9.10: Exercises
Why are sharp knives more effective than dull knives? (Hint: Think about the definition of pressure.)
Why do some small bridges have weight limits that depend on how many wheels or axles the crossing vehicle has?
Why should you roll or belly-crawl rather than walk across a thinly-frozen pond?
A typical barometric pressure in Redding, California, is about 750 mm Hg. Calculate this pressure in atm and kPa.
A typical barometric pressure in Denver, Colorado, is 615 mm Hg. What is this pressure in atmospheres and kilopascals?
A typical barometric pressure in Kansas City is 740 torr. What is this pressure in atmospheres, in millimeters of mercury, and in kilopascals?
Canadian tire pressure gauges are marked in units of kilopascals. What reading on such a gauge corresponds to 32 psi?
During the Viking landings on Mars, the atmospheric pressure was determined to be on the average about 6.50 millibars (1 bar = 0.987 atm). What is that pressure in torr and kPa?
The pressure of the atmosphere on the surface of the planet Venus is about 88.8 atm. Compare that pressure in psi to the normal pressure on earth at sea level in psi.
A medical laboratory catalog describes the pressure in a cylinder of a gas as 14.82 MPa. What is the pressure of this gas in atmospheres and torr?
Consider this scenario and answer the following questions: On a mid-August day in the northeastern United States, the following information appeared in the local newspaper: atmospheric pressure at sea level 29.97 in. Hg, 1013.9 mbar.
- What was the pressure in kPa?
- The pressure near the seacoast in the northeastern United States is usually reported near 30.0 in. Hg. During a hurricane, the pressure may fall to near 28.0 in. Hg. Calculate the drop in pressure in torr.
Why is it necessary to use a nonvolatile liquid in a barometer or manometer?
The pressure of a sample of gas is measured at sea level with a closed-end manometer. The liquid in the manometer is mercury. Determine the pressure of the gas in:
- torr
- Pa
- bar
The pressure of a sample of gas is measured with an open-end manometer, partially shown to the right. The liquid in the manometer is mercury. Assuming atmospheric pressure is 29.92 in. Hg, determine the pressure of the gas in:
- torr
- Pa
- bar
The pressure of a sample of gas is measured at sea level with an open-end mercury manometer. Assuming atmospheric pressure is 760.0 mm Hg, determine the pressure of the gas in:
- mm Hg
- atm
- kPa
The pressure of a sample of gas is measured at sea level with an open-end mercury manometer. Assuming atmospheric pressure is 760 mm Hg, determine the pressure of the gas in:
- mm Hg
- atm
- kPa
How would the use of a volatile liquid affect the measurement of a gas using open-ended manometers vs. closed-end manometers?
Sometimes leaving a bicycle in the sun on a hot day will cause a blowout. Why?
Explain how the volume of the bubbles exhausted by a scuba diver (Figure 9.16) change as they rise to the surface, assuming that they remain intact.
One way to state Boyle’s law is “All other things being equal, the pressure of a gas is inversely proportional to its volume.” (a) What is the meaning of the term “inversely proportional?” (b) What are the “other things” that must be equal?
An alternate way to state Avogadro’s law is “All other things being equal, the number of molecules in a gas is directly proportional to the volume of the gas.” (a) What is the meaning of the term “directly proportional?” (b) What are the “other things” that must be equal?
How would the graph in Figure 9.12 change if the number of moles of gas in the sample used to determine the curve were doubled?
How would the graph in Figure 9.13 change if the number of moles of gas in the sample used to determine the curve were doubled?
In addition to the data found in Figure 9.13, what other information do we need to find the mass of the sample of air used to determine the graph?
Determine the volume of 1 mol of CH 4 gas at 150 K and 1 atm, using Figure 9.12.
Determine the pressure of the gas in the syringe shown in Figure 9.13 when its volume is 12.5 mL, using:
- the appropriate graph
- Boyle’s law
A spray can is used until it is empty except for the propellant gas, which has a pressure of 1344 torr at 23 °C. If the can is thrown into a fire (T = 475 °C), what will be the pressure in the hot can?
What is the temperature of an 11.2-L sample of carbon monoxide, CO, at 744 torr if it occupies 13.3 L at 55 °C and 744 torr?
A 2.50-L volume of hydrogen measured at –196 °C is warmed to 100 °C. Calculate the volume of the gas at the higher temperature, assuming no change in pressure.
A balloon inflated with three breaths of air has a volume of 1.7 L. At the same temperature and pressure, what is the volume of the balloon if five more same-sized breaths are added to the balloon?
A weather balloon contains 8.80 moles of helium at a pressure of 0.992 atm and a temperature of 25 °C at ground level. What is the volume of the balloon under these conditions?
The volume of an automobile air bag was 66.8 L when inflated at 25 °C with 77.8 g of nitrogen gas. What was the pressure in the bag in kPa?
How many moles of gaseous boron trifluoride, BF 3 , are contained in a 4.3410-L bulb at 788.0 K if the pressure is 1.220 atm? How many grams of BF 3 ?
Iodine, I 2 , is a solid at room temperature but sublimes (converts from a solid into a gas) when warmed. What is the temperature in a 73.3-mL bulb that contains 0.292 g of I 2 vapor at a pressure of 0.462 atm?
How many grams of gas are present in each of the following cases?
- 0.100 L of CO 2 at 307 torr and 26 °C
- 8.75 L of C 2 H 4 , at 378.3 kPa and 483 K
- 221 mL of Ar at 0.23 torr and –54 °C
A high altitude balloon is filled with 1.41 10 4 L of hydrogen at a temperature of 21 °C and a pressure of 745 torr. What is the volume of the balloon at a height of 20 km, where the temperature is –48 °C and the pressure is 63.1 torr?
A cylinder of medical oxygen has a volume of 35.4 L, and contains O 2 at a pressure of 151 atm and a temperature of 25 °C. What volume of O 2 does this correspond to at normal body conditions, that is, 1 atm and 37 °C?
A large scuba tank (Figure 9.16) with a volume of 18 L is rated for a pressure of 220 bar. The tank is filled at 20 °C and contains enough air to supply 1860 L of air to a diver at a pressure of 2.37 atm (a depth of 45 feet). Was the tank filled to capacity at 20 °C?
A 20.0-L cylinder containing 11.34 kg of butane, C 4 H 10 , was opened to the atmosphere. Calculate the mass of the gas remaining in the cylinder if it were opened and the gas escaped until the pressure in the cylinder was equal to the atmospheric pressure, 0.983 atm, and a temperature of 27 °C.
While resting, the average 70-kg human male consumes 14 L of pure O 2 per hour at 25 °C and 100 kPa. How many moles of O 2 are consumed by a 70 kg man while resting for 1.0 h?
For a given amount of gas showing ideal behavior, draw labeled graphs of:
- the variation of P with V
- the variation of V with T
- the variation of P with T
- the variation of with V
A liter of methane gas, CH 4 , at STP contains more atoms of hydrogen than does a liter of pure hydrogen gas, H 2 , at STP. Using Avogadro’s law as a starting point, explain why.
The effect of chlorofluorocarbons (such as CCl 2 F 2 ) on the depletion of the ozone layer is well known. The use of substitutes, such as CH 3 CH 2 F( g ), for the chlorofluorocarbons, has largely corrected the problem. Calculate the volume occupied by 10.0 g of each of these compounds at STP:
- CCl 2 F 2 ( g )
- CH 3 CH 2 F( g )
As 1 g of the radioactive element radium decays over 1 year, it produces 1.16 10 18 alpha particles (helium nuclei). Each alpha particle becomes an atom of helium gas. What is the pressure in pascal of the helium gas produced if it occupies a volume of 125 mL at a temperature of 25 °C?
A balloon with a volume of 100.21 L at 21 °C and 0.981 atm is released and just barely clears the top of Mount Crumpit in British Columbia. If the final volume of the balloon is 144.53 L at a temperature of 5.24 °C, what is the pressure experienced by the balloon as it clears Mount Crumpet?
If the temperature of a fixed amount of a gas is doubled at constant volume, what happens to the pressure?
If the volume of a fixed amount of a gas is tripled at constant temperature, what happens to the pressure?
What is the density of laughing gas, dinitrogen monoxide, N 2 O, at a temperature of 325 K and a pressure of 113.0 kPa?
Calculate the density of Freon 12, CF 2 Cl 2 , at 30.0 °C and 0.954 atm.
Which is denser at the same temperature and pressure, dry air or air saturated with water vapor? Explain.
A cylinder of O 2 ( g ) used in breathing by patients with emphysema has a volume of 3.00 L at a pressure of 10.0 atm. If the temperature of the cylinder is 28.0 °C, what mass of oxygen is in the cylinder?
What is the molar mass of a gas if 0.0494 g of the gas occupies a volume of 0.100 L at a temperature 26 °C and a pressure of 307 torr?
What is the molar mass of a gas if 0.281 g of the gas occupies a volume of 125 mL at a temperature 126 °C and a pressure of 777 torr?
How could you show experimentally that the molecular formula of propene is C 3 H 6 , not CH 2 ?
The density of a certain gaseous fluoride of phosphorus is 3.93 g/L at STP. Calculate the molar mass of this fluoride and determine its molecular formula.
Consider this question: What is the molecular formula of a compound that contains 39% C, 45% N, and 16% H if 0.157 g of the compound occupies l25 mL with a pressure of 99.5 kPa at 22 °C?
- Outline the steps necessary to answer the question.
- Answer the question.
A 36.0–L cylinder of a gas used for calibration of blood gas analyzers in medical laboratories contains 350 g CO 2 , 805 g O 2 , and 4,880 g N 2 . At 25 degrees C, what is the pressure in the cylinder in atmospheres?
A cylinder of a gas mixture used for calibration of blood gas analyzers in medical laboratories contains 5.0% CO 2 , 12.0% O 2 , and the remainder N 2 at a total pressure of 146 atm. What is the partial pressure of each component of this gas? (The percentages given indicate the percent of the total pressure that is due to each component.)
A sample of gas isolated from unrefined petroleum contains 90.0% CH 4 , 8.9% C 2 H 6 , and 1.1% C 3 H 8 at a total pressure of 307.2 kPa. What is the partial pressure of each component of this gas? (The percentages given indicate the percent of the total pressure that is due to each component.)
A mixture of 0.200 g of H 2 , 1.00 g of N 2 , and 0.820 g of Ar is stored in a closed container at STP. Find the volume of the container, assuming that the gases exhibit ideal behavior.
Most mixtures of hydrogen gas with oxygen gas are explosive. However, a mixture that contains less than 3.0 % O 2 is not. If enough O 2 is added to a cylinder of H 2 at 33.2 atm to bring the total pressure to 34.5 atm, is the mixture explosive?
A commercial mercury vapor analyzer can detect, in air, concentrations of gaseous Hg atoms (which are poisonous) as low as 2 10 −6 mg/L of air. At this concentration, what is the partial pressure of gaseous mercury if the atmospheric pressure is 733 torr at 26 °C?
A sample of carbon monoxide was collected over water at a total pressure of 756 torr and a temperature of 18 °C. What is the pressure of the carbon monoxide? (See Table 9.2 for the vapor pressure of water.)
In an experiment in a general chemistry laboratory, a student collected a sample of a gas over water. The volume of the gas was 265 mL at a pressure of 753 torr and a temperature of 27 °C. The mass of the gas was 0.472 g. What was the molar mass of the gas?
Joseph Priestley first prepared pure oxygen by heating mercuric oxide, HgO:
- Outline the steps necessary to answer the following question: What volume of O 2 at 23 °C and 0.975 atm is produced by the decomposition of 5.36 g of HgO?
- Answer the question.
Cavendish prepared hydrogen in 1766 by the novel method of passing steam through a red-hot gun barrel:
- Outline the steps necessary to answer the following question: What volume of H 2 at a pressure of 745 torr and a temperature of 20 °C can be prepared from the reaction of 15.O g of H 2 O?
- Answer the question.
The chlorofluorocarbon CCl
2
F
2
can be recycled into a different compound by reaction with hydrogen to produce CH
2
F
2
(
g
), a compound useful in chemical manufacturing:
- Outline the steps necessary to answer the following question: What volume of hydrogen at 225 atm and 35.5 °C would be required to react with 1 ton (1.000 10 3 kg) of CCl 2 F 2 ?
- Answer the question.
Automobile air bags are inflated with nitrogen gas, which is formed by the decomposition of solid sodium azide (NaN 3 ). The other product is sodium metal. Calculate the volume of nitrogen gas at 27 °C and 756 torr formed by the decomposition of 125 g of sodium azide.
Lime, CaO, is produced by heating calcium carbonate, CaCO 3 ; carbon dioxide is the other product.
- Outline the steps necessary to answer the following question: What volume of carbon dioxide at 875 K and 0.966 atm is produced by the decomposition of 1 ton (1.000 10 3 kg) of calcium carbonate?
- Answer the question.
Before small batteries were available, carbide lamps were used for bicycle lights. Acetylene gas, C 2 H 2 , and solid calcium hydroxide were formed by the reaction of calcium carbide, CaC 2 , with water. The ignition of the acetylene gas provided the light. Currently, the same lamps are used by some cavers, and calcium carbide is used to produce acetylene for carbide cannons.
- Outline the steps necessary to answer the following question: What volume of C 2 H 2 at 1.005 atm and 12.2 °C is formed by the reaction of 15.48 g of CaC 2 with water?
- Answer the question.
Calculate the volume of oxygen required to burn 12.00 L of ethane gas, C 2 H 6 , to produce carbon dioxide and water, if the volumes of C 2 H 6 and O 2 are measured under the same conditions of temperature and pressure.
What volume of O 2 at STP is required to oxidize 8.0 L of NO at STP to NO 2 ? What volume of NO 2 is produced at STP?
Consider the following questions:
- What is the total volume of the CO 2 ( g ) and H 2 O( g ) at 600 °C and 0.888 atm produced by the combustion of 1.00 L of C 2 H 6 ( g ) measured at STP?
- What is the partial pressure of H 2 O in the product gases?
Methanol, CH
3
OH, is produced industrially by the following reaction:
Assuming that the gases behave as ideal gases, find the ratio of the total volume of the reactants to the final volume.
What volume of oxygen at 423.0 K and a pressure of 127.4 kPa is produced by the decomposition of 129.7 g of BaO 2 to BaO and O 2 ?
A 2.50-L sample of a colorless gas at STP decomposed to give 2.50 L of N 2 and 1.25 L of O 2 at STP. What is the colorless gas?
Ethanol, C
2
H
5
OH, is produced industrially from ethylene, C
2
H
4
, by the following sequence of reactions:
What volume of ethylene at STP is required to produce 1.000 metric ton (1000 kg) of ethanol if the overall yield of ethanol is 90.1%?
One molecule of hemoglobin will combine with four molecules of oxygen. If 1.0 g of hemoglobin combines with 1.53 mL of oxygen at body temperature (37 °C) and a pressure of 743 torr, what is the molar mass of hemoglobin?
A sample of a compound of xenon and fluorine was confined in a bulb with a pressure of 18 torr. Hydrogen was added to the bulb until the pressure was 72 torr. Passage of an electric spark through the mixture produced Xe and HF. After the HF was removed by reaction with solid KOH, the final pressure of xenon and unreacted hydrogen in the bulb was 36 torr. What is the empirical formula of the xenon fluoride in the original sample? (Note: Xenon fluorides contain only one xenon atom per molecule.)
One method of analyzing amino acids is the van Slyke method. The characteristic amino groups (−NH
2
) in protein material are allowed to react with nitrous acid, HNO
2
, to form N
2
gas. From the volume of the gas, the amount of amino acid can be determined. A 0.0604-g sample of a biological sample containing glycine, CH
2
(NH
2
)COOH, was analyzed by the van Slyke method and yielded 3.70 mL of N
2
collected over water at a pressure of 735 torr and 29 °C. What was the percentage of glycine in the sample?
A balloon filled with helium gas takes 6 hours to deflate to 50% of its original volume. How long will it take for an identical balloon filled with the same volume of hydrogen gas (instead of helium) to decrease its volume by 50%?
Explain why the numbers of molecules are not identical in the left- and right-hand bulbs shown in the center illustration of Figure 9.27.
Starting with the definition of rate of effusion and Graham’s finding relating rate and molar mass, show how to derive the Graham’s law equation, relating the relative rates of effusion for two gases to their molecular masses.
Heavy water, D 2 O (molar mass = 20.03 g mol –1 ), can be separated from ordinary water, H 2 O (molar mass = 18.01), as a result of the difference in the relative rates of diffusion of the molecules in the gas phase. Calculate the relative rates of diffusion of H 2 O and D 2 O.
Which of the following gases diffuse more slowly than oxygen? F 2 , Ne, N 2 O, C 2 H 2 , NO, Cl 2 , H 2 S
During the discussion of gaseous diffusion for enriching uranium, it was claimed that 235 UF 6 diffuses 0.4% faster than 238 UF 6 . Show the calculation that supports this value. The molar mass of 235 UF 6 = 235.043930 + 6 18.998403 = 349.034348 g/mol, and the molar mass of 238 UF 6 = 238.050788 + 6 18.998403 = 352.041206 g/mol.
Calculate the relative rate of diffusion of 1 H 2 (molar mass 2.0 g/mol) compared with 2 H 2 (molar mass 4.0 g/mol) and the relative rate of diffusion of O 2 (molar mass 32 g/mol) compared with O 3 (molar mass 48 g/mol).
A gas of unknown identity diffuses at a rate of 83.3 mL/s in a diffusion apparatus in which carbon dioxide diffuses at the rate of 102 mL/s. Calculate the molecular mass of the unknown gas.
When two cotton plugs, one moistened with ammonia and the other with hydrochloric acid, are simultaneously inserted into opposite ends of a glass tube that is 87.0 cm long, a white ring of NH 4 Cl forms where gaseous NH 3 and gaseous HCl first come into contact. At approximately what distance from the ammonia moistened plug does this occur? (Hint: Calculate the rates of diffusion for both NH 3 and HCl, and find out how much faster NH 3 diffuses than HCl.)
Using the postulates of the kinetic molecular theory, explain why a gas uniformly fills a container of any shape.
Can the speed of a given molecule in a gas double at constant temperature? Explain your answer.
Describe what happens to the average kinetic energy of ideal gas molecules when the conditions are changed as follows:
- The pressure of the gas is increased by reducing the volume at constant temperature.
- The pressure of the gas is increased by increasing the temperature at constant volume.
- The average speed of the molecules is increased by a factor of 2.
The distribution of molecular speeds in a sample of helium is shown in Figure 9.34. If the sample is cooled, will the distribution of speeds look more like that of H 2 or of H 2 O? Explain your answer.
What is the ratio of the average kinetic energy of a SO 2 molecule to that of an O 2 molecule in a mixture of two gases? What is the ratio of the root mean square speeds, u rms , of the two gases?
A 1-L sample of CO initially at STP is heated to 546 K, and its volume is increased to 2 L.
- What effect do these changes have on the number of collisions of the molecules of the gas per unit area of the container wall?
- What is the effect on the average kinetic energy of the molecules?
- What is the effect on the root mean square speed of the molecules?
The root mean square speed of H 2 molecules at 25 °C is about 1.6 km/s. What is the root mean square speed of a N 2 molecule at 25 °C?
Answer the following questions:
- Is the pressure of the gas in the hot-air balloon shown at the opening of this chapter greater than, less than, or equal to that of the atmosphere outside the balloon?
- Is the density of the gas in the hot-air balloon shown at the opening of this chapter greater than, less than, or equal to that of the atmosphere outside the balloon?
- At a pressure of 1 atm and a temperature of 20 °C, dry air has a density of 1.2256 g/L. What is the (average) molar mass of dry air?
- The average temperature of the gas in a hot-air balloon is 1.30 10 2 °F. Calculate its density, assuming the molar mass equals that of dry air.
- The lifting capacity of a hot-air balloon is equal to the difference in the mass of the cool air displaced by the balloon and the mass of the gas in the balloon. What is the difference in the mass of 1.00 L of the cool air in part (c) and the hot air in part (d)?
- An average balloon has a diameter of 60 feet and a volume of 1.1 10 5 ft 3 . What is the lifting power of such a balloon? If the weight of the balloon and its rigging is 500 pounds, what is its capacity for carrying passengers and cargo?
- A balloon carries 40.0 gallons of liquid propane (density 0.5005 g/L). What volume of CO 2 and H 2 O gas is produced by the combustion of this propane?
- A balloon flight can last about 90 minutes. If all of the fuel is burned during this time, what is the approximate rate of heat loss (in kJ/min) from the hot air in the bag during the flight?
Show that the ratio of the rate of diffusion of Gas 1 to the rate of diffusion of Gas 2, is the same at 0 °C and 100 °C.
Graphs showing the behavior of several different gases follow. Which of these gases exhibit behavior significantly different from that expected for ideal gases?
Explain why the plot of PV for CO 2 differs from that of an ideal gas.
Under which of the following sets of conditions does a real gas behave most like an ideal gas, and for which conditions is a real gas expected to deviate from ideal behavior? Explain.
- high pressure, small volume
- high temperature, low pressure
- low temperature, high pressure
Describe the factors responsible for the deviation of the behavior of real gases from that of an
ideal gas.
For which of the following gases should the correction for the molecular volume be largest:
CO, CO 2 , H 2 , He, NH 3 , SF 6 ?
A 0.245-L flask contains 0.467 mol CO 2 at 159 °C. Calculate the pressure:
- using the ideal gas law
- using the van der Waals equation
- Explain the reason for the difference.
- Identify which correction (that for P or V) is dominant and why.
Answer the following questions:
- If XX behaved as an ideal gas, what would its graph of Z vs. P look like?
- For most of this chapter, we performed calculations treating gases as ideal. Was this justified?
- What is the effect of the volume of gas molecules on Z? Under what conditions is this effect small? When is it large? Explain using an appropriate diagram.
- What is the effect of intermolecular attractions on the value of Z? Under what conditions is this effect small? When is it large? Explain using an appropriate diagram.
- In general, under what temperature conditions would you expect Z to have the largest deviations from the Z for an ideal gas?


15.7: Sensory System Learning Activities
Interactive Learning Activity: Practice labeling the parts of the eye, Activity 1.
Query \(\PageIndex{1}\)
Interactive Learning Activity: Practice labeling the parts of the eye, Activity 2.
Query \(\PageIndex{2}\)
Interactive Learning Activity: Practice labeling the parts of the ear.
Query \(\PageIndex{3}\)
Interactive Learning Activity: Study sensory system medical terms discussed in this chapter using these flashcards.
Query \(\PageIndex{4}\)
Interactive Learning Activity: Check your knowledge of sensory system terminology.
Query \(\PageIndex{5}\)
Interactive Learning Activity: Using the sample documentation provided, drag the terms on the right to their appropriate spaces within the documentation.
Query \(\PageIndex{6}\)
Interactive Learning Activity: Using the sample documentation provided, drag the terms on the right to their appropriate spaces within the documentation.
Query \(\PageIndex{7}\)
Interactive Learning Activity: Practice identifying and defining word parts for terms discussed in this chapter.
Query \(\PageIndex{8}\)
You can also print this as a Chapter 15 Practice Worksheet and check your answers with this Answer Key PDF .


2.3: Permutations
In this section, we will discuss the theory of permutations. Before doing so, it is important to note how all of our counting structures will be related to each other. The following flowchart summarizes how each structure is naturally derived from the one before it:
This means rather than focusing on the differences between the different counting structures, we should instead focus on crafting a narrative on how each structure leads to another. In this section, we will discuss how The Generalized Principle of Counting naturally leads to the idea of a permutation.
In order to discuss the theory of permutations, we first define what a permutation is.
Suppose a set has \(n \) distinct elements and an experiment consists of selecting \( k\) of the elements one at a time without replacement. Let each outcome consist of the \( k\) elements in the order selected. Each such outcome is called a permutation of \( n \) elements taken \( k\) at a time or is called an ordered sample without replacement.
Moreover, if the \( k\) elements are selected in the following order: \( s_1, s_2, s_3 \ldots, s_k \), then we denote this permutation of \( n \) elements taken \( k \) at a time by the following notation: \( (s_1, s_2, s_3 \ldots, s_k) \).
There is a lot to take in the above definition and so allow us to highlight the important assumptions above.
In order to have a permutation, the following assumptions must be met:
- We have \(n \) distinct elements.
- We are selecting \( k\) of the elements.
- The selection takes place without replacement (meaning there is no repetition of elements).
- For some reason, we wish to note the order of selection. That is, there has to be something special or unique to being chosen first, and something special or unique to being chosen second, and so on and so forth.
Once all four assumptions are met, we can denote the permutation by writing the elements between parenthesis and placing the elements in the appropriate order.
Here is a classic example of a scenario involving a permutation.
Suppose that in a race between 26 runners, we are asked to guess who receives the gold, silver and bronze medals. Explain why our guesses can be thought of as a permutation of 26 elements taken 3 at a time. Can you write down a few different outcomes?
Solution
In order to explain why our guesses can be thought of as permutations of 26 elements taken 3 at a time, we simply verify that all four assumptions outlined above apply to the given situation.
- We have 26 distinct runners.
- We are selecting 3 of the runners.
- The selection takes place without replacement since once we choose a runner to receive gold, that same runner cannot also get silver. Hence, there is no repetition.
- The order of selection should be noted since there is something special or unique to being first (which is receiving gold), something special or unique to being second (which is receiving silver), and something special or unique to being third (which is receiving bronze).
Hence an outcome can be viewed as a permutation of 26 elements taken 3 at a time.
Now suppose the runners are labeled \(A, B, C, D, \ldots, Z \). Here is an example of a permutation of 26 elements taken 3 at a time: \( (M, A, T ) \). In this outcome, runner \(M \) placed first, runner \(A \) placed second, and runner \(T \) placed third.
Here is another (different!) outcome: \( (T, A, M ) \). This outcome is different because here, runner \(T \) placed first, runner \(A \) placed second, and runner \(M \) placed third.
Finally, although this was not asked in the problem, note that \( (Q, V, Q ) \) is not a possible outcome since in a permutation, there is no repetition.
Informally speaking, a permutation of \( n \) elements taken \( k\) at a time is just simply an ordered \(k\)-tuple without replacement. For instance, a permutation of \( n \) elements taken \(2 \) at a time is just simply an ordered pair \( (x,y) \). Meanwhile, a permutation of \( n \) elements taken \(3 \) at a time is just simply an ordered triple \( (x,y,z) \), so on and so forth.
With this idea in mind, allow to try and count the number of permutations of \( n \) elements taken \( k\) at a time. We first start off with a concrete example and then generalize.
Reconsider the race between 26 runners mentioned in Example 1. If we were asked to guess the outcome on who receives gold, silver and bronze, then how many outcomes are possible?
Solution
By the previous discussion, we know that an outcome can be considered a permutation of 26 elements taken 3 at a time. In this question, we are interested in finding the total number of permutations of 26 elements taken 3 at a time. To do this, we note that an outcome is simply an ordered triple that resembles ( __ , __ , __ ). To find the number of such ordered triples, we can think of this as a three-stage experiment.
- Choose a runner for position 1: 26 choices
- Choose a runner for position 2: 25 choices
- Choose a runner for position 3: 24 choices
By The Generalized Principle of Counting, there are \( 26 \times 25 \times 24 = 15,600 \) different permutations.
With the above example in mind, allow us to consider the more general problem:
How many permutations are there of \( n \) elements taken \( k\) at a time? (Try this on your own before checking the answer!)
- Answer
-
In order to answer this question, we first must understand what the problem is asking us. Based off our intuition of a permutation, this problem is asking us to find the total number of ordered \(k\)-tuples without replacement. This means we must find the number of ways to fill in the following structure: \( ( \underbrace{\_\_}_{1}, \underbrace{\_\_}_{2}, \underbrace{\_\_}_{3}, \ldots, \underbrace{\_\_}_{k-1}, \underbrace{\_\_}_{k} ) \).
We can think of filling in the above spaces as a \(k\)-stage experiment. We proceed in steps:
1. Choose an element for position 1: there are \(n \) choices.
2. Choose an element for position 2: there are \(n - 1 \) choices.
3. Choose an element for position 3: there are \(n - 2 \) choices.
...
...
...
k) Choose an element for position k: there are \(n - (k-1) = n - k + 1 \) choices.
By The Generalized Principle of Counting, the number of permutations of \( n \) elements taken \( k\) at a time is given by \( n \times (n-1) \times (n-2) \times \ldots \times (n - k + 1) \).
The above example proves the following theorem.
The number of permutations of \( n \) elements taken \( k\) at a time is given by \( n \times (n-1) \times (n-2) \times \ldots \times (n - k + 1) \).
Proof: Already done in Example 3.
For convenience, we often denote the expression \( n \times (n-1) \times (n-2) \times \ldots \times (n - k + 1) \) by \( P_{n,k} \) which is read as "the number of permutations of \( n \) elements taken \( k\) at a time". Hence we make the following slight addition to the above theorem:
The number of permutations of \( n \) elements taken \( k\) at a time is given by \( P_{n,k} = n \times (n-1) \times (n-2) \times \ldots \times (n - k + 1) \).
Above, we developed a formula for the number of permutations of \( n \) elements taken \( k\) at a time. Allow us to see what happens if we are asked for the number of permutations of \( n \) elements taken \( n \) at a time.
How many permutations are there of \( n \) elements taken \( n\) at a time? (Try this on your own before checking the answer!)
- Answer
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In order to answer this question, we first must understand what the problem is asking us. Based off our intuition of a permutation, this problem is asking us to find the total number of ordered \(n\)-tuples without replacement. That is, given \(n\) distinct elements, we are asking how many ways are there to select \(n\) elements and order them. This means we must find the number of ways to fill in the following structure: \( ( \underbrace{\_\_}_{1}, \underbrace{\_\_}_{2}, \underbrace{\_\_}_{3}, \ldots, \underbrace{\_\_}_{n-1}, \underbrace{\_\_}_{n} ) \).
We can think of filling in the above spaces as a \(n\)-stage experiment. We proceed in steps:
1. Choose an element for position 1: there are \(n \) choices.
2. Choose an element for position 2: there are \(n - 1 \) choices.
3. Choose an element for position 3: there are \(n - 2 \) choices.
...
...
...
n) Choose an element for position k: there is 1 choice.
By The Generalized Principle of Counting, the number of permutations of \( n \) elements taken \( n\) at a time is given by \( n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \).
We denote the above expression as \( n! \).
Formally, allow us to make the following definition and summarize the above result as a theorem.
Given a positive integer \(n\), we define \(n!\), read as "\(n\) factorial" to be
\[ n! = n(n-1)(n-2) \ldots (3)(2)(1) \nonumber\ \]
For the special case that \(n=0\), we define \( 0! = 1 \).
Given \(n\) distinct objects, there are \(n!\) ways to arrange them.
Proof: Already done in Example 4.
How many ways are there to arrange 6 distinct books on a bookshelf?
Solution
By the above theorem, there are \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \) different arrangements.
We present one last theorem for this section followed by three examples. The following theorem establishes how the number of permutations of \(n\) elements taken \(k\) at a time can be expressed in terms of factorials.
\( \displaystyle P_{n,k} = \frac{n!}{(n-k)!} \)
Proof: The proof is left as an exercise to the reader.
Suppose that from a group of 30 people, a president, vice president, secretary and treasurer are to be chosen. How many different outcomes are possible?
- Solution 1
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For this question, allow us to proceed in steps.
- We first choose a president. There are 30 choices.
- We then choose a vice president. There are 29 choices.
- We then choose a secretary. There are 28 choices.
- We finally choose a treasurer. There are 27 choices.
By The Generalized Principle of Counting, we see that there are \( 30 \times 29 \times 28 \times 27 = 657,720 \) different choices. (Notice that by invoking The Generalized Principle of Counting, we are saying that each step above is different or distinct. This will be important in the next example!)
- Solution 2
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We see that the above question satisfies the four assumptions required to have a permutation. That is, we can think of an outcome as an ordered 4-tuple such as \( ( \underbrace{\_\_}_{pres.}, \underbrace{\_\_}_{vp.}, \underbrace{\_\_}_{sec.}, \underbrace{\_\_}_{trea.} ) \). By our theorem , the number of different permutations of 30 elements taken 4 at a time is given by \( P_{30,4} = 30 \times \times 29 \times 28 \times 27 = 657,720 \) choices.
Suppose that from a group of 30 people, a president, vice president, and two secretaries are to be chosen. How many different outcomes are possible?
Critique This Solution:
For this question, allow us to proceed in steps.
- We first choose a president. There are 30 choices.
- We then choose a vice president. There are 29 choices.
- We then choose a secretary. There are 28 choices.
- We then choose another secretary. There are 27 choices.
By The Generalized Principle of Counting, we see that there are \( 30 \times 29 \times 28 \times 27 = 657,720 \) different choices.
Alternatively, we see that the above question satisfies the four assumptions required to have a permutation. That is, we can think of an outcome as an ordered 4-tuple such as \( ( \underbrace{\_\_}_{pres.}, \underbrace{\_\_}_{vp.}, \underbrace{\_\_}_{sec.}, \underbrace{\_\_}_{sec.} ) \). By our theorem , the number of different permutations of 30 elements taken 4 at a time is given by \( P_{30,4} = 30 \times \times 29 \times 28 \times 27 = 657,720 \) choices. Hence we get the same answer.
Do you think the above solution is correct or incorrect? Why?
- Answer
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Unfortunately, the above solution is incorrect! It is incorrect to use The Generalized Principle of Counting. Recall that The Generalized Principle of Counting requires us to have distinct steps! In the proposed solution, steps 3 and 4 are the same.
Additionally, the second solution in using the permutation is incorrect because we have violated the requirement discussed under the assumptions of a permutation. To have a permutation, there must be something special about the order. Here, there is no difference between the third person being chosen and the fourth and hence we no longer have a permutation.
Thus the proposed solutions are incorrect. In order to correctly answer this problem, we need an additional counting structure that will be discussed in the next section.
In how many different ways can 12 people be seated in a row if two of the people, \(A\) and \(B\), must be seated together?
Critique This Solution:
We have 12 distinct people and we must arrange all 12 people. By our theorem above, there are \( 12! = 479,001,600 \} ways to do so.
Do you think the above solution is correct or incorrect? Why?
- Answer
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Unfortunately, the above solution is incorrect! While the statement itself is true, (there are indeed \12!\) ways to arrange 12 people), the statement does not answer the appropriate question. In \(12!\), we are obtaining all possible arrangements of the 12 people and hence we get arrange arrangements where person \(A)\) is not seated with person \(B\). So how can we fix this?
Here is some advice: whenever a problem requires certain elements to stick together, a good idea is to "bundle" those elements together. For instance, initially we have the following diagram:
\[ \boxed{A} , ~\boxed{B} , ~ \boxed{C} , ~ \boxed{D} , ~ \boxed{E} ,~ \boxed{F} , ~ \boxed{G} , ~ \boxed{H} , ~ \boxed{I} , ~ \boxed{J} , ~ \boxed{K} , ~ \boxed{L} \nonumber\ \]
However, since person \(A\) must be seated with person \(B\), we can regard them as one element. We accomplish this by "bundling" \(A\) and \(B\) together or alternatively gluing \(A\) and \(B\) side-by-side to make sure they are together.  Thus, we proceed in steps:
1) Arrange the 11 blocks above. There are 11! ways to do so.
2) Within every arrangement of the 11 blocks, we can switch persons \(A \) and \(B\). That is, we can either have AB or BA. There are 2 choices here.
By The Basic Principle of Counting, we have a total of \( 11! \times 2 = 79,833,600 \) different arrangements where persons \(A\) and \(B\) are seated together.


12.5: Pluto and Charon
By the end of this section, you will be able to:
- Compare the orbital characteristics of Pluto with those of the planets
- Describe information about Pluto’s surface deduced from the New Horizons images
- Note some distinguishing characteristics of Pluto’s large moon Charon
Pluto is not a moon, but we discuss it here because its size and composition are similar to many moons in the outer solar system. Our understanding of Pluto (and its large moon Charon ) have changed dramatically as a result of the New Horizons flyby in 2015.
Is Pluto a Planet?
Pluto was discovered through a careful, systematic search, unlike Neptune , whose position was calculated from gravitational theory. Nevertheless, the history of the search for Pluto began with indications that Uranus had slight departures from its predicted orbit, departures that could be due to the gravitation of an undiscovered “Planet X.” Early in the twentieth century, several astronomers, most notably Percival Lowell , then at the peak of his fame as an advocate of intelligent life on Mars, became interested in searching for this ninth planet.
Lowell and his contemporaries based their calculations primarily on tiny unexplained irregularities in the motion of Uranus. Lowell’s computations indicated two possible locations for a perturbing Planet X; the more likely of the two was in the constellation Gemini. He predicted a mass for the planet intermediate between the masses of Earth and Neptune (his calculations gave about 6 Earth masses). Other astronomers, however, obtained other solutions from the tiny orbital irregularities, even including one model that indicated two planets beyond Neptune.
At his Arizona observatory, Lowell searched without success for the unknown planet from 1906 until his death in 1916, and the search was not renewed until 1929. In February 1930, a young observing assistant named Clyde Tombaugh (see the Clyde Tombaugh: From the Farm to Fame feature box), comparing photographs he made on January 23 and 29 of that year, found a faint object whose motion appeared to be about right for a planet far beyond the orbit of Neptune (Figure 12.17). The new planet was named for Pluto, the Roman god of the underworld, who dwelt in remote darkness, just like the new planet. The choice of this name, among hundreds suggested, was helped by the fact that the first two letters were Percival Lowell’s initials.
Although the discovery of Pluto appeared initially to be a vindication of gravitational theory similar to the earlier triumph of Adams and Le Verrier in predicting the position of Neptune, we now know that Lowell’s calculations were wrong. When its mass and size were finally measured, it was found that Pluto could not possibly have exerted any measurable pull on either Uranus or Neptune. Astronomers are now convinced that the reported small anomalies in the motions of Uranus are not, and never were, real.
From the time of its discovery, it was clear that Pluto was not a giant like the other four outer solar system planets. For a long time, it was thought that the mass of Pluto was similar to that of Earth, so that it was classed as a fifth terrestrial planet, somehow misplaced in the far outer reaches of the solar system. There were other anomalies, however, as Pluto’s orbit was more eccentric and inclined to the plane of our solar system than that of any other planet. Only after the discovery of its moon Charon in 1978 could the mass of Pluto be measured, and it turned out to be far less than the mass of Earth.
In addition to Charon, Pluto has four much smaller moons. Subsequent observations of Charon showed that this moon is in a retrograde orbit and has a diameter of about 1200 kilometers, more than half the size of Pluto itself (Figure 12.18). This makes Charon the moon whose size is the largest fraction of its parent planet. We could even think of Pluto and Charon as a double world. Seen from Pluto, Charon would be as large as eight full moons on Earth.
To many astronomers, Pluto seemed like the odd cousin that everyone hopes will not show up at the next family reunion. Neither its path around the Sun nor its size resembles either the giant planets or the terrestrial planets. In the 1990s, astronomers began to discover additional small objects in the far outer solar system, showing that Pluto was not unique. We will discuss these trans-neptunian objects later with other small bodies, in the chapter on Comets and Asteroids: Debris of the Solar System. One of them (called Eris ) is nearly the same size as Pluto, and another ( Makemake ) is substantially smaller. It became clear to astronomers that Pluto was so different from the other planets that it needed a new classification. Therefore, it was called a dwarf planet , meaning a planet much smaller than the terrestrial planets. We now know of many small objects in the vicinity of Pluto and we have classified several as dwarf planet s.
A similar history was associated with the discovery of the asteroids . When the first asteroid ( Ceres ) was discovered at the beginning of the nineteenth century, it was hailed as a new planet. In the following years, however, other objects were found with similar orbits to Ceres. Astronomers decided that these should not all be considered planets, so they invented a new class of objects, called minor planets or asteroids. Today, Ceres is also called a dwarf planet. Both minor planets and dwarf planets are part of a whole belt or zones of similar objects (as we will discuss in Comets and Asteroids: Debris of the Solar System).
So, is Pluto a planet? Our answer is yes, but it is a dwarf planet , clearly not in the same league with the eight major planets (four giants and four terrestrials). While some people were upset when Pluto was reclassified, we might point out that a dwarf tree is still a type of tree and (as we shall see) a dwarf galaxy is still a type of galaxy.
Clyde Tombaugh discovered Pluto when he was 24 years old, and his position as staff assistant at the Lowell Observatory was his first paying job. Tombaugh had been born on a farm in Illinois, but when he was 16, his family moved to Kansas. There, with his uncle’s encouragement, he observed the sky through a telescope the family had ordered from the Sears catalog. Tombaugh later constructed a larger telescope on his own and devoted his nights (when he wasn’t too tired from farm work) to making detailed sketches of the planets (Figure 12.19).
In 1928, after a hailstorm ruined the crop, Tombaugh decided he needed a job to help support his family. Although he had only a high school education, he thought of becoming a telescope builder. He sent his planet sketches to the Lowell Observatory, seeking advice about whether such a career choice was realistic. By a wonderful twist of fate, his query arrived just when the Lowell astronomers realized that a renewed search for a ninth planet would require a very patient and dedicated observer.
The large photographic plates (pieces of glass with photographic emulsion on them) that Tombaugh was hired to take at night and search during the day contained an average of about 160,000 star images each. How to find Pluto among them? The technique involved taking two photographs about a week apart. During that week, a planet would move a tiny bit, while the stars remained in the same place relative to each other. A new instrument called a “blink comparator” could quickly alternate the two images in an eyepiece. The stars, being in the same position on the two plates, would not appear to change as the two images were “blinked.” But a moving object would appear to wiggle back and forth as the plates were alternated.
After examining more than 2 million stars (and many false alarms), Tombaugh found his planet on February 18, 1930. The astronomers at the observatory checked his results carefully, and the find was announced on March 13, the 149th anniversary of the discovery of Uranus. Congratulations and requests for interviews poured in from around the world. Visitors descended on the observatory in scores, wanting to see the place where the first new planet in almost a century had been discovered, as well as the person who had discovered it.
In 1932, Tombaugh took leave from Lowell, where he had continued to search and blink, to get a college degree. Eventually, he received a master’s degree in astronomy and taught navigation for the Navy during World War II. In 1955, after working to develop a rocket-tracking telescope, he became a professor at New Mexico State University, where he helped found the astronomy department. He died in 1997; some of his ashes were placed inside the New Horizons spacecraft to Pluto.
The Nature of Pluto
Almost everything we know about Pluto and its moons comes from the New Horizons spacecraft, which flew by in 2015 before traveling on into the outermost parts of the planetary system. Using data from the New Horizons probe, astronomers have measured the diameter of Pluto as 2370 kilometers, only 60 percent as large as our Moon. From the diameter and mass, we find a density of 1.9 g/cm 3 , suggesting that Pluto is a mixture of rocky materials and water ice in about the same proportions as many outer-planet moons.
Parts of Pluto’s surface are highly reflective, and its spectrum demonstrates the presence on its surface of frozen methane, carbon monoxide, and nitrogen. The maximum surface temperature ranges from about 50 K when Pluto is farthest from the Sun to 60 K when it is closest. Even this small difference is enough to cause a partial sublimation (going from solid to gas) of the methane and nitrogen ice. This generates an atmosphere when Pluto is close to the Sun, and it freezes out when Pluto is farther away. Observations of distant stars seen through this thin atmosphere indicate that the surface pressure is about a ten-thousandth of Earth’s. Because Pluto is a few degrees warmer than Triton, its atmospheric pressure is about ten times greater. This atmosphere contains several distinct haze layers, presumably caused by photochemical reactions, like those in Titan’s atmosphere (Figure 12.20).
Reaching Pluto with a spacecraft was a major challenge, especially in an era when reduced NASA budgets could not support large, expensive missions like Galileo and Cassini. Yet like Galileo and Cassini, a Pluto mission would require a nuclear electric system that used the heat from plutonium to generate the energy to power the instruments and keep them operating far from the warmth of the Sun. NASA made available one of the last of its nuclear generators for such a mission. Assuming an affordable but highly capable spacecraft could be built, there was still the problem of getting to Pluto, nearly 5 billion kilometers from Earth, without waiting decades. The answer was to use Jupiter’s gravity to slingshot the spacecraft toward Pluto.
The 2006 launch of New Horizons started the mission with a high speed, and the Jupiter flyby just a year later gave it the required additional boost. The New Horizons spacecraft arrived at Pluto in July 2015, traveling at a relative speed of 14 kilometers per second (or about 50,000 kilometers per hour). With this high speed, the entire flyby sequence was compressed into just one day. Most of the data recorded near closest approach could not be transmitted to Earth until many months later, but when it finally arrived, astronomers were rewarded with a treasure trove of images and data.
Geology of Pluto
Pluto is not the geologically dead world that many anticipated for such a small object—far from it. The division of the surface into areas with different composition and surface texture is apparent in the global color photo shown in Figure 12.21. The reddish color is enhanced in this image to bring out differences in color more clearly. The darker parts of the surface are cratered, but adjacent to them is a nearly featureless light area in the lower right quadrant of this image. This is a huge ice-filled depression called the Sputnik Plains, named for the first artificial Earth satellite. The absence of impact craters suggests a surface no more than 10 million years old. The dark areas show the colors of photochemical haze or smog similar to that in the atmosphere of Titan. The dark material that is staining these old surfaces could come from Pluto’s atmospheric haze or from chemical reactions taking place at the surface due to the action of sunlight.
Figure 12.22 shows some of the remarkable variety of surface features New Horizons revealed. we are finding on Pluto. Toward the right on this image, we see the shoreline of the Sputnik Plains, showing evidence of relatively recent geological activity. The nitrogen and carbon monoxide ice that fills the Sputnik depression shows cells or polygons with an average width of more than 30 kilometers, caused by slow convection in the ice. 2
Figure 12.23 shows another view of the boundary between different types of geology. The width of this image is 250 kilometers, and it shows dark, ancient, heavily cratered terrain; dark, uncratered terrain with a hilly surface; smooth, geologically young terrain; and a small cluster of mountains more than 3000 meters high. In the best images, the light areas of nitrogen ice seem to have flowed much like glaciers on Earth, covering some of the older terrain underneath them.
The isolated mountains in the midst of the smooth nitrogen plains are probably also made of water ice, which is very hard at the temperatures on Pluto and can float on frozen nitrogen. Additional mountains, and some hilly terrain that reminded the mission scientists of snakeskin, are visible in part (b) of Figure 12.23.
A Quick Look at Charon
To add to the mysteries of Pluto, we show in Figure 12.24 one of the best New Horizons images of Pluto’s large moon Charon . Recall from earlier that Charon is roughly half Pluto’s size (its diameter is about the size of Texas). Charon keeps the same side toward Pluto, just as our Moon keeps the same side toward Earth. What is unique about the Pluto-Charon system, however, is that Pluto also keeps its same face toward Charon. Like two dancers embracing, these two constantly face each other as they spin across the celestial dance floor. Astronomers call this a double tidal lock.
What New Horizons showed was another complex world. There are scattered craters in the lower part of the image, but much of the rest of the surface appears smooth. Crossing the center of the image is a belt of rough terrain, including what appear to be tectonic valleys, as if some forces had tried to split Charon apart. Topping off this strange image is a distinctly red polar cap, of unknown composition. Many features on Charon are not yet understood, including what appears to be a mountain in the midst of a low-elevation region.
Footnotes
- 2 In the process of convection, warmer material rises and colder material sinks, transferring energy. You are probably more familiar with convection in warm fluids, like the cells you see when you are heating some kinds of soup, but it can also happen (very slowly) in colder substances.


6.5: Creating the Introduction, Step 3- Relate to Self (Establish Credibility)
Note to Self
After grabbing the audience members’ attention and showing them how the topic relates to them personally, the speaker should relate the topic back to themselves and answer the questions: “Why me? Why now?” To establish credibility as a speaker, this step is absolutely necessary, because audiences want to know up front why they should trust or believe the speaker as he or she delivers all subsequent information to them in the presentation. For example, the speaker may be an expert or an authority on the chosen topic, but he or she may also have nothing more than a vested interest in the topic and a genuine desire to share the information with others.
Scholars and philosopher have conducted a lot of research on Aristotle’s concept called “ethos,” which mean credibility. The late James McCroskey, a communication professor at the University of Alabama– Birmingham, and Jason Teven, a professor at California State University, Fullerton, spent years researching the concept of credibility. They identified three components of individual credibility : competence, trustworthiness, and caring/goodwill (1999). This portion of the introduction should provide three important pieces of information to the audience:
- Home Competence : The degree to which an audience perceives a speaker as being knowledgeable or expert on a given topic. What are the speaker’s credentials, if any, regarding that person’s relationship with the information? Has the person researched the topic extensively or taken coursework in this area of study? Does the speaker’s work experience relate to the topic? Without giving the audience an exhaustive résumé and work history, make this connection clear, yet brief. Explain just enough to establish credibility.
- Home Trustworthiness : The degree to which an audience member perceives a speaker as having honest and sincere qualities. Why does the speaker want to discuss this topic? What vested interest does the speaker have in conveying this information to the audience today?
Note to Self
- Home Caring/Goodwill : The degree to which audience members perceive a speaker as genuinely caring about them.
Note to Self
Review the example below that combines these three aspects:
“ I’m a biology major, so talking with you about the potential for a pandemic relates pretty heavily to my future career goals ( Competence ). Pandemic diseases fascinate me, which is why those goals include pursuing a career in medical research ( Trustworthiness ). I believe that we should all have a basic understanding of what a pandemic flu could do to us as a society, so that we can better prepare ourselves for it, when the next pandemic does strike ( Caring / Goodwill ).”
Be careful with this step, though. Try not to seem boastful or arrogant. Instead of stating, “I have a Ph.D. in this subject, so I’m the most highly educated person in this room,” say something like, “I have extensive training and education on this subject, and since it was the topic of my doctoral studies, I feel well prepared to share some of my findings with you today.” Again, keep this section relatively brief, but long and detailed enough to provide the audience with a reason to trust this particular message.


10.4: The water cycle and climate change
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- Michael E. Ritter
- University of Wisconsin-Stevens Point via The Physical Environment
Changes to the patterns of current precipitation and those projected for the future on account of climate change are discussed in Chapter 6, "Future Geographies: Global Precipitation Patterns". The kind of precipitation is already changing, as northern regions are experiencing more rain than snow. Total snow cover has on average decreased in both hemispheres.The variability of precipitation is changing as well. Many subtropical regions have been or are gripped in devastating droughts. Precipitation is but one element of the hydrologic cycle, that will be affected by climate. The impact of climate change on water resources will have profound effects on ecosystems and humans.
Evapotranspiration
As noted earlier in this chapter, potential evapotranspiration is a measure of water need based primarily on the energy input. Potential evapotranspiration is expected to increase in nearly all regions. As temperature rises the saturation point of air increases, but relative humidity is not expected to change significantly. As a result, the evaporation of water increases yielding a larger amount of water vapor in the air. The increase evaporation of water from the soil places a greater strain on plants as their need for water to maintain growth processes increases. Actual evaporation is projected to increase over most bodies of water, the amount depending on the geographic variation in surface warming.
Plants take in carbon dioxide and release water vapor through their stomata. The increase in atmospheric carbon dioxide has competing roles in the affect of evapotranspiration from vegetation. At higher CO 2 concentrations, stomata need to open less to take in the same amount of carbon dioxide for photosynthesis. However, increased carbon dioxide can act a gaseous fertilizer to spur plant growth and leaf area. Increased leaf area results in more transpiration. The degree to which higher CO 2 concentration have depend on the type of vegetation, nutrient availability, temperature changes, and water availability
Surface water and Runoff
The impact of climate change on river flows and lake levels resulting depend on changes in the amount, type, and timing of precipitation. Changes in evaporation rates also affect river and lake levels. Global scale climate and hydrological models project runoff increases in the wet tropics and high latitudes, and decreases in the mid-latitudes and some parts of the dry tropics. Most studies show that rising temperatures lead to changes in the seasonality of river flow due to changes in the type of precipitation. Higher winter flows occurring in the European Alps, Scandinavia, the Himalayas, western, central, and eastern North America will result from more precipitation falling as rain and earlier snow melt. The frequency of heavy precipitation events are expected to increase in an ever warming world for most regions. This will likely increase the threat for flash flooding and urban flooding.
Ground water
Ground water levels are maintained when aquifers are adequately recharged. How this occurs varies with soil conditions, vegetation and precipitation intensity and timing. Generally speaking, if precipitation intensity is greater than the infiltration capacity of the soil, most water will runoff and not percolate down into the ground water zone. However, more frequent and larger precipitation events projected for some humid region may result in decreased groundwater recharge as the infiltration capacity of the soil is exceeded more often. In semi-arid and arid regions, groundwater level may rise with high-intensity storms as rainfall is ability to infiltrate downward before evaporating.
Soil moisture
The amount of moisture held in the soil depends on when and how much precipitation and evaporation occurs at a place. The resulting changes to the spatial patterns of soil moisture is expected to be quite similar to that of changes in precipitation. Though the amount of change is not well established, projections generally agree on the kind of change that will occur. A reduction in soil moisture in the subtropics and Mediterranean regions will accompany a decrease in precipitation. Projected decreases in snow cover at high latitudes will also result in less soil moisture. Soil moisture will likely increase in East Africa and central Asia where increases precipitation is expected to occur.
Water supply
Population growth, economic development and land use change largely determine the stress put on water supplies. Climate change will be an additional factor that affects stressed water resources. Glacier meltwater provides a source of water for many living in mountainous underdeveloped countries. Recent loss of glacier ice has or will place these people at risk. Peruvian glaciers have lost more than 20% of their mass in the past 35 years, resulting in a 12 percent decrease in runoff to the country’s coastal region, where 60 percent of Peru’s population live.
Agriculture, industry, and urban populations across the earth depend on major river systems to transpiration, irrigation, and drinking water. Millions of acres of farmland and tens of million people rely on the Colorado River system for water. Impoundment of water behind large dams regulate the flow and delivery of water supplies. The timing of when regulated flows deliver water is based on the experience of the past 100 years. Scheduled delivery of water may be missed 60 to 90 percent of the time as human-induced climate change continues to make the Colorado River basin drier.