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General Chemistry--Unit 2

04/25/09

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The Gaseous State


�Before You Begin:

To master this material you will need to be able to perform unit analysis and round answers to the correct number of significant figures. You should review the measurement section from 'Prep Chem' if you haven't already done so.


The Gaseous State

balloonsA gas is a sample of matter that takes the shape of the container and expands to fill the entire volume of the container.

 

Early Experimentation with Gases

When dealing with solids and liquids, as long as you don’t accidentally melt or boil things, the size of a sample doesn’t vary much with temperature or pressure. With gases, since they don’t have definite volume, the size of the sample changes as pressure and temperature change, even if the quantity (mass) of gas does not.  Four variables are needed to specify the size of a sample of gas: mass/moles, volume, temperature and pressure. Note that pressure is force per unit area. In the case of a gas it is the force that a gas exerts on the walls of its container.

Gases are very tricky to handle; they tend to leak out. They are hard to weigh, mainly because it is really hard to get all of the gas out so that you can weigh the empty container. Early experimenters measured changes in the other variables and hoped that the mass remained constant (no leaks) or relied on stoichiometeric principles to deduce the amount of gas. Measuring pressure was somewhat tricky, but not nearly as hard as measuring the temperature. Early thermometers were very unreliable and the earliest, called thermoscopes, did not have a numerical scale. Significant breakthroughs in understanding gas behavior stalled until Fahrenheit proposed a standardized thermometer scale in the early 1700s.

 

How a Barometer Works:

The air around us is the bottom of a layer of gas that is roughly 30 kilometers thick. The Earth’s gravity pulls this gas. This results in a force (the weight of all that gas) applied over an area (the surface of the Earth or your body or anything exposed to the air). Since pressure is force per unit area, the blanket of air surrounding the Earth has a pressure of 100,000 Pa at sea level. This pressure varies with elevation (the layer of air is not so thick in the mountains) and with weather (warm air expands, and, since it is less dense than cold air, it rises, more air flows in to replace it and wind is born).

barometer

A barometer is a device for measures atmospheric pressure. A barometer consists of a reservoir of liquid mercury and an empty glass tube, sealed at one end and open at the other. The liquid mercury is in contact with the atmosphere. The open end of the glass tube touches the surface of the mercury. Air pressure pushes on the surface of the mercury reservoir and forces the mercury to rise in the tube. The height of the column of mercury is proportional to the air pressure. At sea level, it is 76 cm high. Barometers use mercury because it is very dense, for a liquid. If a barometer uses a column of water instead of mercury, it has to be about nine meters high. Check out “Strange Loops” for an interesting account of the history of the barometer.

How a Manometer Works:

A manometer is a device for measuring the pressure of an enclosed gas. An open U-tube manometer is one of the simplest. When your chemistry professor a young and foolish, these manometers were assembled by students in high school chemistry and physics classes. As with the barometer, gas pressure is measured by observing the height of a column of liquid which that pressure of gas can support. As with the barometer, the best liquid is mercury, but, unlike the barometer, these instruments are open. Mercury inevitably gets spilled, accidentally or on purpose, so these have been replaced with electronic pressure gauges in most school laboratories.

 

manometer 

 

A gas is trapped in a u-shaped glass tube by liquid mercury. One end of the tube is open to the atmosphere. Air pressure pushes against the mercury as does the pressure of the trapped gas. Because it is a liquid, the mercury flows until the forces are balanced. The difference in the heights of the mercury columns measured in mm is the same as the difference in pressure between the atmosphere and the trapped gas. In this case, the trapped gas at higher pressure than the atmosphere (it has displaced the mercury toward the opening, so it must be pushing harder than the air).  The difference in height is 47 mmHg. Suppose air pressure today is 765 mmHg. The trapped gas pressure must be 812 mmHg or 1.07 atm. See the ‘Prep Chem’ section on derived units for more information about conversion of pressure units.

The Individual Gas Laws

Historical laws were based on observations named for pioneers in gas experimentation.

The pressure/volume law is named after Robert Boyle, a British chemist, but the same law is called Mariotte’s Law in Europe after Edme Mariotte, a French chemist. Mariotte did similar experiments and reached the same conclusion as Boyle but slightly later. While most U.S. textbooks call the temperature/volume law Charles’ Law after Jacques Alexandre Charles, some give the credit to Joseph Gay-Lussac while others hedge and call it “Charles’ and Gay-Lussac’s Law.” Apparently, Charles did the research first (1787), but Gay-Lussac published first (1802). Credit could also be given to Guillaume Amontons, who, in 1702, invented a thermometer based on gas expansion. It wasn’t a very good thermometer, but his work was a hundred years earlier than Gay-Lussac’s.

The Law of Combining Volumes is sometimes referred to as Gay-Lussac’s Law. Gay-Lussac noted that when gases react to form a gaseous product, the ratio of the gas volumes form small whole numbers (this is similar to Dalton’s Law of Multiple Proportions for similar reasons). Avogadro used this data to support his hypothesis that, if two gases at the same pressure and temperature have the same volume, they must have the same number of particles. This idea lead to Avogadro’s Law, which everybody calls Avogadro’s Law; however, it could just as easily be named Gay-Lussac’s Law.

For most chemistry courses, your instructor will not require that you know which scientist formulated which law, how he did it, and when (especially since there isn’t uniform agreement on who gets credit for which law). All instructors will expect that you understand and remember the principles of all of the laws.

Boyle’s Law, Pressure and Volume

This gas law was the earliest, 1662, because air pumps and manometers were invented before a calibrated thermometer.

If temperature and amount of gas are kept constant, the pressure and volume are inversely proportional. This can be expressed mathematically as

                                                           P is inversely proportional to V
P=c/V
PV=c

 

where P and V are the pressure and volume of a gas and c is a proportionality constant. It is possible to calculate the resulting volume (or pressure) as pressure (or volume) changes if one knows the initial volume and pressure.

                                                              

P1V1=P2V2

In observational terms, if the pressure increases, the volume of a gas must decrease, and vice versa. People who live in the mountains are familiar with this phenomenon. Bags of chips are packaged at close to sea level then shipped up to the mountains where the air pressure is much lower. The air in the sealed bags expands to make little chip filled balloons.

A typical freshman chemistry or physics experiment on Boyle’s Law will have a pressure gauge connected to a large syringe. Students change the volume of air trapped in the syringe by depressing or pulling out the plunger, which changes the pressure of the gas trapped inside. A graph of pressure and the reciprocal of the volume of the gas will be a straight line. In the middle of the seventeenth century, Robert Boyle wasn’t lucky enough to have digital pressure gauges and plastic tubing. He performed his gas experiments using air trapped in a u-shaped glass tube by a mercury plug. Pouring additional mercury into the tube increased the pressure and decreased the volume of the trapped gas.


4Concept Check: 45 mL of air are trapped in a syringe at 1.3 atm of pressure. The gas is compressed by pressing the plunger until the volume is 27 mL. What is the pressure at this volume?

Answer:

                         P=1.3atm(45mL)/27mL=2.2atm

 


 

 

Charles’ Law, Volume and Temperature

This law is much later than Boyle’s Law because we had to wait for reliable thermometers to become available. If pressure and amount of gas are kept constant, the volume and temperature of a gas are directly proportional. This can be expressed mathematically as

                                                            V is proportional to T
V=cT
V/T=c

 

where V and T are volume and temperature, respectively, and c is proportionality constant. It is possible to calculate the resulting volume (or temperature) as temperature (or volume) changes if one knows the initial volume and temperature.

                                                          V1/T1=V2/T2

 

In observational terms, if the temperature increases, the volume of a gas also increases. This is why a cake rises when you bake it. The leavening agents in cake batter produce tiny bubbles of carbon dioxide. When heated, the bubbles expand and the cake gets fluffy and delicious.

A typical freshman chemistry or physics Charles’ Law experiment will have a syringe or piston and a manometer connected to a sealed flask. The flask is submerged in a water bath and slowly heated. The piston allows the volume to change as the air inside the flask warms up. The manometer allows the students to make sure that all the temperature and volume data points are taken at the same pressure, usually atmospheric pressure. A graph of volume on the y-axis and temperature on the x-axis is a straight line with a y-intercept at -273 ºC. This value is set as zero on the Kelvin temperature scale. You need to use the Kelvin scale for all Charles’ Law calculations (actually all gas law problems) because a stray negative sign can imply that a volume is negative, which makes no sense whatsoever. See the ‘Prep Chem’ section on measurement to review conversions, including temperature scale conversions.


4Concept Check: A sample of a gas at 23 ºC is heated to 75 ºC. If the volume starts at 253 mL, what is the final volume?

Answer:

                                                                       

253mL(75+273K)/(23+273K)=297mL

 


The Third Gas Law, P and T

The relationship between pressure and temperature of a gas is sometimes called Gay-Lussac’s Law, but his experiments measured volume and temperature, like Charles’. If volume and amount of gas are kept constant, the pressure and temperature are directly proportional.  This can be expressed mathematically as

                                                           P is proportional to T
P=cT
P/T=c

 where P and T are pressure and temperature, respectively, and c is a proportionality constant. It is possible to calculate the resulting pressure (or temperature) as temperature (or pressure) changes if one knows the initial volume and temperature.

                                                           P1/T1=P2/T2

 

In observational terms, if the temperature of an enclosed gas increases, the pressure also increases. This is why popcorn pops. There is a small amount of water inside the kernel of corn. As the corn heats, the water vaporizes and the pressure of the hot gas rises. Eventually, the husk splits and the foamy corn ‘guts’ burst out. If you look closely at a piece of popcorn, you can see that it is spongy with tiny air bubbles.

 

Combined Gas Law

Since volume is proportional to temperature and inversely proportional to pressure, the three variables can be expressed together for situations in which the amount of gas is kept constant but pressure, volume, and/or temperature vary. This can be expressed mathematically as

                                                                       

V1P1/T1=V2P2/T2

 

If any of the three variables remains constant, this formula reduces to one of the individual gas laws discussed above. Therefore, learning this single formula is more productive than learning each of the individual gas laws.

Note that we use the combined gas law to predict what will happen to a gas if the pressure, volume, and/or temperature of a set amount of gas changes.


4Concept Check: 127 mL of a gas at standard temperature and pressure, STP, are heated to 82 ºC and allowed to expand to 233 mL. What is the resulting pressure?

Answer: For the gas laws, standard temperature and pressure, STP, are 0 ºC and 1 atm. Most chemistry professors will require you memorize STP because it makes typing up quiz questions faster and easier.

 

P2=V1P1/T1(T2/V2)
P=127mL(1.00atm)(355K)/[(273K)(233mL)]=0.709atm

 


Avogadro’s Law, n and V

If pressure and temperature are kept constant, the volume of a gas is directly proportional to number of moles of a gas. This can be expressed mathematically as

                                                              is proportional to n
V=cn

 

where n and V are the volume and the number of moles, respectively, and c is a proportionality constant. In observational terms, if a gas is kept at constant temperature and pressure and more gas is added the volume will increase. This is why a balloon gets bigger as you blow it up. This relationship is true regardless of the identity of the gas. One mole of any ideal gas occupies 22.4 L of volume at STP.

Freshman chemistry and physics students usually don’t do a lab to illustrate Avogadro’s Law. This is because it difficult to weigh a gas, or rather, to weigh the empty container.

Ideal Gas Equation

The individual gas laws are based upon experimental observation. An Ideal Gas is one that obeys the mathematical predictions of these laws. We will discuss what a gas needs in order to be “ideal” in the next section.

Because volume is proportional to temperature, number of moles, and the inverse of pressure, all of these variables can be stated in a single formula as

                                                      V is proportional to T
V is proportional to n
V is proportional to 1/P
so V is proportional to nT/P
PV=nRT where R is a proportionality constant

where P, V, T, and n are pressure, volume, temperature, and number of moles of gas and R is the proportionality constant known as the gas constant. The ideal gas equation is used for situations in which conditions of three of the four gas variables are known in order to find one unknown. Note that this equation is not useful for situations in which the conditions of the gas change.

The value of the gas constant depends on the units of the variables. The temperature must be in Kelvin because the other temperature scales have negative temperature values which might suggest pressure, volume, and number of moles could have negative values, and that doesn’t make sense. The most common form of the gas constant is R = 0.08206 L*atm/mol*K. See the ‘Prep Chem’ unit on measurement for a sample problem involving the units of the gas constant and conversion of pressure and temperature units.

Finding the molar mass of a vapor is a typical freshman chemistry or physics lab experiment illustrating the ideal gas equation. In that experiment, students measure pressure, temperature, and volume to calculate number of moles. They use the mass of the gas and the number of moles to calculate molar mass in grams per mole. But wait! We wrote earlier that it is hard to weigh a gas because it is hard to weigh the empty container. This experiment gets around this difficulty in a very clever way that most students don’t recognize. Students weigh an empty flask which is really full of air. The substance used in the experiment is a liquid at room temperature but boils when placed in a hot water bath. Students start with enough liquid to make five or ten liters of vapor in a small flask. The flask is open to the atmosphere, so the pressure inside the flask has to stay at atmospheric pressure, even though, as the liquid boils, lots of vapor forms. As the amount of vapor rises, the extra vapor AND AIR in the flask leave so that the pressure can stay 1 atm. Since the vapor molecules outnumber the air molecules by several orders of magnitude, the odds are very low that any air is left after several minutes of being flushed by streams of excess vapor. The flask full of hot vapor is placed in a cold water bath. The vapor condenses, and air enters the flask. The mass of the air ends up being subtracted as part of the ‘empty’ container mass. Clever, huh? We would give credit to whoever first thought of this trick, but, the lab is so ubiquitous that we have no idea who to thank.


4Concept Check: 17.5 grams of nitrogen gas at 22 ºC are under 974 mmHg pressure. What is the volume of this sample?

Answer:  Convert the mass of nitrogen to moles using the formula weight. Note that nitrogen gas is composed of nitrogen molecules, N2. Convert temperature to Kelvin and pressure to atmospheres. Use the value R = 0.08206 L*atm/mol*K.

 

V=nRT/P=0.625mol(0.08206L atm/mol K)(295K)/1.28atm=11.8L

 

 



4Concept Check: Students performed a “Molar Mass of an Unknown Vapor” laboratory experiment. After heating an unknown liquid in a boiling water bath until all had vaporized then cooling the flask, they found that 0.355 grams of unknown liquid condensed. The volume of the flask was 177 mL. The temperature of the water bath was 99 ºC. The room air pressure was 779 mmHg. What is the molar mass of the unknown vapor?

 

Answer:

n=PV/RT=1.03atm(0.177L)/[(0.08206L atm/mol K)(372K)]=5.97EE-3mol
M=0.355g/5.97EE-3mol=59.5g/mol

 

  

 



4Concept Check: What is the density of ozone, O3, at STP?

 

Answer: The density is mass per unit volume. Gases have relatively low masses and high volumes, so grams per liters are a convenient set of units. Assume one mole of ozone gas. This is a reasonable assumption, since the density of one mole will be the same as the density of any number of moles. The volume of one mole of any ideal gas at STP is 22.4 L. One mole of ozone gas has a mass of 48.0 g. so the density is 48.0g/22.4L = 2.14 g/L.

What if the conditions were not STP? Density is mass divided by volume. We can solve the ideal gas equation for volume and substitute this into the density formula.

D=m/V
PV=nRT
which rearranges to V=nRT/P
Substitute and simplify:
D=m/[nRT/P]=mP/nRT

                                             

The mass of a substance, m, is equal to the number of moles, n, multiplied by the molar mass, M, or m = nM. If we substitute this into the above equation and cancel, we get                                          

D=mP/nRT
m=nM
substitute and simplify
D=MP/RT

 

 We can use this equation for the density of an ideal gas to find the density of ozone at STP (or any other temperature and pressure):

     D=MP/RT
D=48.0g/mol(1.00atm)/[0.08206L atm/mol K)(273K)]=2.14g/L

 

 


Graham’s Law of Effusion

Diffusion is the process by which a gas will spread throughout a container. A real world example of this process is the way chemistry lab smells sink up the whole building. Diffusion is somewhat difficult to measure without expensive sensors. A related term is effusion, the process by which a gas will leak from a container through a tiny hole. A balloon will gradually deflate due to effusion through microscopic holes in the rubber.

Graham’s Law of Effusion states that a gas effuses at a rate inversely proportional to the square root of its molar mass. In mathematical terms this is

                                                                       

 r is proportional to 1/square root of M

where r is effusion rate and M is molar mass. The smaller the molar mass, the faster the gas will effuse. If we compare the effusion rates of two different gases, we get

 

                                                      r1/r2=square root of (M2/M1)

 

A typical freshman chemistry lab experiment or lecture demonstration over Graham’s Law actually illustrates diffusion. A drop of concentrated hydrochloric acid is placed in one end of a horizontal glass tube and a drop of concentrated ammonium hydroxide is placed in the other end of the tube. Both ends are blocked with a stopper or cotton wool to limit the diffusion to one direction. When the hydrochloric acid vapors meet the ammonium hydroxide vapors, they react to form a white precipitate of ammonium chloride. Students can see the white ring form inside the tube and measure its distance from each end. The precipitate will form closer to the hydrochloric acid end of the tube. It has a higher molar mass and diffuses more slowly than ammonium hydroxide.

 

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