\) amount\(^{-2}\).
Example 4. Consider an equation of the form \[ \Pd{\ln x}{T}{\!p} = \frac {y}{R} \] What are the SI units of \(y\)? \(\ln x\) is dimensionless, so the left side of the equation has the dimensions of \(1/T\), and its SI units are K\(^{-1}\). The SI units of the right side are therefore also K\(^{-1}\). Since \(R\) has the units J K\(^{-1}\) mol\(^{-1}\), the SI units of \(y\) are J K\(^{-2}\) mol\(^{-1}\).
-
Some examples of applying these principles are given here using symbols described in Sec. 1.2.