4.2 Graphical Solutions of Linear Programming
Wouldn’t it be nice if we could simply produce and sell infinitely many units of a product and thus make a never-ending amount of money? In business (and in day-to-day living) we know that some things are just unreasonable or impossible. Instead, our hope is to maximize or minimize some quantity, given a set of constraints.
In order to have a linear programming problem, we must have:
- Constraints, represented as inequalities
- An objective function, which is a function whose value we either want to be a large as possible (want to maximize it) or as small as possible (want to minimize it).
Let’s consider an extension of the example from the end of the last section:
Example of Linear Programming
A company produces a basic and premium version of its product. The basic version requires 20 minutes of assembly and 15 minutes of painting. The premium version requires 30 minutes of assembly and 30 minutes of painting. If the company has staffing for 3,900 minutes of assembly and 3,300 minutes of painting each week. They sell the basic products for a profit of $30 and the premium products for a profit of $40. How many of each version should be produced to maximize profit?
Let b = the number of basic products made, and p = the number of premium products made. Our objective function is what we’re trying to maximize or minimize. In this case, we’re trying to maximize profit. The total profit, P, is:
In the last section, the example developed our constraints. Together, these define our linear programming problem:
Objective function: MAX
Constraints: (We often say “Subject to:” or for short s.t.)
In this section, we will approach this type of problem graphically. We start by graphing the constraints to determine the feasible region – the set of possible solutions. Just showing the solution set where the four inequalities overlap, we see a clear region.
To consider how the objective function connects, suppose we considered all the possible production combinations that gave a profit of P = $3000, so that .That set of combinations would form a line in the graph. Doing the same for a profit of $5000 and $6500 would give additional lines. Graphing those on top of our feasible region, we see a pattern:
Notice that all the constant-profit lines are parallel, and that in general the profit increases as we move up to upper right. Notice also that for a profit of $5000 there are some production levels inside the feasible region for that profit level, but some are outside. That means we could feasibly make $5000 profit by producing, for example, 167 basic items and no premium items, but we can’t make $5000 by producing 125 premium items and no basic items because that falls outside our constraints.
The solution to our linear programming problem will be the largest possible profit that is still feasible. Graphically, that means the line furthest to the upper-right that still touches the feasible region on at least point. That solution is the one below:
This profit line touches the feasible region where and , giving a profit of .
Notice that this is slightly larger than the profit that would be made by completely utilizing all staffing at , , where the profit would be $5600 or at , , which would be and definitely better that (0,0) because that would be $0.
The objective function along with the four corner points above forms a bounded linear programming problem. That is, imagine you are looking at three fence posts connected by fencing (black point and lines, respectively). If you were to put your dog in the middle, you could be sure it would not escape (assuming the fence is tall enough). If this is the case, then you have a bounded linear programming problem. If the dog could walk infinitely in any one direction, then the problem is unbounded.
In the past example, you can see that the line of maximum profit will always touch the boundary of the feasible region. That observation inspires the fundamental theorem of linear programming.
Fundamental Theorem of Linear Programming
- If a solution exists to a bounded linear programming problem, then it occurs at one of the corner points.
- If a feasible region is unbounded, then a maximum value for the objective function does not exist.
- If a feasible region is unbounded, and the objective function has only positive coefficients, then a minimum value exists.
In the last example we solve the problem somewhat intuitively by “sliding” the profit line up. Typically we use a more procedural approach.
Solving a Linear Programming Problem Graphically
- Define the variables
- Define the variable to be optimized. The question asked is a good indicator as to what this will be: Profit, Revenue, Cost are popular choices.
- Write the objective function, as a mathematical equation, like P=10x+15y or C=200a+300b.
- Write the constraints as a system of mathematical inequalities
- Graph the constraint system of inequalities and shade the feasible region
- Identify the corner point by solving systems of linear equation whose intersection represents a corner point.
- Test each corner point in the objective function. The “winning” point is the point that optimizes the objective function (greatest if maximizing, least if minimizing.)
Note that if there is a tie for the winning point in that two points are the max or min, then the objective function is optimized at any point along the line connecting them.
Try it Now 1
Maximize subject to the constraints:
Another Linear Programming Example
A health-food business would like to create a high-potassium blend of dried fruit in the form of a box of 10 fruit bars. It decides to use dried apricots, which have 407 mg of potassium per serving, and dried dates, which have 271 mg of potassium per serving. The company can purchase its fruit through in bulk for a reasonable price. Dried apricots cost $9.99/lb. (about 3 servings) and dried dates cost $7.99/lb. (about 4 servings). The company would like the box of bars to have at least the recommended daily potassium intake of about 4700 mg, and contain at least 1 serving of each fruit. In order to minimize cost, how many servings of each dried fruit should go into the box of bars?
Step 1: Define the variables.
number of servings of dried apricots
number of servings of dried dates
Step 2: We next work on the objective function.
For apricots, there are 3 servings per pound. So the cost per serving is $9.99/3 =$3.33. The cost for servings would thus by 3.33x
For dates, there are 4 servings per pound. This means the cost per serving is $7.99/4=$2.00. The cost for servings would thus be 2y.
Step 4: The total cost, , for apricots and dates would be
Step 4: Normally we would have constraints and since negative servings can’t be used. But in this case, we’re further restricted. In words:
- There just be at least 1 serving of each fruit
- The product must contain at least 4700 mg of potassium
Mathematically,
- Since there must be at least 1 serving of each fruit, then
- There are mg of potassium in servings of apricots and mg of potassium in servings of dates. The sum should be greater than or equal to 4700mg of potassium, or
Thus, we have,
Objective function: MIN
Subject to:
Step 5: We graph the constraints and shade the feasible region:
Step 6: The region is unbounded, but we will be able to find a minimum still. We can see there are two corner points.
The one in the upper left is the intersection of the lines and . Solving for the intersection using substitution:
Point: (1, 15.8)
The one in the lower right is the intersection of the lines and .
Point: (10.9,1)
Step 7: Testing the objective function at each of these corner points:
| Point | Cost, C = 3.33x + 2.00y
|
| (10.9, 1) | C = 3.33(10.9) + 2.00(1) = $38.30 |
| (1, 15.8) | C = 3.33(1) + 2.00(15.8) = $34.96 |
The company can minimize cost by using 1 serving of apricots and 15.8 servings of dates.
Try it Now 2
A company makes two products. Product A requires 3 hours of manufacturing and 1 hour of assembly. Product B requires 4 hours of manufacturing and 2 hours of assembly. There are a total of 84 hours of manufacturing and 32 hours of assembly available. Determine the production to maximize profit if the profit on product A is $50 and the profit on product B is $60.
Yet another Example of Linear Programming
A factory manufactures chairs and tables, each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 40 hours; the second at most 42 hours; and the third at most 25 hours. A chair requires 1 hour of cutting, 2 hours of assembly, and 1 hour of finishing; a table needs 2 hours of cutting, 1 hour of assembly, and 1 hour of finishing. If the profit is 30 for a table, how many units of each should be manufactured to maximize profit?
We begin by defining the variables. Let
c = number of chairs made
t = number of tables made
The profit, P, will be P = 20c + 30t.
For cutting, c chairs will require 1c hours and t tables will require 2t hours. We can use at most 40 hours, so .
For assembly, c chairs will require 2c hours and t tables will require 1t hours. We can use at most 42 hours, so .
For finishing, c chairs will require 1c hours and t tables will require 1t hours. We can use at most 25 hours, so .
Since we can’t produce negative items, .
Graphing the constraints, we can see the feasible region.
There are five corner points for this region.
Point 1: In the lower left, where t = 0 crosses c = 0. Point: (0, 0)
Point 2: In the upper left, where c = 0 crosses .
Using substitution, , so t = 20.
Point: (0, 20)
Point 3: In the lower right, where t = 0 crosses .
Using substitution, , so c = 21.
Point: (21, 0)
Point 4: The solution to the system:
So multiply the second equation by -1 and add:
Substitute to find : so
Point: (10,15)
Point 5: The solution to the system:
So multiply the bottom equation by -1 and add:
Substitute to find : so
Point: (17,8)
Testing the objective function at each of these corner points:
| Point | Profit, P = 20c + 30t
|
| (0, 0) | P = 20(0) + 30(0) = $0 |
| (0, 20) | P = 20(0) + 30(20) = $600 |
| (21, 0) | P = 20(21) + 30(0) = $420 |
| (10, 15) | P = 20(10) + 30(15) = $650 |
| (17, 8) | P = 20(17) + 30(8) = $580 |
The profit will be maximized by producing 10 chairs and 15 tables.
Graphing is not the only way to do Linear Programming and Optimization. There is another method, called The Simplex Method. It is included in the source material for this chapter. Most people use technology for these problems as well and many can be found online. While this books gives only an introduction to Linear Programming, it is used in many business applications.
Try it Now Answers
- Max
subject to:
Graphing the feasible region, we see there are three corner points of potential interest: (0, 9), (5, 0), and the intersection of the two lines at (3, 6). We then evaluate the objective function at each corner point.
| Point | |
| (0, 9) | 81 |
| (5, 0) | 70 |
| (3, 6) | 96 |
is maximized when .
2. Let a be the number of product A produced, and b be the number of product B.
Our goal is to maximize profit:
From manufacturing we get the constraints:
From assembly we get the constraint:
We have corner points at (0,16), (28,0). The third is at the intersection of the two lines. To find that we could multiply the second equation by -2 and add:
The third corner point (20,6).
Evaluating the objective function at each of these points:
| Point | P
|
| (0, 16) | P = 50(0) + 60(16) = 960 |
| (28, 0) | P = 50(28) + 60(0) = 1400 |
| (20, 6) | P = 50(20) + 60(6) = 1360 |
Profit will be maximized by producing 28 units of product A and 0 units of product B.
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The Judiciary
Judiciary Review
Marbury v. Madison (1803)
5 U.S. 137 (1803)
Vote: 4-0
Decision: Dismissed
Majority: Marshall, joined by Paterson, Chase, Washington
Not participating: Cushing, Moore
Mr. Chief Justice MARSHALL delivered the opinion of the Court.
…
- Has the applicant a right to the commission he demands?
- If he has a right, and that right has been violated, do the laws of his country afford him a remedy?
- If they do afford him a remedy, is it a mandamus issuing from this court?
The first object of inquiry is:
- Has the applicant a right to the commission he demands?
…
It appears, from the affidavits, that in compliance with this law, a commission for William Marbury as a justice of peace for the county of Washington was signed by John Adams, then president of the United States after which the seal of the United States was fixed to it, but the commission has never reached the person for whom it was made out.
In order to determine whether he is entitled to this commission, it becomes necessary to enquire whether he has been appointed to the office. For if he has been appointed, the law continues him in office for five years, and he is entitled to the possession of those evidences of office, which, being completed, became his property …
The answer to this question seems an obvious one. The appointment being the sole act of the President, must be completely evidenced, when it is shown that he has done everything to be performed by him …
The last act to be done by the President is the signature of the commission. He has then acted on the advice and consent of the Senate to his own nomination. The time for deliberation has then passed … His judgment, on the advice and consent of the Senate concurring with his nomination, has been made, and the officer is appointed. This appointment is evidenced by an open, unequivocal act, and, being the last act required from the person making it, necessarily excludes the idea of its being, so far as it respects the appointment, an inchoate and incomplete transaction …
The signature is a warrant for affixing the great seal to the commission, and the great seal is only to be affixed to an instrument which is complete. It attests, by an act supposed to be of public notoriety, the verity of the Presidential signature.
It is never to be affixed till the commission is signed, because the signature, which gives force and effect to the commission, is conclusive evidence that the appointment is made …
The transmission of the commission is a practice directed by convenience, but not by law. It cannot therefore be necessary to constitute the appointment, which must precede it and which is the mere act of the President …
It is therefore decidedly the opinion of the Court that, when a commission has been signed by the President, the appointment is made, and that the commission is complete when the seal of the United States has been affixed to it by the Secretary of State …
Mr. Marbury, then, since his commission was signed by the President and sealed by the Secretary of State, was appointed, and as the law creating the office gave the officer a right to hold for five years independent of the Executive, the appointment was not revocable, but vested in the officer legal rights which are protected by the laws of his country.
To withhold the commission, therefore, is an act deemed by the Court not warranted by law, but violative of a vested legal right.
This brings us to the second inquiry, which is:
- If he has a right, and that right has been violated, do the laws of his country afford him a remedy? …
The very essence of civil liberty certainly consists in the right of every individual to claim the protection of the laws whenever he receives an injury. One of the first duties of government is to afford that protection …
The Government of the United States has been emphatically termed a government of laws, and not of men. It will certainly cease to deserve this high appellation if the laws furnish no remedy for the violation of a vested legal right …
It behooves us, then, to inquire whether there be in its composition any ingredient which shall exempt from legal investigation or exclude the injured party from legal redress … Is the act of delivering or withholding a commission to be considered as a mere political act belonging to the Executive department alone, for the performance of which entire confidence is placed by our Constitution in the Supreme Executive, and for any misconduct respecting which the injured individual has no remedy?
That there may be such cases is not to be questioned. but that every act of duty to be performed in any of the great departments of government constitutes such a case is not to be admitted …
If some acts be examinable and others not, there must be some rule of law to guide the Court in the exercise of its jurisdiction.
In some instances, there may be difficulty in applying the rule to particular cases; but there cannot, it is believed, be much difficulty in laying down the rule.
By the Constitution of the United States, the President is invested with certain important political powers, in the exercise of which he is to use his own discretion, and is accountable only to his country in his political character and to his own conscience. To aid him in the performance of these duties, he is authorized to appoint certain officers, who act by his authority and in conformity with his orders.
In such cases, their acts are his acts; and whatever opinion may be entertained of the manner in which executive discretion may be used, still there exists, and can exist, no power to control that discretion. The subjects are political … The acts of such an officer, as an officer, can never be examinable by the Courts.
But when the Legislature proceeds to impose on that officer other duties; when he is directed peremptorily to perform certain acts; when the rights of individuals are dependent on the performance of those acts; he is so far the officer of the law, is amenable to the laws for his conduct, and cannot at his discretion, sport away the vested rights of others …
… where a specific duty is assigned by law, and individual rights depend upon the performance of that duty, it seems equally clear that the individual who considers himself injured has a right to resort to the laws of his country for a remedy …
The question whether a right has vested or not, is, in its nature, judicial, and must be tried by the judicial authority …
That question has been discussed, and the opinion is that the latest point of time which can be taken as that at which the appointment was complete and evidenced was when, after the signature of the President, the seal of the United States was affixed to the commission.
It is then the opinion of the Court:
- That, by signing the commission of Mr. Marbury, the President of the United States appointed him a justice of peace … and that the seal of the United States, affixed thereto by the Secretary of State, is conclusive testimony of the verity of the signature, and of the completion of the appointment, and that the appointment conferred on him a legal right to the office for the space of five years.
- That, having this legal title to the office, he has a consequent right to the commission, a refusal to deliver which is a plain violation of that right, for which the laws of his country afford him a remedy.
It remains to be inquired whether,
- He is entitled to the remedy for which he applies …
This writ, if awarded, would be directed to an officer of government, and its mandate to him would be, to use the words of Blackstone,
“to do a particular thing therein specified, which appertains to his office and duty and which the Court has previously determined or at least supposes to be consonant to right and justice.”
Or, in the words of Lord Mansfield, the applicant, in this case, has a right to execute an office of public concern, and is kept out of possession of that right.
This, then, is a plain case for a mandamus, either to deliver the commission, or a copy of it from the record; and it only remains to be inquired … [w]hether it can [be] issue[d] from the Court …
It has been insisted … that as the original grant of jurisdiction, to the supreme and inferior courts, is general, and the clause, assigning original jurisdiction to the supreme court, contains no negative or restrictive words; the power remains to the Legislature, to assign original jurisdiction to that Court in other cases than those specified in the article which has been recited, provided those cases belong to the judicial power of the United States.
If it had been intended to leave it in the discretion of the Legislature to apportion the judicial power between the Supreme and inferior courts according to the will of that body, it would certainly have been useless to have proceeded further than to have defined the judicial power and the tribunals in which it should be vested. The subsequent part of the section is mere surplusage — is entirely without meaning — if such is to be the construction. If Congress remains at liberty to give this court appellate jurisdiction where the Constitution has declared their jurisdiction shall be original, and original jurisdiction where the Constitution has declared it shall be appellate, the distribution of jurisdiction made in the Constitution, is form without substance.
Affirmative words are often, in their operation, negative of other objects than those affirmed, and, in this case, a negative or exclusive sense must be given to them or they have no operation at all.
It cannot be presumed that any clause in the Constitution is intended to be without effect, and therefore such construction is inadmissible unless the words require it …
When an instrument organizing fundamentally a judicial system divides it into one Supreme and so many inferior courts as the Legislature may ordain and establish, then enumerates its powers, and proceeds so far to distribute them as to define the jurisdiction of the Supreme Court by declaring the cases in which it shall take original jurisdiction, and that in others it shall take appellate jurisdiction, the plain import of the words seems to be that, in one class of cases, its jurisdiction is original, and not appellate; in the other, it is appellate, and not original. If any other construction would render the clause inoperative, that is an additional reason for rejecting such other construction, and for adhering to the obvious meaning.
To enable this court then to issue a mandamus, it must be shown to be an exercise of appellate jurisdiction, or to be necessary to enable them to exercise appellate jurisdiction …
Thus, the particular phraseology of the constitution of the United States confirms and strengthens the principle, supposed to be essential to all written constitutions that a law repugnant to the constitution is void; and that courts, as well as other departments, are bound by that instrument.
The rule must be discharged.
Eakin v. Raub (1825)
12 Sergeant & Rawle 330 (Pa. 1825)
J. Gibson Dissenting
…
But it is said, that without it, the latter act would be unconstitutional; and, instead of controverting this, I will avail myself of it, to express an opinion which I have deliberately formed, on the abstract right of the judiciary to declare an unconstitutional act of the legislature void. It seems to me there is a plain difference, hitherto unnoticed, between acts that are repugnant to the constitution of the particular state, and acts that are repugnant to the constitution of the United States; my opinion being, that the judiciary is bound to execute the former, but not the latter. I shall hereafter attempt to explain this difference, by pointing out the particular provisions in the constitution of the United States on which it depends. I am aware, that a right to declare all unconstitutional acts void, without distinction as to either constitution, is generally held as a professional dogma; but I apprehend rather as a matter of faith than of reason … But I may premise, that it is not a little remarkable, that although the right in question has all along been claimed by the judiciary, no judge has ventured to discuss it, except Chief Justice Marshall, (in Marbury v. Madison) and if the argument of a jurist so distinguished for the strength of his ratiocinative powers to be found inconclusive, it may fairly be set down to the weakness of the position which he attempts to defend … The constitution is a collection of fundamental laws, not to be departed from in practice nor altered by judicial decision, and in the construction of it, nothing would be so alarming as the doctrine of communis error [common error makes law], which offers a ready justification for every usurpation that has not been resisted … Instead, therefore, of resting on the fact, that the right in question has universally been assumed by the American courts, the judge who asserts it ought to be prepared to maintain it on the principles of the constitution …
I begin, then, by observing that in this country, the powers of the judiciary are divisible into those that are POLITICAL and those that are purely CIVIL. Every power by which one organ of the government is enabled to control another, or to exert an influence over its acts, is a political power. The political powers of the judiciary are extraordinary and adventitious; such, for instance, as are derived from certain peculiar provisions in the constitution of the United States, of which hereafter: and they are derived, by direct grant, from the common fountain of all political power. On the other hand, its civil, are its ordinary and appropriate powers; being part of its essence, and existing independently of any supposed grant in the constitution. But where the government exists by virtue of a written constitution, the judiciary does not necessarily derive from that circumstance, any other than its ordinary and appropriate powers. Our judiciary is constructed on the principles of the common law, which enters so essentially into the composition of our social institutions as to be inseparable from them, and to be in fact, the basis of the whole scheme of our civil and political liberty. In adopting any organ or instrument of the common law, we take it with just such powers and capacities as were incident to it at the common law, except where these are expressly, or by necessary implication, abridged or enlarged in the act of adoption; and, that such act is a written instrument, cannot vary its consequences or construction. In the absence of special provision to the contrary, sheriffs, justices of the peace, and other officers whose offices are established in the constitution, exercise no other powers here, than what similar officers do in England; and trial by jury would have been according to the course of the common law, without any declaration to that effect in the constitution. Now, what are the powers of the judiciary at the common law? They are those that necessarily arise out of its immediate business; and they are therefore commensurate only with the judicial execution of the municipal law, or, in other words, with the administration of distributive justice, without extending to anything of a political cast whatever … With us, although the legislature be the depository of only so much of the sovereignty as the people have thought fit to impart, it is nevertheless sovereign within the limit of its powers, and may relatively claim the same pre-eminence here that it may claim elsewhere. It will be conceded, then that the ordinary and essential powers of the judiciary do not extend to the annulling of an act of the legislature …
The constitution of Pennsylvania contains no express grant of political powers to the judiciary. But, to establish a grant by implication, the constitution is said to be a law of superior obligation; and, consequently, that if it were to come into collision with an act of the legislature, the latter would have to give way. This is conceded. But it is a fallacy, to suppose that they can come into collision before the judiciary. What is a constitution? It is an act of extraordinary legislation, by which the people establish the structure and mechanism of their government; and in which they prescribe fundamental rules to regulate the motion of the several parts. What is a statute? It is an act of ordinary legislation, by the appropriate organ of the government; the provisions of which are to be executed by the executive or judiciary, or by officers subordinate to them. The constitution, then, contains no practical rules for the administration of distributive justice, with which alone the judiciary has to do; these being furnished in acts of ordinary legislation, by that organ of the government, which, in this respect, is exclusively the representative of the people; and it is generally true, that the provisions of a constitution are to be carried into effect immediately by the legislature, and only mediately, if at all, by the judiciary … In all other cases, if the act of assembly supposed to be unconstitutional, were laid out of the question, there would remain no rule to determine the point in controversy in the cause, but the statute or common law, as it existed before the act of assembly was passed; and the constitution and act of assembly therefore do not furnish conflicting rules applicable to the point before the court; nor is it at all necessary, that the one or the other of them should give way …
By the third article and second section, appellate jurisdiction of all cases arising under the constitution and laws of the United States, is reserved to the federal judiciary, under such regulations as congress may prescribe; and, in execution of this provision, congress has prescribed regulations for removing into the Supreme Court of the United States, all causes decided by the highest court of judicature of any state, which involve the construction of the constitution, or of any law or treaty of the United States. This is another guard against infraction of the limitations imposed on state sovereignty, and one which is extremely efficient in practice; for reversals of decisions in favour of the constitutionality of acts of assembly, have been frequent on writs of error to the Supreme Court of the United States.
Now, a reversal implies that it was not only the right, but the duty of the inferior court to decide otherwise; for where there is but one way of deciding, there can be no error. But what beneficial result would there be produced by the decision of a state court in favor of a state law palpably unconstitutional? The injured party would have the judgment reversed by the court in the last resort, and the cause would come back with a mandate to decide differently, which the state court dare not disobey; so that nothing would eventually be gained by the party claiming under the law of the state, but on the contrary, he would be burdened with additional costs. I grant, however, that the state judiciary ought not to exercise the power, except in cases free from all doubt, because, as a writ of error to the supreme court of the United States lies to correct an error only in favor of the constitutionality of the state law, an error in deciding against it would be irremediable. Anticipating those who think they perceive in this, exactly what I have censured in those who assume the existence of the same power in respect to laws that are repugnant to the constitution of the state, but restrict the exercise of it to clear cases, I briefly remark, that the instances are not parallel; an error in deciding against the validity of the law, being irreparable in the one, and not so in the other. Unless, then, the respective states are not bound by the engagement, which they have contracted by becoming parties to the constitution of the United States, they are precluded from denying either the right or the duty of their judges, to declare their laws void when they are repugnant to that constitution …
Martin v. Hunter’s Lessee (1816)
14 U.S. 304 (1816)
Vote: 6-0
Decision: Affirmed
Majority: Story, joined by Washington, Johnson, Livingston, Todd, and Duvall
Concurrence: Johnson
Not participating: Marshall
Story J. delivered the opinion of the Court …
This is a writ of error from the Court of Appeals of Virginia, founded upon the refusal of that court to obey the mandate of this Court, requiring the judgment rendered in this very cause, at February term, 1813, to be carried into due execution. The following is the judgment of the court of appeals rendered on the mandate:
“The court is unanimously of opinion, that the appellate power of the Supreme Court of the United States does not extend to this court, under a sound construction of the Constitution of the United States; that so much of the 25th section of the act of Congress to establish the judicial courts of the United States, as extends the appellate jurisdiction of the Supreme Court to this Court, is not in pursuance of the Constitution of the United States; that the writ of error, in this cause, was improvidently allowed under the authority of that act; that the proceedings thereon in the Supreme Court were, coram non judice [law before a court lacking authority to hear and decide the case in question], in relation to this court, and that obedience to its mandate be declined by the court.”
The questions involved in this judgment are of great importance and delicacy. Perhaps it is not too much to affirm that upon their right decision rest some of the most solid principles which have hitherto been supposed to sustain and protect the constitution itself. The great respectability, too, of the Court whose decisions we are called upon to review, and the entire deference which we entertain for the learning and ability of that Court, add much to the difficulty of the task which has so unwelcomely fallen upon us …
Before proceeding to the principal questions, it may not be unfit to dispose of some preliminary considerations which have grown out of the arguments at the bar.
The Constitution of the United States was ordained and established not by the States in their sovereign capacities, but emphatically, as the preamble of the Constitution declares, by “the people of the United States.” There can be no doubt that it was competent to the people to invest the general government with all the powers which they might deem proper and necessary, to extend or restrain these powers according to their own good pleasure, and to give them a paramount and supreme authority … The Constitution was not, therefore, necessarily carved out of existing State sovereignties, nor a surrender of powers already existing in State institutions, for the powers of the States depend upon their own Constitutions, and the people of every State had the right to modify and restrain them according to their own views of the policy or principle. On the other hand, it is perfectly clear that the sovereign powers vested in the State governments by their respective Constitutions remained unaltered and unimpaired except so far as they were granted to the Government of the United States.
These deductions do not rest upon general reasoning, plain and obvious as they seem to be. They have been positively recognised by one of the articles in [the Tenth A]mendment of the Constitution …
The government, then, of the United States can claim no powers which are not granted to it by the Constitution, and the powers actually granted, must be such as are expressly given, or given by necessary implication. On the other hand, this instrument … is to have a reasonable construction, according to the import of its terms, and where a power is expressly given in general terms, it is not to be restrained to particular cases unless that construction grow out of the context expressly or by necessary implication. The words are to be taken in their natural and obvious sense, and not in a sense unreasonably restricted or enlarged.
The Constitution unavoidably deals in general language. It did not suit the purposes of the people, in framing this great charter of our liberties, to provide for minute specifications of its powers or to declare the means by which those powers should be carried into execution. It was foreseen that this would be a perilous and difficult, if not an impracticable, task. The instrument was not intended to provide merely for the exigencies of a few years, but was to endure … It could not be foreseen what new changes and modifications of power might be indispensable to effectuate the general objects of the charter, and restrictions and specifications which at the present might seem salutary might in the end prove the overthrow of the system itself. Hence its powers are expressed in general terms, leaving to the legislature from time to time to adopt its own means to effectuate legitimate objects and to mould and model the exercise of its powers …
With these principles in view, principles in respect to which no difference of opinion ought to be indulged, let us now proceed to the interpretation of the Constitution so far as regards the great points in controversy.
The third article of the constitution is that which must principally attract our attention. The 1st. section declares, “the judicial power of the United States shall be vested in one Supreme Court, and in such other inferior Courts as the Congress may, from time to time, ordain and establish.”
The 2d section declares, that “the judicial power shall extend to all cases in law or equity, arising under this Constitution, the laws of the United States, and the treaties made, or which shall be made, ‘under their authority; to all cases affecting ambassadors, other public ministers and consuls … to controversies between two or more states; between a state and citizens of another state … ”
It then proceeds to declare, that
“in all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be a party the supreme court shall have original jurisdiction. In all the other cases before mentioned, the Supreme Court shall have appellate jurisdiction both as to law and fact, with such exceptions and under such regulations, as the Congress shall make.”
Such is the language of the article creating and defining the judicial power of the United States. It is the voice of the whole American people solemnly declared, in establishing one great department of that Government which was, in many respects, national, and in all, supreme. It is a part of the very same instrument which was to act not merely upon individuals, but upon States, and to deprive them altogether of the exercise of some powers of sovereignty and to restrain and regulate them in the exercise of others …
The object of the Constitution was to establish three great departments of Government — the legislative, the executive, and the judicial departments. The first was to pass laws, the second to approve and execute them, and the third to expound and enforce them. Without the latter, it would be impossible to carry into effect some of the express provisions of the Constitution. How, otherwise, could crimes against the United States be tried and punished? How could causes between two States be heard and determined? The judicial power must, therefore, be vested in some court by Congress; and to suppose that it was not an obligation binding on them, but might, at their pleasure, be omitted or declined, is to suppose that, under the sanction of the Constitution, they might defeat the Constitution itself, a construction which would lead to such a result cannot be sound …
This leads us to the consideration of the great question as to the nature and extent of the appellate jurisdiction of the United States. We have already seen that appellate jurisdiction is given by the Constitution to the Supreme Court in all cases where it has not original jurisdiction, subject, however, to such exceptions and regulations as Congress may prescribe … But the exercise of appellate jurisdiction is far from being limited by the terms of the Constitution to the Supreme Court. There can be no doubt that Congress may create a succession of inferior tribunals, in each of which it may vest appellate as well as original jurisdiction. The judicial power is delegated by the Constitution in the most general terms, and may therefore be exercised by Congress under every variety of form of appellate or original jurisdiction. And as there is nothing in the Constitution which restrains or limits this power, it must therefore, in all other cases, subsist in the utmost latitude of which, in its own nature, it is susceptible.
As, then, by the terms of the Constitution, the appellate jurisdiction is not limited as to the Supreme Court, and as to this Court it may be exercised in all other cases than those of which it has original cognizance, what is there to restrain its exercise over State tribunals in the enumerated cases? The appellate power is not limited by the terms of the third article to any particular Courts. The words are, “the judicial power (which includes appellate power) shall extend to all cases,” &c., and “in all other cases before mentioned, the Supreme Court shall have appellate jurisdiction.” It is the case, then, and not the court, that gives the jurisdiction. If the judicial power extends to the case, it will be in vain to search in the letter of the Constitution for any qualification as to the tribunal where it depends. It is incumbent, then, upon those who assert such a qualification to show its existence by necessary implication. If the text be clear and distinct, no restriction upon its plain and obvious import ought to be admitted …
It must therefore be conceded that the Constitution not only contemplated, but meant to provide for, cases within the scope of the judicial power of the United States which might yet depend before State tribunals. It was foreseen that, in the exercise of their ordinary jurisdiction, State courts would incidentally take cognizance of cases arising under the Constitution, the laws, and treaties of the United States. Yet to all these cases the judicial power, by the very terms of the Constitution, is to extend. It cannot extend by original jurisdiction if that was already rightfully and exclusively attached in the State courts, which (as has been already shown) may occur; it must therefore extend by appellate jurisdiction, or not at all. It would seem to follow that the appellate power of the United States must, in such cases, extend to State tribunals; and if in such cases, there is no reason why it should not equally attach upon all others within the purview of the Constitution.
It has been argued that such an appellate jurisdiction over State courts is inconsistent with the genius of our Governments, and the spirit of the Constitution. That the latter was never designed to act upon State sovereignties, but only upon the people, and that, if the power exists, it will materially impair the sovereignty of the States, and the independence of their courts. We cannot yield to the force of this reasoning; it assumes principles which we cannot admit, and draws conclusions to which we do not yield our assent.
It is a mistake that the Constitution was not designed to operate upon States in their corporate capacities. It is crowded with provisions which restrain or annul the sovereignty of the States in some of the highest branches of their prerogatives …
It is an historical fact that, at the time when the Judiciary Act was submitted to the deliberations of the first Congress, composed, as it was, not only of men of great learning and ability but of men who had acted a principal part in framing, supporting, or opposing that Constitution, the same exposition was explicitly declared and admitted by the friends and by the opponents of that system. It is an historical fact that the Supreme Court of the United States have, from time to time, sustained this appellate jurisdiction in a great variety of cases brought from the tribunals of many of the most important States in the Union, and that no State tribunal has ever breathed a judicial doubt on the subject, or declined to obey the mandate of the Supreme Court until the present occasion. This weight of contemporaneous exposition by all parties, this acquiescence of enlightened State courts, and these judicial decisions of the Supreme Court through so long a period do, as we think, place the doctrine upon a foundation of authority which cannot be shaken without delivering over the subject to perpetual and irremediable doubts.
The next question which has been argued is whether the case at bar be within the purview of the 25th section of the Judiciary Act, so that this Court may rightfully sustain the present writ of error … But it is contended, that the former judgment of this Court was rendered upon a case not within the purview of this section of the Judicial Act, and that, as it was pronounced by an incompetent jurisdiction, it was utterly void, and cannot be a sufficient foundation to sustain any subsequent proceedings. To this argument several answers may be given … The question now litigated is not upon the construction of a treaty, but upon the constitutionality of a statute of the United States, which is clearly within our jurisdiction. In the next place, in ordinary cases a second writ of error has never been supposed to draw in question the propriety of the first judgment, and it is difficult to perceive how such a proceeding could be sustained upon principle. A final judgment of this Court is supposed to be conclusive upon the rights which it decides, and no statute has provided any process by which this Court can revise its own judgments …
We have thus gone over all the principal questions in the cause, and we deliver our judgment with entire confidence that it is consistent with the Constitution and laws of the land …
It is the opinion of the whole Court that the judgment of the Court of Appeals of Virginia, rendered on the mandate in this cause, be reversed, and the judgment of the District Court, held at Winchester, be, and the same is hereby, affirmed.


6.5 The Plantation Colonies
Plantation economies arose first in the West Indies. These produced enormous quantities of products that were essentially new to Europe. Sugar, for example, competed more with honey than with other sources of cane, so there was no displacement of an indigenous European sugar industry to worry about. Likewise the production of tobacco on island colonies added something new to the European marketplace; there were no existing producers of tobacco in Europe who might, for example, block imports of the colonial product. In the 17th century the business model of plantation farming was transported to the mainland of North America and took root from the Carolinas north to Maryland.
Of the colonies associated with the plantation economy, none matters more to the history of Canada than Virginia. First colonized early in the 17th century, it grew rapidly and was aggressively expansive. What emerged was an economy based on private land ownership that resembled in some measure the aristocratic norms of England. Large landholdings conferred power and respectability on owners, all of whom depended on armies of unpaid labour in order to plant, maintain, harvest, and ship their products.
Tobacco was the main crop of the colony from the early 17th century. Initially this labour-intensive plant was managed by indentured servants: English workers and then Africans were recruited for fixed terms as indentured servants. While this indentured servitude was an option for many people — both whites and Africans — in many colonies (including Vancouver Island in the 19th century), in the West Indies and the mainland plantation colonies, it sometimes was a death sentence. Working conditions were poor, and most workers did not adjust well to the climate and the environment. Chattel slavery and African prisoners soon became synonymous, and ubiquitous in the plantation colonies. Aboriginal people were enslaved as well, though most were exported (mainly from North Carolina) to the West Indies to reduce the likelihood of them running away. It has been estimated by one American historian that the number of Aboriginal individuals enslaved between 1670 and 1715 was approximately 24,000 to 50,000.[1]
Growing tobacco required large tracts of land that favoured river transportation over roads. The plantation economy therefore snaked its way deep into the colony’s hinterland and, as the tobacco crop ate into the soil fertility, the leading landowners looked for new territories into which they could expand. Virginia was the first of the plantation colonies to turn its attention to New France as it sought to punch a hole in the Appalachian barrier to settlement in the West, but that did not happen until the mid-18th century.
Remarkably, perhaps, for a colony built on a foundation of inequality, Virginia was the first to experiment successfully with a kind of democratic representative government. The House of Burgesses was established in the 1620s as a legislative body that both advised and, in some measure, directed the governor. In practical terms it became a colony run by the largest and wealthiest planters; offers of free land were taken up by English emigrants but they had to have sufficient capital to make a go of it. These factors — the high costs of plantation farming, the implications of having to purchase and manage a workforce made up of African slaves, and the enormous profits that could be made from tobacco farming — sustained a gentry-style regime with strong common interests and anxieties.
Key Points
- Plantation colonies were typically organized around large estates rather than small holdings in order to better exploit slave labour.
- Colonies like Virginia combined manorial wealth with innovative traditions of democratic government.
- The land hunger that was hardwired into plantation economies led to 18th century pressures to cross the Appalachian Mountains into territories claimed by New France and its Aboriginal allies.
- Alan Gallay, The Indian Slave Trade: The Rise of the English Empire in the American South 1670-1717 (Princeton: Yale University Press, 2002), 298–301. ↵


The Ideal Gas Law
Lumen Learning
Learning Objectives
By the end of this section, you will be able to:
- State the ideal gas law in terms of molecules and in terms of moles.
- Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.
- Use Avogadro’s number to convert between number of molecules and number of moles.
In this section, we continue to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (Most gases, for example nitrogen, N2, and oxygen, O2, are composed of two or more atoms. We will primarily use the term “molecule” in discussing a gas because the term can also be applied to monatomic gases, such as helium.)
Gases are easily compressed. We can see evidence of this in Table 1 in Thermal Expansion of Solids and Liquids, where you will note that gases have the largest coefficients of volume expansion. The large coefficients mean that gases expand and contract very rapidly with temperature changes. In addition, you will note that most gases expand at the same rate, or have the same β. This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates.
The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in Figure 2. Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.
To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into an initially deflated tire. The tire’s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If we continue to pump air into it, the pressure increases. The pressure will further increase when the car is driven and the tires move. Most manufacturers specify optimal tire pressure for cold tires. (See Figure 3.)
At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.
Ideal Gas Law
The ideal gas law states that PV = NkT, where P is the absolute pressure of a gas, V is the volume it occupies, N is the number of atoms and molecules in the gas, and T is its absolute temperature. The constant k is called the Boltzmann constant in honor of Austrian physicist Ludwig Boltzmann (1844–1906) and has the value k = 1.38 × 10−23 J/K.
The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles’ law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle’s law (that for a fixed temperature, the product PV is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of V. The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that N is the total number of atoms and molecules, independent of the type of gas.)
Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure P is essentially equal to atmospheric pressure, and the volume V increases in direct proportion to the number of atoms and molecules N put into the tire. Once the volume of the tire is constant, the equation PV = NkT predicts that the pressure should increase in proportion to the number N of atoms and molecules.
Example 1. Calculating Pressure Changes Due to Temperature Changes: Tire Pressure
Suppose your bicycle tire is fully inflated, with an absolute pressure of 7.00 × 105 Pa (a gauge pressure of just under 90.0 lb/in2) at a temperature of 18.0ºC. What is the pressure after its temperature has risen to 35.0ºC? Assume that there are no appreciable leaks or changes in volume.
Strategy
The pressure in the tire is changing only because of changes in temperature. First we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown.
We know the initial pressure P0 = 7.00 × 105 Pa, the initial temperature T0 = 18.0ºC, and the final temperature Tf = 35.0ºC. We must find the final pressure Pf. How can we use the equation PV = NkT? At first, it may seem that not enough information is given, because the volume V and number of atoms N are not specified. What we can do is use the equation twice: P0V0 = NkT0 and PfVf = NkTf. If we divide PfVf by P0V0 we can come up with an equation that allows us to solve for Pf.
Since the volume is constant, Vf and V0 are the same and they cancel out. The same is true for Nf and N0, and k, which is a constant. Therefore,
We can then rearrange this to solve for Pf: , where the temperature must be in units of kelvins, because T0 and Tf are absolute temperatures.
Solution
Convert temperatures from Celsius to Kelvin:
T0 = (18.0 + 273)K = 291 K
Tf = (35.0 + 273)K = 308 K
Substitute the known values into the equation.
Discussion
The final temperature is about 6% greater than the original temperature, so the final pressure is about 6% greater as well. Note that absolute pressure and absolute temperature must be used in the ideal gas law.
Making Connections: Take-Home Experiment—Refrigerating a Balloon
Inflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why?
Example 2. Calculating the Number of Molecules in a Cubic Meter of Gas
How many molecules are in a typical object, such as gas in a tire or water in a drink? We can use the ideal gas law to give us an idea of how large N typically is.
Calculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP), which is defined to be 0ºC and atmospheric pressure.
Strategy
Because pressure, volume, and temperature are all specified, we can use the ideal gas law PV = NkT, to find N.
Solution
Identify the knowns:
Identify the unknown: number of molecules, N.
Rearrange the ideal gas law to solve for N:
Substitute the known values into the equation and solve for N:
Discussion
This number is undeniably large, considering that a gas is mostly empty space. N is huge, even in small volumes. For example, 1 cm3 of a gas at STP has 2.68 × 1019 molecules in it. Once again, note that N is the same for all types or mixtures of gases.
Moles and Avogadro’s Number
It is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A mole (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called Avogadro’s number (NA), in recognition of Italian scientist Amedeo Avogadro (1776–1856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, contain equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro’s number is NA = 6.02 × 1023 mol−1.
Avogadro’s Number
One mole always contains 6.02 × 1023 particles (atoms or molecules), independent of the element or substance. A mole of any substance has a mass in grams equal to its molecular mass, which can be calculated from the atomic masses given in the periodic table of elements.
NA = 6.02 × 1023 mol−1
Check Your Understanding
The active ingredient in a Tylenol pill is 325 mg of acetaminophen (C8H9NO2). Find the number of active molecules of acetaminophen in a single pill.
Solution
We first need to calculate the molar mass (the mass of one mole) of acetaminophen. To do this, we need to multiply the number of atoms of each element by the element’s atomic mass.
(8 moles of carbon)(12 grams/mole) + (9 moles hydrogen)(1 gram/mole) + (1 mole nitrogen)(14 grams/mole) + (2 moles oxygen)(16 grams/mole) = 151 g
Then we need to calculate the number of moles in 325 mg.
Then use Avogadro’s number to calculate the number of molecules.
N = (2.15 × 10−3 moles)(6.02 × 1023 molecules/mole) = 1.30 × 1021 molecules
Example 3. Calculating Moles per Cubic Meter and Liters per Mole
Calculate the following:
- The number of moles in 1.00 m3 of gas at STP
- The number of liters of gas per mole.
Strategy and Solution
- We are asked to find the number of moles per cubic meter, and we know from Example 2 that the number of molecules per cubic meter at STP is 2.68 × 1025. The number of moles can be found by dividing the number of molecules by Avogadro’s number. We let n stand for the number of moles,
- Using the value obtained for the number of moles in a cubic meter, and converting cubic meters to liters, we obtain
Discussion
This value is very close to the accepted value of 22.4 L/mol. The slight difference is due to rounding errors caused by using three-digit input. Again this number is the same for all gases. In other words, it is independent of the gas.
The (average) molar weight of air (approximately 80% N2 and 20% O2 is M = 28.8 g. Thus the mass of one cubic meter of air is 1.28 kg. If a living room has dimensions 5 m × 5 m × 3 m, the mass of air inside the room is 96 kg, which is the typical mass of a human.
Check Your Understanding
The density of air at standard conditions (P = 1 atm and T = 20ºC) is 1.28 kg/m3. At what pressure is the density 0.64 kg/m3 if the temperature and number of molecules are kept constant?
Solution
The best way to approach this question is to think about what is happening. If the density drops to half its original value and no molecules are lost, then the volume must double. If we look at the equation PV = NkT, we see that when the temperature is constant, the pressure is inversely proportional to volume. Therefore, if the volume doubles, the pressure must drop to half its original value, and Pf = 0.50 atm.
The Ideal Gas Law Restated Using Moles
A very common expression of the ideal gas law uses the number of moles, n, rather than the number of atoms and molecules, N. We start from the ideal gas law, PV = NkT, and multiply and divide the equation by Avogadro’s number NA. This gives .
Note that is the number of moles. We define the universal gas constant R=NAk, and obtain the ideal gas law in terms of moles.
Ideal Gas Law (in terms of moles)
The ideal gas law (in terms of moles) is PV = nRT.
The numerical value of R in SI units is R = NAk = (6.02 × 1023 mol−1)(1.38 × 10−23 J/K) = 8.31 J/mol · K.
In other units,
R = 1.99 cal/mol · K
R = 0.0821 L · atm/mol · K
You can use whichever value of R is most convenient for a particular problem.
Example 4. Calculating Number of Moles: Gas in a Bike Tire
How many moles of gas are in a bike tire with a volume of 2.00 × 10−3 m3(2.00 L), a pressure of 7.00 × 105 Pa (a gauge pressure of just under 90.0 lb/in2), and at a temperature of 18.0ºC?
Strategy
Identify the knowns and unknowns, and choose an equation to solve for the unknown. In this case, we solve the ideal gas law, PV = nRT, for the number of moles n.
Solution
Identify the knowns:
Rearrange the equation to solve for n and substitute known values.
Discussion
The most convenient choice for R in this case is 8.31 J/mol · K, because our known quantities are in SI units. The pressure and temperature are obtained from the initial conditions in Example 1, but we would get the same answer if we used the final values.
The ideal gas law can be considered to be another manifestation of the law of conservation of energy (see Conservation of Energy). Work done on a gas results in an increase in its energy, increasing pressure and/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas’s atoms and molecules.
The Ideal Gas Law and Energy
Let us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.
The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in PV = NkT is NkT. This term is roughly the amount of translational kinetic energy of N atoms or molecules at an absolute temperature T, as we shall see formally in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature. The left-hand side of the ideal gas law is PV, which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands—something we explore in Heat and Heat Transfer Methods—similar to what occurs in gasoline or steam engines and turbines.
Problem-Solving Strategy: The Ideal Gas Law
Step 1. Examine the situation to determine that an ideal gas is involved. Most gases are nearly ideal.
Step 2. Make a list of what quantities are given, or can be inferred from the problem as stated (identify the known quantities). Convert known values into proper SI units (K for temperature, Pa for pressure, m3 for volume, molecules for N, and moles for n).
Step 3. Identify exactly what needs to be determined in the problem (identify the unknown quantities). A written list is useful.
Step 4. Determine whether the number of molecules or the number of moles is known, in order to decide which form of the ideal gas law to use. The first form is PV = NkT and involves N, the number of atoms or molecules. The second form is PV = nRT and involves n, the number of moles.
Step 5. Solve the ideal gas law for the quantity to be determined (the unknown quantity). You may need to take a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.
Step 6. Substitute the known quantities, along with their units, into the appropriate equation, and obtain numerical solutions complete with units. Be certain to use absolute temperature and absolute pressure.
Step 7. Check the answer to see if it is reasonable: Does it make sense?
Check Your Understanding
Liquids and solids have densities about 1000 times greater than gases. Explain how this implies that the distances between atoms and molecules in gases are about 10 times greater than the size of their atoms and molecules.
Solution
Atoms and molecules are close together in solids and liquids. In gases they are separated by empty space. Thus gases have lower densities than liquids and solids. Density is mass per unit volume, and volume is related to the size of a body (such as a sphere) cubed. So if the distance between atoms and molecules increases by a factor of 10, then the volume occupied increases by a factor of 1000, and the density decreases by a factor of 1000.
Section Summary
- The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.
- The ideal gas law can be written in terms of the number of molecules of gas: PV = NkT, where P is pressure, V is volume, T is temperature, N is number of molecules, and k is the Boltzmann constant k = 1.38 × 10–23 J/K.
- A mole is the number of atoms in a 12-g sample of carbon-12.
- The number of molecules in a mole is called Avogadro’s number NA, NA = 6.02 × 1023 mol−1.
- A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.
- The ideal gas law can also be written and solved in terms of the number of moles of gas: PV = nRT, where n is number of moles and R is the universal gas constant, R = 8.31 J/mol ⋅ K.
- The ideal gas law is generally valid at temperatures well above the boiling temperature.
Conceptual Questions
Find out the human population of Earth. Is there a mole of people inhabiting Earth? If the average mass of a person is 60 kg, calculate the mass of a mole of people. How does the mass of a mole of people compare with the mass of Earth?
Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?
A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?
Problems & Exercises
- The gauge pressure in your car tires is 2.50 × 105 N/m2 at a temperature of 35.0ºC when you drive it onto a ferry boat to Alaska. What is their gauge pressure later, when their temperature has dropped to –40.0ºC?
- Convert an absolute pressure of 7.00 × 105 N/m2 to gauge pressure in lb/in2. (This value was stated to be just less than 90.0 lb/in2 in Example 4. Is it?)
- Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of 20.0ºC. (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is 60.0ºC (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks. (b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. What will the actual final pressure be, taking this into account? Is this a negligible difference?
- Large helium-filled balloons are used to lift scientific equipment to high altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of 10.0ºC and rises to an altitude where its volume is twenty times the original volume and its temperature is –50.0ºC? (b) What is the gauge pressure? (Assume atmospheric pressure is constant.)
- Confirm that the units of nRT are those of energy for each value of R: (a) 8.31 J/mol ⋅ K, (b) 1.99 cal/mol ⋅ K, and (c) 0.0821 L ⋅ atm/mol ⋅ K.
- In the text, it was shown that N/V = 2.68 × 1025 m−3 for gas at STP. (a) Show that this quantity is equivalent to N/V = 2.68 × 1019 cm−3, as stated. (b) About how many atoms are there in one μm3 (a cubic micrometer) at STP? (c) What does your answer to part (b) imply about the separation of atoms and molecules?
- Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at 37.0ºC (body temperature).
- An airplane passenger has 100 cm3 of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to 7.50 × 104 N/m2?
- (a) What is the volume (in km3) of Avogadro’s number of sand grains if each grain is a cube and has sides that are 1.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages 100 m in width and 10.0 m in depth? Neglect air spaces between grains.
- An expensive vacuum system can achieve a pressure as low as 1.00 × 10–7 N/m2 at 20ºC. How many atoms are there in a cubic centimeter at this pressure and temperature?
- The number density of gas atoms at a certain location in the space above our planet is about 1.00 × 1011 m−3, and the pressure is 2.75 × 10–10 N/m2 in this space. What is the temperature there?
- A bicycle tire has a pressure of 7.00 × 105 N/m2 at a temperature of 18.0ºC and contains 2.00 L of gas. What will its pressure be if you let out an amount of air that has a volume of 100cm3 at atmospheric pressure? Assume tire temperature and volume remain constant.
- A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of 1.40 × 107 N/m2 and a temperature of 25.0ºC. Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperature (–78.5ºC) to reduce the leak rate and pressure so that it can be safely repaired. (a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank appear to be a practical solution?
- Find the number of moles in 2.00 L of gas at 35.0ºC and under 7.41 × 107 N/m2 of pressure.
- Calculate the depth to which Avogadro’s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.
- (a) What is the gauge pressure in a 25.0ºC car tire containing 3.60 mol of gas in a 30.0 L volume? (b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and 25.0ºC? Assume the temperature returns to 25.0ºC and the volume remains constant.
- (a) In the deep space between galaxies, the density of atoms is as low as 106 atoms/m3, and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in m3) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?
Glossary
ideal gas law: the physical law that relates the pressure and volume of a gas to the number of gas molecules or number of moles of gas and the temperature of the gas
Boltzmann constant: k, a physical constant that relates energy to temperature; k = 1.38 × 10–23 J/K
Avogadro’s number: NA, the number of molecules or atoms in one mole of a substance; NA = 6.02 × 1023 particles/mole
mole: the quantity of a substance whose mass (in grams) is equal to its molecular mass
Selected Solutions to Problems & Exercises
1. 1.62 atm
3. (a) 0.136 atm; (b) 0.135 atm. The difference between this value and the value from part (a) is negligible.
5. (a) ;
(b) ;
(c)
7. 7.86 × 10−2 mol
9. (a) 6.02 × 105 km3; (b) 6.02 × 108 km
11. −73.9ºC
13. (a) 9.14 × 106 N/m2; (b) 8.23 × 106 N/m2; (c) 2.16 K; (d) No. The final temperature needed is much too low to be easily achieved for a large object.
15. 41 km
17. (a) 3.7 × 10−17 Pa; (b) 6.0 × 1017 m3; (c) 8.4 × 102 km


1 Chapter 1-Identifying Thickeners and Gelling Agents
Learning Objectives
- What is Thickening?
- What is Gelling?
- What is the difference between the two?
- The functions of all thickening and gelling agents.
Thickening: Thickening happens when water and other molecules in a substance slowly bump around and entangle. Both polysaccharides and proteins have many molecules that link together to form chains, and these chains entangle with one another and moisture molecules to create a slow-moving substance (jam/preserves). Thickening also happens when moisture is trapped in swollen starch granules (pastry cream). And lastly, air can be trapped in liquid (foams-whipped cream) or in fat (emulsions-mayonnaise) to create thickening.
(Figoni, p. 324)
Gelling is the process by which water and other molecules in a substance entangle and can not move around at all! The same polysaccharides and proteins entangle so tightly, that the product is completely set (gelled substance-jello). Even though a gel seems to act like a solid, they are mostly liquid, but the liquid is so well trapped by the entangled polysaccharide and protein chains, that the liquid acts as a solid.
(Figoni, p. 324)
Some thickening and gelling agents can do both! That is, some thickeners will produce a gelled product if used in a high enough quantity, and some gelling agents will merely thicken a product if used in a low enough quantity. For example, gelatin, cornstarch, and pectin can do both!
Pectin is often used to assist in thickening jams.
Cornstarch can set a cream pie filling that holds its shape when cut.
While each type of thickener and gelling agent has its own set of specific functions and uses, all of these do have some functions in common:
- Structure-thickening and gelling agents give a product more or less texture.
- Stability-thickening and gelling agents provide overall stability to a product.


5 Pelvis, Urinary System, Reproductive System | Radiology of the Pelvis
Learning Objectives:
By the end of this lab, students will be able to:
- Identify and describe the anatomical structures of the kidney.
- Identify the structures of the urinary system and its blood supply.
- Identify the anatomy of the female and male reproductive systems.
- Identify the structures of the pelvis using multiple imaging modalities.
Terms to Know
|
Vasculature
Kidney
Urinary Tract
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Female Reproductive System
Male Reproductive System
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Introduction
In this lab, you will explore the organs of the urinary system and the male and female reproductive systems. You will also observe the abdominal aorta and the arteries that branch off of the aorta along its path through the abdominal cavity.
Lab Activities
Activity 1: Urinary System
Watch the videos on Canvas exploring the kidneys, ureters, and bladder. Also, observe the structures of the urinary system in the Abrahams’ and Netter’s (kidney & bladder) atlases.
- Examine the kidneys. The kidneys function to filter our blood, keeping nutrients and fluids in the blood while removing waste products from the blood. Observe the following features:
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- Renal cortex
- Renal pyramids
- Renal medulla (Darker area made up of the renal pyramids)
- Major calyx
- Minor calyx
- Renal pelvis
- Renal column
-
- Examine the hilum of the kidney. This is where the renal arteries enter the kidneys, and the renal veins and ureters exit the kidneys. Also, observe the renal arteries branching from the abdominal aorta and the renal veins draining into the inferior vena cava.
- Observe the ureters leaving the kidney and traveling inferiorly. The ureters enter the inferior posterior aspect of the bladder at an angle. As the bladder fills with urine, the pressure on the ureters within the wall of the bladder increases and prevents the backflow of urine into the ureters.
- Note the size of the ureters and imagine passing a kidney stone through these structures. The ureters are highly innervated, and passing kidney stones are excruciating. If the stone is too large, it cannot pass through these narrow structures and remains in the kidney.
- In atlas images, observe the urethra. The male urethra has three parts: the prostatic urethra, membranous urethra, and spongy urethra. The female urethra is much shorter and is positioned anterior to the vagina.
Activity 2: female and male reproductive systems
Watch the videos on Canvas exploring the female reproductive system and identify these structures in the Abrahams’ and Netter’s atlases.
- Observe the muscular uterus. It has been cut in the coronal plane (roughly) and opened in the tissue video. The uterus houses the embryo/fetus as it develops during the 40 weeks of pregnancy. You can also see the cervix in cross-section and follow it inferiorly to observe the opening to the vagina.
- Observe the uterine (fallopian) tubes and ovaries. Follow the uterine tubes and observe the fimbriae near the ovary. These finger-like projections function to gently sweep the oocyte from the ovary into the uterine tube, as these two structures are not connected.
- In the atlas images, observe the external genitalia of the female reproductive system, including the labia majora, labia minora, and clitoris.
- In sagittal cross-sectional images, you can observe the vesicouterine pouch anterior to the uterus, between the uterus and the bladder, and the rectouterine pouch posterior to the uterus, between the uterus and the rectum. Fluids and infections within the abdominal cavity can accumulate in these pouches, especially in the rectouterine pouch. The rectouterine pouch is also the preferred place for peritoneal dialysis, a form of dialysis used in end-stage renal (kidney) failure patients.
- Using the 3D image below, can you identify the ovaries, uterine tubes, uterus, vagina, labia majora, labia minora, and clitoris? (not all structures have annotations, and there are more annotations than you need to know)
Watch the videos on Canvas exploring the male reproductive system and identify these structures in the Abrahams’ and Netter’s atlases.
- Observe the gland just inferior to the bladder. This is the prostate gland. Observe the tube-like structures coming from the area of the prostate gland. These are the right and left ductus (vas) deferens. The ductus deferens carries sperm from the epididymis in the testes to the ejaculatory duct.
- Follow the ductus deferens towards the bladder and notice that it is increasing in size as it reaches its end. This is called the ampulla. The ampulla then meets the seminal vesicle, which you can observe next to the ampulla, to form the short ejaculatory duct. Then the contents of the ejaculatory duct enter the prostatic urethra. The prostatic urethra is surrounded by the prostate gland.
- Observe the spermatic cord. The spermatic cord contains the ductus (vas) deferens, as well as the testicular artery and pampiniform plexus of veins.
- Observe the testes. Notice the epididymis as well as the origin of the ductus deferens.
- Observe the anatomy of the penis, including the glans, external urethral orifice, corpora cavernosa, and corpus spongiosum.
- Using the 3D image below, can you identify the spermatic cord (housing the testicular artery, pampiniform plexus, and ductus (vas) defferens), testes, epididymis, ductus deferens, ampulla, seminal vesicles, prostate gland, penis, glans, corpora cavernosa, and corpus spongiosum?
Activity 3: Aorta and its branches in the abdominopelvic cavity
Watch the videos on Canvas exploring the aorta and its branches. Also, observe these structures in this image from the Netter’s atlas.
- Observe the aorta. In the tissue videos, the aorta has been opened.
- Ascending aorta: the portion of the aorta that carries blood superiorly as it leaves the heart. This is difficult to see on these aortas, but it can be seen in the hearts.
- Arch of the aorta: the portion of the aorta the forms an arch between the ascending and descending aorta. Though the aorta has been cut open, you can generally still appreciate the arch of the aorta on this tissue.
- Descending aorta: the portion of the aorta carrying blood inferiorly. The portion of the descending aorta that is located in the thorax is called the thoracic aorta, while the portion located in the abdomen is called the abdominal aorta.
- In the superior abdominal region of the descending aorta, you will observe two midline arteries branching, one just superior to the other. These are the celiac trunk (superior) and the superior mesenteric artery (inferior). These are usually located just superior to where the renal arteries (going to the kidneys) branch. Three branches of the celiac trunk supply the digestive tract from the inferior esophagus through the first half of the duodenum of the small intestines, as well as the spleen and liver: the common hepatic artery, splenic artery, and left gastric artery. You observed these branches in the previous lab. The superior mesenteric artery supplies the second half of the duodenum of the small intestines to the first 2/3 of the transverse colon. The superior mesenteric artery angles slightly downward, while the celiac trunk branches are roughly perpendicular to the aorta.
- Move inferiorly and observe the inferior mesenteric artery. This artery branches at an angle downward (inferiorly). The inferior mesenteric artery supplies the digestive tract from the distal 1/3 of the transverse colon to the rectum.
- At the most inferior aspect of the aorta, observe how it branches into the right and left common iliac arteries. The common iliac arteries then branch into the internal iliac artery, which supplies pelvic and gluteal structures and the external iliac artery, which supplies the lower extremity.
- Also, observe the inferior vena cava ascending next to the aorta. Its walls are much thinner than the aorta. This vessel carries blood from the legs and abdomen back to the heart.
- Using the 3D image below, can you identify the descending aorta and associated branches. The 3D image is unique because it is a case of an aneurism in the aorta just proximal to the split into the common iliac arteries. You will also notice the inferior mesenteric artery is displaced due to the aneurism.
Activity 4: Cross-sectional Anatomy of the urinary and reproductive systems
Watch the video on Canvas that reviews the cross-sectional anatomy of the urinary and reproductive systems. Like the organs of the digestive system in the abdominal cavity, we often view these organs in cross-section via CT or MRI images. If you can understand the cross-sectional anatomy in the color images in the video, it will be easier to understand what you see in the radiology images.
Activity 5: Pelvic Radiology (uRINARY AND REPRODUCTIVE SYSTEMS)
View the radiology video in Canvas presenting on the radiology of the pelvic cavity, specifically the urinary and reproductive organs. You will also see the anatomy of the kidneys through radiology images in the abdominal cavity. The slides will also be posted as a study tool. Observe the following structures in the radiology images.
- Kidneys
- Bladder
- Ureters
- Uterus
- Fallopian Tubes
- Renal Arteries
- Descending Aorta
- Common Iliac Artery
WRAP UP AND Clinical Connection
For today’s clinical connection, we will discuss the clinical evaluation of the abdomen – utilizing palpation, auscultation, and percussion. Auscultation is performed by listening to the abdominal cavity with a stethoscope for altered bowel sounds (auscultation means listening to the sounds made by various body structures). Percussion is performed by tapping on the surface of the abdomen and listening to hear a dull or resonant sound. Palpation is the physical touching of the abdomen to evaluate for areas of rigidity, softness, and pain/rebound tenderness. Watch this video to learn more about how a clinician performs a physical exam of the abdomen.