4.5: Chapter 4 Review
Graphing Linear Inequalities
- Graph the linear inequality: \(y<\frac{3}{5}x+2\).
- Graph the linear inequality: \(y\geq −\frac{1}{3}x−2\).
- Graph the linear inequality: \(2x−4y\geq −12\).
- Graph the linear inequality: \(x+5y<−5\).
Graphing Systems of Linear Inequalities
-
Determine whether each ordered pair is a solution to the system of inequalities.
\(\left\{\begin{array} {l} 7x+2y>14\\5x−y\leq 8\end{array}\right.\)
ⓐ \((2, 3)\)
ⓑ \((7, −1)\)
-
Graph \(\left\{\begin{array} {l} y<2x−1\\y\leq −\frac{1}{2}x+4\end{array}\right.\)
-
Graph \(\left\{\begin{array} {l} 3x−y\geq 6\\y\geq −\frac{1}{2}x\end{array}\right.\)
-
Graph \(\left\{\begin{array} {l} 2x+2y>−4\\−x+3y\geq 9\end{array}\right.\)
-
Graph \(\left\{\begin{array} {l} y\leq −\frac{1}{4}x−2\\x+4y<6\end{array}\right.\)
Maximizing and Minimizing with Linear Programming
- Mr. Shoemacher has $20,000 to invest in two types of mutual funds: a High-Yield Fund and an Equity Fund. The High-Yield fund has an annual yield of 12%, while the Equity fund earns 8%. He would like to invest at least $3000 in the High-Yield fund and at least $4000 in the Equity fund. How much should he invest in each to maximize his annual yield, and what is the maximum yield?
- Dr. Lum teaches part-time at two community colleges, Hilltop College and Serra College. Dr. Lum can teach up to 5 classes per semester. For every class he teaches at Hilltop College, he needs to spend 3 hours per week preparing lessons and grading papers. For each class at Serra College, he must do 4 hours of work per week. He has determined that he cannot spend more than 18 hours per week preparing lessons and grading papers. If he earns $6,000 per class at Hilltop College and $7,500 per class at Serra College, how many classes should he teach at each college to maximize his income, and what will be his income?
- Mr. Shamir employs two part-time typists, Inna and Jim, for his typing needs. Inna charges $15 an hour and can type 6 pages an hour, while Jim charges $18 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each typist to minimize his typing costs, and what will be the total cost?
- Mr. Boutros wants to invest up to $20,000 in two stocks, Cal Computers and Texas Tools. The Cal Computers stock is expected to yield a 16% annual return, while the Texas Tools stock promises a 12% yield. Mr. Boutros would like to earn at least $2,880 this year. According to Value Line Magazine's safety index (1 highest to 5 lowest), Cal Computers has a safety number of 3 and Texas Tools has a safety number of 2. How much money should he invest in each to minimize the safety number? Note: A lower safety number means less risk.
- A store sells two types of copy machines: compact (low capacity) and standard (which takes more space). The store can sell up to 90 copiers a month. A maximum of 1080 cubic feet of storage space is available. A compact copier requires 6 cu. ft. of storage space, and a standard copier requires 18 cu. ft.. The compact and standard copy machines take, respectively, 1 and 1.5 sales hours of labor. A maximum of 99 hours of labor is available. The profit from each of these copiers is $60 and $80, respectively, how many of each type should be sold to maximize profit, and what is the maximum profit?
- A company manufactures two types of cell phones, a Basic model and a Pro model. The Basic model generates a profit of $100 per phone and the Pro model has a profit of $150 per phone. On the assembly line the Basic phone requires 7 hours, while the Pro model takes 11 hours. The Basic phone requires one hour and the Pro phone needs 3 hours for finishing, which includes loading software. Both phones require one hour for testing. On a particular production run the company has available 1,540 work hours on the assembly line, 360 work hours for finishing, and 200 work hours in the testing department. How many cell phones of each type should be produced to maximize profit, and what is that maximum profit?
- John wishes to choose a combination of two types of cereals for breakfast - Cereal A and Cereal B. A small box (one serving) of Cereal A costs $0.50 and contains 10 units of vitamins, 5 units of minerals, and 15 calories. A small box(one serving) of Cereal B costs $0.40 and contains 5 units of vitamins, 10 units of minerals, and 15 calories. John wants to buy enough boxes to have at least 500 units of vitamins, 600 units of minerals, and 1200 calories. How many boxes of each food should he buy to minimize his cost, and what is the minimum cost?
- Jessica needs at least 60 units of vitamin A, 40 units of vitamin B, and 140 units of vitamin C each week. She can choose between Costless brand or Savemore brand tablets. A Costless tablet costs 5 cents and contains 3 units of vitamin A, 1 unit of vitamin B, and 2 units of vitamin C. A Savemore tablet costs 7 cents and contains 1 unit of A, 1 of B, and 5 of C. How many tablets of each kind should she buy to minimize cost, and what is the minimum cost?
- A small company manufactures two products: A and B. Each product requires three operations: Assembly, Finishing and Testing. Product A requires 1 hour of Assembly, 3 hours of Finishing, and 1 hour of Testing. Product B requires 3 hours of Assembly, 1 hour of Finishing, and 1 hour of Testing. The total work-hours available per week in the Assembly division is 60, in Finishing is 60, and in Testing is 24. Each item of product A has a profit of $50, and each item of Product B has a profit of $75. How many of each should be made to maximize profit? What is the maximum profit?
- A factory manufactures two products, A and B. Each product requires the use of three machines, Machine I, Machine II, and Machine III. The time requirements and total hours available on each machine are listed below.
|
Machine I |
Machine II |
Machine III |
|
|---|---|---|---|
|
Product A |
1 |
2 |
4 |
|
Product B |
2 |
2 |
2 |
|
Total hours |
70 |
90 |
160 |
If product A generates a profit of $60 per unit and product B a profit of $50 per unit, how many units of each product should be manufactured to maximize profit, and what is the maximum profit?
- A company produces three types of shoes, formal, casual, and athletic, at its two factories, Factory I and Factory II. The company must produce at least 6000 pairs of formal shoes, 8000 pairs of casual shoes, and 9000 pairs of athletic shoes. Daily production of each factory for each type of shoe is:
|
Factory I |
Factory II |
|
|---|---|---|
|
Formal |
100 |
100 |
|
Casual |
100 |
200 |
|
Athletic |
300 |
100 |
Operating Factory I costs $1500 per day and it costs $2000 per day to operate Factory II. How many days should each factory operate to complete the order at a minimum cost, and what is the minimum cost?
- A professor gives two types of quizzes, objective and recall. He plans to give at least 15 quizzes this quarter. The student preparation time for an objective quiz is 15 minutes and for a recall quiz 30 minutes. The professor would like a student to spend at least 5 hours (300 minutes) preparing for these quizzes above and beyond the normal study time. The average score on an objective quiz is 7, and on a recall type 5, and the professor would like the students to score at least 85 points on all quizzes. It takes the professor one minute to grade an objective quiz, and 1.5 minutes to grade a recall type quiz. How many of each type should he give in order to minimize his grading time?
- A company makes two mixtures of nuts: Mixture A and Mixture B. Mixture A contains 30% peanuts, 30% almonds and 40% cashews and sells for $5 per pound. Mixture B contains 30% peanuts, 60% almonds and 10% cashews and sells for $3 a pound. The company has 540 pounds of peanuts, 900 pounds of almonds, 480 pounds of cashews. How many pounds of each of mixtures A and B should the company make to maximize profit, and what is the maximum profit?


26.1B: Evolution of Gymnosperms
- Explain how and why gymnosperms became the dominant plant group during the Permian period
Evolution of Gymnosperms
The fossil plant Elkinsia polymorpha , a “seed fern” from the Devonian period (about 400 million years ago) is considered the earliest seed plant known to date. Seed ferns produced their seeds along their branches without specialized structures. What makes them the first true seed plants is that they developed structures called cupules to enclose and protect the ovule (the female gametophyte and associated tissues) which develops into a seed upon fertilization. Seed plants resembling modern tree ferns became more numerous and diverse in the coal swamps of the Carboniferous period. This appears to have been the result of a whole genome duplication event around 319 million years ago.
Fossil records indicate the first gymnosperms (progymnosperms) most likely originated in the Paleozoic era, during the middle Devonian period about 390 million years ago. Following the wet Mississippian and Pennsylvanian periods, which were dominated by giant fern trees, the Permian period was dry. This gave a reproductive edge to seed plants, which are better adapted to survive dry spells. The Ginkgoales, a group of gymnosperms with only one surviving species, the Gingko biloba, were the first gymnosperms to appear during the lower Jurassic. Gymnosperms expanded in the Mesozoic era (about 240 million years ago), supplanting ferns in the landscape, and reaching their greatest diversity during this time. It has been suggested that during the mid-Mesozoic era, pollination of some extinct groups of gymnosperms was performed by extinct species of scorpionflies that had a specialized proboscis for feeding on pollination drops. The scorpionflies probably engaged in pollination mutualisms with gymnosperms, long before the similar and independent coevolution of nectar-feeding insects on angiosperms.
The Jurassic period was as much the age of the cycads (palm-tree-like gymnosperms) as the age of the dinosaurs. Gingkoales and the more familiar conifers also dotted the landscape. Although angiosperms (flowering plants) are the major form of plant life in most biomes, gymnosperms still dominate some ecosystems, such as the taiga (boreal forests) and the alpine forests at higher mountain elevations because of their adaptation to cold and dry growth conditions.
Key Points
- Seed ferns were the first seed plants, protecting their reproductive parts in structures called cupules.
- Seed ferns gave rise to the gymnosperms during the Paleozoic Era, about 390 million years ago.
- Gymnosperms include the gingkoes and conifers and inhabit many ecosystems, such as the taiga and the alpine forests, because they are well adapted for cold weather.
- True seed plants became more numerous and diverse during the Carboniferous period around 319 million years ago; an explosion that appears to be due to a whole genome duplication event.
Key Terms
- cupule : any small structure shaped like a cup
- gymnosperm : any plant, such as a conifer, whose seeds are not enclosed in an ovary
- mutualism : any interaction between two species that benefits both


10.5: Review and Additional Resources
Review
Review and assess your learning. Start with the "Important Terms and Concepts" to ensure you know the terminology related to the topic of the chapter and concepts discussed. Move on to the "Review Questions" to answer critical thinking questions about concepts and processes discussed in the chapter. Finally, test your overall understanding by taking the "Self-assessment quiz".
-
- hydrosphere
- often called the "water sphere" as it includes all the earth's water found in the oceans, glaciers, streams, lakes, the soil, groundwater, and in the air.
-
- hydrologic cycle
- aka water cycle. the cycling of water through the earth system. It is a cycle of energy as well.
-
- soil water
- the water that is immediately available to plants.
-
- groundwater
- occupies the zone of saturation. Found in aquifers and replenished by percolation of water from the zone of aeration downward to the zone of saturation, or in the recharge zone of a confined aquifer.
-
- evaporation
- the phase change of liquid water into a vapor (gas)
-
- latent energy
- "locked up" in the water molecule when water undergoes the phase change from a liquid to a gas.
-
- precipitation
- droplets of water that fall from the clouds to the earth
-
- interception
- Precipitation that collects on the leaves or stems of plants
-
- infiltration
- refers to water that penetrates into the surface of soil.
-
- zone of aeration
- aka "unsaturated zone", includes soil water zone. See image under "zone of saturation"
-
- zone of saturation
- includes ground water
-
- groundwater zone
- included in zone of saturation. See image under "zone of saturation"
-
- water table
- divides the zone of aeration from the zone of saturation. The height of the water table will fluctuate with precipitation, increasing in elevation during wet periods and decreasing during dry.
-
- hygroscopic water
- a microscopic film of water surrounding soil particles
-
- capillary water
- held by cohesive forces between the films of hygroscopic water
-
- wilting point
- At this point the plant cannot pull water from the plant-rooting zone and it wilts
-
- gravity water
- water moved through the soil by the force of gravity
-
- field capacity
- The amount of water held in the soil after excess water has drained
-
- available water
- The difference between the wilting point and the field capacity
-
- percolation
- the movement of water downward through the soil
-
- seepage
- water that moves downward through the soil toward a stream channel of large body of water
-
- aquifer
- bodies of earth material that have the ability to hold and transmit water
-
- aquiclude
- dense impermeable layers of earth material
-
- hydraulic gradient
- the difference in elevation between two points on the water table divided by the horizontal distance between them
-
- groundwater flow rate
- permeability X hydraulic gradient
-
- runoff
- precipitation that moves across the surface
-
- unconfined flow
- moves across the surface in broad sheets of water often creating sheet erosion
-
- confined flow
- water confined to channels
-
- depression storage
- Water that became trapped in depressions
-
- stream discharge
- the volume of water passing through a particular cross-section of a stream in a unit of time
-
- stream hydrograph
- illustrates the relationship between discharge and runoff
-
- water balance
- an accounting of the inputs and outputs of water
-
- precipitation
- makes up the primarily supply of water to the surface.
-
- actual evapotranspiration
- the amount of water delivered to the air from evaporation and transpiration
-
- evaporation
- the phase change from a liquid to a gas releasing water from a wet surface into the air above
-
- transpiration
- represents a phase change when water is released into the air by plants
-
- evapotranspiration
- combined transfer of water into the air by evaporation and transpiration
-
- potential evapotranspiration
- the amount of water that would be evaporated under an optimal set of conditions, among which is an unlimited supply of water
-
- soil moisture storage
- the amount of water held in the soil at any particular time
-
- change in soil moisture storage
- the amount of water that is being added to or removed from what is stored
-
- deficit
- demand for water exceeds that which is actually available; when potential evapotranspiration exceeds actual evapotranspiration (PE>AE)
-
- surplus
- when P exceeds PE and the soil is at its field capacity (saturated); we have more water than we actually need to use given the environmental conditions at a place
-
- soil moisture recharge
- Precipitation satisfies the need for water with water left over and begins to replenish the soil moisture
-
- soil moisture utilization
- precipitation is no longer is able to meet the demands of potential evapotranspiration. Plants must extract water that is stored in the soil from the previous months
Briefly describe what the hydrologic cycle is.
- Answer
-
The hydrologic cycle or water cycle is the pathway through which water moves in the Earth system. It is a cycle of energy as well as moisture.
Compare and contrast soil water and groundwater.
- Answer
-
Soil water is held in the soil moisture zone that lies in the zone of aeration. Groundwater is held in the zone of saturation. Soil water is directly available for plants to use, ground water is not.
How do plants affect the hydrologic cycle?
- Answer
-
Plants affect the water cycle though by extracting water from the soil moisture zone and passing it to the atmosphere. Water moves as through fall through plant canopy.
What impact does soil texture have on field capacity?
- Answer
-
The field capacity is the maximum amount of water held in the soil after it has bee drained by gravity. Field capacity is higher in fine textured soils because there is more pore space per unit volume than for coarse textured soils.
What affects the permeability of subsurface warth materials?
- Answer
-
Permeability is the ability for water to move through earth material. The connectivity of pore spaces largely controls permeability. Large, well-connected pore space results in greater permeability. Thus, coarse soils are more permeable than fine textured soils.
How does soil texture affect available water?
- Answer
-
Finer textured soils hold more water and thus have more available water than coarse textured soils.
Compare and contrast an aquiclude with an aquifier.
- Answer
-
An aquifer is a body of earth material able to hold and transmit groundwater in economical amounts. An aquiclude is far less permeable and cannot transmit water through it.
Describe the effect of urbanization on a stream hydrograph.
- Answer
-
Urbanization can decrease the lag time between maximum precipitation and runoff, and increase and steepen the recessional limb of a hydrograph.
What is potential evapotranspiration?
- Answer
-
Evapotranspiration is the amount of water evaporated and transpired under an unlimited supply of water. Fundamentally it is determined by energy input to the environment. Evapotranspiration can be thought of as "water need”.
Under what conditions does a soil water deficit occur?
- Answer
-
A soil water deficit occurs when potential evapotranspiration exceeds precipitation and the soil water storage is zero (dry soil).
Under what conditions does a soil water surplus occur?
- Answer
-
A soil water surplus occurs when precipitation exceeds potential evapotranspiration and the soil is at field capacity.
-
In soil moisture budgeting, potential evapotranspiration is largely dependent on
- soil texture
- energy input
- plant type
- all the above
-
The largest store of fresh water in the hydrosphere is
- the ocean
- glaciers
- the Great Lakes
- ground water
-
Over pumping of ground water could lead to
- lowered water tables
- land subsidence
- decreased aquifer permeability
- all the above
-
Infiltration of water into the soil
- is higher for coarse textured soil
- is higher when the soil is dry
- is higher for vegetated surfaces
- is affected by all the above
-
Which of the following soil texture classes would have the largest available water?
- sand
- silt
- clay
- loam
-
The water held that is "bound" the tightest to soil particles is
- hydroscopic water
- capillary water
- gravity water
- pore water
-
The process whereby water drips from leaf-to-leaf finally making it to the ground is called
- through flow
- stem flow
- through fall
- interception
-
The zone of saturation is the
- soil water zone
- intermediate zone
- aeration zone
- ground water zone
-
The point at which plants can no longer extract water from the soil is called the
- field capacity point
- dryness point
- wilting point
- none of the above
-
If PE is greater than AE and P, and the soil is dry, then
- soil water recharge is likely to occur
- soil water surplus is likely to occur
- soil water utilization is likely to occur
- soil water deficit is likely to occur
- Answer
-
- B
- B
- D
- D
- D
- A
- A
- D
- C
- D
Additional Resources
Use these resources to further explore the world of geography
Focus on The Physical Environment : " The Rise and Fall of Africa’s Great Lake " NASA Earth Observatory
Connections : " Could California’s drought make residents sick? " NewsHour (PBS) 09/03/2016 report
Physical Geography Today : Ground Water Climate Response Network - USGS
World of Change : Evaporation of the Aral Sea ( NASA Earth Observatory )
Multimedia
"The Desert Springs of Mexico's Cuatro Cienegas", (8:56)


2.2: Affine Properties of Lorentz Geometry (Part 1)
Parallelism and measurement
We think of a frame of reference as a body of measurements or possible measurements to be made by some observer. Ordered geometry lacks measure. The following argument shows that merely by adding a notion of parallelism to our geometry, we automatically gain a system of measurement.
We only expect Lorentz frames to be local, but we do need them to be big enough to cover at least some amount of spacetime. If Betty does an Eötvös experiment by releasing a pencil and a lead ball side by side, she is essentially trying to release them at the same event A, so that she can observe them later and determine whether their world-lines stay right on top of one another at point B. That was all that was required for the Eötvös experiment, but in order to set up a Lorentz frame we need to start dealing with objects that are not right on top of one another. Suppose we release two lead balls in two different locations, at rest relative to one another. This could be the first step toward adding measurement to our geometry, since the balls mark two points in space that are separated by a certain distance, like two marks on a ruler, or the goals at the ends of a soccer field. Although the balls are separated by some finite distance, they are still close enough together so that if there is a gravitational field in the area, it is very nearly the same in both locations, and we expect the distance defined by the gap between them to stay the same. Since they are both subject only to gravitational forces, their world-lines are by definition straight lines (geodesics). The goal here is to end up with some kind of coordinate grid defining a (t,x) plane, and on such a grid, the two balls' world-lines are vertical lines. If we release them at events P and Q, then observe them again later at R and S, PQRS should form a rectangle on such a plot. In the figure, the irregularly spaced tick marks along the edges of the rectangle are meant to suggest that although ordered geometry provides us with a well-defined ordering along these lines, we have not yet constructed a complete system of measurement.
The depiction of PQSR as a rectangle, with right angles at its vertices, might lead us to believe that our geometry would have something like the concept of angular measure referred to in Euclid's E4, equality of right angles. But this is too naive even for the Euclidean 3+1 spacetime of Newton and Galileo. Suppose we switch to a frame that is moving relative to the first one, so that the balls are not at rest. In the Euclidean spacetime, time is absolute, so events P and Q would remain simultaneous, and so would R and S; the top and bottom edges PQ and RS would remain horizontal on the plot, but the balls' world-lines PR and QS would become slanted. The result would be a parallelogram. Since observers in different states of motion do not agree on what constitutes a right angle, the concept of angular measure is clearly not going to be useful here. Similarly, if Euclid had observed that a right angle drawn on a piece of paper no longer appeared to be a right angle when the paper was turned around, he would never have decided that angular measure was important enough to be enshrined in E4.
In the context of relativity, where time is not absolute, there is not even any reason to believe that different observers must agree on the simultaneity of PQ and RS. Our observation that time flows differently depending on the observer's state of motion tells us specifically to expect this not to happen when we switch to a frame moving to the relative one. Thus in general we expect that PQRS will be distorted into a form like the one shown in Figure 2.1.3. We do expect, however, that it will remain a parallelogram; a Lorentz frame is one in which the gravitational field, if any, is constant, so the properties of spacetime are uniform, and by symmetry the new frame should still have PR=QS and PQ=RS.
With this motivation, we form the system of affine geometry by adding the following axioms to set O1-O4. 1 The notation [PQRS] means that events P, Q, S, and R form a parallelogram, and is defined as the statement that the lines determined by PQ and RS never meet at a point, and similarly for PR and QS.
- A1 Constructibility of parallelograms: Given any P, Q, and R, there exists S such that [PQRS], and if P, Q, and R are distinct then S is unique.
- A2 Symmetric treatment of the sides of a parallelogram: If [PQRS], then [QRSP], [QPSR], and [PRQS].
- A3 Lines parallel to the same line are parallel to one another: If [ABCD] and [ABEF], then [CDEF].
Note
The axioms are summarized for convenient reference in Appendix C . This formulation is essentially the one given by Penrose, The Road to Reality, in section 14.1.
The following theorem is a stronger version of Playfair's axiom E5, the interpretation being that affine geometry describes a spacetime that is locally flat.
- Theorem: Given any line ℓ and any point P not on the line, there exists a unique line through P that is parallel to ℓ.
This is stronger than E5, which only guarantees uniqueness, not existence. Informally, the idea here is that A1 guarantees the existence of the parallel, and A3 makes it unique. 2
proof
Pick any two distinct points A and B on ℓ, and construct the uniquely determined parallelogram [ABPQ] (axiom A1). Points P and Q determine a line (axiom O1), and this line is parallel to ℓ (definition of the parallelogram). To prove that this line is unique, we argue by contradiction. Suppose some other parallel m to exist. If m crosses the infinite line BQ at some point Z, then both [ABPQ] and [ABPZ], so by A1, Q=Z, so the ℓ and m are the same. The only other possibility is that m is parallel to BQ, but then the following chain of parallelisms holds: PQ || AB || m || BQ. By A3, lines parallel to another line are parallel to each other, so PQ || BQ, but this is a contradiction, since they have Q in common.
Although these new axioms do nothing more than to introduce the concept of parallelism lacking in ordered geometry, it turns out that they also allow us to build up a concept of measurement. Let ℓ be a line, and suppose we want to define a number system on this line that measures how far apart events are. Depending on the type of line, this could be a measurement of time, of spatial distance, or a mixture of the two. First we arbitrarily single out two distinct points on ℓ and label them 0 and 1. Next, pick some auxiliary point q 0 not lying on ℓ. By A1, construct the parallelogram 01q 0 q 1 . Next construct q 0 1q 1 2. Continuing in this way, we have a scaffolding of parallelograms adjacent to the line, determining an infinite lattice of points 1, 2, 3, . . . on the line, which represent the positive integers. Fractions can be defined in a similar way. For example, \(\frac{1}{2}\) is defined as the point such that when the initial lattice segment 0 \(\frac{1}{2}\) is extended by the same construction, the next point on the lattice is 1.
The continuously varying variable constructed in this way is called an affine parameter . The time measured by a free-falling clock is an example of an affine parameter, as is the distance measured by the tick marks on a free-falling ruler. Since light rays travel along geodesics, the wave crests on a light wave can even be used analogously to the ruler’s tick marks.
Example 1: centroids
The affine parameter can be used to define the centroid of a set of points. In the simplest example, finding the centroid of two points, we simply bisect the line segment as described above in the construction of the number \(\frac{1}{2}\). Similarly, the centroid of a triangle can be defined as the intersection of its three medians, the lines joining each vertex to the midpoint of the opposite side.
Example 2: Conservation of momentum
In nonrelativistic mechanics, the concept of the center of mass is closely related to the law of conservation of momentum. For example, a logically complete statement of the law is that if a system of particles is not subjected to any external force, and we pick a frame in which its center of mass is initially at rest, then its center of mass remains at rest in that frame. Since centroids are well defined in affine geometry, and Lorentz frames have affine properties, we have grounds to hope that it might be possible to generalize the definition of momentum relativistically so that the generalized version is conserved in a Lorentz frame. On the other hand, we don’t expect to be able to define anything like a global Lorentz frame for the entire universe, so there is no such natural expectation of being able to define a global principle of conservation of momentum. This is an example of a general fact about relativity, which is that conservation laws are difficult or impossible to formulate globally.
Although the affine parameter gives us a system of measurement for free in a geometry whose axioms do not even explicitly mention measurement, there are some restrictions:
- The affine parameter is defined only along straight lines, i.e., geodesics. Alice’s clock defines an affine parameter, but Betty’s does not, since it is subject to nongravitational forces.
- We cannot compare distances along two arbitrarily chosen lines, only along a single line or two parallel lines.
- The affine parameter is arbitrary not only in the choice of its origin 0 (which is to be expected in any case, since any frame of reference requires such an arbitrary choice) but also in the choice of scale. For example, there is no fundamental way of deciding how fast to make a clock tick.
We will eventually want to lift some of these restrictions by adding to our kit a tool called a metric, which allows us to define distances along arbitrary curves in space time, and to compare distances in different directions. The affine parameter, however, will not be entirely superseded. In particular, we’ll find that the metric has a couple of properties that are not as nice as those of the affine parameter. The square of a metric distance can be negative, and the metric distance measured along a light ray is precisely zero, which is not very useful.
Exercise \(\PageIndex{1}\)
Self-check: By the construction of the affine parameter above, affine distances on the same line are comparable. By another construction, verify the claim made above that this can be extended to distances measured along two different parallel lines.
Example 3: Area and volume
It is possible to define area and volume in affine geometry. This is a little surprising, since distances along different lines are not even comparable. However, we are already accustomed to multiplying and dividing numbers that have different units (a concept that would have given Euclid conniptions), and the situation in affine geometry is really no different. To define area, we extend the one-dimensional lattice to two dimensions. Any planar figure can be superimposed on such a lattice, and dissected into parallelograms, each of which has a standard area.
Example 4: Area on a graph of v versus t
If an object moves at a constant velocity v for time t, the distance it travels can be represented by the area of a parallelogram in an affine plane with sides having lengths v and t. These two lengths are measured by affine parameters along two different directions, so they are not comparable. For example, it is meaningless to ask whether 1 m/s is greater than, less than, or equal to 1 s. If we were graphing velocity as a function of time on a conventional Cartesian graph, the v and t axes would be perpendicular, but affine geometry has no notion of angular measure, so this is irrelevant here.
Exercise \(\PageIndex{2}\)
Self-check: If multiplication is defined in terms of affine area, prove the commutative property ab = ba and the distributive rule a(b + c) = ab + bc from axioms A1-A3.


People as Sources
People don’t just create the sources we use. They are actually sources them-selves. Most of us use people as sources all the time in our private lives, such when we ask a friend for a restaurant recommendation or ask whether a movie is worth watching. But you probably aren’t using people as sources very often in your assignments–unless you are a journalism major, of course.
In fact, research indicates that employers such as Battelle, Nationwide Insurance, Microsoft, the FBI, the Smithsonian, the Port of Los Angeles, SS&G Financial Services, and Marriott International have been dissatisfied with their new hires’ inability to gather information by talking with real people. They’ve found new hires unwilling or unprepared to ask the experienced employee down the hall or the expert across town for information to solve a problem. For instance, the study linked to above quotes one employer as saying about new hires:
Here’s something we’re targeting in interviews now—the big thing is they believe the computer is their workspace, so basic interactions between people are lost. They won’t get up and walk over and ask someone a question. They are less comfortable and have some lack of willingness to use people as sources and also have a lack of awareness that people are a valid source of information…
So getting some experience using people as sources is likely to help you not just with a current research assignment but with your work in the future.
Important: Who’s an “Expert”?
Experts aren’t only researchers with Ph.D.s doing academic work. The question when trying to decide who can be a source is really always, who can speak with authority about any part of the subject? And the answer to that question is always contextual, a kind of “it depends.”
People can speak with authority for different reasons. According to the framework for information literacy, they can have subject expertise (say, having done scholarship in the field), societal position (maybe a public office or other relevant work title), or special experience (say, living or working in a particular situation of interest or having participated in an historical event).
For instance, people who have had firsthand experience living or working with a situation (say, a survivor of school shooting if your topic is on that subject) you are studying can have a unique perspective unavailable elsewhere. And it’s that up-close, firsthand view of the situation that gives them the authority that you and your audience respond to.
Of course, such sources have to be evaluated just like any other. Could they be biased? Like any source, yes. We just have to keep that possible bias in mind as we use the information from such a source. That’s part of exercising the critical thinking that research assignments are famous for producing.
Potentially biased or not, sometimes a source’s firsthand experience can’t be beat. And recognizing what they offer can help us open up to diverse ideas and worldviews that we would otherwise miss. Don’t be surprised if this kind of source takes you off in completely new directions with your assignment, ones that turn out to be much more interesting than those you were following before. For many researchers, finding sources that really open up a topic like that is one of the most rewarding—and fun—things about doing research.
Some Examples of People as Sources
|
Research Question
|
Potential Person as Source
|
Potential Person as Source
|
|
How are tools originally developed for medicine, geology, and manufacturing used to explore paintings and sculptures?
|
An art conservator who uses those tools that you read about in the newspaper or other source
|
The person who invented one of the tools on the floor of the factory where he works
|
|
Why do most people who qualify for food at foodbanks not ask for food?
|
A local food bank director
|
A person (perhaps a fellow student) who qualifies but does not ask for food at a food bank
|
|
How and why do city and county governments brand themselves?
|
An official in such a city or county who has been involved in branding decisions
|
The director of a company that designs branding for cities and counties
|
You can interview a person as a source on the phone, in email, with Skype, or face-to-face. You’ll need to:
-
Pay attention when reading other sources so you can identify whom to contact and know what they could have to offer.
-
Prepare by learning enough about your topic so you can ask appropriate questions, know what your expert has done in relation to that topic so you don’t seem ignorant of their contribution, and know how to contact them. You might also want to do a practice interview with a friend.
-
Contact your source to see if they are willing to talk with you and when that would be convenient. Then follow through.
Use good interview techniques, such as trying to put them at ease, using active listening techniques to encourage them to talk, asking follow up questions, and thanking them at the end of the interview.
TIP: How to identify researchers at OSU to interview.
-
Search the database
Scopus
for your topic. Once you have some results, use the Affiliation option among Define Results options on the left to limit your search to Ohio State University. You may also want to limit by the Year option.
-
Pull up relevant articles’ records in Scopus (you can sort by times cited) and then identify which of the authors were at the time at OSU. (Articles may have multiple authors–sometimes over a dozen or more if they are in science).
-
Go to Find People (on the OSU navigation bar on all OSU pages) and search for the OSU person’s name to see if they’re still at OSU and get their contact information. In many cases, they will still be at OSU, especially for fairly recent articles, because OSU updates author profiles regularly. Note that researchers may or may not be faculty–some may be staff or even students.
Citing People as Sources
Like other sources, people should be cited in your research final product, depending on the citation style you’re using. For instance, in APA style, interviews, e-mail, and other personal communication should not appear in the reference list but should be in your main text only like this: (A. Authorslastname, personal communication, July 29, 2018).
See
Purdue Online Writing Lab
(OWL) for information on how to handle interviews and other communication with people in other styles.


2 .
A nurse is providing discharge instructions for a patient with a BMI of 31. The patient was admitted for an appendectomy and holds a job as a full-time construction worker and part-time mover. Which should be included in the patient’s postoperative plan and why: decrease or increase caloric intake?
A nurse on the medical unit is caring for four patients and just received the vital sign data from the patient care tech. The patient care tech used a digital vital sign machine and a standard adult cuff for all the patients but made sure to clean it between each use.
-
Patient 1: A patient admitted with kidney stones awaiting stent placement. T 99.1, P 104, RR 20, BP 136/82, Oxygen saturation 100 percent on room air
-
Patient 2: A patient admitted with heart failure with pulmonary edema. T 97.1, P 110, RR 24, BP 88/42, Oxygen saturation 91 percent on 2 L of oxygen
-
Patient 3: A patient admitted with chronic renal failure and atherosclerosis. T 98.2, P 90, RR 18, BP 175/101, Oxygen saturation 97 percent on room air
-
Patient 4: An older adult patient admitted with failure to thrive. T 96.9, P 68, RR 18, BP 92/49, Oxygen saturation 96 percent on room air
5 .
Who should the nurse see first and why?


17.5: Collision Theory
- Use the postulates of collision theory to explain the effects of physical state, temperature, and concentration on reaction rates
- Define the concepts of activation energy and transition state
- Use the Arrhenius equation in calculations relating rate constants to temperature
We should not be surprised that atoms, molecules, or ions must collide before they can react with each other. Atoms must be close together to form chemical bonds. This simple premise is the basis for a very powerful theory that explains many observations regarding chemical kinetics, including factors affecting reaction rates. Collision theory is based on the following postulates:
- The rate of a reaction is proportional to the rate of reactant collisions: \[\mathrm{reaction\: rate ∝ \dfrac{\#\,collisions}{time}} \nonumber \]
- The reacting species must collide in an orientation that allows contact between the atoms that will become bonded together in the product.
- The collision must occur with adequate energy to permit mutual penetration of the reacting species’ valence shells so that the electrons can rearrange and form new bonds (and new chemical species).
We can see the importance of the two physical factors noted in postulates 2 and 3, the orientation and energy of collisions, when we consider the reaction of carbon monoxide with oxygen:
\[\ce{2CO}(g)+\ce{O2}(g)⟶\ce{2CO2}(g) \nonumber \]
Carbon monoxide is a pollutant produced by the combustion of hydrocarbon fuels. To reduce this pollutant, automobiles have catalytic converters that use a catalyst to carry out this reaction. It is also a side reaction of the combustion of gunpowder that results in muzzle flash for many firearms. If carbon monoxide and oxygen are present in sufficient quantity, the reaction is spontaneous at high temperature and pressure.
The first step in the gas-phase reaction between carbon monoxide and oxygen is a collision between the two molecules:
\[\ce{CO}(g)+\ce{O2}(g)⟶\ce{CO2}(g)+\ce{O}(g) \nonumber \]
Although there are many different possible orientations the two molecules can have relative to each other, consider the two presented in Figure \(\PageIndex{1}\). In the first case, the oxygen side of the carbon monoxide molecule collides with the oxygen molecule. In the second case, the carbon side of the carbon monoxide molecule collides with the oxygen molecule. The second case is clearly more likely to result in the formation of carbon dioxide, which has a central carbon atom bonded to two oxygen atoms \(\ce{(O=C=O)}\). This is a rather simple example of how important the orientation of the collision is in terms of creating the desired product of the reaction.
If the collision does take place with the correct orientation, there is still no guarantee that the reaction will proceed to form carbon dioxide. Every reaction requires a certain amount of activation energy for it to proceed in the forward direction, yielding an appropriate activated complex along the way. As Figure \(\PageIndex{2}\) demonstrates, even a collision with the correct orientation can fail to form the reaction product. In the study of reaction mechanisms, each of these three arrangements of atoms is called a proposed activated complex or transition state .
In most circumstances, it is impossible to isolate or identify a transition state or activated complex. In the reaction between carbon monoxide and oxygen to form carbon dioxide, activated complexes have only been observed spectroscopically in systems that utilize a heterogeneous catalyst. The gas-phase reaction occurs too rapidly to isolate any such chemical compound.
Collision theory explains why most reaction rates increase as concentrations increase. With an increase in the concentration of any reacting substance, the chances for collisions between molecules are increased because there are more molecules per unit of volume. More collisions mean a faster reaction rate, assuming the energy of the collisions is adequate.
Activation Energy and the Arrhenius Equation
The minimum energy necessary to form a product during a collision between reactants is called the activation energ y (\(E_a\)). The kinetic energy of reactant molecules plays an important role in a reaction because the energy necessary to form a product is provided by a collision of a reactant molecule with another reactant molecule. (In single-reactant reactions, activation energy may be provided by a collision of the reactant molecule with the wall of the reaction vessel or with molecules of an inert contaminant.) If the activation energy is much larger than the average kinetic energy of the molecules, the reaction will occur slowly: Only a few fast-moving molecules will have enough energy to react. If the activation energy is much smaller than the average kinetic energy of the molecules, the fraction of molecules possessing the necessary kinetic energy will be large; most collisions between molecules will result in reaction, and the reaction will occur rapidly.
Figure \(\PageIndex{3}\) shows the energy relationships for the general reaction of a molecule of \(A\) with a molecule of \(B\) to form molecules of \(C\) and \(D\):
\[A+B⟶C+D \nonumber \]
The figure shows that the energy of the transition state is higher than that of the reactants \(A\) and \(B\) by an amount equal to \(E_a\), the activation energy. Thus, the sum of the kinetic energies of \(A\) and \(B\) must be equal to or greater than E a to reach the transition state. After the transition state has been reached, and as \(C\) and \(D\) begin to form, the system loses energy until its total energy is lower than that of the initial mixture. This lost energy is transferred to other molecules, giving them enough energy to reach the transition state. The forward reaction (that between molecules \(A\) and \(B\)) therefore tends to take place readily once the reaction has started. In Figure \(\PageIndex{3}\), \(ΔH\) represents the difference in enthalpy between the reactants (\(A\) and \(B\)) and the products (\(C\) and \(D\)). The sum of \(E_a\) and \(ΔH\) represents the activation energy for the reverse reaction:
\[C+D⟶A+B \nonumber \]
We can use the Arrhenius equation to relate the activation energy and the rate constant, k , of a given reaction:
\[k=Ae^{−E_a/RT} \label{Arrhenius} \]
In this equation,
- \(R\) is the ideal gas constant, which has a value 8.314 J/mol/K,
- \(T\) is temperature on the Kelvin scale,
- \(E_a\) is the activation energy in joules per mole,
- \(e\) is the constant 2.7183, and
- \(A\) is a constant called the frequency factor , which is related to the frequency of collisions and the orientation of the reacting molecules.
Both postulates of the collision theory of reaction rates are accommodated in the Arrhenius equation. The frequency factor A is related to the rate at which collisions having the correct orientation occur. The exponential term, \(e^{−E_a/RT}\), is related to the fraction of collisions providing adequate energy to overcome the activation barrier of the reaction.
At one extreme, the system does not contain enough energy for collisions to overcome the activation barrier. In such cases, no reaction occurs. At the other extreme, the system has so much energy that every collision with the correct orientation can overcome the activation barrier, causing the reaction to proceed. In such cases, the reaction is nearly instantaneous.
The Arrhenius equation (Equation \ref{Arrhenius}) describes quantitatively much of what we have already discussed about reaction rates. For two reactions at the same temperature, the reaction with the higher activation energy has the lower rate constant and the slower rate. The larger value of \(E_a\) results in a smaller value for \(e^{−E_a/RT}\), reflecting the smaller fraction of molecules with enough energy to react. Alternatively, the reaction with the smaller \(E_a\) has a larger fraction of molecules with enough energy to react. This will be reflected as a larger value of \(e^{−E_a/RT}\), a larger rate constant, and a faster rate for the reaction.
An increase in temperature has the same effect as a decrease in activation energy. A larger fraction of molecules has the necessary energy to react (Figure \(\PageIndex{4}\)), as indicated by an increase in the value of \(e^{−E_a/RT}\). The rate constant is also directly proportional to the frequency factor, \(A\). Hence a change in conditions or reactants that increases the number of collisions with a favorable orientation for reaction results in an increase in \(A\) and, consequently, an increase in \(k\).
A convenient approach to determining \(E_a\) for a reaction involves the measurement of \(k\) at different temperatures and using of an alternate version of the Arrhenius equation that takes the form of linear equation:
\[\begin{align*}
\ln k&=\left(\dfrac{−E_a}{R}\right)\left(\dfrac{1}{T}\right)+\ln A\\
y&=mx+b
\end{align*} \nonumber \]
Thus, a plot of \(\ln k\) versus \(\dfrac{1}{T}\) gives a straight line with the slope \(\dfrac{-E_\ce{a}}{R}\), from which E a may be determined. The intercept gives the value of \(\ln A\). This is sometimes call an Arrhenius Plot.
Determination of E a The variation of the rate constant with temperature for the decomposition of HI( g ) to H 2 ( g ) and I 2 ( g ) is given here. What is the activation energy for the reaction?
\[\ce{2HI}(g)⟶\ce{H2}(g)+\ce{I2}(g) \nonumber \]
| T (K) | k (L/mol/s) |
|---|---|
| 555 | 3.52 × 10 −7 |
| 575 | 1.22 × 10 −6 |
| 645 | 8.59 × 10 −5 |
| 700 | 1.16 × 10 −3 |
| 781 | 3.95 × 10 −2 |
Solution
Values of \(\dfrac{1}{T}\) and ln k are:
| \(\mathrm{\dfrac{1}{T}\:(K^{−1})}\) | ln k |
|---|---|
| 1.80 × 10 −3 | −14.860 |
| 1.74 × 10 −3 | −13.617 |
| 1.55 × 10 −3 | −9.362 |
| 1.43 × 10 −3 | −6.759 |
| 1.28 × 10 −3 | −3.231 |
Figure \(\PageIndex{5}\) is a graph of ln k versus \(\dfrac{1}{T}\). To determine the slope of the line, we need two values of ln k , which are determined from the line at two values of \(\dfrac{1}{T}\) (one near each end of the line is preferable). For example, the value of ln k determined from the line when \(\dfrac{1}{T}=1.25×10^{−3}\) is −2.593; the value when \(\dfrac{1}{T}=1.78×10^{−3}\) is −14.447.
The slope of this line is given by the following expression:
\[\begin{align*}\ce{Slope}&=\dfrac{Δ(\ln k)}{Δ\left(\dfrac{1}{T}\right)}\\
&=\mathrm{\dfrac{(−14.447)−(−2.593)}{(1.78×10^{−3}\:K^{−1})−(1.25×10^{−3}\:K^{−1})}}\\
&=\mathrm{\dfrac{−11.854}{0.53×10^{−3}\:K^{−1}}=2.2×10^4\:K}\\
&=−\dfrac{E_\ce{a}}{R}
\end{align*} \nonumber \]
Thus:
\[ \begin{align*} E_\ce{a} &=\mathrm{−slope×\mathit R=−(−2.2×10^4\:K×8.314\: J\: mol^{−1}\:K^{−1})} \\[4pt] &=\mathrm{1.8×10^5\:J\: mol^{−1}} \end{align*} \nonumber \]
In many situations, it is possible to obtain a reasonable estimate of the activation energy without going through the entire process of constructing the Arrhenius plot. The Arrhenius equation:
\[\ln k=\left(\dfrac{−E_\ce{a}}{R}\right)\left(\dfrac{1}{T}\right)+\ln A \nonumber \]
can be rearranged as shown to give:
\[\dfrac{Δ(\ln k)}{Δ\left(\dfrac{1}{T}\right)}=−\dfrac{E_\ce{a}}{R} \nonumber \]
or
\[\ln\dfrac{k_1}{k_2}=\dfrac{E_\ce{a}}{R}\left(\dfrac{1}{T_2}−\dfrac{1}{T_1}\right) \nonumber \]
This equation can be rearranged to give a one-step calculation to obtain an estimate for the activation energy:
\[E_\ce{a}=−R\left( \dfrac{\ln k_2−\ln k_1}{\left(\dfrac{1}{T_2}\right)−\left(\dfrac{1}{T_1}\right)}\right ) \nonumber \]
Using the experimental data presented here, we can simply select two data entries. For this example, we select the first entry and the last entry:
| T (K) | k (L/mol/s) | \(\dfrac{1}{T}\:(K^{-1})\) | ln k |
|---|---|---|---|
| 555 | 3.52 × 10 −7 | 1.80 × 10 −3 | −14.860 |
| 781 | 3.95 × 10 −2 | 1.28 × 10 −3 | −3.231 |
After calculating \(\dfrac{1}{T}\) and ln k , we can substitute into the equation:
\[E_\ce{a}=\mathrm{−8.314\:J\:mol^{−1}\:K^{−1}\left(\dfrac{−3.231−(−14.860)}{1.28×10^{−3}\:K^{−1}−1.80×10^{−3}\:K^{−1}}\right)} \nonumber \]
and the result is E a = 185,900 J/mol.
This method is very effective, especially when a limited number of temperature-dependent rate constants are available for the reaction of interest.
The rate constant for the rate of decomposition of N 2 O 5 to NO and O 2 in the gas phase is 1.66 L/mol/s at 650 K and 7.39 L/mol/s at 700 K:
\[\ce{2N2O5}(g)⟶\ce{4NO}(g)+\ce{3O2}(g) \nonumber \]
Assuming the kinetics of this reaction are consistent with the Arrhenius equation, calculate the activation energy for this decomposition.
- Answer
-
113,000 J/mol
Summary
Chemical reactions require collisions between reactant species. These reactant collisions must be of proper orientation and sufficient energy in order to result in product formation. Collision theory provides a simple but effective explanation for the effect of many experimental parameters on reaction rates. The Arrhenius equation describes the relation between a reaction’s rate constant and its activation energy, temperature, and dependence on collision orientation.
Key Equations
- \(k=Ae^{−E_a/RT}\)
- \(\ln k=\left(\dfrac{−E_\ce{a}}{R}\right)\left(\dfrac{1}{T}\right)+\ln A\)
- \(\ln\dfrac{k_1}{k_2}=\dfrac{E_\ce{a}}{R}\left(\dfrac{1}{T_2}−\dfrac{1}{T_1}\right)\)
Glossary
- activated complex
- (also, transition state) unstable combination of reactant species representing the highest energy state of a reaction system
- activation energy ( E a )
- energy necessary in order for a reaction to take place
- Arrhenius equation
- mathematical relationship between the rate constant and the activation energy of a reaction
- collision theory
- model that emphasizes the energy and orientation of molecular collisions to explain and predict reaction kinetics
- frequency factor ( A )
- proportionality constant in the Arrhenius equation, related to the relative number of collisions having an orientation capable of leading to product formation


10.1: Radiation and Electromagnetic waves
- Explain the basic behavior of waves.
- Identify the forms of radiation present in the electromagnetic spectrum.
- Distinguish forms of radiation based on wavelength, frequency, and energy
- Describe the wave nature of light
- Use appropriate equations to calculate related light-wave properties such as period, frequency, wavelength, and energy
- Describe the particle nature of light
The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms and was able to demonstrate that white light consists of the individual colors of the rainbow combined together. Newton explained his optics findings in terms of a "corpuscular" view of light, in which light was composed of streams of extremely tiny particles travelling at high speeds according to Newton's laws of motion. Others in the seventeenth century, such as Christiaan Huygens , had shown that optical phenomena such as reflection and refraction could be equally well explained in terms of light as waves travelling at high speed through a medium called "luminiferous aether" that was thought to permeate all space. Early in the nineteenth century, Thomas Young demonstrated that light passing through narrow, closely spaced slits produced interference patterns that could not be explained in terms of Newtonian particles but could be easily explained in terms of waves. Later in the nineteenth century, after James Clerk Maxwell developed his theory of electromagnetic radiation and showed that light was the visible part of a vast spectrum of electromagnetic waves, the particle view of light became thoroughly discredited. By the end of the nineteenth century, scientists viewed the physical universe as roughly comprising two separate domains: matter composed of particles moving according to Newton's laws of motion, and electromagnetic radiation consisting of waves governed by Maxwell's equations. Today, these domains are referred to as classical mechanics and classical electrodynamics (or classical electromagnetism). Although there were a few physical phenomena that could not be explained within this framework, scientists at that time were so confident of the overall soundness of this framework that they viewed these aberrations as puzzling paradoxes that would ultimately be resolved somehow within this framework. As we shall see, these paradoxes led to a contemporary framework that intimately connects particles and waves at a fundamental level called wave-particle duality, which has superseded the classical view.
Visible light and other forms of electromagnetic radiation play important roles in chemistry, since they can be used to infer the energies of electrons within atoms and molecules. Much of modern technology is based on electromagnetic radiation. For example, radio waves from a mobile phone, X-rays used by dentists, the energy used to cook food in your microwave, the radiant heat from red-hot objects, and the light from your television screen are forms of electromagnetic radiation that all exhibit wavelike behavior.
Waves
A wave is an oscillation or periodic movement that can transport energy from one point in space to another. Common examples of waves are all around us. Shaking the end of a rope transfers energy from your hand to the other end of the rope, dropping a pebble into a pond causes waves to ripple outward along the water's surface, and the expansion of air that accompanies a lightning strike generates sound waves (thunder) that can travel outward for several miles. In each of these cases, kinetic energy is transferred through matter (the rope, water, or air) while the matter remains essentially in place. An insightful example of a wave occurs in sports stadiums when fans in a narrow region of seats rise simultaneously and stand with their arms raised up for a few seconds before sitting down again while the fans in neighboring sections likewise stand up and sit down in sequence. While this wave can quickly encircle a large stadium in a few seconds, none of the fans actually travel with the wave-they all stay in or above their seats.
Waves need not be restricted to travel through matter. As Maxwell showed, electromagnetic waves consist of an electric field oscillating in step with a perpendicular magnetic field, both of which are perpendicular to the direction of travel. These waves can travel through a vacuum at a constant speed of 2.998 × 10 8 m/s, the speed of light (denoted by c ).
All waves, including forms of electromagnetic radiation, are characterized by, a wavelength (denoted by λ , the lowercase Greek letter lambda), a frequency (denoted by ν , the lowercase Greek letter nu), and an amplitude . As can be seen in Figure \(\PageIndex{1}\), the wavelength is the distance between two consecutive peaks or troughs in a wave (measured in meters in the SI system). Electromagnetic waves have wavelengths that fall within an enormous range-wavelengths of kilometers (10 3 m) to picometers (10 −12 m) have been observed. The frequency is the number of wave cycles that pass a specified point in space in a specified amount of time (in the SI system, this is measured in seconds). A cycle corresponds to one complete wavelength. The unit for frequency, expressed as cycles per second [s −1 ], is the hertz (Hz) . Common multiples of this unit are megahertz, (1 MHz = 1 × 10 6 Hz) and gigahertz (1 GHz = 1 × 10 9 Hz). The amplitude corresponds to the magnitude of the wave's displacement and so, in Figure, this corresponds to one-half the height between the peaks and troughs. The amplitude is related to the intensity of the wave, which for light is the brightness, and for sound is the loudness.
The product of a wave's wavelength ( λ ) and its frequency ( ν ), λν , is the speed of the wave. Thus, for electromagnetic radiation in a vacuum:
\[c=\mathrm{2.998×10^8\,ms^{−1}}=λν \label{6.2.1} \]
Wavelength and frequency are inversely proportional: As the wavelength increases, the frequency decreases. The inverse proportionality is illustrated in Figure \(\PageIndex{2}\). This figure also shows the electromagnetic spectrum , the range of all types of electromagnetic radiation. Each of the various colors of visible light has specific frequencies and wavelengths associated with them, and you can see that visible light makes up only a small portion of the electromagnetic spectrum. Because the technologies developed to work in various parts of the electromagnetic spectrum are different, for reasons of convenience and historical legacies, different units are typically used for different parts of the spectrum. For example, radio waves are usually specified as frequencies (typically in units of MHz), while the visible region is usually specified in wavelengths (typically in units of nm or angstroms).
A sodium streetlight gives off yellow light that has a wavelength of 589 nm (1 nm = 1 × 10 −9 m). What is the frequency of this light?
Solution
We can rearrange the Equation \ref{6.2.1} to solve for the frequency:
\[ \nu=\dfrac{c}{λ} \nonumber \]
Since c is expressed in meters per second, we must also convert 589 nm to meters.
\[ \nu=\mathrm{\left(\dfrac{2.998×10^8\:\cancel{m}s^{−1}}{589\cancel{nm}}\right)\left(\dfrac{1×10^9\cancel{nm}}{1\cancel{m}}\right)=5.09×10^{14}\,s^{−1}} \nonumber \]
One of the frequencies used to transmit and receive cellular telephone signals in the United States is 850 MHz. What is the wavelength in meters of these radio waves?
- Answer
-
0.353 m = 35.3 cm
Wireless Communication
Many valuable technologies operate in the radio (3 kHz-300 GHz) frequency region of the electromagnetic spectrum. At the low frequency (low energy, long wavelength) end of this region are AM (amplitude modulation) radio signals (540-2830 kHz) that can travel long distances. FM (frequency modulation) radio signals are used at higher frequencies (87.5-108.0 MHz). In AM radio, the information is transmitted by varying the amplitude of the wave (Figure \(\PageIndex{5}\)). In FM radio, by contrast, the amplitude is constant and the instantaneous frequency varies.
Other technologies also operate in the radio-wave portion of the electromagnetic spectrum. For example, 4G cellular telephone signals are approximately 880 MHz, while Global Positioning System (GPS) signals operate at 1.228 and 1.575 GHz, local area wireless technology (Wi-Fi) networks operate at 2.4 to 5 GHz, and highway toll sensors operate at 5.8 GHz. The frequencies associated with these applications are convenient because such waves tend not to be absorbed much by common building materials.
One particularly characteristic phenomenon of waves results when two or more waves come into contact: They interfere with each other. Figure \(\PageIndex{5}\) shows the interference patterns that arise when light passes through narrow slits closely spaced about a wavelength apart. The fringe patterns produced depend on the wavelength, with the fringes being more closely spaced for shorter wavelength light passing through a given set of slits. When the light passes through the two slits, each slit effectively acts as a new source, resulting in two closely spaced waves coming into contact at the detector (the camera in this case). The dark regions in Figure \(\PageIndex{5}\) correspond to regions where the peaks for the wave from one slit happen to coincide with the troughs for the wave from the other slit (destructive interference), while the brightest regions correspond to the regions where the peaks for the two waves (or their two troughs) happen to coincide (constructive interference). Likewise, when two stones are tossed close together into a pond, interference patterns are visible in the interactions between the waves produced by the stones. Such interference patterns cannot be explained by particles moving according to the laws of classical mechanics.
Because the wavelengths of X-rays (10-10,000 picometers [pm]) are comparable to the size of atoms, X-rays can be used to determine the structure of molecules. When a beam of X-rays is passed through molecules packed together in a crystal, the X-rays collide with the electrons and scatter. Constructive and destructive interference of these scattered X-rays creates a specific diffraction pattern. Calculating backward from this pattern, the positions of each of the atoms in the molecule can be determined very precisely. One of the pioneers who helped create this technology was Dorothy Crowfoot Hodgkin.
She was born in Cairo, Egypt, in 1910, where her British parents were studying archeology. Even as a young girl, she was fascinated with minerals and crystals. When she was a student at Oxford University, she began researching how X-ray crystallography could be used to determine the structure of biomolecules. She invented new techniques that allowed her and her students to determine the structures of vitamin B 12 , penicillin, and many other important molecules. Diabetes, a disease that affects 382 million people worldwide, involves the hormone insulin. Hodgkin began studying the structure of insulin in 1934, but it required several decades of advances in the field before she finally reported the structure in 1969. Understanding the structure has led to better understanding of the disease and treatment options.
Not all waves are travelling waves. Standing waves (also known as stationary waves ) remain constrained within some region of space. As we shall see, standing waves play an important role in our understanding of the electronic structure of atoms and molecules. The simplest example of a standing wave is a one-dimensional wave associated with a vibrating string that is held fixed at its two end points. Figure \(\PageIndex{6}\) shows the four lowest-energy standing waves (the fundamental wave and the lowest three harmonics) for a vibrating string at a particular amplitude. Although the string's motion lies mostly within a plane, the wave itself is considered to be one dimensional, since it lies along the length of the string. The motion of string segments in a direction perpendicular to the string length generates the waves and so the amplitude of the waves is visible as the maximum displacement of the curves seen in Figure \(\PageIndex{6}\). The key observation from the figure is that only those waves having an integer number, n, of half-wavelengths between the end points can form. A system with fixed end points such as this restricts the number and type of the possible waveforms. This is an example of quantization , in which only discrete values from a more general set of continuous values of some property are observed. Another important observation is that the harmonic waves (those waves displaying more than one-half wavelength) all have one or more points between the two end points that are not in motion. These special points are nodes . The energies of the standing waves with a given amplitude in a vibrating string increase with the number of half-wavelengths n . Since the number of nodes is n – 1, the energy can also be said to depend on the number of nodes, generally increasing as the number of nodes increases.
An example of two-dimensional standing waves is shown in Figure \(\PageIndex{7}\) which shows the vibrational patterns on a flat surface. Although the vibrational amplitudes cannot be seen like they could in the vibrating string, the nodes have been made visible by sprinkling the drum surface with a powder that collects on the areas of the surface that have minimal displacement. For one-dimensional standing waves, the nodes were points on the line, but for two-dimensional standing waves, the nodes are lines on the surface (for three-dimensional standing waves, the nodes are two-dimensional surfaces within the three-dimensional volume). Because of the circular symmetry of the drum surface, its boundary conditions (the drum surface being tightly constrained to the circumference of the drum) result in two types of nodes: radial nodes that sweep out all angles at constant radii and, thus, are seen as circles about the center, and angular nodes that sweep out all radii at constant angles and, thus, are seen as lines passing through the center. The upper left image in Figure \(\PageIndex{7}\) shows two radial nodes, while the image in the lower right shows the vibrational pattern associated with three radial nodes and two angular nodes.
Blackbody Radiation and the Ultraviolet Catastrophe
The last few decades of the nineteenth century witnessed intense research activity in commercializing newly discovered electric lighting. This required obtaining a better understanding of the distributions of light emitted from various sources being considered. Artificial lighting is usually designed to mimic natural sunlight within the limitations of the underlying technology. Such lighting consists of a range of broadly distributed frequencies that form a continuous spe ctrum. Figure \(\PageIndex{8}\) shows the wavelength distribution for sunlight. The most intense radiation is in the visible region, with the intensity dropping off rapidly for shorter wavelength ultraviolet (UV) light, and more slowly for longer wavelength infrared (IR) light.
In Figure \(\PageIndex{8}\), the solar distribution is compared to a representative distribution, called a blackbody spectrum, that corresponds to a temperature of 5250 °C. The blackbody spectrum matches the solar spectrum quite well. A blackbody is a convenient, ideal emitter that approximates the behavior of many materials when heated. It is “ideal” in the same sense that an ideal gas is a convenient, simple representation of real gases that works well, provided that the pressure is not too high nor the temperature too low. A good approximation of a blackbody that can be used to observe blackbody radiation is a metal oven that can be heated to very high temperatures. The oven has a small hole allowing for the light being emitted within the oven to be observed with a spectrometer so that the wavelengths and their intensities can be measured. Figure \(\PageIndex{8}\) shows the resulting curves for some representative temperatures. Each distribution depends only on a single parameter: the temperature. The maxima in the blackbody curves, λ max , shift to shorter wavelengths as the temperature increases, reflecting the observation that metals being heated to high temperatures begin to glow a darker red that becomes brighter as the temperature increases, eventually becoming white hot at very high temperatures as the intensities of all of the visible wavelengths become appreciable. This common observation was at the heart of the first paradox that showed the fundamental limitations of classical physics that we will examine.
Physicists derived mathematical expressions for the blackbody curves using well-accepted concepts from the theories of classical mechanics and classical electromagnetism. The theoretical expressions as functions of temperature fit the observed experimental blackbody curves well at longer wavelengths, but showed significant discrepancies at shorter wavelengths. Not only did the theoretical curves not show a peak, they absurdly showed the intensity becoming infinitely large as the wavelength became smaller, which would imply that everyday objects at room temperature should be emitting large amounts of UV light. This became known as the “ultraviolet catastrophe” because no one could find any problems with the theoretical treatment that could lead to such unrealistic short-wavelength behavior. Finally, around 1900, Max Planck derived a theoretical expression for blackbody radiation that fit the experimental observations exactly (within experimental error). Planck developed his theoretical treatment by extending the earlier work that had been based on the premise that the atoms composing the oven vibrated at increasing frequencies (or decreasing wavelengths) as the temperature increased, with these vibrations being the source of the emitted electromagnetic radiation. But where the earlier treatments had allowed the vibrating atoms to have any energy values obtained from a continuous set of energies (perfectly reasonable, according to classical physics), Planck found that by restricting the vibrational energies to discrete values for each frequency, he could derive an expression for blackbody radiation that correctly had the intensity dropping rapidly for the short wavelengths in the UV region.
\[E=nhν,\:n=1,2,3,\:. . . \nonumber \]
The quantity h is a constant now known as Planck's constant, in his honor. Although Planck was pleased he had resolved the blackbody radiation paradox, he was disturbed that to do so, he needed to assume the vibrating atoms required quantized energies, which he was unable to explain. The value of Planck's constant is very small, 6.626 × 10 −34 joule seconds (J s), which helps explain why energy quantization had not been observed previously in macroscopic phenomena.
The Photoelectric Effect
The next paradox in the classical theory to be resolved concerned the photoelectric effect (Figure \(\PageIndex{10}\)). It had been observed that electrons could be ejected from the clean surface of a metal when light having a frequency greater than some threshold frequency was shone on it. Surprisingly, the kinetic energy of the ejected electrons did not depend on the brightness of the light, but increased with increasing frequency of the light. Since the electrons in the metal had a certain amount of binding energy keeping them there, the incident light needed to have more energy to free the electrons. According to classical wave theory, a wave's energy depends on its intensity (which depends on its amplitude), not its frequency. One part of these observations was that the number of electrons ejected within in a given time period was seen to increase as the brightness increased. In 1905, Albert Einstein was able to resolve the paradox by incorporating Planck's quantization findings into the discredited particle view of light (Einstein actually won his Nobel prize for this work, and not for his theories of relativity for which he is most famous).
Einstein argued that the quantized energies that Planck had postulated in his treatment of blackbody radiation could be applied to the light in the photoelectric effect so that the light striking the metal surface should not be viewed as a wave, but instead as a stream of particles (later called photons ) whose energy depended on their frequency, according to Planck's formula, E = hν (or, in terms of wavelength using c = νλ , \(E=\dfrac{hc}{λ}\)). Electrons were ejected when hit by photons having sufficient energy (a frequency greater than the threshold). The greater the frequency, the greater the kinetic energy imparted to the escaping electrons by the collisions. Einstein also argued that the light intensity did not depend on the amplitude of the incoming wave, but instead corresponded to the number of photons striking the surface within a given time period. This explains why the number of ejected electrons increased with increasing brightness, since the greater the number of incoming photons, the greater the likelihood that they would collide with some of the electrons.
With Einstein's findings, the nature of light took on a new air of mystery. Although many light phenomena could be explained either in terms of waves or particles, certain phenomena, such as the interference patterns obtained when light passed through a double slit, were completely contrary to a particle view of light, while other phenomena, such as the photoelectric effect, were completely contrary to a wave view of light. Somehow, at a deep fundamental level still not fully understood, light is both wavelike and particle-like. This is known as wave-particle duality .
When we see light from a neon sign, we are observing radiation from excited neon atoms. If this radiation has a wavelength of 640 nm, what is the energy of the photon being emitted?
Solution
We use the part of Planck's equation that includes the wavelength, λ , and convert units of nanometers to meters so that the units of λ and c are the same.
\[\begin{align*} E&=\dfrac{hc}{λ} \\[4pt] &=\mathrm{\dfrac{(6.626×10^{−34}\:J\cancel{s})(2.998×10^{8}\:m\cancel{s}^{−1})}{(640\cancel{nm})\left(\dfrac{1\:m}{10^9\cancel{nm}}\right)}}\\[4pt] &=\mathrm{3.10×10^{−19}\:J} \end{align*} \nonumber \]
The microwaves in an oven are of a specific frequency that will heat the water molecules contained in food. (This is why most plastics and glass do not become hot in a microwave oven-they do not contain water molecules.) This frequency is about 3 × 10 9 Hz. What is the energy of one photon in these microwaves?
- Answer
-
2 × 10 −24 J
Identify which of the following statements are false and, where necessary, change the italicized word or phrase to make them true, consistent with Einstein's explanation of the photoelectric effect.
- Increasing the brightness of incoming light increases the kinetic energy of the ejected electrons.
- Increasing the wavelength of incoming light increases the kinetic energy of the ejected electrons.
- Increasing the brightness of incoming light increases the number of ejected electrons.
- Increasing the frequency of incoming light can increase the number of ejected electrons.
Solution
- False. Increasing the brightness of incoming light has no effect on the kinetic energy of the ejected electrons. Only energy, not the number or amplitude, of the photons influences the kinetic energy of the electrons.
- False. Increasing the frequency of incoming light increases the kinetic energy of the ejected electrons. Frequency is proportional to energy and inversely proportional to wavelength. Frequencies above the threshold value transfer the excess energy into the kinetic energy of the electrons.
- True. Because the number of collisions with photons increases with brighter light, the number of ejected electrons increases.
- True with regard to the threshold energy binding the electrons to the metal. Below this threshold, electrons are not emitted and above it they are. Once over the threshold value, further increasing the frequency does not increase the number of ejected electrons
Calculate the threshold energy in kJ/mol of electrons in aluminum, given that the lowest frequency photon for which the photoelectric effect is observed is \(9.87 \times 10^{14}\; Hz\).
- Answer
-
\(3.94 \: kJ/mol\)
Summary
Light and other forms of electromagnetic radiation move through a vacuum with a constant speed, c , of 2.998 × 10 8 m s −1 . This radiation shows wavelike behavior, which can be characterized by a frequency, ν , and a wavelength, λ , such that c = λν . Light is an example of a travelling wave. Other important wave phenomena include standing waves, periodic oscillations, and vibrations. Standing waves exhibit quantization, since their wavelengths are limited to discrete integer multiples of some characteristic lengths. Electromagnetic radiation that passes through two closely spaced narrow slits having dimensions roughly similar to the wavelength will show an interference pattern that is a result of constructive and destructive interference of the waves. Electromagnetic radiation also demonstrates properties of particles called photons. The energy of a photon is related to the frequency (or alternatively, the wavelength) of the radiation as E = hν (or \(E=\dfrac{hc}{λ}\)), where h is Planck's constant. That light demonstrates both wavelike and particle-like behavior is known as wave-particle duality. All forms of electromagnetic radiation share these properties, although various forms including X-rays, visible light, microwaves, and radio waves interact differently with matter and have very different practical applications. Electromagnetic radiation can be generated by exciting matter to higher energies, such as by heating it. The emitted light can be either continuous (incandescent sources like the sun) or discrete (from specific types of excited atoms). Continuous spectra often have distributions that can be approximated as blackbody radiation at some appropriate temperature.


1.6: New Data and Experimentation
We must trust to nothing but facts: these are presented to us by nature and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation.
—ANTOINE LAVOISIER1
The Crazy Philosopher’s Evidence
As you will remember, Johnson thought he had discovered evidence that there was a glitch in his iPod software. His schematized argument was as follows:
e 1 . Johnson went to a Pink Martini concert, planning to ask for a specific encore.
e 2 . “Que Sera Sera” was played during the concert.
e 3 . He never got a chance to ask for “Lilly.”
e 4 . On the ride home the next morning, he set his iPod to play all thirty-six of the Pink Martini songs.
e 5 . He set the iPod to “Shuffle Songs.”
e 6 . He listened to all thirty-six songs.
e 7 . The last two songs played were “Lilly” and “Que Sera Sera”—the imagined encore from the night before!
e 8 . “Lilly” and “Que Sera Sera” are the two Pink Martini songs he listens to most often.
t 0 . There is a glitch in the iPod software—rather than playing the songs in completely “random” order, it is weighing things according to how often songs are listened to.
There are thirty-six Pink Martini songs in Johnson’s iPod. What are the odds of his imagined encore occurring on the drive home? Let’s spend just a minute and figure that out. “Lilly” came up as the next to the last song played. The odds of this happening are straightforward. Any one of thirty-six songs could have come up here, so the odds are 1/36. But to have the encore, you had to also have “Que Sera Sera” come up last. So what are the odds of that happening? It’s actually easy to figure out. We already know the odds of “Lilly,” so it’s a question of “Lilly” and “Que Sera Sera.” Since “Lilly” has already been played, the odds of “Que Sera Sera” are 1/35, and the odds of “Lilly” and “Que Sera Sera” are 1/36 × 1/35, or 1/1,260. But of course, I would have also had my encore if the last two songs had been “Que Sera Sera” and then “Lilly.” The odds of this happening figure out exactly the same—1/1,260. So the odds of my encore popping up—“Lilly” and “Que Sera Sera” or “Que Sera Sera” and “Lilly” are 1/1,260 + 1/1,260, or 1/630.
Certainly, one thing that would explain that 1/630 shot coming up on the ride home is that my imagined encore was composed of my two favorite (and most listened to) Pink Martini songs, and the program was illegitimately taking this into account in generating the “random” play order. But I hope it’s obvious by now, it’s easy enough to think of lots of rival explanations.
t 1 . This was just a true, 1/630 coincidence.
t 2 . This is not a software glitch; the iPod software is designed to do exactly this.
t 3 . The iPod software is illegitimately weighing things, not by number of times played, but something else—length of the songs, where they occur in the album, and so on.
t 4 . The philosopher set his iPod incorrectly.
t 5 . The philosopher dozed in and out on the drive home and only thought that these two songs came up last.
t 6 . The problem is in Johnson’s iPod—the hardware, not the software.
My students have been worrying about what happened for the last several years on quizzes, ever since this really happened on a drive back from the Oregon League of Cities. They pretty generally rank the coincidence hypothesis as a much better explanation, though they are often surprised once they see the math that the odds are really 1/630. They also don’t seem to have too much confidence in their professor, since explanations such as t 4 and t 5 are consistently ranked ahead of t 0 . So according to the inference-to-the-best-explanation recipe, these students are committed to saying that Johnson’s evidence for the glitch theory is pretty weak.
Why Don’t You Just Test It?
I’ve told you this little anecdote for two very different reasons. One, of course, is I wanted a little exercise that would allow you to apply the inference-to-the-best-explanation test from chapter 5 to an argument. The other, though, is to tell you about a very common feature that my students have felt compelled to add to their discussions. There is almost a sense of frustration or least the need to lecture their professor. They suggest, indeed insist on, a very simple test of the glitch hypothesis.
Look, isn’t there an obvious way to settle this matter? Turn off the iPod, reset everything, play Pink Martini’s songs again and see what happens. What is being proposed here is a classic little experiment—the kind of thing that some philosophers and scientists say is the defining condition of real science. I hope to convince you in the next couple of chapters that there is something brilliantly right about this claim but, at the same time, dangerously misleading.
A Pretty Picture of Science
Here is an idealization about the natural sciences. The scientist is really smart and is trained to go about her business in a very special, almost ritualized, way. She goes out and observes the world. Being smart and being trained to be a careful observer, she notices things. Sometimes she is puzzled by the things she observes and she asks questions, Why am I observing this? She starts looking for an explanation . Being smart and creative she thinks about this really hard and comes up with a possible answer—a hypothesis or a theory . This is all fine and good, but according to the pretty picture, it’s only now that the rules of science kick in. It’s not good enough to just have a theory; the theory must now be tested. The scientist must devise an experiment and let the results of the experiment determine the fate of her theory.
Bear with me for a bit of technical stuff in symbolic logic. Logicians talk about conditionals , “if . . . then” sentences. There are two valid inferences that follow directly from a true conditional.
- 1. If the figure is a plane right triangle, then the interior angles total 180°.
- 2. The figure is a plane right triangle.
- 3. The interior angles total 180°.
This inference is called modes ponens . A kind of mirror image inference is called modes tollens .
- 1. If the figure is a plane right triangle, then the interior angles total 180°.
- 2. The interior angles do not total 180°.
- 3. The figure is not a plane right triangle.
Finally, there is a tempting inference that is not valid but is rather a logical fallacy, affirming the consequent .
- 1. If the figure is a plane right triangle, then the interior angles total 180º.
- 2. The interior angles total 180º.
- 3. The figure is a plane right triangle.
You can easily spot the fallacy by noting that the figure might total 180° because it’s a triangle, but, at the same time, not be a right triangle but rather, say, an equilateral triangle.
OK, so what does all this have to do with the pretty picture of science and maybe Johnson’s iPod? Well, suppose the conditional sets up something we might expect to see in an experimental circumstance, given the theory we are testing is true.
- 1. If the theory is true, we will see . . . in the experiment.
By the inference of modes tollens , we will be able to falsify the theory by disconfirming it in an experiment.
- 1. If the theory is true, we will see . . . in the experiment.
- 2. We do not see . . . in the experiment.
- 3. The theory is not true.
Experiments, according to the pretty picture, provide tests that can show us that theories are false. They cannot, however, show us that theories are true. Remember, it is a fallacy to affirm the consequent.
- 1. If the theory is true, we will see . . . in the experiment.
- 2. We see . . . in the experiment.
- 3. The theory is true.
A Better, But Untidy, Picture of Scientific Disconfirmation
Now, the theory about the iPod hardly counts as deeply scientific, but suppose we imagine an experiment nonetheless. The conditional that sets all this up looks something such as the following:
- 1. If there is a glitch in the software, so that when the iPod is set to play all the songs by an artist and is set to “shuffle” these songs, then rather than playing them in random order, it will play the most often listened to tracks last.
I could test my theory by reprogramming everything with the Pink Martini tracks, but since I’ve offered a general theory, let’s test it with a different artist. I have lots of Lucinda Williams’s albums, and I’m certain I listen to two of her songs, “Right in Time” and “Essence,” the most. So if I set my iPod to play all her tracks and to shuffle them, I am predicting that the two songs will be played last.
 We can imagine four different outcomes to the experiment. Focusing on the last two songs, we might observe any of the following.
e n a . The two songs come up as the last two played.
e n b . Neither song is in the last two.
e n c . Only “Right in Time” is in the last two.
e n d . Only “Essence” is in the last two.
Options e n c and e n d are interesting and deserve further study, but let’s set them to the side and focus on the “pure” experimental outcomes. According to the pretty picture, e n b conclusively establishes that the glitch theory is false. But isn’t that a little extreme? We’ve already honed our skills at rival explanations—surely we can imagine scenarios where the glitch hypothesis is (was) true but neither song played last.
t 1 . Between the drive home and the experiment, iTunes downloaded a newer (debugged) version of the software.
t 2 . The glitch only occurs in playlists shorter than fifty songs.
t 3 . There is a countervailing glitch when any of the songs are classified as “country.”
It’s doubtful in the extreme that a negative experimental outcome can falsify a theory, though it certainly can provide strong evidence that there is something wrong with the theory.
The problem here goes back to the original conditional that set up the experiment in the first place. Remember the difference between a sound argument and a valid one? The if . . . then sentence that gets our inference going in the first place states an absolute connection between the glitch theory and the predicted outcome of the experiment. But the rival explanations we have just considered above seem to show that this connection is not so absolute after all. Almost always the conditional that sets up our experiment contains what Larry Wright calls a weasel word . A more modest, but also more accurate, statement of the predicted experimental outcome will look more like this:
If the theory in question is true, then all things being equal we will see . . . in our experiment.
We predict that we will observe an as-yet-undiscovered planet at such-and-such location in the night sky, but certainly not if the observatory is socked in by clouds. We expect the solution to turn a certain color in our chemistry experiment but not if the test tube is contaminated.
When we include this suppressed, but understood, ceteris paribus clause,2 our inference looks a little more problematic.
- 1. If there is a glitch in the software, so that when the iPod is set to play all the songs by an artist and is set to “shuffle” these songs, then, all things being equal , rather than playing them in random order, it will play the most often listened to tracks last.
- 2. “Essence” and “Right in Time” did not play last.
Two valid conclusions can be derived from these premises. One, of course, is that the glitch hypothesis is mistaken. But as a matter of pure logic, it is equally legitimate to infer that all things in our experimental circumstances were not equal.
Does any of this mean that the “scientific method” and the requirement that we experimentally test our theories is a waste of time? Nothing could be further from the truth. Let’s go back to our original “evidence” for the glitch theory but add to it the new data from our experiment.
e 1 . Johnson went to a Pink Martini concert, planning to ask for a specific encore.
e 2 . “Que Sera Sera” was played during the concert.
e 3 . He never got a chance to ask for “Lilly.”
e 4 . On the ride home the next morning, he set his iPod to play all thirty-six of the Pink Martini songs.
e 5 . He set the iPod to “Shuffle Songs.”
e 6 . He listened to all thirty-six songs.
e 7 . The last two songs played were “Lilly” and “Que Sera Sera”—the imagined encore from the night before!
e 8 . “Lilly” and “Que Sera Sera” are the two Pink Martini songs he listens to most often.
e 9 . When Johnson tried the “shuffle all songs” routine for Lucinda Williams, his most listened to songs did not come up last.
t 0 . There is a glitch in the iPod software—rather than playing the songs in completely “random” order, it is weighing things according to how often songs are listened to.
We’ve already imagined some rivals to e 9 , but I assume that you would all agree with me that t 0 has been seriously weakened by our experiment and that the random fluke hypothesis or the operator error rivals look even better.
The moral here is straightforward. When a theory suggests that we can expect to see something as yet undiscovered and we go out and look for this thing but don’t find it, this is highly relevant new data that almost always hurts the status of the original explanation as being the best explanation of everything, including, of course, the experimental results.
A Better, But Untidy, Picture of Scientific Confirmation
None of what I have just told you is earthshaking nor is it unknown by careful scientists and philosophers. Still, the pretty picture, partly because it is so pretty, can allow us to lose sight of the subtleties of experimental design and protocol. Maybe even more problematic for the pretty picture is the evidential value of experimental confirmation.
Suppose I program my iPod to play all 116 Lucinda Williams tracks. I set the iPod to shuffle the songs and then sit back for a really long time and wait to see what the last two songs are. Sure enough, up pops “Essence” and “Right in Time” as the last two played. What do you think of my glitch hypothesis now?
According to the pretty picture, my theory has been put to the test and perhaps surprisingly, has survived the test. But it would be the fallacy of affirming the consequent to say that the experiment has confirmed my theory. We’ve already seen that if confirmation means “logically derived” from the experimental setup and results, that’s exactly right. But none of this means that the experiment hasn’t produced very strong evidence that the songs are not playing in purely random order.3
What is the best explanation of e 1 through e 8 when we add the positive experimental result below?
e 9 . When Johnson tried the “shuffle all songs” routine for Lucinda Williams, his most listened to songs did indeed come up last.
All the rivals we thought of with Pink Martini are still possible, but hardly any seem plausible any longer. One of the most seriously misleading features of the pretty picture is that it sets up an asymmetry between experimental confirmation and disconfirmation. We’ve seen why as a matter of deductive logic this asymmetry exists. But no such asymmetry exists when we see experimental results as additional data that the tested theory and its rivals must explain.
The Significance of New Data
One of the remarkable things about the natural sciences is that we can devise experiments and go looking for highly relevant new data. But new data can cause us to rethink our evidence or feel even more confident about it in any of the arguments we’ve been thinking about, not just the scientific ones. If we find out that Dick’s been in the hospital with pneumonia and that he loaned his car to his buddy, Sam, things are going to seem much more promising for Dick and Jane. And if we find a copy of Sarah’s midterm on Charlie’s laptop, the case for cheating is obviously strengthened.
Three very important things follow from all this. The first is that evidence evaluation is always relative to what we presently know. If we learn new things and assemble them in new arguments, there will be times when our original conclusion will be strengthened, times when it will be weakened, and times when it will be pretty much left untouched. The second is that new data are always possible. The fact that we could imagine rival explanations means that we can imagine new evidence for these rivals. But this last fact leads to our third moral. Just because new data are possible, it does not mean that our assessment of the current evidence is unreliable. If all the rivals are farfetched, then the chances of finding new data that supports them are pretty slim. We do, of course, need a certain kind of intellectual modesty. We concede that things could change on the basis of new discoveries. But at the same time, for some kinds of evidence, we can be pretty confident that they won’t change .
EXERCISES
- 1. According to the “pretty picture of science,” why is it possible to disconfirm a scientific theory but never confirm one?
- 2. What kind of new data would strengthen Connie’s evidence about what happened at the record hop? What kind of new data would weaken her theory?
QUIZ SIX
For the past few years, I have been forming an uncharitable hypothesis about one of my colleagues. He is Professor Hide-Smith-Jones, who teaches in the Department of Hermeneutic Metaphysics. I believe that he virtually gives away grades and demands almost no work from his students. His courses are wildly popular with students and have very high enrollments. What started my suspicions was a number of students who complained about the workload in my courses, who I later discovered were all hermeneutic metaphysics majors. A couple of my online students explicitly compared my course to Hide-Smith-Jones’s courses, accusing me of being unfair and unreasonable. This past weekend, I went into the university’s database and looked at the transcripts for all my advisees in the past five years. Many of them had taken at least one course with Hide-Smith-Jones. I discovered that on average, the grades they earned in his courses were .78 grade points higher than their total grade point averages.
- 1. Use the tools of inference to the best explanation to assess the quality evidence we have for Johnson’s theory that Hide-Smith-Jones is an easy grader who doesn’t demand much from his students.
- 2. Explain a test or experiment that could be conducted to test Johnson’s hypothesis.
- 3. Using inference to the best explanation, show how new data could be discovered that would either help (confirm) or hurt (disconfirm) Johnson’s theory.
Notes
1. Antoine Lavoisier, Elements of Chemistry , trans. Robert Kerr (Edinburgh, Scotland: Dover, 1790), xiii–xvii, www.iupui.edu/~histwhs/H374.dir/H374.webreader/Lavoisier.elements.html.
2. The online Merriam-Webster Dictionary defines ceteris paribus as “if all other relevant things, factors, or elements remain unaltered.”
3. It is, of course, true that devices such as iPods do not truly generate anything randomly. But their random number generating algorithms simulate randomness for all practical purposes.