Kelter P., Mosher M., Scott A. Chemistry. The Practical Science

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65. Consider the calculation completed in question 63. How

many grams of silver must be used to cover the area with a

thin ﬁlm that is one atom thick?...three atoms thick?

66. Consider the copper-coated notecard in question 64. How

thick must the thin ﬁlm be to increase the mass of the note-

card by 5.00 g?

Chemical Applications and Practice

67. A recent advance in thin ﬁlms has involved the use of tita-

nium dioxide on window glass. The ﬁlm helps keep dust

from depositing on the windows. The exact technique for

applying the thin ﬁlm of TiO

is proprietary information

(meaning that the information is not available to the public).

But the process involves heating the glass and passing the ﬁlm

material as a gas stream over the glass. The resulting interac-

tion forms TiO

as a ﬁlm during cooling. Which of the three

techniques described in this chapter most resembles this

process?

68. The thin ﬁlm in the application described in problem 67

is approximately 5.00 × 10

angstroms thick (1 ang-

strom = 1 × 10

−10

m). Describe how this measurement is

made. How much additional mass does the thin ﬁlm add to

a 2.500-ft

piece of glass? (Assume the density of the ﬁlm is

4.26 g/cm

Section 13.6

On the Horizon—What Does the Future Hold?

Skill Review

69. Uses for zeolites, both naturally formed from clays and spe-

cially synthesized, vary greatly but typically depend on what

property of these versatile substances?

70. Recycling is an important part of the process of conserving

and reusing resources. List two common items that are typi-

cally recycled.

71. Instead being thrown away, your old TV can be recycled.

What types of materials make up the TV? Which of these can

be recycled?

72. Many used automobiles can be recycled. What are some

of the materials that could be harvested from a modern

automobile?

73. Name and brieﬂy describe the three classes of biopolymers.

74. Describe the chemical and physical makeup of an aerogel.

Provide at least one application for an aerogel.

Chemical Applications and Practices

75. Zeolite catalysts may be employed in the production of the

hydrocarbons used in gasoline. Would this be an example of

a heterogeneous or a homogeneous catalyst? How can a cata-

lyst be used to reduce waste?

76. Chemical companies now advertise many products, such as

pharmaceuticals, as containing one chiral isomer. What is the

advantage of using a chiral catalyst rather than a nonchiral

catalyst when synthesizing a product that must be only one

type of isomer?

77. In terms of density, insulation, and resistance to cracking,

what advantages does an aerogel offer over other ceramic

materials?

578 Chapter 13 Modern Materials

78. One of the guidelines for practicing Green Chemistry is to

reduce the amount of chemical waste that occurs in manu-

facturing compounds. From where does this waste arise?

What is typically done with chemical waste in “nongreen”

processes?

Comprehensive Problems

79. How do molecular and ionic solids, in both the molten and

the dissolved state, differ in their ability to conduct an electric

current?

80. When selecting molecules that make good candidates for liq-

uid crystal displays, chemists search for (compact; long rigid)

molecules with (strong; weak) dipole moments. Explain why

each of your circled choices is appropriate for liquid crystals.

81. Metals such as silver, chromium, and platinum are known for

their shiny appearance. Most metals, in their pure state, also

have this appearance. Use the delocalized electron model to

explain this property of metals.

82. Superconductor materials are able to conduct electricity,

typically with pairs of electrons, as a consequence of the

arrangement of sheets or layers within their structure.

These ceramic materials are continually being studied to

make it possible to have superconducting ceramics at

higher temperatures. Use Internet resources to obtain an

example of the chemical composition and structure of a

ceramic superconductor.

83. Use the Internet or other sources to ﬁnd out how Charles

Goodyear used crosslinking with sulfur to change the prop-

erties of some rubber substances he was working with.

84. Use the Internet to ﬁnd the twelve principles of Green Chem-

istry. Which of these principles are used in your chemistry

laboratory?

85. Replacing hazardous pesticides with less persistent (but

just as effective) compounds is a common goal. What are

the hazards associated with pesticides if they persist in the

environment?

Thinking Beyond the Calculation

86. A chemist prepares a polymer from methyl acrylate (also

known as methyl propenoate). The resulting polymer was

made by dissolving a small amount of the monomer in solu-

tion and then forming the polymer. The polymer was very

gel-like in nature, with a spongy appearance.

Acrylic acid MethanolMethyl acrylate

a. Draw any potential resonance structures for the monomer.

b. Draw the polymer that would result from the poly-

merization of methyl acrylate.

c. What uses would such a polymer ﬁnd?

d. If the monomer is treated with water and acid, it reacts

to form acrylic acid and methanol. How many grams of

methanol would be obtained from the reaction of 1.50 g

of methyl acrylate with excess water?

e. What properties that are not found in a polymer of methyl

acrylate would you expect a polymer of acrylic acid to

possess?

579

Thermodynamics:

A Look at

Why Reactions

Happen

Sugary between-meal snacks are a good

source of energy. Each contains sucrose,

which is broken down in the body into

glucose and fructose. Both of these mol-

ecules undergo complete oxidation and

provide energy for the body.

Contents and Selected Applications

Chemical Encounters: Bioenergetics

14.1 Probability as a Predictor of Chemical Behavior

14.2 Why Do Chemical Reactions Happen? Entropy and the Second

Law of Thermodynamics

Chemical Encounters: Glycolysis

14.3 Temperature and Spontaneous Processes

Chemical Encounters: Psychrotrophs and Psychrophiles

14.4 Calculating Entropy Changes in Chemical Reactions

Chemical Encounters: More Glycolysis

14.5 Free Energy

Chemical Encounters: Pyruvate and Lactate

14.6 When G = 0; A Taste of Equilibrium

Chemical Encounters: ATP Formation

Go to college.hmco.com/pic/kelterMEE for online learning resources.

After a hearty breakfast, we leave for work full

of energy and ready to conquer the day. However,

the midmorning hours can be difﬁcult to get through

because our energy level drops a couple of hours after we

eat. This is especially true if we had a big bowl of sugar-coated,

sugar-injected cereal for breakfast. To make matters worse, the mid-

afternoon hours are no easier—they seem to be the longest of the day.

Why does this happen and what can we do to give ourselves that “burst of

energy” we need when we feel so tired? Enter the snack. Whether it takes the form

of a chocolate bar, a donut, or a bottle of juice, it has the effect of raising our energy

level. Just like breakfast, lunch, and dinner, the snack provides our bodies with a

source of glucose. How does the consumption of glucose give us energy?

Antoine Lavoisier (the scientist we met in Chapter 3, whose measurements led

to the formulation of the law of conservation of mass before he was guillotined in the

French Revolution) noticed that living things consume foods and transform them

into the energy that maintains life. Lavoisier’s views on the process seem rudimen-

tary given our modern understanding, but they were quite revolutionary in his day.

The addition of glucose to living cells is an example of this food-to-energy transfor-

mation process. To the biochemist, this is part of the broader ﬁeld of

bioenergetics,

the study of the energy changes that occur within a living cell.

In this chapter we will discuss some of the chemical reactions in the ﬁeld of

bioenergetics. Although we’ll soon introduce terms such as entropy, spontaneity, free

energy, and equilibrium, the underlying concept of probability demands our imme-

diate attention. Chemists beneﬁt from understanding these topics. They use proba-

bility in many ways: to locate electrons, to determine the macroscopic properties of

compounds and mixtures, and to predict the outcome of chemical reactions. We will

use probabilities to discover why chemical processes occur—including the chemical

transformations of the body and their relationship to our ability to live.

Application

HEMICAL

NCOUNTERS:

Bioenergetics

14.1 Probability as a Predictor

of Chemical Behavior

If you entered a classroom or movie theater and noticed all the males seated on

one side and all the females on the other, as shown in Figure 14.1, would you

think that some announcement, rule, or social convention had dictated that

arrangement? Perhaps. But this seating pattern could also have emerged from

purely random choices. In fact, there are numerous ways in which a room full of

people could be sitting, and “separate seating” is just one of many possibilities.

Let’s shift our focus to the room in which you are now sitting as you read this.

As we know, the air in the room is a mixture of primarily oxygen and nitrogen

gases. Oxygen and nitrogen molecules do not chemically interact with each other,

so they can occupy essentially any unfurnished position in the room. Is it possible

that these gases could be arranged such that all of the oxygen molecules were in

one corner of the room and all of the nitrogen molecules in another, as shown in

Figure 14.2? Possible? Yes. Likely? No. Is there something that these two seemingly

unrelated situations—the room of people and the room of molecules—share?

The short answer is yes, the two situations have a lot in common. To under-

stand how this could be so, let’s introduce some common terminology to help in

our discussion. We refer to the

macrostate of a system (whether seated people or

FIGURE 14.1

An interesting arrangement of the students

in a classroom.

580

molecular positions in a room) when we want to take a snapshot of the overall sit-

uation. The macrostate of the theater seating is that it is sorted by gender. The dis-

persed oxygen and nitrogen molecules in the room exert a certain pressure and

have a certain composition; this is all part of the macrostate of the gases in the

room.

If we wanted to describe the individuals in the theater, we might discuss which

seat they are in, which way they are facing, and whether their arms or ankles are

crossed. Each of these individual descriptions is referred to as a

microstate. For in-

stance, each arrangement of people in the theater indicates a different microstate.

How does this apply to molecules? The total of the microstates in a system deﬁnes,

or describes, the macrostate of the system. For individual molecules, the microstate

can include their translational motion (changing from one location to another),

their vibrational–rotational state (this involves the atomic movements around a

chemical bond), and their electron conﬁguration (ground state versus the excited

state, even the oxidized state).

How might you describe the macrostate of the gases in your room? You could

measure their temperature and pressure. You could also assess the number of

molecules of each component in the room. Given the huge number of molecules

of gas in the room, we would predict an even larger number of microstates for the

gas. We mentioned the hypothetical condition of all the oxygen molecules being

in one corner. To bring about that unlikely macrostate of air, the oxygen mole-

cules (being restricted to the corner) would have to assume a limited number of

the multitude of possible microstates in the room.

Why have you never encountered the macrostate that has all the oxygen mol-

ecules stuck in one corner of the room? One answer can be found in looking at

14.1 Probability as a Predictor of Chemical Behavior 581

= O

= N

FIGURE 14.2

Is this arrangement of oxygen molecules

within a room possible?

A macrostate is a snapshot description of

an overall situation.

A microstate is a snapshot description of a particular representation

of individual particles. Each of these three pictures includes a differ-

ent microstate for a single person in a movie theater.

Two individual macrostates.

the probability that such an event could occur. Out of all the possible macrostates,

what are the chances that “all O

on one side, all N

on the other side” would occur?

In short, there isn’t a very good probability that this situation would happen.

(Read on for a more detailed answer!)

Think back to the seating choices for a person entering a room. Which situa-

tion has more choices, the strict division by gender or “open seating”? Which

situation would you consider to have the greater number of microstates that sat-

isfy the conditions for a particular macrostate? Ludwig Boltzmann (1844–1906),

an Austrian mathematician/physicist, dealt with this idea, along with its powerful

implications, mathematically. We can get a sense of what he did by looking inside

the tanks of oxygen that scuba divers and mountain climbers carry with them as

they set off on their quests. Look closely at Figure 14.3. This photo, taken of

mountain climbers George Leigh Mallory and Andrew Irvine before their fateful

June 1924 attempt to scale the summit of Mt. Everest, shows the climbers each

with two tanks of oxygen strapped to their backs. Let’s assume that these tanks are

connected by a valve and that one of the tanks is empty and the other full. What

will happen to the distribution of the gas if the valve is opened? The ﬁnal volume

of the two tanks, which we’ll call V

, is double the initial volume of just

one tank, V

. We can also express this by saying that the ratio

= 2. How many

ways can the individual molecules be distributed within the two tanks? (That is,

how many “microstates” are there?) To make things a little simpler, let’s assume

that the tanks contain only two molecules of a gas. If we call the molecules A and

B, we can have a total of four microstates, as shown in Figure 14.4. We will call the

ﬁnal number of microstates W

(for the German word Wahrscheinlichkeit, “prob-

ability”), and W

= 4 in this case. The initial number of microstates, W

, is equal

to 1, representing A and B both present in one tank, before the valve was opened

allowing the gases to spread apart. This means that for two molecules in two

tanks, the ratio

How is the number of microstates,

, related to the volume,

? For N

molecules,





582 Chapter 14 Thermodynamics: A Look at Why Reactions Happen

FIGURE 14.4

The possible arrangements of two

molecules into two tanks.

Microstate Tank 1 Tank 2

1AB

2AB

3AB

4BA

FIGURE 14.3

George Mallory and Andrew Irvine on Mt.

Everest pack heavy cylinders of oxygen

on their backs. This photo, taken in 1924

during their ascent to the summit, is the

last known image of the pair.

14.1 Probability as a Predictor of Chemical Behavior 583

That is the equation that Boltzmann derived. Using our numbers,

4 = 2

What if our tanks contained three molecules (N =3) instead of two?





= 2

= 8

In this case, there would be eight possible microstates. More molecules in the

tanks make available more microstates—more ways in which the molecules can

distribute themselves among the two tanks. This means that the probability of

all of the molecules being in only one tank—a single microstate among all

other possibilities—becomes smaller. This is the key point: As the number of

molecules increases, the likelihood of their all being in one tank decreases sharply,

and the probability of the gases tending toward an equal distribution increases.

Let’s look further into this point and discuss its vital implications.

Let’s put four oxygen molecules in the full tank. Here Boltzmann’s equa-

tion indicates that there should be 16 (2

= 2

= 16) different microstates of

the oxygen molecules in the two-tank system. This is exactly what we predict

by drawing each of the arrangements as in Figure 14.5. But if we look closely,

we see that some of the microstates would give the same macrostate for the

system. Figure 14.5 shows that only 5 unique macrostates could arise from the

16 different microstates. Only 6 of these 16 microstates describe the equal dis-

tribution of the gases. The probability is therefore 6/16, or 37.5%, that the

four molecules will arrange themselves into the two tanks equally. The proba-

bility of exactly equal distribution has diminished. However, there are many

more microstates at or near equal distribution than microstates that are all on

one side of either tank. Only 2 of the 16 microstates, or 12.5%, describe all of

the molecules being on one side or the other.

As illustrated in Figure 14.6, greatly increasing the number of molecules in

our experiment (from 8 to 32 to 128) also greatly increases the number of mi-

crostates that are available. As a result, the probability of equal or nearly equal

distribution of the molecules between the two tanks increases. At the same time,

the probability of all of the molecules being in one tank or the other is ap-

proaching zero. For even small real-world-sized samples of molecules (such as

0.01 mol or 0.1 mol), equal or nearly equal distribution of the gases becomes the

most likely outcome. The bottom line is that probability—a mathematical

construct—governs physical behavior, and in this case, probability says that

the gases will spread from one tank to occupy both tanks evenly.

5 different macrostates

16 different

microstates

FIGURE 14.5

Microstates describing the possible

arrangements of four molecules

between two tanks.

EXERCISE 14.1 Probability

A valve connects two glass jars. One of the jars contains six atoms of gaseous he-

lium, and the other is empty. Determine the probability of the macrostate in which

each jar will ﬁll equally (that is, each jar will contain three atoms of helium) after the

valve is opened. For the same system, what is the probability of ﬁnding four mole-

cules in the left-hand jar and two in the right? What about the probability of ﬁnd-

ing all six molecules in the left-hand jar?

Solution

Because there are six molecules in this example, there are 2

= 64 possible mi-

crostates. Drawing each of them out and grouping them into similar macrostates

leads us to the conclusion that there are only 7 macrostates (see Figure 14.7). There

are 20 microstates that indicate equal distribution, so the probability of equal dis-

tribution of the six molecules of gas is 20/64, or 31%. The probability of ﬁnding two

molecules of gas in the right-hand jar and four in the jar on the left is 15/64, or 23%.

The probability of ﬁnding all of the molecules in the left-hand jar is 1/64, or 1.6%.

584 Chapter 14 Thermodynamics: A Look at Why Reactions Happen

Number of molecules in the left cylinder

Probability

0 8 16 24 32 40 48 56 64 72 80 88

128 molecules

96 104 112 120 128

Number of molecules in the left cylinder

Probability

0246810121416182022

32 molecules

24 26 28 30 32

Number of molecules in the left cylinder

Probability

012345678

8 molecules

FIGURE 14.6

Probability of distribution of a gas between two cylinders.As

the number of molecules increases in the system, the tanks’

containing equal numbers of molecules becomes the more likely

event. Note that the probability of equal distribution is a bell-

shaped (Gaussian) curve.

6 different

microstate

arrangements

15 different

microstate

arrangements

20 different

microstate

arrangements

7 different

macrostates

15 different

microstate

arrangements

6 different

microstate

arrangements

1 microstate

arrangement

1 microstate

arrangement

FIGURE 14.7

Probability distribution for six molecules

of gas in two tanks.

ATP

NADH NADH

ATP

CHO

CHOH

COO

–

COO

–

CHO

CHOH

PRACTICE 14.1

Let’s use the same valve and glass jar set-up, with one jar containing eight atoms

of gaseous helium. Then the valve is opened. Determine the probability of the

macrostate that shows three atoms of He in one jar and ﬁve atoms of He in the

other jar.

See Problems 1–4, 7, and 8.

14.2 Why Do Chemical Reactions Happen?

Entropy and the Second Law of

Thermodynamics

When sucrose enters the body, it is broken down into two simpler molecules,

glucose and fructose. Glucose and fructose are structural isomers, both with the

formula C

(s) + H

O(l) n C

(aq) + C

(aq)

Sucrose Fructose Glucose

Both molecules are used by the body to generate energy. For example, glucose in

a cell undergoes

glycolysis, the series of ten chemical transformations, shown in

Figure 14.8, that produces two molecules of pyruvate. The pyruvate is further

converted, releasing energy via a different set of transformations. In addition,

during glycolysis two new energy storage molecules known as ATP (adenosine

triphosphate, Figure 14.9) are produced. Glucose provides a major source of

energy for living cells via the glycolysis pathway, which is the sole source of meta-

bolic energy in some mammalian tissues and cell types. Many anaerobic (non-

oxygen-consuming) microorganisms depend entirely on glycolysis for energy to

carry out other biological reactions and survive.

Glycolysis is an example of a

spontaneous process. The rusting of iron nails,

the melting of ice in a glass, and the decay of wood buried in moist soil are also

spontaneous processes. That is, these processes occur without continuous outside

intervention. Say a brand new deck of cards falls off the kitchen table. The princi-

ples of probability say that there are many more ways in which the cards can land

out of order than ways in which they can land sequentially. Furthermore,

some of the cards will land face up and some face down. This disorder hap-

pens without our assistance—the process occurs spontaneously.

The reverse of a spontaneous process is known as a

nonspontaneous

process

. Nonspontaneous processes do not occur without continuous outside

14.2 Why Do Chemical Reactions Happen? Entropy and the Second Law of Thermodynamics 585

Application

HEMICAL

ENCOUNTERS:

Glycolysis

FIGURE 14.8

Glucose enters the glycolysis pathway on

its way to complete oxidation. The reac-

tions along the pathway produce two

molecules of pyruvate and energy.

Adenosine triphosphate (ATP)



O H

FIGURE 14.9

FIGURE 14.10

A green patina on the bronze Silent

Witness Memorial near Gander,

Newfoundland, Canada.

intervention. A rusty nail does not revert to a polished iron nail without continu-

ous help. Decayed wood buried in moist soil does not re-form freshly cut pieces

of wood, and a deck of cards spread out on the ground does not leap into nu-

merical and suit-based order. Just like these transformations, the reactions in

chemistry can be considered either spontaneous or nonspontaneous.

What happens to a copper or bronze statue when it is exposed to the

environment?

As illustrated by the Newfoundland memorial erected to the

memory of those lost in the worst plane crash in Canadian history (see Fig-

ure 14.10), a green patina forms on the surface. This patina is a complex mixture

of colored compounds, such as antlerite (Cu

(OH)

, blue-green), brochantite

(Cu

(SO

)(OH)

, pale green), chalcanthite (CuSO

·5H

O, blue-green), cuprite

(Cu

O, dark red), and tenorite (CuO, black). All of these compounds include

either Cu

or Cu

, so they are oxidation products of the copper metal. Although

the chemistry is much more complex, we can represent this patina by showing its

formation as the simple oxidation of copper:

2Cu(s) +O

(g) n 2CuO(s)

A chemical reaction causes the green patina to form without our assistance, so

this is a spontaneous process. The reverse process,

2CuO(s) n 2Cu(s) + O

(g)

is nonspontaneous; it does not occur without some outside assistance.The reverse

reaction of a spontaneous process is always nonspontaneous.

The spontaneous oxidation of metals due to exposure to the environment is

quite common. Table 14.1 lists some of the patinas that form on other metals.

Does the spontaneity of the oxidation of copper imply anything about the speed

of the process?

The short answer is no. It takes years for the green patina on a new

copper roof to form completely. Under standard conditions, the conversion of di-

amond into graphite is also a spontaneous process, but luckily for people with

diamond jewelry, the rate of this reaction is incredibly (almost immeasurably)

slow. Rust forming on a nail and the decay of a buried log, even though they are

spontaneous, are also slow processes. Biochemical reactions of glucose, on the

other hand, are spontaneous and very rapid. The combustion of methane used to

heat a pot of soup on the stovetop is even faster.

Thermodynamics, the study of the

changes in energy in a reaction (see also Chapter 5), determines whether a process

is possible. In Chapter 15, we’ll study the chemical kinetics of these processes to

determine how fast, and by what mechanism, they occur.

586 Chapter 14 Thermodynamics: A Look at Why Reactions Happen

A disordered pile of playing cards.

Patinas That Spontaneously Form on Metals

Element Symbol

Metal or Composition Natural Color Patina Color

Aluminum Al Silvery white Light gray

Brass copper and zinc Gold Dark brown to black

Bronze copper and tin Yellow to olive brown Dark brown to black

Copper Cu Light red brown Green

Iron Fe Lustrous silvery white Reddish brown

Silver Ag White to gray Black

TABLE 14.1

Application

Video Lesson: Spontaneous

Processes

Visualization: Spontaneous

Reaction of Phosphorus

(Barking Dogs)

Glucose

EXERCISE 14.2 Spontaneity in Common Processes

Which of these processes are spontaneous?

a. Ice melting in a hot oven

b. Carbon dioxide and water reacting at STP to form methane and oxygen

c. A basketball player jumping to dunk a basketball

d. NaOH(aq) and HCl(aq) reacting when combined in a beaker

Solution

The processes described in parts (a) and (d) are spontaneous, because they happen

without continuous outside intervention. The combination of sodium hydroxide

and hydrochloric acid is often used in chemical analysis precisely because the reac-

tion is not only spontaneous but also rapid. We will have much more to say about

reaction speed in the next chapter, in which we discuss chemical kinetics. The

process described in part (b) is nonspontaneous, which suggests that the reverse

process, the combustion of methane, is spontaneous. The basketball player in part

dunk the basketball. This process is nonspontaneous.

PRACTICE 14.2

Indicate whether each of these processes is spontaneous or nonspontaneous.

a. Potassium and water reacting c. A puddle evaporating from the sidewalk

b. Leaves falling from a tree d. Photosynthesis

See Problems 15–16.

Entropy

The overall catabolism (the biological degradation of molecules to provide

smaller molecules and energy to an organism) of glucose via glycolysis is a spon-

taneous process similar to the combustion of glucose. When glucose is burned in

air, six molecules of oxygen gas and one molecule of glucose combine to produce

six molecules of water vapor and six molecules of carbon dioxide gas.

(s) + 6O

(g) n 6CO

(g) + 6H

O(g)

When this reaction occurs, the number of gaseous molecules increases, which

corresponds to a dramatic increase in the number of microstates that describe the

system. Because of this increase, the probability that the reaction will produce

products is greater than the probability that carbon dioxide and water will spon-

taneously form glucose and oxygen. In short, an increase in the number of micro-

states favors spontaneous reactions. We can describe this principle in more

practical terms by introducing entropy.

Entropy (S) can be thought of as a measure of how the energy and matter of

a system are distributed throughout the system. Investigations related to the

concept of entropy began in the 1820s and 1830s with Nicolas Léonard Sadi

Carnot (1796–1832) and Benoit Paul Emile Clapeyron (1799–1864). However,

the concept wasn’t mathematically developed until Rudolf Julius Emmanuel

Clausius (1822–1888) worked on it in 1865. And although Clausius properly

illustrated entropy, its relationship to the molecular level wasn’t illuminated until

Boltzmann did so several decades later.

Entropy isn’t an easy concept to master, but we can gain important insight by

considering the probabilities that have been the focus of our discussion. If the

multiplicity—the number of microstates—increases, the number of ways in

14.2 Why Do Chemical Reactions Happen? Entropy and the Second Law of Thermodynamics 587

Visualization: Entropy

Video Lesson: Entropy and the

Second Law of Thermodynamics

Tutorial: Positional Entropy