3.1: Basics of Probability
We see probabilities almost every day in our lives. When you pick up the newspaper or read the news on the internet, you most likely encounter probability. There is a 60% chance of rain today, or a poll shows that 52% of voters approve of the President's job performance. Probabilities are essential in sports, gaming, and gambling establishments, but probabilities are also used to make business decisions, figure out insurance premiums, and determine the price of raffle tickets. In its most general sense, probability provides us a way to measure the chance or likelihood that something will happen.
Terminology Used in Probability
Before discussing how to find probabilities, we need to familiarize ourselves with some basic terminology. When studying probability, we consider an random experiment to be an activity or an operation that gives a result that can be observed but not predicted ahead of time. If we roll a pair of dice, pick a card from a deck of playing cards, spin a spinner, or randomly select a person and observe their hair color, we are executing an experiment and observing the result.
Any possible result of conducting an experiment is called an outcome. For the experiment of flipping a coin, there are only two outcomes: head or tail . For the experiment of rolling a single die, there are 6 outcomes: 1, 2, 3, 4, 5 , or 6 . For an experiment, this collection of all possible outcomes is called the sample space.
An event is a collection of outcomes from an experiment. In some instances events contain only one outcome while at other times an event may contain more than one outcome. Consider the event of rolling a single dice. The event " rolling a 3 " contains only the outcome \(\{3\}\) while the event " rolling an even number " contains the outcomes \(\{2,4,6\}\).
- A random experiment is an activity or operation with a result that cannot be predicted ahead of time.
- Any result from conducting an experiment is called an outcome .
- The sample space of an experiment is the set of all its possible outcomes.
- An event is a subset of the sample space and describes a collection of outcomes.
These terms are best illustrated with some examples.
 When we roll it, only one outcome will occur, but we are unsure which outcome. There are six possible outcomes and so the sample space is \(S=\{1, 2, 3, 4, 5, 6\}\).
'"Rolling a 2" is an event that contains only one outcome \(\{2\}\).
"Rolling a number greater than 2" is a another event that contains multiple outcomes \(\{3, 4, 5, 6\}\).
Two pennies are tossed at the same time. Both pennies may land heads up (which we write as HH ), or the first penny might land heads up and the second one tails up (which we write as HT ), and so on. Write the sample space for the experiment and list the outcomes in the event " getting at least one heads ."
Solution
The sample space for this experiment is \(S = \{HH, HT, TH, TT\}\).
If we define event \(A\) as " getting at least one heads, " the outcomes in event \(A\) can be written as \(A = \{HH, HT, TH\}\).

- Give the sample space for how the coin and die could land.
- Give the outcomes in event \(A\): rolls an odd number.
- Give the outcomes in the event \(B\): gets tails and rolls an even number.
- Answer
-
- \(\{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 \}\)
- \(\{H1, H3, H5, T1, T3, T5\}\)
- \(\{T2, T4, T6 \}\)
Probability is one way to measure the chance or the likelihood that an event will occur. Probability is usually denoted in function notation by \(P\), and the event is denoted by a capital letter such as \(A, B, C,\) etc . The mathematical notation that indicates the probability that event A happens is \(P(A)\).
Probability is a numerical measure of the chance or the likelihood that an event will occur.
We will examine two types of probability in this chapter: empirical probability and theoretical probability . But, we will begin with theoretical probability and discuss the connection and differences between the two types later in this section.
Theoretical Probability
We turn to the concept of theoretical probability and relate it to the sample space, outcomes, and events. We define theoretical probability of an event as the ratio of the number of outcomes in the event to the number of outcomes in the sample space of an experiment. For this definition of probability, it is very important that the outcomes in the sample space must be equally likely to occur.
A theoretical probability is based on a mathematical model where the number of outcomes in the event is compared with the number of outcomes in the sample space of an experiment. A formula for the theoretical probability of event \(E\) is
\(P(E) = \dfrac{\text{number of outcomes* in event E}}{\text{number of outcomes* in the sample space }S}\)
*where the outcomes are equally likely.
Let's apply this formula in some relatively simple examples.
 Find the probability of each event:
- rolling a 5
- rolling an even number
- rolling a number greater than 4
- rolling a 7
- rolling a number less than 7
Solution
There are 6 possible equally likely outcomes in the sample space for this experiment: \(S=\{1, 2, 3, 4, 5, 6\}\).
- There is only 1 outcome in the event " rolling a 5 ": \(\{5\}\)
\(P(\text{rolling a 5}) = \frac{\text{number of ways to roll a 5}}{\text{number of outcomes in the sample space}} = \frac{1}{6}\)
- There are 3 outcomes in the event " rolling an even number ": \(\{2, 4, 6\}\).
\(P(\text{rolling an even number}) = \frac{\text{number of ways to roll an even number}}{\text{number of outcomes in the sample space}} = \frac{3}{6}=\frac{1}{2}\)
- There are 2 outcomes in the event " rolling a number greater than 4 ": \(\{5, 6\}\).
\(P(\text{rolling a number greater than 4}) = \frac{\text{number of ways to roll a number greater than 4}}{\text{number of outcomes in the sample space}} = \frac{2}{6}=\frac{1}{3}\)
- There are no outcomes in the event " rolling a 7 ": \(\{ \; \; \}\).
\(P(\text{rolling a 7}) = \frac{\text{number of ways to roll a 7}}{\text{number of outcomes in the sample space}} = \frac{0}{6}=0\)
- There are 6 outcomes in the event " rolling a number less than 7 ": \(\{1, 2, 3, 4, 5, 6\}\).
\(P(\text{rolling a number less than 7}) = \frac{\text{number of ways to roll a number less than 7}}{\text{number of outcomes in the sample space}} = \frac{6}{6}=1\)
The previous example illustrates some important properties about values that can be legitimate probabilities.
- The number of outcomes in an event can never be lower than 0. So, the smallest a probability can be is 0. If the probability of an event is 0 (such as the probability of rolling a 7 in Example 3), we say that event is impossible .
- Furthermore, the number of outcomes in an event can never be more than the number of outcomes in the sample space. Therefore, the largest a probability can be is 1. If the probability of an event is 1 (such as the probability of rolling a number less than 7 in Example 3), we say that event is certain .
- The previous two facts tell us that the probability of any event must always fall between 0 and 1, inclusive. In the course of this chapter, if you compute a probability and get an answer that is negative or greater than 1, you have made a mistake and should recheck your work.
An event that cannot occur has a probability of 0. This event is impossible.
An event that must occur has a probability of 1. This event is certain .
The probability of any event must be between 0 and 1, inclusive. That is, \(0 \leq P(E) \leq 1\).
Recall the experiment in Example 2: Two pennies are tossed simultaneously and how they land is recorded. Find the probability getting each result:
- exactly two heads
- exactly one head
- at least one head
- more than 2 heads
Solution
Recall the the sample space for this experiment is \(S = \{HH, HT, TH, TT\}\).
- The event " exactly two heads " occurs in only 1 outcome \(\{HH\}\): \(P(\text{exactly two heads}) = \frac{1}{4}\)
- The event " exactly one head " occurs in 2 outcomes, \(\{HT,TH\}\): \(P(\text{exactly one head}) = \frac{2}{4}= \frac{1}{2}\)
- The event " at least one head " occurs in 3 outcomes, \(\{HH,HT,TH\}\): \(P(\text{at least 1 head}) = \frac{3}{4}\)
- The event " more than 2 heads " occurs in 0 outcomes, \(\{ \; \;\}\): \(P(\text{more than 2 heads}) = \frac{0}{4}= 0\)
Gabe performs an experiment of flipping a coin and then rolling a regular six-sided die. (See Try it Now 1 .) Find the probability of each event:
- Gabe rolls an odd number.
- Gabe gets tails and rolls an even number.
- Gabe rolls a number than is less than 10.
- Answer
-
- \(\frac{6}{12}=\frac{1}{2}\)
- \(\frac{3}{12}=\frac{1}{4}\)
- \(\frac{12}{12}=1\)
A small bookcase contains five math books, three English books, and seven science books - all the same size. One book is chosen at random. What is the probability that a math book is chosen?
Solution
There are 15 possible equally likely books that could be selected, so the number of possible outcomes in the sample space is 15. Of these 15 outcomes, 5 are in the event " math book ," so the probability that the book will be a math book is \(\frac{5}{15}=\frac{1}{3}\).
Let's say you have a container of 20 cherries, 14 sweet and 6 sour. If you pick a cherry at random, what is the probability that it will be sweet?
- Answer
-
There are 20 possible cherries that could be picked, so the number of possible outcomes in the sample space is 20. Of these 20 possible outcomes, 14 are in the event "sweet," so the probability that the cherry will be sweet is \(\frac{14}{20}=\frac{7}{10}\).
Find the probability of drawing each of these cards on the first draw from a standard deck of playing cards:
- a red card
- a club
- a King
- a King of clubs
Solution
To assist in analyzing the outcomes and sample space, here is a diagram of a deck of cards. Counting the cards, there are 52 equally likely outcomes in the sample space.
[1] ""English pattern" playing cards deck laid out, in SVG vector format" (opens in new window) by Dmitry Fomin (opens in new window) is in the Public Domain, CC0 (opens in new window) .
- There are 26 outcomes in the event " drawing a red card. " (13 hearts, 13 diamonds). \(P(\text{red card} = \frac{26}{52} = \frac{1}{2}\).
- There are 13 outcomes in the event "d rawing a club ." \(P(\text{club}= \frac{13}{52} = \frac{1}{4}\).
- There are 4 outcomes in the event " drawing a King ." \(P(\text{King}) = \frac{4}{52}= \frac{1}{12}\).
- There is only 1 outcome in the event " d rawing a King of clubs ." \(P(\text{King of clubs}) = \frac{1}{52}\)
Find the probability of drawing each of these cards on the first draw from a standard deck of playing cards:
- a 3 card
- a red 3 card
- a 3 of diamonds
- a red face card (Jack, Queen, or King)
- Answer
-
- \(\frac{4}{52}=\frac{1}{13}\)
- \(\frac{2}{52}=\frac{1}{26}\)
- \(\frac{1}{52}\)
- \(\frac{12}{52}=\frac{3}{13}\)
Empirical Probability
Up to this point, we have been finding probability theoretically by counting the equally likely outcomes in the sample space and in the event. For example, the theoretical probability of a coin ‘ landing heads up’ is \(\frac{1}{2}\) because there are two equally likely outcomes in the sample space and only one of those outcomes is in the event ‘ landing heads up.’
However, if you flip a coin 10 times, you may very well observe that the coin lands heads up 40%, 60%, or even 70% of the time! This discrepancy is perfectly natural and expected when conducting experiments, and it is important to recognize it. This brings us to the difference between theoretical probability and empirical probability .
- Theoretical probability is the calculated chance that an event occurs if the same experiment were completed an infinite number of times. We sometimes use our knowledge of sets and set operations when finding theoretical probabilities.
- Empirical probability is the calculated chance that an event occurs based on how often the outcome has occurred in the past over a limited number of trials. Generally speaking, when we determine the chance of something happening in the future by studying past results we are using empirical probability. Because empirical probability is determined by actual observations of an experiment it is also known as experimental probability .
- A theoretical probability is based on a mathematical model where all outcomes are equally likely to occur.
- An empirical probability is based on collected data and is the relative frequency of the event occurring.
Lawrence is playing with a standard 52-card deck and wants to find the probability of selecting a Queen from the deck.
The theoretical probability that Lawrence pulls a single Queen is \(\frac{4}{52} = \frac{1}{13} \approx 0.0769\) or about \(7.69\%\).
If Lawrence decides to try it out 25 times and pulls a Queen at random 3 times in 25 trials of “pull a card, record it, put it back,” the empirical probability of pulling a Queen is \(\frac{3}{25} = 0.12\), or about 12%.
A game uses a spinner like the one shown below. When spun, the arrow lands on either a white, yellow, blue, or red region. Let’s describe how we could use empirical probability to investigate the chance of landing on the white region.
The regions do not appear to be the same size, so the four color outcomes are not equally likely. We cannot apply our theoretical probability formula here unless we are told the areas of the regions or the measures of the central angles in the circular spinner. Instead, we can spin the spinner many times and record the results. Assume these hypothetical results have occurred:
| Color | Number of Spins |
|---|---|
| White | 18 |
| Red | 12 |
| Blue | 6 |
| Yellow | 4 |
Using the data in the table, the spinner landed on ' white ' 18 times out of a total of \(18+12+6+4=40\) times. The empirical probability that the spinner lands on white is \( \frac{18}{40} = \frac{9}{20} = 0.45 = 45\% \). This seems reasonable as the white region makes up slightly less than half the spinner.
Often times, we calculate empirical probabilities from data that have already been collected and organized for us.
In a given week, a veterinarian recorded how many times she treated each type of animal. What is the probability that the next animal she treats is a cat?
| Animal | Number Treated |
|---|---|
| Dog | 40 |
| Cat | 32 |
| Bird | 9 |
| Rabbit | 7 |
| Iguana | 2 |
Solution
The total number of trials (animals treated) is \(40 + 32+9+7+2=90\). Of these, 32 were cats. The fraction of trials that resulted in the event " cat " gives the empirical probability: \(P(\text{cat})=\frac{32}{90}=\frac{16}{45} \approx 0.3556 \text{ or } 35.56\%\).
A survey was conducted to determine the programs that students enrolled in MAT 1130 were taking. The results are shown in the table. What is the probability that a student in MAT 1130 is in the Computer Information Technology program?
| Major | English | Art | Nursing | Culinary Arts & Hospitality | Computer Information Technology |
|---|---|---|---|---|---|
| Number of Students | 3 | 8 | 2 | 12 | 10 |
- Answer
-
\(\frac{10}{35}=\frac{2}{7} \approx 0.2857 \text{ or } 28.57\%\)
The table shows the results of a survey where each person was asked 1) Do you have cable TV? and 2) Did you go on vacation in the past year? .
| Took a Vacation | No Vacation | Total | |
|---|---|---|---|
| Has Cable TV | 97 | 38 | 135 |
| Doesn't Have Cable TV | 14 | 17 | 31 |
| Total | 111 | 55 | 166 |
Use the table to find the probability of each event:
- \(P( \text{a person takes a vacation})\)
- \(P( \text{a person does not have cable TV})\)
- \(P( \text{a person takes a vacation and has cable TV})\)
- \(P( \text{a person takes a vacation or has cable TV })\)
Solution
There are 166 people surveyed so the number of trials of this experiment is 166. We need to determine the number of outcomes in each event and then find its relative frequency.
- The total in the first column shows 111 people took a vacation last year:
\(P(\text{a person takes a vacation}) = \frac{111}{166} \approx 0.6687 \text{ or } 66.87\%\).
- The total in the second row shows 31 people do not have cable TV:
\(P(\text{a person does not have cable TV}) =\frac{31}{166} \approx 0.1867 \text{ or } 18.67\%\).
- There are 97 people who did both of these things:
\(P(\text{a person takes a vacation and has cable TV}) = \frac{97}{166} \approx 0.5843 \text{ or } 58.43\%\).
- There are \(97+38+14=149\) people who did one or the other or both of these things:
\(P(\text{a person takes a vacation or has cable TV}) = \frac{149}{166} \approx 0.8976 \text{ or } 89.76\%\).
Nate recorded the topping and type of crust for all the pizzas he sold yesterday.
| Pepperoni | Sausage | Veggie | Total | |
|---|---|---|---|---|
| Thick Crust | 10 | 15 | 5 | 30 |
| Thin Crust | 24 | 16 | 10 | 50 |
| Total | 34 | 31 | 15 | 80 |
Use the table to find the probability of each event the next time Nate sells a pizza.
- \(P( \text{sausage topping})\)
- \(P( \text{sausage topping or thick crust})\)
- \(P( \text{veggie topping and thin crust})\)
- \(P( \text{topping contains meat })\)
- Answer
-
- \(\frac{31}{80} = 0.3875 \text{ or } 38.75\%\)
- \(\frac{46}{80} = 0.575 \text{ or } 57.5\%\)
- \(\frac{10}{80} = 0.125 \text{ or } 12.5\%\)
- \(\frac{65}{80} = 0.8125 \text{ or } 81.25\%\)
The Law of Large Numbers
Flipping a coin is often used to randomly make a decision when there are only two choices. For example, you may flip a coin to decide whether you have steak or fish for dinner. Or, a referee uses a coin flip to decide which football team receives the ball prior to kickoff. The reason why a coin flip seems fair in these circumstances is that most of us agree that the probability of getting heads (and tails) on a coin is \(\frac{1}{2} = 0.5 = 50\%\). But what does this mean in practice?
Does that mean if we flip a coin twice we will get heads exactly once? If a coin is tossed 10 times, will we necessarily get heads five times? Most of intuitively know the answer is no. Indeed, if we flip a coin 10 times we might find that it lands on heads 7 times. So what does it mean to say that the probability of heads on a fair coin is \(\frac{1}{2}\)?
To investigate this question, consider the table showing results that may happen when a coin is toss several times. The top row shows the number times the coin has been tossed. The next row shows the number of heads that have occurred. The bottom row shows the empirical probability which is the ratio of the number of heads observed to the number trials.
| Number of Trials | 10 | 20 | 30 | 40 | 50 |
|---|---|---|---|---|---|
| Number of Heads Observed | 7 | 13 | 17 | 22 | 26 |
| Empirical Probability of Heads | \(\frac{7}{10}=0.7\) | \(\frac{13}{20}=0.65\) | \(\frac{17}{30} \approx 0.57\) | \(\frac{22}{40}=0.55\) | \(\frac{26}{50}=0.52\) |
Notice that as the number of trials increases, the empirical probability gets closer to 0.5, which is what we expect to happen theoretically. In fact, if we kept increasing the number of trials, we would find that the empirical probability would eventually be very, very close to \(\frac{1}{2}= 0.5\).
This relationship between empirical probability and theoretical probability can be summarized by the Law of Large Numbers . The probability of an event applies to a large number of trials, not a single or a few trials. We should not be surprised that the empirical probability calculated from only a few trials is different from the theoretical probability. It is only empirical probability calculated over the long-run that gives an accurate probability.
The Law of Large Numbers states that we can only expect the empirical probability of an event to approximate its true probability when the number of trials of the experiment is large.
Stanley picks a card from a standard deck of cards and gets heart. He returns the card to the deck, picks a second time, and gets another heart. Stanley repeats this process a total of five times and gets a heart in four of those trials. Should Stanley conclude that the probability of selecting a heart is \(\frac{4}{5} = 0.80 = 80\%\)?
Solution
No. Stanley's reasoning is incorrect. He should not make a conclusion about the true probability of selecting a heart based only on five trials. If Stanley wants to estimate the chance of selecting a heart using empirical probability, he should perform the experiment a lot more than five times.


15.4: Drugs that Affect the Autonomic System
Learning Objectives
- List the classes of pharmaceuticals that interact with the autonomic nervous system
- Differentiate between cholinergic and adrenergic compounds
- Differentiate between sympathomimetic and sympatholytic drugs
- Relate the consequences of nicotine abuse with respect to autonomic control of the cardiovascular system
An important way to understand the effects of native neurochemicals in the autonomic system is in considering the effects of pharmaceutical drugs. This can be considered in terms of how drugs change autonomic function. These effects will primarily be based on how drugs act at the receptors of the autonomic system neurochemistry. The signaling molecules of the nervous system interact with proteins in the cell membranes of various target cells. In fact, no effect can be attributed to just the signaling molecules themselves without considering the receptors. A chemical that the body produces to interact with those receptors is called an endogenous chemical , whereas a chemical introduced to the system from outside is an exogenous chemical . Exogenous chemicals may be of a natural origin, such as a plant extract, or they may be synthetically produced in a pharmaceutical laboratory.
Broad Autonomic Effects
One important drug that affects the autonomic system broadly is not a pharmaceutical therapeutic agent associated with the system. This drug is nicotine. The effects of nicotine on the autonomic nervous system are important in considering the role smoking can play in health.
All ganglionic neurons of the autonomic system, in both sympathetic and parasympathetic ganglia, are activated by ACh released from preganglionic fibers. The ACh receptors on these neurons are of the nicotinic type, meaning that they are ligand-gated ion channels. When the neurotransmitter released from the preganglionic fiber binds to the receptor protein, a channel opens to allow positive ions to cross the cell membrane. The result is depolarization of the ganglia. Nicotine acts as an ACh analog at these synapses, so when someone takes in the drug, it binds to these ACh receptors and activates the ganglionic neurons, causing them to depolarize.
Ganglia of both divisions are activated equally by the drug. For many target organs in the body, this results in no net change. The competing inputs to the system cancel each other out and nothing significant happens. For example, the sympathetic system will cause sphincters in the digestive tract to contract, limiting digestive propulsion, but the parasympathetic system will cause the contraction of other muscles in the digestive tract, which will try to push the contents of the digestive system along. The end result is that the food does not really move along and the digestive system has not appreciably changed.
The system in which this can be problematic is in the cardiovascular system, which is why smoking is a risk factor for cardiovascular disease. First, there is no significant parasympathetic regulation of blood pressure. Only a limited number of blood vessels are affected by parasympathetic input, so nicotine will preferentially cause the vascular tone to become more sympathetic, which means blood pressure will be increased. Second, the autonomic control of the heart is special. Unlike skeletal or smooth muscles, cardiac muscle is intrinsically active, meaning that it generates its own action potentials. The autonomic system does not cause the heart to beat, it just speeds it up (sympathetic) or slows it down (parasympathetic). The mechanisms for this are not mutually exclusive, so the heart receives conflicting signals, and the rhythm of the heart can be affected (Figure \(\PageIndex{1}\)).
Sympathetic Effect
The neurochemistry of the sympathetic system is based on the adrenergic system. Norepinephrine and epinephrine influence target effectors by binding to the α-adrenergic or β-adrenergic receptors. Drugs that affect the sympathetic system affect these chemical systems. The drugs can be classified by whether they enhance the functions of the sympathetic system or interrupt those functions. A drug that enhances adrenergic function is known as a sympathomimetic drug , whereas a drug that interrupts adrenergic function is a sympatholytic drug .
Sympathomimetic Drugs
When the sympathetic system is not functioning correctly or the body is in a state of homeostatic imbalance, these drugs act at postganglionic terminals and synapses in the sympathetic efferent pathway. These drugs either bind to particular adrenergic receptors and mimic norepinephrine at the synapses between sympathetic postganglionic fibers and their targets, or they increase the production and release of norepinephrine from postganglionic fibers. Also, to increase the effectiveness of adrenergic chemicals released from the fibers, some of these drugs may block the removal or reuptake of the neurotransmitter from the synapse.
A common sympathomimetic drug is phenylephrine, which is a common component of decongestants. It can also be used to dilate the pupil and to raise blood pressure. Phenylephrine is known as an α 1 -adrenergic agonist , meaning that it binds to a specific adrenergic receptor, stimulating a response. In this role, phenylephrine will bind to the adrenergic receptors in bronchioles of the lungs and cause them to dilate. By opening these structures, accumulated mucus can be cleared out of the lower respiratory tract. Phenylephrine is often paired with other pharmaceuticals, such as analgesics, as in the “sinus” version of many over-the-counter drugs, such as Tylenol Sinus ® or Excedrin Sinus ® , or in expectorants for chest congestion such as in Robitussin CF ® .
A related molecule, called pseudoephedrine, was much more commonly used in these applications than was phenylephrine, until the molecule became useful in the illicit production of amphetamines. Phenylephrine is not as effective as a drug because it can be partially broken down in the digestive tract before it is ever absorbed. Like the adrenergic agents, phenylephrine is effective in dilating the pupil, known as mydriasis (Figure \(\PageIndex{2}\)). Phenylephrine is used during an eye exam in an ophthalmologist’s or optometrist’s office for this purpose. It can also be used to increase blood pressure in situations in which cardiac function is compromised, such as under anesthesia or during septic shock.
Other drugs that enhance adrenergic function are not associated with therapeutic uses, but affect the functions of the sympathetic system in a similar fashion. Cocaine primarily interferes with the uptake of dopamine at the synapse and can also increase adrenergic function. Caffeine is an antagonist to a different neurotransmitter receptor, called the adenosine receptor. Adenosine will suppress adrenergic activity, specifically the release of norepinephrine at synapses, so caffeine indirectly increases adrenergic activity. There is some evidence that caffeine can aid in the therapeutic use of drugs, perhaps by potentiating (increasing) sympathetic function, as is suggested by the inclusion of caffeine in over-the-counter analgesics such as Excedrin ® .
Sympatholytic Drugs
Drugs that interfere with sympathetic function are referred to as sympatholytic, or sympathoplegic, drugs. They primarily work as an antagonist to the adrenergic receptors. They block the ability of norepinephrine or epinephrine to bind to the receptors so that the effect is “cut” or “takes a blow,” to refer to the endings “-lytic” and “-plegic,” respectively. The various drugs of this class will be specific to α-adrenergic or β-adrenergic receptors, or to their receptor subtypes.
Possibly the most familiar type of sympatholytic drug are the β-blockers. These drugs are often used to treat cardiovascular disease because they block the β-receptors associated with vasoconstriction and cardioacceleration. By allowing blood vessels to dilate, or keeping heart rate from increasing, these drugs can improve cardiac function in a compromised system, such as for a person with congestive heart failure or who has previously suffered a heart attack. A couple of common versions of β-blockers are metoprolol, which specifically blocks the β 2 -receptor, and propanolol, which nonspecifically blocks β-receptors. There are other drugs that are α-blockers and can affect the sympathetic system in a similar way.
Other uses for sympatholytic drugs are as antianxiety medications. A common example of this is clonidine, which is an α-agonist. The sympathetic system is tied to anxiety to the point that the sympathetic response can be referred to as “fight, flight, or fright.” Clonidine is used for other treatments aside from hypertension and anxiety, including pain conditions and attention deficit hyperactivity disorder.
Parasympathetic Effects
Drugs affecting parasympathetic functions can be classified into those that increase or decrease activity at postganglionic terminals. Parasympathetic postganglionic fibers release ACh, and the receptors on the targets are muscarinic receptors. There are several types of muscarinic receptors, M1–M5, but the drugs are not usually specific to the specific types. Parasympathetic drugs can be either muscarinic agonists or antagonists, or have indirect effects on the cholinergic system. Drugs that enhance cholinergic effects are called parasympathomimetic drugs , whereas those that inhibit cholinergic effects are referred to as anticholinergic drugs .
Pilocarpine is a nonspecific muscarinic agonist commonly used to treat disorders of the eye. It reverses mydriasis, such as is caused by phenylephrine, and can be administered after an eye exam. Along with constricting the pupil through the smooth muscle of the iris, pilocarpine will also cause the ciliary muscle to contract. This will open perforations at the base of the cornea, allowing for the drainage of aqueous humor from the anterior compartment of the eye and, therefore, reducing intraocular pressure related to glaucoma.
Atropine and scopolamine are part of a class of muscarinic antagonists that come from the Atropa genus of plants that include belladonna or deadly nightshade (Figure \(\PageIndex{3}\)). The name of one of these plants, belladonna, refers to the fact that extracts from this plant were used cosmetically for dilating the pupil. The active chemicals from this plant block the muscarinic receptors in the iris and allow the pupil to dilate, which is considered attractive because it makes the eyes appear larger. Humans are instinctively attracted to anything with larger eyes, which comes from the fact that the ratio of eye-to-head size is different in infants (or baby animals) and can elicit an emotional response. The cosmetic use of belladonna extract was essentially acting on this response. Atropine is no longer used in this cosmetic capacity for reasons related to the other name for the plant, which is deadly nightshade. Suppression of parasympathetic function, especially when it becomes systemic, can be fatal. Autonomic regulation is disrupted and anticholinergic symptoms develop. The berries of this plant are highly toxic, but can be mistaken for other berries. The antidote for atropine or scopolamine poisoning is pilocarpine.
| Sympathetic and Parasympathetic Effects of Different Drug Types | ||||
|---|---|---|---|---|
| Drug type | Example(s) | Sympathetic effect | Parasympathetic effect | Overall result |
| Nicotinic agonists | Nicotine | Mimic ACh at preganglionic synapses, causing activation of postganglionic fibers and the release of norepinephrine onto the target organ | Mimic ACh at preganglionic synapses, causing activation of postganglionic fibers and the release of ACh onto the target organ | Most conflicting signals cancel each other out, but cardiovascular system is susceptible to hypertension and arrhythmias |
| Sympathomimetic drugs | Phenylephrine | Bind to adrenergic receptors or mimics sympathetic action in some other way | No effect | Increase sympathetic tone |
| Sympatholytic drugs | β-blockers such as propanolol or metoprolol; α-agonists such as clonidine | Block binding to adrenergic drug or decrease adrenergic signals | No effect | Increase parasympathetic tone |
| Parasymphatho-mimetics/muscarinic agonists | Pilocarpine | No effect, except on sweat glands | Bind to muscarinic receptor, similar to ACh | Increase parasympathetic tone |
| Anticholinergics/muscarinic antagonists | Atropine, scopolamine, dimenhydrinate | No effect | Block muscarinic receptors and parasympathetic function | Increase sympathetic tone |
DISORDERS OF THE...
Autonomic Nervous System
Approximately 33 percent of people experience a mild problem with motion sickness, whereas up to 66 percent experience motion sickness under extreme conditions, such as being on a tossing boat with no view of the horizon. Connections between regions in the brain stem and the autonomic system result in the symptoms of nausea, cold sweats, and vomiting.
The part of the brain responsible for vomiting, or emesis, is known as the area postrema. It is located next to the fourth ventricle and is not restricted by the blood–brain barrier, which allows it to respond to chemicals in the bloodstream—namely, toxins that will stimulate emesis. There are significant connections between this area, the solitary nucleus, and the dorsal motor nucleus of the vagus nerve. These autonomic system and nuclei connections are associated with the symptoms of motion sickness.
Motion sickness is the result of conflicting information from the visual and vestibular systems. If motion is perceived by the visual system without the complementary vestibular stimuli, or through vestibular stimuli without visual confirmation, the brain stimulates emesis and the associated symptoms. The area postrema, by itself, appears to be able to stimulate emesis in response to toxins in the blood, but it is also connected to the autonomic system and can trigger a similar response to motion.
Autonomic drugs are used to combat motion sickness. Though it is often described as a dangerous and deadly drug, scopolamine is used to treat motion sickness. A popular treatment for motion sickness is the transdermal scopolamine patch. Scopolamine is one of the substances derived from the Atropa genus along with atropine. At higher doses, those substances are thought to be poisonous and can lead to an extreme sympathetic syndrome. However, the transdermal patch regulates the release of the drug, and the concentration is kept very low so that the dangers are avoided. For those who are concerned about using “The Most Dangerous Drug,” as some websites will call it, antihistamines such as dimenhydrinate (Dramamine®) can be used.
Watch this video to learn about the side effects of 3-D movies. As discussed in this video, movies that are shot in 3-D can cause motion sickness, which elicits the autonomic symptoms of nausea and sweating. The disconnection between the perceived motion on the screen and the lack of any change in equilibrium stimulates these symptoms. Why do you think sitting close to the screen or right in the middle of the theater makes motion sickness during a 3-D movie worse?
Chapter Review
The autonomic system is affected by a number of exogenous agents, including some that are therapeutic and some that are illicit. These drugs affect the autonomic system by mimicking or interfering with the endogenous agents or their receptors. A survey of how different drugs affect autonomic function illustrates the role that the neurotransmitters and hormones play in autonomic function. Drugs can be thought of as chemical tools to effect changes in the system with some precision, based on where those drugs are effective.
Nicotine is not a drug that is used therapeutically, except for smoking cessation. When it is introduced into the body via products, it has broad effects on the autonomic system. Nicotine carries a risk for cardiovascular disease because of these broad effects. The drug stimulates both sympathetic and parasympathetic ganglia at the preganglionic fiber synapse. For most organ systems in the body, the competing input from the two postganglionic fibers will essentially cancel each other out. However, for the cardiovascular system, the results are different. Because there is essentially no parasympathetic influence on blood pressure for the entire body, the sympathetic input is increased by nicotine, causing an increase in blood pressure. Also, the influence that the autonomic system has on the heart is not the same as for other systems. Other organs have smooth muscle or glandular tissue that is activated or inhibited by the autonomic system. Cardiac muscle is intrinsically active and is modulated by the autonomic system. The contradictory signals do not just cancel each other out, they alter the regularity of the heart rate and can cause arrhythmias. Both hypertension and arrhythmias are risk factors for heart disease.
Other drugs affect one division of the autonomic system or the other. The sympathetic system is affected by drugs that mimic the actions of adrenergic molecules (norepinephrine and epinephrine) and are called sympathomimetic drugs. Drugs such as phenylephrine bind to the adrenergic receptors and stimulate target organs just as sympathetic activity would. Other drugs are sympatholytic because they block adrenergic activity and cancel the sympathetic influence on the target organ. Drugs that act on the parasympathetic system also work by either enhancing the postganglionic signal or blocking it. A muscarinic agonist (or parasympathomimetic drug) acts just like ACh released by the parasympathetic postganglionic fiber. Anticholinergic drugs block muscarinic receptors, suppressing parasympathetic interaction with the organ.
Interactive Link Questions
Watch this video to learn about the side effects of 3-D movies. As discussed in this video, movies that are shot in 3-D can cause motion sickness, which elicits the autonomic symptoms of nausea and sweating. The disconnection between the perceived motion on the screen and the lack of any change in equilibrium stimulates these symptoms. Why do you think sitting close to the screen or right in the middle of the theater makes motion sickness during a 3-D movie worse?
Answer: When the visual field is completely taken up by the movie, the brain is confused by the lack of vestibular stimuli to match the visual stimuli. Sitting to the side, or so that the edges of the screen can be seen, will help by providing a stable visual cue along with the magic of the cinematic experience.
Review Questions
Q. A drug that affects both divisions of the autonomic system is going to bind to, or block, which type of neurotransmitter receptor?
A. nicotinic
B. muscarinic
C. α-adrenergic
D. β-adrenergic
Answer: A
Q. A drug is called an agonist if it ________.
A. blocks a receptor
B. interferes with neurotransmitter reuptake
C. acts like the endogenous neurotransmitter by binding to its receptor
D. blocks the voltage-gated calcium ion channel
Answer: C
Q. Which type of drug would be an antidote to atropine poisoning?
A. nicotinic agonist
B. anticholinergic
C. muscarinic agonist
D. α-blocker
Answer: C
Q. Which kind of drug would have anti-anxiety effects?
A. nicotinic agonist
B. anticholinergic
C. muscarinic agonist
D. α-blocker
Answer: D
Q. Which type of drug could be used to treat asthma by opening airways wider?
A. sympatholytic drug
B. sympathomimetic drug
C. anticholinergic drug
D. parasympathomimetic drug
Answer: B
Critical Thinking Questions
Q. Why does smoking increase the risk of heart disease? Provide two reasons based on autonomic function.
A. Blood vessels, and therefore blood pressure, are primarily influenced by only the sympathetic system. There is no parasympathetic influence on blood pressure, so nicotine activation of autonomic ganglia will preferentially increase blood pressure. Also, cardiac muscle tissue is only modulated by autonomic inputs, so the conflicting information from both sympathetic and parasympathetic postganglionic fibers will cause arrhythmias. Both hypertension and arrhythmias are cardiac risk factors.
Q. Why might topical, cosmetic application of atropine or scopolamine from the belladonna plant not cause fatal poisoning, as would occur with ingestion of the plant?
A. Drops of these substances into the eyes, as was once done cosmetically, blocks the muscarinic receptors in the smooth muscle of the iris. The concentration of this direct application is probably below the concentration that would cause poisoning if it got into the bloodstream. The possibility of that concentration being wrong and causing poisoning is too great, however, for atropine to be used as a cosmetic.
Glossary
- agonist
- any exogenous substance that binds to a receptor and produces a similar effect to the endogenous ligand
- antagonist
- any exogenous substance that binds to a receptor and produces an opposing effect to the endogenous ligand
- anticholinergic drugs
- drugs that interrupt or reduce the function of the parasympathetic system
- endogenous chemical
- substance produced and released within the body to interact with a receptor protein
- exogenous chemical
- substance from a source outside the body, whether it be another organism such as a plant or from the synthetic processes of a laboratory, that binds to a transmembrane receptor protein
- mydriasis
- dilation of the pupil; typically the result of disease, trauma, or drugs
- parasympathomimetic drugs
- drugs that enhance or mimic the function of the parasympathetic system
- sympatholytic drug
- drug that interrupts, or “lyses,” the function of the sympathetic system
- sympathomimetic drug
- drug that enhances or mimics the function of the sympathetic system
Contributors and Attributions
-
OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


4.8: Learning Activities
Exercises
(Answers to the exercises are located in the Answer Key at the back of the book).
Case Study #1
Autumn, age 32, has a history of Diabetes Mellitus Type II and has been admitted to the hospital with a left lower leg wound that developed cellulitis. She has been receiving antibiotic therapy in the hospital for the past two days through a right upper arm PICC line and is now ready for discharge. When at home, she will continue to receive cefazolin 500 mg IV every 8 hours for the next 14 days.
1. What will you provide for patient education for Autumn regarding her PICC line?
2. What are the maintenance care priorities for care of the PICC line?
3. Are there any specific concerns related to Autumn’s need for a PICC line that should be monitored or addressed?
4. What is the purpose of the PICC line?
5. How often should a PICC line be assessed?
6. How does the dressing get changed for a PICC line?
7. What makes a PICC line different from a peripheral IV and from an implanted port?
An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://wtcs.pressbooks.pub/nursingadvancedskills/?p=458#h5p-10
An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://wtcs.pressbooks.pub/nursingadvancedskills/?p=458#h5p-11
An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://wtcs.pressbooks.pub/nursingadvancedskills/?p=458#h5p-12


8.4: Globalization and Health
Epidemiologic Transitions
New Infectious Diseases
A disease that emerges within a population that is new or the number of infectious cases within a population or geographic area rapidly increases. Since 1940 over 300 new infectious diseases have been discovered, some of the most well known being severe acute respiratory syndrome (SARS), methicillin-resistant staph (MRSA), and Human Immunodeficiency Virus (HIV). Many drug-resistant strains of diseases are becoming more common and 71% of these new infectious diseases start in the wildlife. As globalization increases, infectious diseases will continue to affect a larger and wider population.[4] Classification of disease is as follows; an epidemic is a local outbreak of a rare disease. When this disease spreads through many human populations across a large region, it is then classified as a pandemic.
Medical Pluralism
The integration of biomedicine and other forms of health care. Examples of medical pluralism include taking antibiotics and vaccines upon acute trauma or infection, as well as relaxation rituals to decrease stress and improve mental health. Medical pluralism includes involving different wellness techniques to improve, maintain, and prevent overall well-being. In earlier times, medical pluralism was considered a war zone and was seen as "other" or alternative medicines therapies that were trying to take over. Today, medical pluralism is seen as a positive thing as it can provide multiple solutions/treatments to something [5]
Diseases of Development
The main causes of illness and death in developed countries are cancer and diseases of the respiratory, cardiovascular and nervous systems. In the developing world, communicable diseases are the main problem, with deaths occurring primarily due to respiratory infections, HIV/AIDS, infections at birth, diarrhoeal disease and tropical diseases such as malaria. Failure to use existing treatments effectively, inadequate or non-existent interventions, and insufficient knowledge of disease all contribute to damaged health.
Globalization
Malaria
Malaria is a disease that is spread by mosquitoes. Mosquitoes pick up malarial parasites from the blood of infected. While only one species of mosquito can carry the parasite, there exist four types of the malarial parasite leading to four types of malaria. Symptoms include fever, shivering, pain in the joints, headache, vomiting, convulsions, and coma. If an infected person is not treated, he or she can die.
In recent years, globalization has increases the spread of malaria through travel, war, and urbanization. Persons traveling to countries which have a larger rate of malaria cases can become infected and carry the disease back to their country, and malaria-carrying mosquitos can stow away on international flights to bring the disease far from infected areas. War time refugees who spend long periods exposed to the elements and cross borders fleeing violence are more likely to come in contact with malaria-carrying mosquitoes as well.
HIV and AIDS
AIDS (Acquired Immune Deficiency Syndrome) is an incurable disease that attacks the patient’s immune system. AIDS is caused by infection with HIV (Human Immunodeficiency Virus), which is transmitted through the exchange of bodily fluids by sexual contact, sharing needles, and coming into contact with infected blood. HIV/AIDS is a global disease where its highest rates are in poor countries that lack a developed health service. While the lifespan of an infected person can be extended via anti-retroviral drugs, the disease is ultimately fatal.
Despite efforts in numerous countries, awareness and prevention programs have been unsuccessful in reducing the numbers of new HIV cases in many parts of the world where poverty and social mores increase peoples’ risk of infection. Even in countries where the epidemic has a high impact, such as South Africa and Swaziland, a large portion of the population do not believe they are at risk of infection. While initially, HIV prevention focused on preventing sexual transmission of HIV through behavior change, in more recent years it has become evident that HIV prevention requires interventions that take into account underlying socio-cultural, economic, political, and legal factors.
Smallpox
Smallpox is a highly contagious disease caused by the Variola virus. Symptoms of the disease include hemorrhaging, blindness, back ache, and vomiting. The virus attacks the skin cells after an incubation period, causing pimples and pustules to form that can further spread the disease. Smallpox is easily transmitted through airborne pathways such as coughing or sneezing, as well as through contaminated bedding and clothing.
Smallpox was a common disease in the 15 th century in Eurasia, being spread by explorers and invaders such as Columbus. During the 16 th century, Spanish soldiers introduced smallpox by contact with the Aztec natives in Tenochtitlan, causing a devastating epidemic that killed thousands. In 1617, smallpox reached Massachusetts and spread to Boston by 1638. Persons who fled after an outbreak in 1721 spread the disease to the other thirteen colonies. A vaccine was developed in the 18 th century, and by 1979 the disease has since been completely eradicated globally.


3.5: Polyatomic Ions
The compound NaOH has wide industrial use and is the active ingredient in drain cleaners. Based on the discussion in the previous section, we would expect NaOH to be an ionic compound because it contains sodium, a Group 1A metal. Hydrogen and oxygen, however, are nonmetals, and we would expect these to bond together covalently. This compound, called sodium hydroxide, is an example of an ionic compound formed between a metal ion (sodium) and a polyatomic ion (HO – ). Charged groups of atoms, like HO – , that are bonded together covalently are called polyatomic ions . Within an ionic compound a polyatomic behaves as a single unit forming salts with other cations or anions.
Using the rules described in Section 3.2 , we can draw a Lewis diagram for HO – . Oxygen has six valence electrons and hydrogen has one, for a total of seven. The central atom in our structure will be hydrogen (it is to the left of oxygen in the periodic table). Next, because this is a polyatomic ion with a single negative charge, we add the extra electron to the central atom, pair the electrons and then draw the two atoms bonded together. Next, the six remaining electrons are distributed around the oxygen to form an octet. Finally, the polyatomic ion is enclosed in brackets with the charge as a superscript to show that the ion behaves as a single unit.
Polyatomic ions are very common in chemistry. It is essential that you memorize these and be able to correlate the name, the composition and the charge for each of them, as they will be discussed freely throughout the remainder of the course and you will be expected to know these in General Chemistry.
Construct a Lewis diagram for the polyatomic ion CO 3 2 – .
Solution
Oxygen has six valence electrons and carbon has four; therefore in CO 3 2 – there will be a total of 22 valence electrons, plus two additional electrons from the 2- charge. The central atom in our structure will be carbon (it is to the left of oxygen in the periodic table). Next, we draw the carbon (our central atom) with its’ four electrons and add the additional two electrons from the charge. The three oxygens are placed around the carbon and the electrons are arranged to form the three covalent bonds. Next, the 18 remaining electrons are distributed around the oxygens so that they all have a full octet. The carbon, however, is only surrounded by six electrons. To remedy this, we move one electron pair in to form a double bond to one of the oxygen atoms. Finally, the polyatomic ion is enclosed in brackets with the charge as a superscript.
We need to understand that the process of placing electrons into a particular bond in a compound is an artificial aspect of building Lewis diagrams. In fact, the electrons are added to the polyatomic ion, but it is impossible to know exactly where they went.
- Draw Lewis diagrams for NO 2 - and NH 4 + .


11.4: Enthalpies of Solution and Dilution
The processes of solution (dissolution) and dilution are related. The IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry , 3rd edition, RSC Publishing, Cambridge, 2007, Sec. 2.11.1) recommends the abbreviations sol and dil for these processes.
For an electrolyte solute, a plot of \(\Del H\m\solmB\) versus \(m\B\) has a limiting slope of \(+\infty\) at \(m\B{=}0\), whereas the limiting slope of \(\Del H\m\solmB\) versus \(\sqrt{m\B}\) is finite and can be predicted from the Debye–Hückel limiting law. Accordingly, a satisfactory procedure is to plot \(\Del H\m\solmB\) versus \(\sqrt{m\B}\), perform a linear extrapolation of the experimental points to \(\sqrt{m\B}{=}0\), and then shift the origin to the extrapolated intercept. The result is a plot of \(\varPhi_L\) versus \(\sqrt{m\B}\). An example for aqueous NaCl solutions is shown in Fig. 11.10(a).
We can also evaluate \(\varPhi_L\) from experimental enthalpies of dilution. From Eqs. 11.4.10 and 11.4.22, we obtain the relation \begin{equation} \varPhi_L(m\B'')-\varPhi_L(m\B') = \Del H\m(\tx{dil, \(m\B'{\ra}m\B''\)}) \tag{11.4.25} \end{equation} We can measure the enthalpy changes for diluting a solution of initial molality \(m\B'\) to various molalities \(m\B''\), plot the values of \(\Del H\m(\tx{dil, \(m\B'{\ra}m\B''\)})\) versus \(\sqrt{m\B}\), extrapolate the curve to \(\sqrt{m\B}{=}0\), and shift the origin to the extrapolated intercept, resulting in a plot of \(\varPhi_L\) versus \(\sqrt{m\B}\).
In order to be able to use Eq. 11.4.23, we need to relate the derivative \(\dif\varPhi_L/\dif m\B\) to the slope of the curve of \(\varPhi_L\) versus \(\sqrt{m\B}\). We write \begin{equation} \dif \sqrt{m\B} = \frac{1}{2\sqrt{m\B}}\dif m\B \qquad \dif m\B = 2\sqrt{m\B} \dif\sqrt{m\B} \tag{11.4.26} \end{equation} Substituting this expression for \(\dif m\B\) into Eq. 11.4.23, we obtain the following operational equation for evaluating \(L\B\) from the plot of \(\varPhi_L\) versus \(\sqrt{m\B}\): \begin{gather} \s{ L\B = \varPhi_L + \frac{\sqrt{m\B}}{2} \frac{\dif\varPhi_L}{\dif\sqrt{m\B}} } \tag{11.4.27} \cond{(constant \(T\) and \(p\))} \end{gather}
The value of \(\varPhi_L\) goes to zero at infinite dilution. When the solute is an electrolyte, the dependence of \(\varPhi_L\) on \(m\B\) in solutions dilute enough for the Debye–Hückel limiting law to apply is given by \begin{gather} \s{ \varPhi_L = C_{\varPhi_L}\sqrt{m\B} } \tag{11.4.28} \cond{(very dilute solution)} \end{gather} For aqueous solutions of a 1:1 electrolyte at \(25\units{\(\degC\)}\), the coefficient \(C_{\varPhi_L}\) has the value \begin{equation} C_{\varPhi_L} = 1.988\timesten{3}\units{J kg\(^{1/2}\) mol\(^{-3/2}\)} \tag{11.4.29} \end{equation} (The fact that \(C_{\varPhi_L}\) is positive means, according to Eq. 11.4.25, that dilution of a very dilute electrolyte solution is an exothermic process.) \(C_{\varPhi_L}\) is equal to the limiting slope of \(\varPhi_L\) versus \(\sqrt{m\B}\), of \(\Del H\m\solmB\) versus \(\sqrt{m\B}\), and of \(\Del H\m(\tx{dil, \(m\B'{\ra}m\B''\)})\) versus \(\sqrt{m'\B}\). The value given by Eq. 11.4.29 can be used for extrapolation of measurements at \(25\units{\(\degC\)}\) and low molality to infinite dilution.
Equation 11.4.28 can be derived as follows. For simplicity, we assume the pressure is the standard pressure \(p\st\). At this pressure \(H\B^\infty\) is the same as \(H\B\st\), and Eq. 11.4.17 becomes \(L\B=H\B-H\B\st\). From Eqs. 12.1.3 and 12.1.6 in the next chapter, we can write the relations \begin{equation} H\B=-T^2\bPd{(\mu\B/T)}{T}{p,\allni} \qquad H\B\st=-T^2\frac{\dif(\mu\mbB\st/T)}{\dif T} \tag{11.4.30} \end{equation} Subtracting the second of these relations from the first, we obtain \begin{equation} H\B-H\B\st = -T^2\bPd{(\mu\B-\mu\mbB\st)/T}{T}{p,\allni} \tag{11.4.31} \end{equation} The solute activity on a molality basis, \(a\mbB\), is defined by \(\mu\B-\mu\mbB\st=RT\ln a\mbB\). The activity of an electrolyte solute at the standard pressure, from Eq. 10.3.10, is given by \(a\mbB = (\nu_{+}^{\nu_{+}} \nu_{-}^{\nu_{-}}) \g_{\pm}^{\nu} (m\B/m\st)^{\nu}\). Accordingly, the relative partial molar enthalpy of the solute is related to the mean ionic activity coefficient by \begin{equation} L\B=-RT^2\nu\Pd{\ln\g_{\pm}}{T}{\!\!p,\allni} \tag{11.4.32} \end{equation}
We assume the solution is sufficiently dilute for the mean ionic activity coefficient to be adequately described by the Debye–Hückel limiting law, Eq. 10.4.8: \(\ln\g_{\pm} = -A\subs{DH}\left|z_+z_-\right|\sqrt{I_m}\), where \(A\subs{DH}\) is a temperature-dependent quantity defined in Sec. 10.4. Then Eq. 11.4.32 becomes \begin{gather} \s{ L\B=RT^2\nu\left|z_+z_-\right|\sqrt{I_m}\Pd{A\subs{DH}}{T}{\!\!p,\allni} } \tag{11.4.33} \cond{(very dilute solution)} \end{gather} Substitution of the expression given by Eq. 10.4.9 for \(I_m\) in a solution of a single completely-dissociated electrolyte converts Eq. 11.4.33 to \begin{gather} \s{ L\B = \left[ \frac{RT^2}{\sqrt{2}}\Pd{\rho\A^*A\subs{DH}}{T}{p,\allni} \left(\nu\left|z_+z_-\right|\right)^{3/2} \right]\sqrt{m\B} = C_{L\B}\sqrt{m\B} } \tag{11.4.34} \cond{(very dilute solution)} \end{gather} The coefficient \(C_{L\B}\) (the quantity in brackets) depends on \(T\), the kind of solvent, and the ion charges and number of ions per solute formula unit, but not on the solute molality.
Let \(C_{\varPhi_L}\) represent the limiting slope of \(\varPhi_L\) versus \(\sqrt{m\B}\). In a very dilute solution we have \(\varPhi_L = C_{\varPhi_L}\sqrt{m\B}\), and Eq. 11.4.27 becomes \begin{equation} L\B = \varPhi_L + \frac{\sqrt{m\B}}{2} \frac{\dif\varPhi_L}{\dif\sqrt{m\B}} = C_{\varPhi_L}\sqrt{m\B} + \frac{\sqrt{m\B}}{2} C_{\varPhi_L} \tag{11.4.35} \end{equation} By equating this expression for \(L\B\) with the one given by Eq. 11.4.34 and solving for \(C_{\varPhi_L}\), we obtain \(C_{\varPhi_L}=(2/3)C_{L\B}\) and \(\varPhi_L = (2/3)C_{L\B}\sqrt{m\B}\).


10.5: Case Study - Yosemite National Park
Yosemite National Park is an American National Park in California, surrounded on the southeast by Sierra National Forest and on the northwest by Stanislaus National Forest. The park is managed by the National Park Service and covers an area of 759,620 acres (1,187 sq mi; 3,074 km2) and sits in four counties – centered in Tuolumne and Mariposa, extending north and east to Mono and south to Madera County. Designated a World Heritage Site in 1984, Yosemite is internationally recognized for its granite cliffs, waterfalls, clear streams, giant sequoia groves, lakes, mountains, meadows, glaciers, and biological diversity. Almost 95 percent of the park is designated wilderness. Yosemite is one of the largest and least fragmented blocks in the Sierra Nevada, and the park supports a diversity of plants and animals.
The geology of the Yosemite area is characterized by granite rocks and remnants of older rock. About 10 million years ago, the Sierra Nevada was uplifted and tilted to form its unique slopes, which increased the steepness of stream and riverbeds, resulting in the formation of deep, narrow canyons. About two million years ago glaciers formed at higher elevations which eventually melted and moved downslope, cutting, and sculpting the U-shaped valley that attracts so many visitors to its scenic vistas.
European American settlers first entered Yosemite Valley itself in 1851. There are earlier instances of other travelers entering the Valley, but James D. Savage is credited with discovering the area that became Yosemite National Park. Despite Savage and other white men claiming their discovery of Yosemite, the region and Valley itself has been inhabited for nearly 4,000 years, although humans may have visited the area as early as 10,000 years ago.
Yosemite was critical to the development of the national park idea. Galen Clark and others lobbied to protect Yosemite Valley from development, ultimately leading to President Abraham Lincoln signing the Yosemite Grant of 1864 which declared Yosemite as federally preserved land. It was not until 1890 when John Muir led a successful movement which had Congress establish Yosemite Valley and its surrounding areas as a National Park. This helped pave the way for the National Park System. Yosemite draws about four million visitors each year, and most visitors spend most of their time in the seven square miles (18 km2) of Yosemite Valley. The park set a visitation record in 2016, surpassing five million visitors for the first time in its history. The park began requiring reservations to access the park during peak periods starting in 2020 as a response to the rise in visitors.
Geology
The first rocks were laid down in Precambrian times, when the area around Yosemite National Park was on the edge of a very young North American continent. The sediment that formed the area first settled in the waters of a shallow sea, and compressive forces from a subduction zone in the mid-Paleozoic fused the seabed rocks and sediments, appending them to the continent. Heat generated from the subduction created island arcs of volcanoes that were also thrust into the ancient area. In time, the igneous and sedimentary rocks of the area were later heavily metamorphosed.
Most of the rock now exposed in the park is granitic, having been formed 210 to 80 million years ago as igneous diapirs 6 miles (10 km) below the surface. Over time, most of the overlying rock uplifted along with the rest of the Sierra Nevada, was removed by erosion. This exposed the granitic rock to much lower pressure, which caused the rock to expand, joint and exfoliate.
Glaciations
Starting about 2 years ago a series of glaciations further modified the area by accelerating mass wasting through ice-wedging, glacial plucking, scouring/abrasion, and the release of pressure after the retreat of each glaciation. The glaciers tended to strip and transport topsoil and talus piles far down glacial valleys, while less-severe glaciations deposited a great deal of glacial till further up in the valleys.
At least 4 major glacial periods have occurred in the Sierra Nevada; locally called the Sherwin (also called the pre-Tahoe), Tahoe, Tenaya, and Tioga. The Sherwin glaciers were the largest, filling Yosemite and other valleys, while later stages produced much smaller glaciers. The Sherwin may have lasted almost 300 thousand years and ended about 1 million years ago. A Sherwin-age glacier was almost surely responsible for the major excavation and shaping of Yosemite Valley and other canyons in the area.
The Tahoe, Tenaya, and Tioga stages were part of the Wisconsinan glaciation. The Tahoe glacial stage is thought to have reached its maximum extent around 70,000 to 130,000 years ago; little is known about the more recent Tenaya. Evidence also suggests that the most recent local glacial stage, the Tioga, started about 28,000 cal (calibrated Radiocarbon dating) years ago, reached its maximum extent 20,000 to 25,000 cal years ago, and ended by ~15,000 cal yr ago. Glaciers reformed in the highest cirques during a minor late-glacial readvance, the Recess Peak event, between about 14,200 and 13,100 years ago.
Glacial systems reached depths of up to 4000 feet (1200 m) and left their marks in the Yosemite area. The longest glacier in the Yosemite area ran down the Grand Canyon of the Tuolumne River for 60 miles (95 km), passing well beyond Hetch Hetchy Valley. Merced Glacier flowed out of Yosemite Valley and into the Merced River Gorge. Lee Vining Glacier carved Lee Vining Canyon and emptied into Lake Russell (the much-enlarged ice age version of Mono Lake). Only the highest peaks, such as Mount Dana and Mount Conness, were not covered by glaciers. Retreating glaciers often left recessional moraines that impounded lakes such as Lake Yosemite (a shallow lake that periodically covered much of the floor of Yosemite Valley).
Some domes in the park were covered by glaciers and modified into roche moutonnées, which are characterized by having a smooth, rounded side and a steep face. The rounded side was where the glacier flowed over the dome and the steep side is where the glacier flowed away from it. The steepness is caused by glacial plucking of rock along fracture joints. Good examples in the park are Liberty Cap, Lembert Dome, Mount Broderick and Half Dome (figure10.7).
Half Dome was created by a different process, since erosion acting on jointing planes was still the major factor. Half Dome is a quartz monzonite batholith at the eastern end of Yosemite Valley in Yosemite National Park, California. It is a well-known rock formation in the park, named for its distinct shape.  It stands at nearly 8,800 feet above sea level and is composed of quartz monzonite, an igneous rock that solidified several thousand feet within the Earth. At its core are the remains of a magma chamber that cooled slowly and crystallized beneath the Earth's surface. The solidified magma chamber was then exposed and cut in half by erosion, therefore leading to the geographic name Half Dome.
Cultural History & Influence
Historically attested Native American populations, such as the Sierra Miwok, Mono and Paiute, belong to the Uto-Aztecan and Utian phyla. In the mid-19th century, a band of Native Americans called the Ahwahnechee lived in Yosemite Valley. The California Gold Rush greatly increased the number of non-indigenous people in the region. Tensions between Native Americans and white settlers escalated into the Mariposa War. As part of this conflict, settler James Savage led the Mariposa Battalion into Yosemite Valley in 1851, in pursuit of Ahwaneechees led by Chief Tenaya. The California state military forces burned the tribe's villages, destroyed their food stores, killed the chief's sons, and forced the tribe out of Yosemite. Accounts from the Mariposa Battalion, especially from Dr. Lafayette Bunnell, popularized Yosemite Valley as a scenic wonder.
Native Americans of Yosemite
Humans may have lived in the Yosemite area as long as 10,000 years ago. Habitation of the Yosemite Valley proper can be traced to about 3,000 years ago, when vegetation and game in the region was like that present today; the western slopes of the Sierra Nevada had acorns, deer, and salmon, while the eastern Sierra had pinyon nuts and obsidian. Native American groups traveled between these two regions to trade and raid.
Archaeologists divide the pre-European American contact period of the region into three cultural phases. The Crane Flat phase lasted from 1000 BCE to 500 CE and is marked by hunting with the atlatl and the use of grinding stones. The Tamarack phase lasted from 500 until 1200, marked by a shift to using smaller rock points, indicating development and use of the bow and arrow. The Mariposa phase lasted from 1200 until contact with European Americans.
Trade between tribes became more widespread during the Mariposa phase, and the diet continued to improve. Paiutes, Miwok, and Monos visited the area to trade; one major trading route went over Mono Pass and through Bloody Canyon to Mono Lake in Eastern California, (often acorns from the west with obsidian from the eastern side of the Sierra).
Paiutes were the primary inhabitants of the Yosemite area and the foothills to the east during the Mariposa and historic phases. The Central Sierra Miwoks lived along the drainage area of the Tuolumne and Stanislaus Rivers, while the Paiutes inhabited the upper drainage of the Merced and Chowchilla Rivers.
The Paiutes of Yosemite Valley called themselves the Ah-wah-ne-chee, meaning "dwellers in Ahwahnee.” The Ahwahneechees were decimated by a disease in about 1800, and left the valley, although about 200 returned under the leadership of Tenaya, son of an Ahwahneechee chief.
Displaced Native Americans from the Californian coast moved to the Sierra Nevada during the early-to-mid-19th century, bringing with them their knowledge of Spanish food, technology, and clothing. Joining forces with the other tribes in the area, they raided land grant ranchos on the coast and drove herds of horses to the Sierra, where horse meat became a major new food source.
Then Mariposa Battalion & The First Tourists
The first non-Native Americans to see Yosemite Valley were probably members of the 1833 Joseph Walker Party, which was the first to cross the Sierra Nevada from east to west. The first descriptions of Yosemite, however, came nearly 20 years later. The 1849 California Gold Rush led to conflicts between miners and Native Americans, and the state formed the volunteer Mariposa Battalion as a punitive expedition against the Native Americans living in the Yosemite area. In 1851, the Battalion was led by Major Jim Savage, whose trading post on the Merced River the Awaneechee had raided. This and other missions (the Mariposa Wars) resulted in Chief Teneiya and the Awaneechee spending months on a reservation in the San Joaquin Valley. The band returned the next year to the Valley but took refuge among the Mono Paiutes after further conflicts with miners. Most of the Awaneechee (along with Teneiya) were chased back to the Valley and killed by the Paiutes after violating hospitality by stealing horses.
While the members of that first expedition of the Mariposa Battalion had heard rumors of what could be found up the Merced River, none were prepared for what they saw March 27, 1851, from what is now called Old Inspiration Point (close to the better-visited Tunnel View). Dr. Lafayette Bunnell later wrote:
“The grandeur of the scene was but softened by the haze that hung over the valley – light as gossamer – and by the clouds which partially dimmed the higher cliffs and mountains. This obscurity of vision but increased the awe with which I beheld it, and as I looked, a peculiar, exalted sensation seemed to fill my whole being, and I found my eyes in tears with emotion.”
Camping that night on the Valley floor, the group agreed with the suggestion of Dr. Bunnell to call it "Yo-sem-i-ty", mistakenly believing that was the native name. The term is from the Southern Sierra Miwok word Yohhe'meti, meaning "they are killers," which referred to the inhabitants of the place.
James Hutchings—who organized the first tourist party to the Valley in 1855—and artist Thomas Ayers generated much of the earliest publicity about Yosemite, creating articles and entire magazine issues about the Valley. Ayres' highly detailed angularly exaggerated artwork and his written accounts were distributed nationally, and an art exhibition of his drawings was held in New York City.
Two of Hutchings’ first group of tourists, Milton, and Houston Mann built the first toll route into the valley, with the development of the first hotels in the area and other trails quickly following. Many of the early pioneers in the valley of European descent, and a few Native Americans, are buried in Yosemite Cemetery.
The First Park
The work of Ayres gave Easterners an appreciation for Yosemite Valley and started a movement to preserve it. Influential figures such as Galen Clark, clergyman Thomas Starr King and leading landscape architect Frederick Law Olmsted were among those who urged Senator John Conness of California to try to preserve Yosemite.
President Abraham Lincoln signed a bill on June 30, 1864, granting Yosemite Valley and the Mariposa Grove of giant sequoias to the State of California "for public use, resort and recreation," the two tracts "shall be inalienable for all time". This was the first time in history that a federal government had set aside scenic lands simply to protect them and to allow for their enjoyment by all people.
Simply designating an area, a park isn't sufficient to protect it. California did not set up an administration for the park until 1866 when the state appointed Galen Clark as the park's guardian. An 11-year struggle followed to resolve homesteading claims in the valley. The challenge of increasing tourism, with the need to first build stagecoach roads, then the Yosemite Valley Railroad, along with hotels and other facilities in and around the Valley was met during the rest of the 19th century. But much environmental damage was caused to the valley itself at that time. The problems that Yosemite Park had under state control was one of the factors in establishing Yellowstone National Park as the first completely national park in 1872.
Due to the difficulty of traveling there, early visitors to the valley came for several weeks to a couple of months, often bringing entire families with many possessions. Early hotels were therefore set up for extended stays and catered primarily to wealthy patrons who could spend extended periods away from home. One of these hotels—the Wawona Hotel, built in the 1880s—still operates.
After the Valley became a park, the surrounding territory was still subject to logging, mining, and grazing. John Muir publicized the damage to the subalpine meadows that surround the Valley and in 1890, the government created a national park that included a much larger territory—enclosing Yosemite Valley and Mariposa Grove.
In May of 1903, U.S. President Theodore Roosevelt camped with John Muir near Glacier Point for three days. During that trip, Muir convinced Roosevelt to take control of the valley and the grove away from California and give it to the federal government. On June 11, 1906, Roosevelt signed a bill that did precisely that, and the superintendent's headquarters was moved from Wawona to Yosemite Valley.
To secure congressional and State of California approval for the plan, the size of the park was reduced by more than 500 square miles (1,300 km2), which excluded natural wonders such as the Devils Postpile and prime wildlife habitat. The park was again reduced in size in 1906, when logging began in an area around Wawona. Acting superintendent Major H. C. Benson said in 1908 that game is on the decrease. Each reduction of the park has cut another portion of the winter resort of game. The various changes meant that the park was reduced to two-thirds of its original size.
About 12,000 acres between the Tuolumne and the Merced big tree groves were added to the park in 1930 through land purchases by the federal government and matching funds provided by industrialist John D. Rockefeller. Another 8,765 acres near Wawona were added in 1932. The Carl Inn Tract, close to the Rockefeller purchase, was secured in 1937 and 1939.