13.2: Content Analysis
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Content analysis is the systematic analysis of the content of a text—e.g., who says what, to whom, why, and to what extent and with what effect—in a quantitative or qualitative manner. Content analysis is typically conducted as follows. First, when there are many texts to analyse—e.g., newspaper stories, financial reports, blog postings, online reviews, etc.—the researcher begins by sampling a selected set of texts from the population of texts for analysis. This process is not random, but instead, texts that have more pertinent content should be chosen selectively. Second, the researcher identifies and applies rules to divide each text into segments or ‘chunks’ that can be treated as separate units of analysis. This process is called unitising . For example, assumptions, effects, enablers, and barriers in texts may constitute such units. Third, the researcher constructs and applies one or more concepts to each unitised text segment in a process called coding . For coding purposes, a coding scheme is used based on the themes the researcher is searching for or uncovers as they classify the text. Finally, the coded data is analysed, often both quantitatively and qualitatively, to determine which themes occur most frequently, in what contexts, and how they are related to each other.
A simple type of content analysis is sentiment analysis —a technique used to capture people’s opinion or attitude toward an object, person, or phenomenon. Reading online messages about a political candidate posted on an online forum and classifying each message as positive, negative, or neutral is an example of such an analysis. In this case, each message represents one unit of analysis. This analysis will help identify whether the sample as a whole is positively or negatively disposed, or neutral towards that candidate. Examining the content of online reviews in a similar manner is another example. Though this analysis can be done manually, for very large datasets—e.g., millions of text records—natural language processing and text analytics based software programs are available to automate the coding process, and maintain a record of how people’s sentiments fluctuate with time.
A frequent criticism of content analysis is that it lacks a set of systematic procedures that would allow the analysis to be replicated by other researchers. Schilling (2006) addressed this criticism by organising different content analytic procedures into a spiral model. This model consists of five levels or phases in interpreting text: convert recorded tapes into raw text data or transcripts for content analysis, convert raw data into condensed protocols, convert condensed protocols into a preliminary category system, use the preliminary category system to generate coded protocols, and analyse coded protocols to generate interpretations about the phenomenon of interest.
Content analysis has several limitations. First, the coding process is restricted to the information available in text form. For instance, if a researcher is interested in studying people’s views on capital punishment, but no such archive of text documents is available, then the analysis cannot be done. Second, sampling must be done carefully to avoid sampling bias. For instance, if your population is the published research literature on a given topic, then you have systematically omitted unpublished research or the most recent work that is yet to be published.


6.8: Working With a Thesis
After doing some researching and brainstorming, you have finally chosen one main idea (or closely related group of ideas) out of all of your possible ideas to focus on. Congratulations! Now, you need to decide what you want to say about it. If applicable, you may want to develop a working thesis to guide your drafting process.
A thesis is the controlling idea of a text (often an argument). Depending on the type of text you are creating, your writing may develop, explore multiple angles, and/or support your thesis.
But how do you know, before getting any of the paper written, exactly what your thesis is and how your sources will support it? Often, you can’t. The closest you can get in these cases is a working thesis , which is a best guess at what the thesis is likely to be, based on the information you are working with at this time. The main idea of it may not change, but the specifics are probably going to be tweaked a bit as you complete a draft and continue to do research.
Student Tip
Is a Thesis Always Needed?
Not all assignments will require you to develop a thesis, but some will, especially those involving persuasive, critical, and opinion types of writing. A thesis may not be appropriate for reflective writing, annotated bibliographies, and some analytical writing – you will still have a main idea for these types of writing. In general, if you need to argue something or take a stand or position, you will need a thesis statement. When in doubt, discuss it with your instructor.
Attribution statement
Other than the figure and student tip, content for this page was adapted (with editorial changes) from:
The Word on College Reading and Writing by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted. Download for free at: https://openoregon.pressbooks.pub/wrd/


2.1: Understanding Decimals
Numbers written with decimals is another way of expressing fractions. However, decimals have a distinct differentiation with fractions in that decimals are based on 10. In that, one decimal place to the right of the whole number indicates tenths , two decimal places to the right of the whole number indicates hundredths , three decimal places to the right of the whole number indicates thousandths , etc.
Example \(\PageIndex{1}\)
0.1 = tenths place
1.01 = hundredths place
0.001 = thousandths place
1.01 = ten thousandths place
Decimals can easily be written as fractions by using the number to the right of the decimal point as the numerator and then using a 10, 100, 1,000, 10,000, etc. as the denominator. Determining which “base-ten” number to use as the denominator is determined by the number of digits there are to the right of the decimal point.
Example \(\PageIndex{2}\)
\(\begin{array}{ll}
0.1 \rightarrow \dfrac{1}{10} & 1 \text { tenth } \\
0.01 \rightarrow \dfrac{1}{100} & 1 \text { hundredth } \\
0.001 \rightarrow \dfrac{1}{1,000} & 1 \text { thousandth } \\
0.0001 \rightarrow \dfrac{1}{10,000} & 1 \text { ten-thousandth }
\end{array}\)
Exercise 2.1
Write the following decimals as fractions and reduce if necessary.
- 0.2 =
- 0.103 =
- 0.13 =
- 0.02 =
- 0.1234 =
- 0.0023 =
- 0.0101 =
- 0.1010 =
- 0.020 =
- 0.0202 =
- 0.1000 =
- 0.4500 =
Exercise 2.1.1
Write the following expressions as decimals and fractions.
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Decimal |
Fraction |
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7 tenths |
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1,000 ten-thousandths |
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475 thousandths |
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32 hundredths |
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12 thousandths |
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2,345 ten-thousandths |
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3 hundredths |
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10 thousandths |
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132 ten-thousandths |
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4,002 ten-thousandths |
Digits to the left of the decimal indicate a whole number. Whenever there is a whole number with a decimal fraction the expression is pronounced using “and.”
Example:
12.3 = Twelve and three tenths = \(12 \dfrac{3}{10}\)
100.07 = One hundred and seven hundredths = \(100 \dfrac{7}{100}\)
3,005.023 = Three thousand five and twenty-three thousandths = \(3,005 \dfrac{23}{1,000}\)
Exercise 2.1.2
Write the following expressions as a decimal and a fraction. Remember to reduce if necessary.
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Decimal |
Fraction |
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Two and five hundredths |
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Thirteen and two hundred thousandths |
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One thousand three and two ten-thousandths |
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Fifty-five and fifty hundredths |
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One and fourteen hundredth |
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Eleven thousand four and twelve thousandths |
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Ten and two tenths |
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Four hundred one and four ten-thousandths |
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Five thousand two hundred eight and one thousand ten-thousandths |


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determinants
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factors that influence health; can include events, characteristics, or other entities that change a health condition or the level of a health problem
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disability-adjusted life years (DALYs)
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estimates the number of healthy years lost across the population due to premature death and disability from a select condition
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disparities
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inequalities or differences
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downstream
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describes interventions that focus on the behavior of individuals to modify the risk of disease, prevent illness, or manage chronic conditions
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general mortality
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the number of deaths across a large population, such as a geographic region or country, reports of which include information about leading causes of death across the population and estimates of years of potential life lost when people die prematurely
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global nursing
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the use of evidence-based nursing processes to promote sustainable planetary health and equity for all people
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health care continuum
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the spectrum of health care services delivered in public health, acute care, ambulatory care, and long-term care settings
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health care utilization
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the extent to which a population accesses and uses different health care services
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health inequities
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avoidable inequalities related to health that stem from a form of injustice
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infant mortality
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the rate of infants who die before their first birthday
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life expectancy
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the average number of years a newborn is expected to live if current death rates remain constant
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midstream
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describes interventions that occur within a specific organization
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morbidity rate
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the prevalence or incidence of a specific health condition, disease, or diagnosis within the population
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outcomes
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in a population health context, all possible results that may come from exposure to interventions aiming to prevent or treat a condition
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population
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a group of people that may receive care in various settings at the local, regional, national, and global levels
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population health
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an approach to supporting the health of people through research, data analysis, and health programming that considers the impact of public policy and environmental, social, behavioral, and other factors that might facilitate or hinder health for all
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quality of life
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indication of enjoyment and satisfaction people have, given their multidimensional experiences in health, relationships, activities, and life circumstances
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risk factors
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aspects of personal behaviors, lifestyle choices, exposures, or attributes that generally increase the likelihood or severity of acquiring a health condition
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social determinants of health
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conditions and social factors that affect outcomes and risk related to health, functioning, and quality of life
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transitional care
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efforts aimed at coordinating health care as clients receive services in various areas of the health care continuum
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upstream
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describes interventions that involve enacting policies that change regulations, increase access, or provide economic incentives to impact health across a population


12: The Sixties
Demonstrators march from Selma to Montgomery, Alabama, in 1965 to champion African American civil rights. Library of Congress .
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- 12.3: Kennedy and Cuba
- The decade’s political landscape began with a watershed presidential election. Americans were captivated by the 1960 race between Republican vice president Richard Nixon and Democratic senator John F. Kennedy, two candidates who pledged to move the nation forward and invigorate an economy experiencing the worst recession since the Great Depression. Kennedy promised to use federal programs to strengthen the economy and address pockets of longstanding poverty.
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- 12.5: Lyndon Johnson's Great Society
- The Great Society’s legislation was breathtaking in scope, and many of its programs and agencies are still with us today. Most importantly, the Civil Rights Act of 1964 and the Voting Rights Act of 1965 codified federal support for many of the civil rights movement’s goals by prohibiting job discrimination, abolishing the segregation of public accommodations, and providing vigorous federal oversight of southern states’ election laws in order to guarantee minority access to the ballot.


19.1: Glaciation
A glacier is a natural accumulation of land ice showing movement at some time. Many times during Earth's history, great ice sheets waxed and waned over the surface. What caused these periods of glaciation is still not fully understood and no single reason will probably be found.
Causes of glaciation
The onset of a period or stage of glaciation is due to a change in Earth temperature and circulation. It is generally accepted that a global decrease of 4 o to 5 o C, especially during the summer, and a substantial increase in the amount of snowfall in subarctic and arctic regions is necessary for the onset of a glacial episode. Several theories have been proposed for such a change in climate -- reductions in solar radiation due to meteorite collisions with the Earth, increased volcanism, the shifting location of continents, and the uplift of vast mountain regions. Milutin Milankovitch proposed one of the most significant theories to account for climate change by variations in Earth orbit. Changes in the eccentricity of earth orbit, the degree of deviation of the orbit from a perfect circular path, is thought to cause the necessary change in insolation to decrease global temperatures. Recall that the Earth's orbit is elliptical, but over periods of 100,000 years the shape varies. The changes in orbit have been correlated with ocean sediments that record the history of glacial stages. The cyclical nature of warming and cooling correspond well with the estimated dates of glacial and interglacial periods. In addition to the change in orbit, the Earth "wobbles" on its axis which alters the amount of insolation reaching the surface of the Earth. [For more about the causes and stages of glaciation in earth history.
Video : Milankovitch cycles precession and obliquity: How changes in Earth's rotation can effect Earth's seasons and climate
Video : Is An Ice Age Coming? (Courtesy National Geographic)
Anatomy of a Glacier
Whatever the cause, the main reason glacial advances are initiated is that winter accumulation exceeds the summer loss of snow over a long period of time. Snow metamorphoses into glacial ice under the increasing pressure of accumulated layers of snow. It first changes to a granular form called firn , and ultimately into ice. Glacial ice sometimes looks blue because it absorbs all colors of the visible light spectrum except blue, which it transmits and hence its blue appearance. Glacier ice may also appear white because some ice is fractured with pockets of air that indiscriminately scatters the visible light spectrum.
The mass balance of a glacier determines if it will advance across the surface or not. The mass balance is determined by the amount of gain and loss of ice from the glacier. The mass balance is positive when it accumulates more ice than it loses. A glacier has a negative mass balance if it loses more ice than it gains.
Glaciers form in the zone of accumulation , the portion of the glacier over which accumulation exceeds ablation. Ablation is the loss of ice (or snow) from the glacier. Ablation includes sublimation, wind erosion, melting, and evaporation. The zone of accumulation for the large continental ice sheets resides at high latitudes. For mountain glaciers, the zone of accumulation is at a high altitude where temperatures are cold prevent complete summertime melt. The zone of ablation is where loss of ice mass is greater than accumulation. The boundary between these two zones is the firn or equilibrium line . If accumulation exceeds ablation the glacier will grow. If ablation exceeds accumulation, the glacier will retreat by melting in place. You can approximate the location of the equilibrium line by examining an aerial photograph. The glacier looks dirtier below the equilibrium line as glacial sediment is exposed on top of the ice. Above the line it is more white because fresh snow usually covers the surface. Listen to a glacier refreezing (Antarctica 2000).
Video : "Glacier Power" Go inside a glacier to to help understand power of a glacier (Courtesy of National Geographic)
Glacier Movement
Once the ice reaches a thickness of about 20 meters (66 ft) it will begin to move under the pressure of its weight. Glaciers move across the surface by internal deformation and basal slip . Under the weight of accumulating ice,the ice is deformed and begins to move by pseudo-plastic flow. Glaciers slip over the surface lubricated by meltwater at their base. Generally speaking, flow velocity in a glacier is greatest near the surface of the ice and decreases towards the bottom. The surface moves faster than the base does due to internal deformation and basal slipping. The actual forward movement of a particle of ice in the glacier is about 1,000 feet per year. A typical glacier will move at about 10 inches a day, though some move more rapidly like Greenland's Jakobshavn glacier. [ View " Fastest Glacier " from Nova scienceNow ] Variations in the speed of the ice caused by surface irregularities results in differential expansion and compression of the ice and the development of crevasses . A deadly situation for hikers, crevasses can open and close with little warning.
A glacial surge occurs as an abrupt movement that can cover tens of meters per day. The exact cause is not well-known, but may result from water pressure building at the base which "floats" the glacier. In 1986 the Hubbard Glacier surged across the mouth of the Russell fjord in Alaska cutting it off from Yakut Bay. Glacier movement exceeded 112 feet per day, compared to a normal rate of 10 inches per day! The Black Rapids Glacier, AK in \(\PageIndex{4}\) is a surging glacier. Tributary glaciers enter the main valley from the left. Massive amounts of glacial sediment have been deposited along the sides of the glacier and up against the valley walls. The tell-tale sign of a surging glacier are the looped moraines that snake their way across the ice.
Types of glaciers
Two major categories of glaciers are alpine and continental glaciers. Alpine glaciers are those that form at high altitudes in mountainous regions where the temperature is favorable for their formation. There are four types of alpine glaciers, cirque, valley, piedmont, and ice field shown. They range from glaciers a few hundred square meters to vast expanses that smother whole mountain ranges. Cirque glaciers form in bowl-shaped depressions, called cirques , scoured into the side of a mountain. Weathering and erosion Â loosen bedrock at the head of of a preexisting stream valley as ice accumulates. As erosion proceeds, a depression forms in which the cirque glacier continues to grow. Valley glaciers fill former stream valleys with ice as they push outward from their zone of accumulation. Cirque glaciers that grow beyond the limits of the depression in which they form spill out to become valley glaciers. Piedmont glaciers form when valley glaciers flow out of canyons onto the lowland adjacent to a mountain front. Ice fields are confined to large basins or plateaus having tall mountain peaks called nunataks protruding above the ice. Ice fields feed alpine valley glaciers.
Continental glaciers are vast, continuous masses of ice that originate in high latitudes and cover portions of a continent or island Figure \(\PageIndex{7}\)). Continental glaciers flow outwards in all directions from thicker ice near the center of the ice mass towards the thinner periphery of the ice mass. Continental glaciers are considered "unconfined glaciers" because they flow over a landscape and generally are not affected by it.
There are three types of continental glaciers, ice sheets, ice caps, and outlet. An ice sheet covers an area larger than 50,000 square kilometers (19,305 sq. mil.). Greenland and Antarctica and are covered by massive ice sheets today. The Greenland ice sheet occupies 1,710,000 square kilometers (660,000 sq. mi), nearly 80% of Greenland. The ice sheet is over 3 km (1.9 mi) at its thickest point. The Antarctic ice sheet covers nearly 98% of the continent. With almost 14 million square kilometers (5.4 million sq. miles) of ice, the Antarctic ice sheet is the largest single mass of ice on Earth. The ice sheet has a maximum thickness of 4.78 kilometers (2.97 miles).
An ice cap is a dome-shaped mass of ice of less than 50,000 square kilometers (approximately 19,000 square miles) and usually covering a highland area. Though covering a mountainous region, an ice cap is categorized as a type of continental glacier because it is not constrained by the topography. Think of an ice cap as a precursor to an ice sheet. The Vatnajokull ice cap in Iceland, the large mass of ice in Figure B, sits on top of three active volcanoes. The bright green lakes and coastal waters are the result of very fine sediment eroded by glaciers and washed out in glacial meltwater. Outlet glaciers are similar to valley glaciers as they are confined by the topography but originate from an ice cap.
Video : Antarctica 2009 (Courtesy NASA). The Nov. 4, 2009 Operation Ice Bridge flight criss-crossed the Antarctic Peninsula to survey glaciers flowing down mountain valleys to feed the floating ice shelves.


2.5.1: Hamlet- Act 1, Scenes 1 and 2
Hamlet
By William Shakespeare
Edited by Barbara A. Mowat and Paul Werstine
with Michael Poston and Rebecca Niles
Folger Shakespeare Library
https://shakespeare.folger.edu/shakespeares-works/hamlet/
Created on Apr 23, 2016, from FDT version 0.9.2.
Characters in the Play
THE GHOST
HAMLET, Prince of Denmark, son of the late King Hamlet and Queen Gertrude
QUEEN GERTRUDE, widow of King Hamlet, now married to Claudius
KING CLAUDIUS, brother to the late King Hamlet
OPHELIA
LAERTES, her brother
POLONIUS, father of Ophelia and Laertes, councillor to King Claudius
REYNALDO, servant to Polonius
HORATIO, Hamlet’s friend and confidant
Courtiers at the Danish court:
VOLTEMAND
CORNELIUS
ROSENCRANTZ
GUILDENSTERN
OSRIC
Gentlemen
A Lord
Danish soldiers:
FRANCISCO
BARNARDO
MARCELLUS
FORTINBRAS, Prince of Norway
A Captain in Fortinbras’s army
Ambassadors to Denmark from England
Players who take the roles of Prologue, Player King, Player Queen, and Lucianus in The Murder of Gonzago
Two Messengers
Sailors
Gravedigger
Gravedigger’s companion
Doctor of Divinity
Attendants, Lords, Guards, Musicians, Laertes’s Followers, Soldiers, Officers
ACT 1
Scene 1
Enter Barnardo and Francisco, two sentinels.
BARNARDO Who’s there?
FRANCISCO
Nay, answer me. Stand and unfold yourself.
BARNARDO Long live the King!
FRANCISCO Barnardo?
BARNARDO He. 5
FRANCISCO
You come most carefully upon your hour.
BARNARDO
’Tis now struck twelve. Get thee to bed, Francisco.
FRANCISCO
For this relief much thanks. ’Tis bitter cold,
And I am sick at heart.
BARNARDO Have you had quiet guard? 10
FRANCISCO Not a mouse stirring.
BARNARDO Well, good night.
If you do meet Horatio and Marcellus,
The rivals of my watch, bid them make haste.
Enter Horatio and Marcellus.
FRANCISCO
I think I hear them.—Stand ho! Who is there? 15
HORATIO Friends to this ground.
MARCELLUS And liegemen to the Dane.
FRANCISCO Give you good night.
MARCELLUS
O farewell, honest soldier. Who hath relieved
you? 20
FRANCISCO
Barnardo hath my place. Give you good night.
Francisco exits.
MARCELLUS Holla, Barnardo.
BARNARDO Say, what, is Horatio there?
HORATIO A piece of him.
BARNARDO
Welcome, Horatio.—Welcome, good Marcellus. 25
HORATIO
What, has this thing appeared again tonight?
BARNARDO I have seen nothing.
MARCELLUS
Horatio says ’tis but our fantasy
And will not let belief take hold of him
Touching this dreaded sight twice seen of us. 30
Therefore I have entreated him along
With us to watch the minutes of this night,
That, if again this apparition come,
He may approve our eyes and speak to it.
HORATIO
Tush, tush, ’twill not appear. 35
BARNARDO Sit down awhile,
And let us once again assail your ears,
That are so fortified against our story,
What we have two nights seen.
HORATIO Well, sit we down, 40
And let us hear Barnardo speak of this.
BARNARDO Last night of all,
When yond same star that’s westward from the pole
Had made his course t’ illume that part of heaven
Where now it burns, Marcellus and myself, 45
The bell then beating one—
Enter Ghost.
MARCELLUS
Peace, break thee off! Look where it comes again.
BARNARDO
In the same figure like the King that’s dead.
MARCELLUS, to Horatio
Thou art a scholar. Speak to it, Horatio.
BARNARDO
Looks he not like the King? Mark it, Horatio. 50
HORATIO
Most like. It harrows me with fear and wonder.
BARNARDO
It would be spoke to.
MARCELLUS Speak to it, Horatio.
HORATIO
What art thou that usurp’st this time of night,
Together with that fair and warlike form 55
In which the majesty of buried Denmark
Did sometimes march? By heaven, I charge thee,
speak.
MARCELLUS
It is offended.
BARNARDO See, it stalks away. 60
HORATIO
Stay! speak! speak! I charge thee, speak!
Ghost exits.
MARCELLUS ’Tis gone and will not answer.
BARNARDO
How now, Horatio, you tremble and look pale.
Is not this something more than fantasy?
What think you on ’t? 65
HORATIO
Before my God, I might not this believe
Without the sensible and true avouch
Of mine own eyes.
MARCELLUS Is it not like the King?
HORATIO As thou art to thyself. 70
Such was the very armor he had on
When he the ambitious Norway combated.
So frowned he once when, in an angry parle,
He smote the sledded Polacks on the ice.

HORATIO
In what particular thought to work I know not,
But in the gross and scope of mine opinion
This bodes some strange eruption to our state. 80
MARCELLUS
Good now, sit down, and tell me, he that knows,
Why this same strict and most observant watch
So nightly toils the subject of the land,
And why such daily cast of brazen cannon
And foreign mart for implements of war, 85
Why such impress of shipwrights, whose sore task
Does not divide the Sunday from the week.
What might be toward that this sweaty haste
Doth make the night joint laborer with the day?
Who is ’t that can inform me? 90
HORATIO That can I.
At least the whisper goes so: our last king,
Whose image even but now appeared to us,
Was, as you know, by Fortinbras of Norway,
Thereto pricked on by a most emulate pride, 95
Dared to the combat; in which our valiant Hamlet
(For so this side of our known world esteemed him)
Did slay this Fortinbras, who by a sealed compact,
Well ratified by law and heraldry,
Did forfeit, with his life, all those his lands 100
Which he stood seized of, to the conqueror.
Against the which a moiety competent
Was gagèd by our king, which had returned
To the inheritance of Fortinbras
Had he been vanquisher, as, by the same comart 105
And carriage of the article designed,
His fell to Hamlet. Now, sir, young Fortinbras,
Of unimprovèd mettle hot and full,
Hath in the skirts of Norway here and there
Sharked up a list of lawless resolutes 110
For food and diet to some enterprise
That hath a stomach in ’t; which is no other
(As it doth well appear unto our state)
But to recover of us, by strong hand
And terms compulsatory, those foresaid lands 115
So by his father lost. And this, I take it,
Is the main motive of our preparations,
The source of this our watch, and the chief head
Of this posthaste and rummage in the land.
BARNARDO
I think it be no other but e’en so. 120
Well may it sort that this portentous figure
Comes armèd through our watch so like the king
That was and is the question of these wars.
HORATIO
A mote it is to trouble the mind’s eye.
In the most high and palmy state of Rome, 125
A little ere the mightiest Julius fell,
The graves stood tenantless, and the sheeted dead
Did squeak and gibber in the Roman streets;
As stars with trains of fire and dews of blood,
Disasters in the sun; and the moist star, 130
Upon whose influence Neptune’s empire stands,
Was sick almost to doomsday with eclipse.
And even the like precurse of feared events,
As harbingers preceding still the fates
And prologue to the omen coming on, 135
Have heaven and Earth together demonstrated
Unto our climatures and countrymen.
Enter Ghost.
But soft, behold! Lo, where it comes again!
I’ll cross it though it blast me.—Stay, illusion!
It spreads his arms.
If thou hast any sound or use of voice, 140
Speak to me.
If there be any good thing to be done
That may to thee do ease and grace to me,
Speak to me.
If thou art privy to thy country’s fate, 145
Which happily foreknowing may avoid,
O, speak!
Or if thou hast uphoarded in thy life
Extorted treasure in the womb of earth,
For which, they say, you spirits oft walk in death, 150
Speak of it.
Stay and speak!—Stop it, Marcellus.
MARCELLUS
Shall I strike it with my partisan?
HORATIO Do, if it will not stand.
BARNARDO ’Tis here. 155
HORATIO ’Tis here.
Ghost exits.
MARCELLUS ’Tis gone.
We do it wrong, being so majestical,
To offer it the show of violence,
For it is as the air, invulnerable, 160
And our vain blows malicious mockery.

HORATIO
And then it started like a guilty thing
Upon a fearful summons. I have heard
The cock, that is the trumpet to the morn, 165
Doth with his lofty and shrill-sounding throat
Awake the god of day, and at his warning,
Whether in sea or fire, in earth or air,
Th’ extravagant and erring spirit hies
To his confine, and of the truth herein 170
This present object made probation.
MARCELLUS
It faded on the crowing of the cock.
Some say that ever ’gainst that season comes
Wherein our Savior’s birth is celebrated,
This bird of dawning singeth all night long; 175
And then, they say, no spirit dare stir abroad,
The nights are wholesome; then no planets strike,
No fairy takes, nor witch hath power to charm,
So hallowed and so gracious is that time.
HORATIO
So have I heard and do in part believe it. 180
But look, the morn in russet mantle clad
Walks o’er the dew of yon high eastward hill.
Break we our watch up, and by my advice
Let us impart what we have seen tonight
Unto young Hamlet; for, upon my life, 185
This spirit, dumb to us, will speak to him.
Do you consent we shall acquaint him with it
As needful in our loves, fitting our duty?
MARCELLUS
Let’s do ’t, I pray, and I this morning know
Where we shall find him most convenient. 190
They exit.
Scene 2
Flourish. Enter Claudius, King of Denmark, Gertrude the
Queen, the Council, as Polonius, and his son Laertes,
Hamlet, with others, among them Voltemand and
Cornelius.
KING
Though yet of Hamlet our dear brother’s death
The memory be green, and that it us befitted
To bear our hearts in grief, and our whole kingdom
To be contracted in one brow of woe,
Yet so far hath discretion fought with nature 5
That we with wisest sorrow think on him
Together with remembrance of ourselves.
Therefore our sometime sister, now our queen,
Th’ imperial jointress to this warlike state,
Have we (as ’twere with a defeated joy, 10
With an auspicious and a dropping eye,
With mirth in funeral and with dirge in marriage,
In equal scale weighing delight and dole)
Taken to wife. Nor have we herein barred
Your better wisdoms, which have freely gone 15
With this affair along. For all, our thanks.
Now follows that you know. Young Fortinbras,
Holding a weak supposal of our worth
Or thinking by our late dear brother’s death
Our state to be disjoint and out of frame, 20
Colleaguèd with this dream of his advantage,
He hath not failed to pester us with message
Importing the surrender of those lands
Lost by his father, with all bonds of law,
To our most valiant brother—so much for him. 25
Now for ourself and for this time of meeting.
Thus much the business is: we have here writ
To Norway, uncle of young Fortinbras,
Who, impotent and bedrid, scarcely hears
Of this his nephew’s purpose, to suppress 30
His further gait herein, in that the levies,
The lists, and full proportions are all made
Out of his subject; and we here dispatch
You, good Cornelius, and you, Voltemand,
For bearers of this greeting to old Norway, 35
Giving to you no further personal power
To business with the King more than the scope
Of these dilated articles allow.
Giving them a paper.
Farewell, and let your haste commend your duty.
CORNELIUS/VOLTEMAND
In that and all things will we show our duty. 40
KING
We doubt it nothing. Heartily farewell.
Voltemand and Cornelius exit.
And now, Laertes, what’s the news with you?
You told us of some suit. What is ’t, Laertes?
You cannot speak of reason to the Dane
And lose your voice. What wouldst thou beg, 45
Laertes,
That shall not be my offer, not thy asking?
The head is not more native to the heart,
The hand more instrumental to the mouth,
Than is the throne of Denmark to thy father. 50
What wouldst thou have, Laertes?
LAERTES My dread lord,
Your leave and favor to return to France,
From whence though willingly I came to Denmark
To show my duty in your coronation, 55
Yet now I must confess, that duty done,
My thoughts and wishes bend again toward France
And bow them to your gracious leave and pardon.
KING
Have you your father’s leave? What says Polonius?
POLONIUS
Hath, my lord, wrung from me my slow leave 60
By laborsome petition, and at last
Upon his will I sealed my hard consent.
I do beseech you give him leave to go.
KING
Take thy fair hour, Laertes. Time be thine,
And thy best graces spend it at thy will.— 65
But now, my cousin Hamlet and my son—
HAMLET, aside
A little more than kin and less than kind.
KING
How is it that the clouds still hang on you?
HAMLET
Not so, my lord; I am too much in the sun.
QUEEN
Good Hamlet, cast thy nighted color off, 70
And let thine eye look like a friend on Denmark.
Do not forever with thy vailèd lids
Seek for thy noble father in the dust.
Thou know’st ’tis common; all that lives must die,
Passing through nature to eternity. 75
HAMLET
Ay, madam, it is common.
QUEEN If it be,
Why seems it so particular with thee?
HAMLET
“Seems,” madam? Nay, it is. I know not “seems.”
’Tis not alone my inky cloak, good mother, 80
Nor customary suits of solemn black,
Nor windy suspiration of forced breath,
No, nor the fruitful river in the eye,
Nor the dejected havior of the visage,
Together with all forms, moods, shapes of grief, 85
That can denote me truly. These indeed “seem,”
For they are actions that a man might play;
But I have that within which passes show,
These but the trappings and the suits of woe.
KING
’Tis sweet and commendable in your nature, 90
Hamlet,
To give these mourning duties to your father.
But you must know your father lost a father,
That father lost, lost his, and the survivor bound
In filial obligation for some term 95
To do obsequious sorrow. But to persever
In obstinate condolement is a course
Of impious stubbornness. ’Tis unmanly grief.
It shows a will most incorrect to heaven,
A heart unfortified, a mind impatient, 100
An understanding simple and unschooled.
For what we know must be and is as common
As any the most vulgar thing to sense,
Why should we in our peevish opposition
Take it to heart? Fie, ’tis a fault to heaven, 105
A fault against the dead, a fault to nature,
To reason most absurd, whose common theme
Is death of fathers, and who still hath cried,
From the first corse till he that died today,
“This must be so.” We pray you, throw to earth 110
This unprevailing woe and think of us
As of a father; for let the world take note,
You are the most immediate to our throne,
And with no less nobility of love
Than that which dearest father bears his son 115
Do I impart toward you. For your intent
In going back to school in Wittenberg,
It is most retrograde to our desire,
And we beseech you, bend you to remain
Here in the cheer and comfort of our eye, 120
Our chiefest courtier, cousin, and our son.

I pray thee, stay with us. Go not to Wittenberg.
HAMLET
I shall in all my best obey you, madam.
KING
Why, ’tis a loving and a fair reply. 125
Be as ourself in Denmark.—Madam, come.
This gentle and unforced accord of Hamlet
Sits smiling to my heart, in grace whereof
No jocund health that Denmark drinks today
But the great cannon to the clouds shall tell, 130
And the King’s rouse the heaven shall bruit again,
Respeaking earthly thunder. Come away.
Flourish. All but Hamlet exit.
HAMLET
O, that this too, too sullied flesh would melt,
Thaw, and resolve itself into a dew,
Or that the Everlasting had not fixed 135
His canon ’gainst self-slaughter! O God, God,
How weary, stale, flat, and unprofitable
Seem to me all the uses of this world!
Fie on ’t, ah fie! ’Tis an unweeded garden
That grows to seed. Things rank and gross in nature 140
Possess it merely. That it should come to this:
But two months dead—nay, not so much, not two.
 Heaven and Earth,
Must I remember? Why, she would hang on him
As if increase of appetite had grown
By what it fed on. And yet, within a month
(Let me not think on ’t; frailty, thy name is woman!), 150
A little month, or ere those shoes were old
With which she followed my poor father’s body,
Like Niobe, all tears—why she, even she
(O God, a beast that wants discourse of reason
Would have mourned longer!), married with my 155
uncle,
My father’s brother, but no more like my father
Than I to Hercules. Within a month,
Ere yet the salt of most unrighteous tears
Had left the flushing in her gallèd eyes, 160
She married. O, most wicked speed, to post
With such dexterity to incestuous sheets!
It is not, nor it cannot come to good.
But break, my heart, for I must hold my tongue.
Enter Horatio, Marcellus, and Barnardo.
HORATIO Hail to your Lordship. 165
HAMLET I am glad to see you well.
Horatio—or I do forget myself!
HORATIO
The same, my lord, and your poor servant ever.
HAMLET
Sir, my good friend. I’ll change that name with you.
And what make you from Wittenberg, Horatio?— 170
Marcellus?
MARCELLUS My good lord.
HAMLET
I am very glad to see you. To Barnardo. Good
even, sir.—
But what, in faith, make you from Wittenberg? 175
HORATIO
A truant disposition, good my lord.
HAMLET
I would not hear your enemy say so,
Nor shall you do my ear that violence
To make it truster of your own report
Against yourself. I know you are no truant. 180
But what is your affair in Elsinore?
We’ll teach you to drink deep ere you depart.
HORATIO
My lord, I came to see your father’s funeral.
HAMLET
I prithee, do not mock me, fellow student.
I think it was to see my mother’s wedding. 185
HORATIO
Indeed, my lord, it followed hard upon.
HAMLET
Thrift, thrift, Horatio. The funeral baked meats
Did coldly furnish forth the marriage tables.
Would I had met my dearest foe in heaven
Or ever I had seen that day, Horatio! 190
My father—methinks I see my father.
HORATIO
Where, my lord?
HAMLET In my mind’s eye, Horatio.
HORATIO
I saw him once. He was a goodly king.
HAMLET
He was a man. Take him for all in all, 195
I shall not look upon his like again.
HORATIO
My lord, I think I saw him yesternight.
HAMLET Saw who?
HORATIO
My lord, the King your father.
HAMLET The King my father? 200
HORATIO
Season your admiration for a while
With an attent ear, till I may deliver
Upon the witness of these gentlemen
This marvel to you.
HAMLET For God’s love, let me hear! 205
HORATIO
Two nights together had these gentlemen,
Marcellus and Barnardo, on their watch,
In the dead waste and middle of the night,
Been thus encountered: a figure like your father,
Armed at point exactly, cap-à-pie, 210
Appears before them and with solemn march
Goes slow and stately by them.  This to me
In dreadful secrecy impart they did,
And I with them the third night kept the watch,
Where, as they had delivered, both in time,
Form of the thing (each word made true and good), 220
The apparition comes. I knew your father;
These hands are not more like.
HAMLET But where was this?
MARCELLUS
My lord, upon the platform where we watch.
HAMLET
Did you not speak to it? 225
HORATIO My lord, I did,
But answer made it none. Yet once methought
It lifted up its head and did address
Itself to motion, like as it would speak;
But even then the morning cock crew loud, 230
And at the sound it shrunk in haste away
And vanished from our sight.
HAMLET ’Tis very strange.
HORATIO
As I do live, my honored lord, ’tis true.
And we did think it writ down in our duty 235
To let you know of it.
HAMLET Indeed, sirs, but this troubles me.
Hold you the watch tonight?

HAMLET From top to toe?
ALL My lord, from head to foot.
HAMLET Then saw you not his face?
HORATIO
O, yes, my lord, he wore his beaver up. 245
HAMLET What, looked he frowningly?
HORATIO
A countenance more in sorrow than in anger.
HAMLET Pale or red?
HORATIO
Nay, very pale.
HAMLET And fixed his eyes upon you? 250
HORATIO
Most constantly.
HAMLET I would I had been there.
HORATIO It would have much amazed you.
HAMLET Very like. Stayed it long?
HORATIO
While one with moderate haste might tell a 255
hundred.
BARNARDO/MARCELLUS Longer, longer.
HORATIO
Not when I saw ’t.
HAMLET His beard was grizzled, no?
HORATIO
It was as I have seen it in his life, 260
A sable silvered.
HAMLET I will watch tonight.
Perchance ’twill walk again.
HORATIO I warrant it will.
 I pray you all,
If you have hitherto concealed this sight,
Let it be tenable in your silence still;
And whatsomever else shall hap tonight, 270
Give it an understanding but no tongue.
I will requite your loves. So fare you well.
Upon the platform, ’twixt eleven and twelve,
I’ll visit you.
ALL Our duty to your Honor. 275
HAMLET
Your loves, as mine to you. Farewell.
All but Hamlet exit.
My father’s spirit—in arms! All is not well.
I doubt some foul play. Would the night were come!
Till then, sit still, my soul. Foul deeds will rise,
Though all the earth o’erwhelm them, to men’s 280
eyes.
He exits.
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3.1: Prelude to The First Law of Thermodynamics
Heat is the transfer of energy due to a temperature difference between two systems. Heat describes the process of converting from one form of energy into another. A car engine, for example, burns gasoline. Heat is produced when the burned fuel is chemically transformed into mostly \(\ce{CO2}\) and \(\ce{H2O}\), which are gases at the combustion temperature. These gases exert a force on a piston through a displacement, doing work and converting the piston’s kinetic energy into a variety of other forms—into the car’s kinetic energy; into electrical energy to run the spark plugs, radio, and lights; and back into stored energy in the car’s battery.
Energy is conserved in all processes, including those associated with thermodynamic systems. The roles of heat transfer and internal energy change vary from process to process and affect how work is done by the system in that process. We will see that the first law of thermodynamics explains that a change in the internal energy of a system comes from changes in heat or work. Understanding the laws that govern thermodynamic processes and the relationship between the system and its surroundings is therefore paramount in gaining scientific knowledge of energy and energy consumption.


15.10B: Aflatoxin Poisoning
- Summarize the causes and effects of alfatoxin poisoning
Aflatoxins are categorized as mycotoxins that are typically produced by species of Aspergillus . Aflatoxins, although only synthesized by a few Aspergillus species, are considered to be one of the most important mycotoxins identified to date. Both Aspergillus flavus and Aspergillus parasiticus are well known for their production of aflatoxins. Aflatoxins are most commonly transmitted to humans through the diet. Aflatoxins grow on whole grains and contaminate food supplies during processing, storage, or transport when there are favorable conditions for mold growth, specifically for the Aspergillus species.
Aflatoxin poisoning, or aflatoxicosis, occurs when there is ingestion of aflatoxin contaminated foods. Upon ingestion or exposure to aflatoxin, it is common to see injury to the liver. Aflatoxicosis is a primarily hepatic disease, as the liver is the target organ for this toxin in mammals. Although the liver demonstrates the ability to metabolize the ingested aflatoxins, the intermediate formed is a reactive epoxide or a less harmful hydroxylated form referred to as M1. There have been various studies stating that metabolic activation of aflatoxins is required for the aflatoxin to exert its carcinogenic effects. These metabolites are harmful to the liver and have been implicated in liver cancer development. The aflatoxins produced by these Aspergillus species have been show to produce adducts (altered forms of DNA). These adducts are now used as a diagnostic factor to test for aflatoxin exposure by testing blood and urine.
Aflatoxin poisoning or aflatoxicosis is rarely diagnosed in developed countries but continues to be an issue in underdeveloped countries. In developed countries, commercial crops are screened for the presence of aflatoxins. However, in underdeveloped or developing countries, screening methods are lacking, or are in the process of being introduced. Interestingly, a rise in homegrown food has been correlated with a slight increase in aflatoxin exposure via diet.
Aflatoxin poisoning can be diagnosed as either acute or chronic. In cases of acute aflatoxicosis, an individual has been exposed to moderate to high levels of aflatoxins. Acute aflatoxicosis is characterized by symptoms such as hemorrhaging; acute liver damage and issues with digestion; and absorption and metabolism of nutrients. In cases of chronic aflatoxicosis, an individual has been exposed to low to moderate levels of aflatoxins. Chronic aflatoxicosis is characterized by symptoms such as dysfunctional food conversion and slow growth rates.
Key Points
- Aspergillus flavus and Aspergillus parasiticus are two commonly known species of Aspergillus that can aflatoxin poisoning.
- Aflatoxin poisoning can result in either acute aflatoxicosis, which is a result of moderate to high level ingestion; or chronic aflatoxicosis, which results from ingestion of low to moderate levels of aflatoxins.
- Aspergillus species which cause aflatoxin poisoning are often found in crops in underdeveloped countries, due to lack of detection techniques and inadequate harvesting and storage.
Key Terms
- mycotoxins : a substance produced by a mold or fungus.


1.1: What is Electromagnetics?
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The topic of this book is applied engineering electromagnetics. This topic is often described as “the theory of electromagnetic fields and waves,” which is both true and misleading. The truth is that electric fields, magnetic fields, their sources, waves, and the behavior these waves are all topics covered by this book. The misleading part is that our principal aim shall be to close the gap between basic electrical circuit theory and the more general theory that is required to address certain topics that are of broad and common interest in the field of electrical engineering. (For a preview of topics where these techniques are required, see the list at the end of this section.) In basic electrical circuit theory, the behavior of devices and systems is abstracted in such a way that the underlying electromagnetic principles do not need to be considered. Every student of electrical engineering encounters this, and is grateful since this greatly simplifies analysis and design. For example, a resistor is commonly defined as a device which exhibits a particular voltage
\[V = IR \nonumber \]
in response to a current \(I\), and the resistor is therefore completely described by the value \(R\). This is an example of a “lumped element” abstraction of an electrical device. Much can be accomplished knowing nothing else about resistors; no particular knowledge of the physical concepts of electrical potential, conduction current, or resistance is required. However, this simplification makes it impossible to answer some frequently-encountered questions. Here are just a few:
- What determines R? How does one go about designing a resistor to have a particular resistance?
- Practical resistors are rated for power-handling capability; e.g., discrete resistors are frequently identified as “1/8-W,” “1/4-W,” and so on. How does one determine this, and how can this be adjusted in the design?
- Practical resistors exhibit significant reactance as well as resistance. Why? How is this determined? What can be done to mitigate this?
- Most things which are not resistors also exhibit significant resistance and reactance – for example, electrical pins and interconnects. Why? How is this determined? What can be done to mitigate this?
The answers to the these questions must involve properties of materials and the geometry in which those materials are arranged . These are precisely the things that disappear in lumped element device models, so it is not surprising that such models leave us in the dark on these issues. It should also be apparent that what is true for the resistor is also going to be true for other devices of practical interest, including capacitors (and devices unintentionally exhibiting capacitance), inductors (and devices unintentionally exhibiting inductance), transformers (and devices unintentionally exhibiting mutual impedance), and so on. From this perspective, electromagnetics may be viewed as a generalization of electrical circuit theory that addresses these considerations. Conversely basic electric circuit theory may be viewed a special case of electromagnetic theory that applies when these considerations are not important. Many instances of this “electromagnetics as generalization” vs. “lumped-element theory as special case” dichotomy appear in the study of electromagnetics.
There is more to the topic, however. There are many devices and applications in which electromagnetic fields and waves are primary engineering considerations that must be dealt with directly. Examples include electrical generators and motors; antennas; printed circuit board stackup and layout; persistent storage of data (e.g., hard drives); fiber optics; and systems for radio, radar, remote sensing, and medical imaging. Considerations such as signal integrity and electromagnetic compatibility (EMC) similarly require explicit consideration of electromagnetic principles.
Although electromagnetic considerations pertain to all frequencies, these considerations become increasingly difficult to avoid with increasing frequency. This is because the wavelength of an electromagnetic field decreases with increasing frequency. When wavelength is large compared to the size of the region of interest (e.g., a circuit), then analysis and design is not much different from zero-frequency (“DC”) analysis and design.
For example, the free space wavelength at 3 MHz is about 100 m, so a planar circuit having dimensions 10 cm × 10 cm is just 0.1% of a wavelength across at this frequency. Although an electromagnetic wave may be present, it has about the same value over the region of space occupied by the circuit. In contrast, the free space wavelength at 3 GHz is about 10 cm, so the same circuit is one full wavelength across at this frequency. In this case, different parts of this circuit observe the same signal with very different magnitude and phase.
Some of the behaviors associated with non-negligible dimensions are undesirable, especially if not taken into account in the design process. However, these behaviors can also be exploited to do some amazing and useful things – for example, to launch an electromagnetic wave (i.e., an antenna) or to create filters and impedance matching devices consisting only of metallic shapes, free of discrete capacitors or inductors.
Electromagnetic considerations become not only unavoidable but central to analysis and design above a few hundred MHz, and especially in the millimeter-wave, infrared (IR), optical, and ultraviolet (UV) bands. The discipline of electrical engineering encompasses applications in these frequency ranges even though – ironically – such applications may not operate according to principles that can be considered “electrical”! Nevertheless, electromagnetic theory applies.
Another common way to answer the question “What is electromagnetics?” is to identify the topics that are commonly addressed within this discipline. Here’s a list of topics – some of which have already been mentioned – in which explicit consideration of electromagnetic principles is either important or essential:
- Antennas
- Coaxial cable
- Design and characterization of common discrete passive components including resistors, capacitors, inductors, and diodes
- Distributed (e.g., microstrip) filters
- Electromagnetic compatibility (EMC)
- Fiber optics
- Generators
- Magnetic resonance imaging (MRI)
- Magnetic storage (of data)
- Microstrip transmission lines
- Modeling of non-ideal behaviors of discrete components
- Motors
- Non-contact sensors
- Photonics
- Printed circuit board stackup and layout
- Radar
- Radio wave propagation
- Radio frequency electronics
- Signal integrity
- Transformers
- Waveguides
Summary
Applied engineering electromagnetics is the study of those aspects of electrical engineering in situations in which the electromagnetic properties of materials and the geometry in which those materials are arranged is important. This requires an understanding of electromagnetic fields and waves, which are of primary interest in some applications.
Finally, here are two broadly-defined learning objectives that should now be apparent: (1) Learn the techniques of engineering analysis and design that apply when electromagnetic principles are important, and (2) Better understand the physics underlying the operation of electrical devices and systems, so that when issues associated with these physical principles emerge one is prepared to recognize and grapple with them.


1.1: Why It Matters- Introduction to Retailing
When is the last time you went to your favorite retailer? What store was it? Is it a clothing store, or is it a grocery store? Maybe you prefer to shop online. Why? Have you wondered about the journey those products take before making it out to the shelves? There are so many aspects to the retail experience and process for us to examine and better understand.
Just how wide is the scope of the retail industry? What drives the retail industry and what changes are on the horizon for the future that will make an impact to this industry? Let’s dig a little deeper into the scope and impact of retail to our economy and why it’s important for you to understand. We’ll put some numbers behind this information so you can see just how big this industry is and the far reaching implications of retailers such as Toys R Us closing its doors.
Today you might be a part-time sales associate, a store manager who oversees a million-dollar store, or in one of the other many sectors of the business. However, tomorrow you could be one of the buyers in the corporate office trying to understand how to leverage your customer’s needs to drive sales. At each level you will make key decisions that will impact the retail environment that range from pricing, to assortment planning, and even human resources activities.
Before we dive into the inner workings of the retail industry and why it matters to you — both as a consumer and a professional in that industry — here is a brief video on what some consider the future of retail. Considering where retail is headed helps us look at the current and past landscape of retail with a critical eye.
You can view the transcript for “The Future of Retail” (opens in new window).
Contributors and Attributions
- clothing store. Authored by : Free-Photos. Provided by : Pixabay. Located at : https://pixabay.com/en/store-clothing-shop-bouique-984393/ . License : CC0: No Rights Reserved
- The Future of Retail video. Located at : https://youtu.be/t49JkakYAoE . License : All Rights Reserved
- National Retail Federation & impact on global economy. Provided by : National Retail Federation. Located at : https://nrf.com/sites/default/files/Documents/retail%20library/Retail%27s%20Impact%20-%20Printable%20Highlights.pdf . License : Public Domain: No Known Copyright


18.8: Applications of Electrostatics
Learning Objectives
By the end of this section, you will be able to:
- Name several real-world applications of the study of electrostatics.
The study of electrostatics has proven useful in many areas. This module covers just a few of the many applications of electrostatics.
The Van de Graaff Generator
Van de Graaff generators (or Van de Graaffs) are not only spectacular devices used to demonstrate high voltage due to static electricity—they are also used for serious research. The first was built by Robert Van de Graaff in 1931 (based on original suggestions by Lord Kelvin) for use in nuclear physics research. Figure \(\PageIndex{1}\) shows a schematic of a large research version. Van de Graaffs utilize both smooth and pointed surfaces, and conductors and insulators to generate large static charges and, hence, large voltages.
A very large excess charge can be deposited on the sphere, because it moves quickly to the outer surface. Practical limits arise because the large electric fields polarize and eventually ionize surrounding materials, creating free charges that neutralize excess charge or allow it to escape. Nevertheless, voltages of 15 million volts are well within practical limits.
TAKE-HOME EXPERIMENT: ELECTROSTATICS AND HUMIDITY
Rub a comb through your hair and use it to lift pieces of paper. It may help to tear the pieces of paper rather than cut them neatly. Repeat the exercise in your bathroom after you have had a long shower and the air in the bathroom is moist. Is it easier to get electrostatic effects in dry or moist air? Why would torn paper be more attractive to the comb than cut paper? Explain your observations.
Xerography
Most copy machines use an electrostatic process called xerography —a word coined from the Greek words xeros for dry and graphos for writing. The heart of the process is shown in simplified form in Figure \(\PageIndex{2}\). A selenium-coated aluminum drum is sprayed with positive charge from points on a device called a corotron. Selenium is a substance with an interesting property—it is a photoconductor. That is, selenium is an insulator when in the dark and a conductor when exposed to light.
In the first stage of the xerography process, the conducting aluminum drum is grounded so that a negative charge is induced under the thin layer of uniformly positively charged selenium. In the second stage, the surface of the drum is exposed to the image of whatever is to be copied. Where the image is light, the selenium becomes conducting, and the positive charge is neutralized. In dark areas, the positive charge remains, and so the image has been transferred to the drum.
The third stage takes a dry black powder, called toner, and sprays it with a negative charge so that it will be attracted to the positive regions of the drum. Next, a blank piece of paper is given a greater positive charge than on the drum so that it will pull the toner from the drum. Finally, the paper and electrostatically held toner are passed through heated pressure rollers, which melt and permanently adhere the toner within the fibers of the paper.
Laser Printers
Laser printers use the xerographic process to make high-quality images on paper, employing a laser to produce an image on the photoconducting drum as shown in Figure \(\PageIndex{3}\). In its most common application, the laser printer receives output from a computer, and it can achieve high-quality output because of the precision with which laser light can be controlled. Many laser printers do significant information processing, such as making sophisticated letters or fonts, and may contain a computer more powerful than the one giving them the raw data to be printed.
Ink Jet Printers and Electrostatic Painting
The ink jet printer , commonly used to print computer-generated text and graphics, also employs electrostatics. A nozzle makes a fine spray of tiny ink droplets, which are then given an electrostatic charge (Figure \(\PageIndex{4}\)). Once charged, the droplets can be directed, using pairs of charged plates, with great precision to form letters and images on paper. Ink jet printers can produce color images by using a black jet and three other jets with primary colors, usually cyan, magenta, and yellow, much as a color television produces color (this is more difficult with xerography, requiring multiple drums and toners).
Electrostatic painting employs electrostatic charge to spray paint onto odd-shaped surfaces. Mutual repulsion of like charges causes the paint to fly away from its source. Surface tension forms drops, which are then attracted by unlike charges to the surface to be painted. Electrostatic painting can reach those hard-to-get at places, applying an even coat in a controlled manner. If the object is a conductor, the electric field is perpendicular to the surface, tending to bring the drops in perpendicularly. Corners and points on conductors will receive extra paint. Felt can similarly be applied.
Smoke Precipitators and Electrostatic Air Cleaning
Another important application of electrostatics is found in air cleaners, both large and small. The electrostatic part of the process places excess (usually positive) charge on smoke, dust, pollen, and other particles in the air and then passes the air through an oppositely charged grid that attracts and retains the charged particles (Figure \(\PageIndex{5}\)). Large electrostatic precipitators are used industrially to remove over 99% of the particles from stack gas emissions associated with the burning of coal and oil. Home precipitators, often in conjunction with the home heating and air conditioning system, are very effective in removing polluting particles, irritants, and allergens.
PROBLEM-SOLVING STRATEGIES FOR ELECTROSTATICS
- Examine the situation to determine if static electricity is involved. This may concern separated stationary charges, the forces among them, and the electric fields they create.
- Identify the system of interest. This includes noting the number, locations, and types of charges involved.
- Identify exactly what needs to be determined in the problem (identify the unknowns). A written list is useful. Determine whether the Coulomb force is to be considered directly—if so, it may be useful to draw a free-body diagram, using electric field lines.
- Make a list of what is given or can be inferred from the problem as stated (identify the knowns). It is important to distinguish the Coulomb force \(\mathbf{F}\) from the electric field \(\mathbf{E}\), for example.
- Solve the appropriate equation for the quantity to be determined (the unknown) or draw the field lines as requested.
- Examine the answer to see if it is reasonable: Does it make sense? Are units correct and the numbers involved reasonable?
The following worked example illustrates how this strategy is applied:
Example \(\PageIndex{1}\): Acceleration of a Charged Drop of Gasoline
If steps are not taken to ground a gasoline pump, static electricity can be placed on gasoline when filling your car’s tank. Suppose a tiny drop of gasoline has a mass of \(4.00\times 10^{-15}kg\) and is given a positive charge of \(3.20\times 10^{-19}C\). (a) Find the weight of the drop. (b) Calculate the electric force on the drop if there is an upward electric field of strength \(3.00\times 10^{5} N/C\) due to other static electricity in the vicinity. (c) Calculate the drop’s acceleration.
Strategy
To solve an integrated concept problem, we must first identify the physical principles involved and identify the chapters in which they are found.
- Part (a) of this example asks for weight. This is a topic of dynamics and is defined in Dynamics: Force and Newton’s Laws of Motion .
- Part (b) deals with electric force on a charge, a topic of Electric Charge and Electric Field . Part (c) asks for acceleration, knowing forces and mass. These are part of Newton’s laws, also found in Dynamics: Force and Newton’s Laws of Motion .
The following solutions to each part of the example illustrate how the specific problem-solving strategies are applied. These involve identifying knowns and unknowns, checking to see if the answer is reasonable, and so on.
Solution for (a)
Weight is mass times the acceleration due to gravity, as first expressed in
\[w=mg.\]
Entering the given mass and the average acceleration due to gravity yields
\[\begin{align*} w &=(4.00\times 10^{-15}kg)(9.80m/s^{2}) \\[5pt] &= 3.92\times 10^{-14}N.\end{align*} \]
Discussion for (a)
This is a small weight, consistent with the small mass of the drop.
Solution for (b)
The force an electric field exerts on a charge is given by rearranging the following equation:
\[F=qE. \nonumber \]
Here we are given the charge (\(3.20\times 10^{-19}C\) is twice the fundamental unit of charge) and the electric field strength, and so the electric force is found to be
\[\begin{align*} F&=(3.20\times 10^{-19}C)(3.00\times 10^{5}N/C) \\[5pt] &=9.60\times 10^{-14}N. \end{align*} \]
Discussion for (b)
While this is a small force, it is greater than the weight of the drop.
Solution for (c)
The acceleration can be found using Newton’s second law, provided we can identify all of the external forces acting on the drop. We assume only the drop’s weight and the electric force are significant. Since the drop has a positive charge and the electric field is given to be upward, the electric force is upward. We thus have a one-dimensional (vertical direction) problem, and we can state Newton’s second law as
\[a=\dfrac{F_{net}}{m}. \nonumber\]
where \(F_{net}=F-w\). Entering this and the known values into the expression for Newton’s second law yields
\[\begin{align*} a &=\dfrac{F-w}{m} \\[5pt] &=\dfrac{9.60\times 10^{-14}N-3.92\times 10^{-14}N}{4.00\times 10^{-15}kg} \\[5pt] &= 14.2m/s^{2}. \end{align*} \]
Discussion for (c)
This is an upward acceleration great enough to carry the drop to places where you might not wish to have gasoline.
This worked example illustrates how to apply problem-solving strategies to situations that include topics in different chapters. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknown using familiar problem-solving strategies. These are found throughout the text, and many worked examples show how to use them for single topics. In this integrated concepts example, you can see how to apply them across several topics. You will find these techniques useful in applications of physics outside a physics course, such as in your profession, in other science disciplines, and in everyday life. The following problems will build your skills in the broad application of physical principles.
UNREASONABLE RESULTS
The Unreasonable Results exercises for this module have results that are unreasonable because some premise is unreasonable or because certain of the premises are inconsistent with one another. Physical principles applied correctly then produce unreasonable results. The purpose of these problems is to give practice in assessing whether nature is being accurately described, and if it is not to trace the source of difficulty.
PROBLEM-SOLVING STRATEGY
To determine if an answer is reasonable, and to determine the cause if it is not, do the following.
- Solve the problem using strategies as outlined above. Use the format followed in the worked examples in the text to solve the problem as usual.
- Check to see if the answer is reasonable. Is it too large or too small, or does it have the wrong sign, improper units, and so on?
- If the answer is unreasonable, look for what specifically could cause the identified difficulty. Usually, the manner in which the answer is unreasonable is an indication of the difficulty. For example, an extremely large Coulomb force could be due to the assumption of an excessively large separated charge.
Summary
- Electrostatics is the study of electric fields in static equilibrium.
- In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink-jet printers and electrostatic air filters.
Glossary
- Van de Graaff generator
- a machine that produces a large amount of excess charge, used for experiments with high voltage
- electrostatics
- the study of electric forces that are static or slow-moving
- photoconductor
- a substance that is an insulator until it is exposed to light, when it becomes a conductor
- xerography
- a dry copying process based on electrostatics
- grounded
- connected to the ground with a conductor, so that charge flows freely to and from the Earth to the grounded object
- laser printer
- uses a laser to create a photoconductive image on a drum, which attracts dry ink particles that are then rolled onto a sheet of paper to print a high-quality copy of the image
- ink-jet printer
- small ink droplets sprayed with an electric charge are controlled by electrostatic plates to create images on paper
- electrostatic precipitators
- filters that apply charges to particles in the air, then attract those charges to a filter, removing them from the airstream


4.4: Homogeneous ODEs
We now study solutions of the homogeneous, constant coefficient ode, written as \[\label{eq:1}a\overset{..}{x}+b\overset{.}{x}+cx=0,\] with \(a,\: b,\) and \(c\) constants. Such an equation arises for the charge on a capacitor in an unpowered \(RLC\) electrical circuit, or for the position of a freely-oscillating frictional mass on a spring, or for a damped pendulum. Our solution method finds two linearly independent solutions to \(\eqref{eq:1}\), multiplies each of these solutions by a constant, and adds them. The two free constants can then be used to satisfy two given initial conditions.
Because of the differential properties of the exponential function, a natural ansatz, or educated guess, for the form of the solution to \(\eqref{eq:1}\) is \(x = e^{rt}\), where \(r\) is a constant to be determined. Successive differentiation results in \(\overset{.}{x}= re^{rt}\) and \(\overset{..}{x}= r^2 e^{rt}\), and substitution into \(\eqref{eq:1}\) yields \[\label{eq:2}ar^2e^{rt}+bre^{rt}+ce^{rt}=0.\]
Our choice of exponential function is now rewarded by the explicit cancelation in \(\eqref{eq:2}\) of \(e^{rt}\). The result is a quadratic equation for the unknown constant \(r\):
\[\label{eq:3}ar^2+br+c=0.\]
Our ansatz has thus converted a differential equation into an algebraic equation. Equation \(\eqref{eq:3}\) is called the characteristic equation of \(\eqref{eq:1}\). Using the quadratic formula, the two solutions of the characteristic equation \(\eqref{eq:3}\) are given by \[r_{\pm}=\frac{1}{2a}\left(-b\pm\sqrt{b^2-4ac}\right).\nonumber\]
There are three cases to consider:
- if \(b^2 − 4ac > 0\), then the two roots are distinct and real;
- if \(b^2 − 4ac < 0\), then the two roots are distinct and complex conjugates of each other;
- if \(b^2 − 4ac = 0\), then the two roots are degenerate and there is only one real root. We will consider these three cases in turn.
Distinct Real Roots
When \(r_+\neq r_−\) are real roots, then the general solution to \(\eqref{eq:1}\) can be written as a linear superposition of the two solutions \(e^{r_+t}\) and \(e^{r_−t}\); that is, \[x(t)=c_1e^{r_+t}+c_2e^{r_-t}.\nonumber\]
The unknown constants \(c_1\) and \(c_2\) can then be determined by the given initial conditions \(x(t_0) = x_0\) and \(\overset{.}{x}(t_0) = u_0\). We now present two examples.
Solve \(\overset{..}{x}+5\overset{.}{x}+6x=0\) with \(x(0)=2\), \(\overset{.}{x}(0)=3\), and find the maximum value attained by \(x\).
Solution
We take as our ansatz \(x = e^{rt}\) and obtain the characteristic equation \[r^2+5r+6=0,\nonumber\] which factors to \[(r+3)(r+2)=0.\nonumber\]
The general solution to the ode is thus \[x(t)=c_1e^{-2t}+c_2e^{-3t}.\nonumber\]
The solution for \(\overset{.}{x}\) obtained by differentiation is \[\overset{.}{x}(t)=-2c_1e^{-2t}-3c_2e^{-3t}.\nonumber\]
Use of the initial conditions then results in two equations for the two unknown constant \(c_1\) and \(c_2\):
\[\begin{aligned}c_1+c_2&=2, \\ -2c_1-3c_2&=3.\end{aligned}\]
Adding three times the first equation to the second equation yields \(c_1 = 9\); and the first equation then yields \(c_2 = 2 − c_1 = −7\). Therefore, the unique solution that satisfies both the ode and the initial conditions is \[\begin{aligned}x(t)&=9e^{-2t}-7e^{-3t} \\ &=9e^{-2t}\left(1-\frac{7}{9}e^{-t}\right).\end{aligned}\]
Note that although both exponential terms decay in time, their sum increases initially since \(\overset{.}{x}(0) > 0\). The maximum value of \(x\) occurs at the time \(t_m\) when \(\overset{.}{x}= 0\), or \[t_m=\ln (7/6).\nonumber\]
The maximum \(x_m = x(t_m)\) is then determined to be \[x_m=108/49.\nonumber\]
Solve \(\overset{..}{x}-x=0\) with \(x(0)=x_0\), \(\overset{.}{x}(0)=u_0\).
Solution
Again our ansatz is \(x = e^{rt}\), and we obtain the characteristic equation \[r^2-1=0,\nonumber\] with solution \(r_± = ±1\). Therefore, the general solution for \(x\) is \[x(t)=c_1e^t+c_2e^{-t},\nonumber\] and the derivative satisfies \[\overset{.}{x}(t)=c_1e^t-c_2e^{-t}.\nonumber\]
Initial conditions are satisfied when \[\begin{aligned}c_1+c_2&=x_0, \\ c_1-c_2&=u_0.\end{aligned}\]
Adding and subtracting these equations, we determine \[c_1=\frac{1}{2}(x_0+u_0),\quad c_2=\frac{1}{2}(x_0-u_0),\nonumber\] so that after rearranging terms \[x(t)=x_0\left(\frac{e^t+e^{-t}}{2}\right)+u_0\left(\frac{e^t-e^{-t}}{2}\right).\nonumber\]
The terms in parentheses are the usual definitions of the hyperbolic cosine and sine functions; that is, \[\cosh t=\frac{e^t+e^{-t}}{2},\quad\sinh t=\frac{e^t-e^{-t}}{2}. \nonumber\]
Our solution can therefore be rewritten as \[x(t)=x_0\cosh t+u_0\sinh t.\nonumber\]
Note that the relationships between the trigonometric functions and the complex exponentials were given by \[\cos t=\frac{e^{it}+e^{-it}}{2},\quad\sin t=\frac{e^{it}-e^{-it}}{2i},\nonumber\] so that \[\cosh t=\cos it,\quad\sinh t=-i\sin it.\nonumber\]
Also note that the hyperbolic trigonometric functions satisfy the differential equations \[\frac{d}{dt}\sinh t=\cosh t,\quad\frac{d}{dt}\cosh t=\sinh t,\nonumber\] which though similar to the differential equations satisfied by the more commonly used trigonometric functions, is absent a minus sign.
Distinct Complex-Conjugate Roots
We now consider a characteristic equation \(\eqref{eq:3}\) with \(b^2 − 4ac < 0\), so the roots occur as complex-conjugate pairs. With \[\lambda =-\frac{b}{2a},\quad\mu =\frac{1}{2a}\sqrt{4ac-b^2},\nonumber\] the two roots of the characteristic equation are \(\lambda +i\mu\) and \(\lambda -i\mu\). We have thus found the following two complex exponential solutions to the differential equation:
\[Z_1(t)=e^{\lambda t}e^{i\mu t},\quad Z_2(t)=e^{\lambda t}e^{-i\mu t}.\nonumber\]
Applying the principle of superposition, any linear combination of \(Z_1\) and \(Z_2\) is also a solution to the second-order ode.
Recall that if \(z = x + iy\), then \(\text{Re }z = (z + \overline{z})/2\) and \(\text{Im }z = (z − \overline{z})/2i\). We can therefore form two different linear combinations of \(Z_1(t)\) and \(Z_2(t)\) that are real, namely \(X_1(t) = \text{Re }Z_1(t)\) and \(X_2(t) = \text{Im }Z_1(t)\). We have \[X_1(t)=e^{\lambda t}\cos\mu t,\quad X_2(t)=e^{\lambda t}\sin\mu t.\nonumber\]
Having found these two real solutions, \(X_1(t)\) and \(X_2(t)\), we can then apply the principle of superposition a second time to determine the general solution for \(x(t)\):
\[\label{eq:4}x(t)=e^{\lambda t}(A\cos\mu t+B\sin\mu t).\]
It is best to memorize this result. The real part of the roots of the characteristic equation goes into the exponential function; the imaginary part goes into the argument of cosine and sine.
Solve \(\overset{..}{x}+x=0\) with \(x(0)=x_0\) and \(\overset{.}{x}(0)=u_0\).
Solution
The characteristic equation is \[r^2+1=0,\nonumber\] with roots \[r_{\pm}=\pm i.\nonumber\]
The general solution of the ode is therefore \[x(t)=A\cos t+B\sin t.\nonumber\]
The derivative is \[\overset{.}{x}(t)=-A\sin t+B\cos t.\nonumber\]
Applying the initial conditions:
\[x(0)=A=x_0,\quad\overset{.}{x}(0)=B=u_0;\nonumber\] so that the final solution is \[x(t)=x_0\cos t+u_0\sin t.\nonumber\]
Recall that we wrote the analogous solution to the ode \(\overset{..}{x}− x = 0\) as \(x(t) = x_0 \cosh t + u_0 \sinh t\).
Solve \(\overset{..}{x}+\overset{.}{x}+x=0\) with \(x(0)=1\) and \(\overset{.}{x}(0)=0\).
Solution
The characteristic equation is \[r^2+r+1=0,\nonumber\] with roots \[r_{\pm}=-\frac{1}{2}\pm i\frac{\sqrt{3}}{2}.\nonumber\]
The general solution of the ode is therefore \[x(t)=e^{-\frac{1}{2}t}\left(A\cos\frac{\sqrt{3}}{2}t+B\sin\frac{\sqrt{3}}{2}t\right).\nonumber\]
The derivative is \[\overset{.}{x}(t)=-\frac{1}{2}e^{-\frac{1}{2}t}\left(A\cos\frac{\sqrt{3}}{2}t+B\sin\frac{\sqrt{3}}{2}t\right)+\frac{\sqrt{3}}{2}e^{-\frac{1}{2}t}\left(-A\sin\frac{\sqrt{3}}{2}t+B\cos\frac{\sqrt{3}}{2}t\right).\nonumber\]
Applying the initial conditions \(x(0) = 1\) and \(\overset{.}{x}(0) = 0\):
\[\begin{aligned}A&=1, \\ \frac{-1}{2}A+\frac{\sqrt{3}}{2}B&=0;\end{aligned}\] or \[A=1,\quad B=\frac{\sqrt{3}}{3}.\nonumber\]
Therefore, \[x(t)=e^{-\frac{1}{2}t}\left(\cos\frac{\sqrt{3}}{2}t+\frac{\sqrt{3}}{3}\sin\frac{\sqrt{3}}{2}t\right).\nonumber\]
Repeated Roots
Finally, we consider the characteristic equation, \[ar^2+br+c=0,\nonumber\] with \(b^2-4ac=0\). The degenerate root is then given by \[r=-\frac{b}{2a},\nonumber\] yielding only a single solution to the ode:
\[\label{eq:5}x_1(t)=\exp\left(-\frac{bt}{2a}\right).\]
To satisfy two initial conditions, a second independent solution must be found with nonzero Wronskian, and apparently this second solution is not of the form of our ansatz \(x = \exp (rt)\).
One method to determine this missing second solution is to try the ansatz \[\label{eq:6}x(t)=y(t)x_1(t),\] where \(y(t)\) is an unknown function that satisfies the differential equation obtained by substituting \(\eqref{eq:6}\) into \(\eqref{eq:1}\). This standard technique is called the reduction of order method and enables one to find a second solution of a homogeneous linear differential equation if one solution is known. If the original differential equation is of order \(n\), the differential equation for \(y = y(t)\) reduces to an order one lower, that is, \(n − 1\).
Here, however, we will determine this missing second solution through a limiting process. We start with the solution obtained for complex roots of the characteristic equation, and then arrive at the solution obtained for degenerate roots by taking the limit \(\mu\to 0\).
Now, the general solution for complex roots was given by \(\eqref{eq:4}\), and to properly limit this solution as \(\mu\to 0\) requires first satisfying the specific initial conditions \(x(0) = x_0\) and \(\overset{.}{x}(0) = u_0\). Solving for \(A\) and \(B\), the general solution given by \(\eqref{eq:4}\) becomes the specific solution \[x(t;\mu )=e^{\lambda t}\left(\overset{.}{x}_0\cos\mu t+\frac{u_0-\lambda x_0}{\mu}\sin\mu t\right).\nonumber\]
Here, we have written \(x = x(t;\mu )\) to show explicitly that \(x\) depends on \(\mu\).
Taking the limit as \(\mu\to 0\), and using \(\lim_{\mu\to 0}\mu^{−1} \sin \mu t = t\), we have \[\underset{\mu\to 0}{\lim}x(t;\mu )=e^{\lambda t}(x_0+(u_0-\lambda x_0)t).\nonumber\]
The second solution is observed to be a constant, \(u_0 − \lambda x_0\), times \(t\) times the first solution, \(e^{\lambda t}\). Our general solution to the ode \(\eqref{eq:1}\) when \(b^2 − 4ac = 0\) can therefore be written in the for \[x(t)=(c_1+c_2t)e^{rt},\nonumber\] where \(r\) is the repeated root of the characteristic equation. The main result to be remembered is that for the case of repeated roots, the second solution is \(t\) times the first solution.
Solve \(\overset{..}{x}+2\overset{.}{x}+x=0\) with \(x(0)=1\) and \(\overset{.}{x}(0)=0\).
Solution
The characteristic equation is \[\begin{aligned}r^2+2r+1&=(r+1)^2 \\ &=0,\end{aligned}\] which has a repeated root given by \(r = −1\). Therefore, the general solution to the ode is \[x(t)=c_1e^{-t}+c_2te^{-t},\nonumber\] with derivative \[\overset{.}{x}(t)=-c_1e^{-t}+c_2e^{-t}-c_2te^{-t}.\nonumber\]
Applying the initial conditions, we have \[\begin{aligned}c_1&=1, \\ -c_1+c_2&=0,\end{aligned}\] so that \(c_1=c_2=1\). Therefore, the solution is \[x(t)=(1+t)e^{-t}.\nonumber\]


13.6: Humidity, Evaporation, and Boiling
Learning Objectives
By the end of this section, you will be able to:
- Explain the relationship between vapor pressure of water and the capacity of air to hold water vapor.
- Explain the relationship between relative humidity and partial pressure of water vapor in the air.
- Calculate vapor density using vapor pressure.
- Calculate humidity and dew point.
The expression “it’s not the heat, it’s the humidity” makes a valid point. We keep cool in hot weather by evaporating sweat from our skin and water from our breathing passages. Because evaporation is inhibited by high humidity, we feel hotter at a given temperature when the humidity is high. Low humidity, on the other hand, can cause discomfort from excessive drying of mucous membranes and can lead to an increased risk of respiratory infections.
Figure \(\PageIndex{1}\): Dew drops like these, on a banana leaf photographed just after sunrise, form when the air temperature drops to or below the dew point. At the dew point, the rate at which water molecules join together is greater than the rate at which they separate, and some of the water condenses to form droplets. (credit: Aaron Escobar, Flickr)
When we say humidity, we really mean relative humidity . Relative humidity tells us how much water vapor is in the air compared with the maximum possible. At its maximum, denoted as saturation , the relative humidity is 100%, and evaporation is inhibited. The amount of water vapor in the air depends on temperature. For example, relative humidity rises in the evening, as air temperature declines, sometimes reaching the dew point . At the dew point temperature, relative humidity is 100%, and fog may result from the condensation of water droplets if they are small enough to stay in suspension. Conversely, if you wish to dry something (perhaps your hair), it is more effective to blow hot air over it rather than cold air, because, among other things, the increase in temperature increases the energy of the molecules, so the rate of evaporation increases.
The amount of water vapor in the air depends on the vapor pressure of water. The liquid and solid phases are continuously giving off vapor because some of the molecules have high enough speeds to enter the gas phase; see Figure (a). If a lid is placed over the container, as in Figure (b), evaporation continues, increasing the pressure, until sufficient vapor has built up for condensation to balance evaporation. Then equilibrium has been achieved, and the vapor pressure is equal to the partial pressure of water in the container. Vapor pressure increases with temperature because molecular speeds are higher as temperature increases. Table gives representative values of water vapor pressure over a range of temperatures.
Relative humidity is related to the partial pressure of water vapor in the air. At 100% humidity, the partial pressure is equal to the vapor pressure, and no more water can enter the vapor phase. If the partial pressure is less than the vapor pressure, then evaporation will take place, as humidity is less than 100%. If the partial pressure is greater than the vapor pressure, condensation takes place. In everyday language, people sometimes refer to the capacity of air to “hold” water vapor, but this is not actually what happens. The water vapor is not held by the air. The amount of water in air is determined by the vapor pressure of water and has nothing to do with the properties of air.
| Temperature \(^oC\) | Vapor Pressure \((Pa)\) | Saturation vapor density \(g/m^3\) |
| −50 | 4.0 | 0.039 |
| -20 | \(1.04 \times 10^2\) | 0.89 |
| -10 | \(2.60 \times 10^2\) | 2.36 |
| 0 | \(6.10 \times 10^2\) | 4.84 |
| 5 | \(8.68 \times 10^2\) | 6.80 |
| 10 | \(1.19 \times 10^3\) | 9.40 |
| 15 | \(1.69 \times 10^3\) | 12.8 |
| 20 | \(2.33 \times 10^3\) | 17.2 |
| 25 | \(3.17 \times 10^3\) | 23.0 |
| 30 | \(4.24 \times 10^3\) | 30.4 |
| 37 | \(6.31 \times 10^3\) | 44.0 |
| 40 | \(7.34 \times 10^3\) | 51.1 |
| 50 | \(1.23 \times 10^4\) | 82.4 |
| 60 | \(1.99 \times 10^4\) | 130 |
| 70 | \(3.12 \times 10^4\) | 197 |
| 80 | \(4.73 \times 10^4\) | 294 |
| 90 | \(7.01 \times 10^4\) | 418 |
| 95 | \(8.59 \times 10^4\) | 505 |
| 100 | \(1.01 \times 10^5\) | 598 |
| 120 | \(1.99 \times 10^5\) | 1095 |
| 150 | \(4.76 \times 10^5\) | 2430 |
| 200 | \(1.55 \times 10^6\) | 7090 |
| 220 | \(2.32 \times 10^6\) | 10,200 |
Saturation Vapor Density of Water
Example \(\PageIndex{1}\): Calculating Density Using Vapor Pressure
Table gives the vapor pressure of water at .\(20.0^oC\) as \(2.33 \times 10^3 \, Pa\). Use the ideal gas law to calculate the density of water vapor in \(g/m^3\) that would create a partial pressure equal to this vapor pressure. Compare the result with the saturation vapor density given in the table.
Strategy
To solve this problem, we need to break it down into a two steps. The partial pressure follows the ideal gas law,
\[PV = nRT,\]
where \(n\) is the number of moles. If we solve this equation for \(n/V\)
to calculate the number of moles per cubic meter, we can then convert this quantity to grams per cubic meter as requested. To do this, we need to use the molecular mass of water, which is given in the periodic table.
Solution
1. Identify the knowns and convert them to the proper units:
- temperature \(T = 20^oC = 203 \, K\)
- vapor pressure \(P\) of water at \(20^oC\) is \(2.33 \times 10^3 \, Pa\)
- molecular mass of water is \(18.0 \, g/m\)
2. Solve the ideal gas law for \(n?V\).
\[\dfrac{n}{V} = \dfrac{P}{RT}\]
3. Substitute known values into the equation and solve for \(n/V\).
\[\dfrac{n}{V} = \dfrac{P}{RT} = \dfrac{2.33 \times 10^3 \, Pa}{(8.31 \, J/mol \cdot K)(293 \, K)} = 0.957 \, mol/m^3\]
4. Convert the density in moles per cubic meter to grams per cubic meter.
\[\rho = \left(0.957\dfrac{mol}{m^3}\right)\left(\dfrac{18.0 \, g}{mol}\right) = 17.2 \, g/m^3\]
Discussion
The density is obtained by assuming a pressure equal to the vapor pressure of water at \(20.0^oC\). The density found is identical to the value in Table , which means that a vapor density of \(17.2 \, g/m^3\) at \(20.9^oC\) creates a partial pressure of \(2.33 \times 10^3 \, Pa\), equal to the vapor pressure of water at that temperature. If the partial pressure is equal to the vapor pressure, then the liquid and vapor phases are in equilibrium, and the relative humidity is 100%. Thus, there can be no more than 17.2 g of water vapor per \(m^3\) at \(20.0^oC\), so that this value is the saturation vapor density at that temperature. This example illustrates how water vapor behaves like an ideal gas: the pressure and density are consistent with the ideal gas law (assuming the density in the table is correct). The saturation vapor densities listed in Table are the maximum amounts of water vapor that air can hold at various temperatures.
Percent Relative Humidity
We define percent relative humidity as the ratio of vapor density to saturation vapor density, or
\[percent \, relative \, humidity = \dfrac{vapor \, density}{saturation \, vapor \, density} \times 100\]
We can use this and the data in Table to do a variety of interesting calculations, keeping in mind that relative humidity is based on the comparison of the partial pressure of water vapor in air and ice.
We can use this and the data in Table to do a variety of interesting calculations, keeping in mind that relative humidity is based on the comparison of the partial pressure of water vapor in air and ice.
Exampl \(\PageIndex{2}\): Calculating Humidity and Dew Point
(a) Calculate the percent relative humidity on a day when the temperature is \(25.0^oC\) and the air contains 9.40 g of vapor per \(m^3\). (b) At what temperature will this air reach 100% relative humidity (the saturation density)? This temperature is the dew point. (c) What is the humidity when the air temperature is \(25.0^oC\) and the dew point is \(-10.0^oC\)?
Strategy and Solution
(a) Percent relative humidity is defined as the ratio of vapor density to saturation vapor density.
\[percent \, relative \, humidity = \dfrac{vapor \, density}{saturation \, vapor \, density} \times 100\]
The first is given to be \(9.40 \, g/m^3\), and the second is found in Table to be \(23.0 \, g/m^3\). Thus,
\[percent \, relative \, humidity = \dfrac{9.40 \, g/m^3}{23.0 \, g/m^3} \times 100 = 40.9 \%\]
(b) The air contains \(9.40 \, g/m^3\) of water vapor. The relative humidity will be 100% at a temperature where \(9.40 \, g/m^3\) is the saturation density. Inspection of Table reveals this to be the case at \(10.0^oC\),
where the relative humidity will be 100%. That temperature is called the dew point for air with this concentration of water vapor.
(c) Here, the dew point temperature is given to be \(10.0^oC\). Using Table , we see that the vapor density is \(2.36 \, g/m^3\), because this value is the saturation vapor density at \(-10.0^oC\). The saturation vapor density at \(25.0^oC\) is seen to be \(23.0 \, g/m^3\). Thus, the relative humidity at \(25.0^oC\) is
\[percent \, relative \, humidity = \dfrac{2.36 \, g/m^3}{23.0 \, g/m^3} \times 100 = 10.3 \%\]
Discussion
The importance of dew point is that air temperature cannot drop below \(10.0^oC\) in part (b), or \(-10.0^oC\) in part (c), without water vapor condensing out of the air. If condensation occurs, considerable transfer of heat occurs (discussed in Heat and Heat Transfer Methods ), which prevents the temperature from further dropping. When dew points are below \(0^oC\), freezing temperatures are a greater possibility, which explains why farmers keep track of the dew point. Low humidity in deserts means low dew-point temperatures. Thus condensation is unlikely. If the temperature drops, vapor does not condense in liquid drops. Because no heat is released into the air, the air temperature drops more rapidly compared to air with higher humidity. Likewise, at high temperatures, liquid droplets do not evaporate, so that no heat is removed from the gas to the liquid phase. This explains the large range of temperature in arid regions.
Why does water boil at \(100^oC\)? You will note from Table that the vapor pressure of water at \(100^oC\) is \(1.01 \times 10^5 \, Pa\), or 1.00 atm. Thus, it can evaporate without limit at this temperature and pressure. But why does it form bubbles when it boils? This is because water ordinarily contains significant amounts of dissolved air and other impurities, which are observed as small bubbles of air in a glass of water. If a bubble starts out at the bottom of the container at \(20^oC\), it contains water vapor (about 2.30%). The pressure inside the bubble is fixed at 1.00 atm (we ignore the slight pressure exerted by the water around it). As the temperature rises, the amount of air in the bubble stays the same, but the water vapor increases; the bubble expands to keep the pressure at 1.00 atm. At \(100^oC\), water vapor enters the bubble continuously since the partial pressure of water is equal to 1.00 atm in equilibrium. It cannot reach this pressure, however, since the bubble also contains air and total pressure is 1.00 atm. The bubble grows in size and thereby increases the buoyant force. The bubble breaks away and rises rapidly to the surface—we call this boiling! (See Figure .)
Check Your Understanding
- Freeze drying is a process in which substances, such as foods, are dried by placing them in a vacuum chamber and lowering the atmospheric pressure around them. How does the lowered atmospheric pressure speed the drying process, and why does it cause the temperature of the food to drop?
[Hide Solution]
Decreased the atmospheric pressure results in decreased partial pressure of water, hence a lower humidity. So evaporation of water from food, for example, will be enhanced. The molecules of water most likely to break away from the food will be those with the greatest velocities. Those remaining thus have a lower average velocity and a lower temperature. This can (and does) result in the freezing and drying of the food; hence the process is aptly named freeze drying.
PHET EXPLORATIONS: STATES OF MATTER
Watch different types of molecules form a solid, liquid, or gas. Add or remove heat and watch the phase change. Change the temperature or volume of a container and see a pressure-temperature diagram respond in real time. Relate the interaction potential to the forces between molecules.
Summary
- Relative humidity is the fraction of water vapor in a gas compared to the saturation value.
- The saturation vapor density can be determined from the vapor pressure for a given temperature.
- Percent relative humidity is defined to be \[percent \, relative \, humidity = \dfrac{vapor \, density}{saturation \, vapor \, density} \times 100\%\]
- The dew point is the temperature at which air reaches 100% relative humidity.
Glossary
- dew point
- the temperature at which relative humidity is 100%; the temperature at which water starts to condense out of the air
- saturation
- the condition of 100% relative humidity
- percent relative humidity
- the ratio of vapor density to saturation vapor density
- relative humidity
- the amount of water in the air relative to the maximum amount the air can hold