9.3 Gas Laws
The “pneumatic” era of chemistry began with the discovery of the vacuum around 1650 which clearly established that gases are a form of matter. The ease with which gases could be studied soon led to the discovery of numerous empirical (experimentally-discovered) laws that proved fundamental to the later development of chemistry and led indirectly to the atomic view of matter. These laws are so fundamental to all of natural science and engineering that everyone learning these subjects needs to be familiar with them.
1 Pressure-volume relations: Boyle’s law
Robert Boyle (1627-91) showed that the volume of air trapped by a liquid in the closed short limb of a J-shaped tube decreased in exact proportion to the pressure produced by the liquid in the long part of the tube. The trapped air acted much like a spring, exerting a force opposing its compression. Boyle called this effect “the spring of the air“, and published his results in a pamphlet of that title.
The difference between the heights of the two mercury columns gives the pressure (76 cm = 1 atm), and the volume of the air is calculated from the length of the air column and the tubing diameter.
Some of Boyle’s actual data are shown in the table.
Figure 9.21 Pressure determination using manometer
Boyle’s law states that the pressure of a gas, held at a constant temperature, varies inversely with its volume:
It is very important that you understand how these two equations express the green-highlighted statement of the law.
If you feel the need to memorize Eqn 1-2, you probably lack this understanding; please study the paragraph that follows.
PV = constant (1-1) must know!
or, equivalently,
P1V1 = P2V2 (1-2)
Boyle’s law is a relation of inverse proportionality; any change in the pressure is exactly compensated by an opposing change in the volume. As the pressure decreases toward zero, the volume will increase without limit. Conversely, as the pressure is increased, the volume decreases, but can never reach zero. There will be a separate P-V plot for each temperature; a single P-V plot is therefore called an isotherm.
Please take the time to understand this kind of plot which governs any relationship of inverse proportionality, and why the curves never reach the x– and y axes. You should be able to sketch out such a plot when given the value of any one (x,y) pair.
Shown here are some isotherms for one mole of an ideal gas at several different temperatures. Each plot has the shape of a hyperbola— the locus of all points having the property x y = a, where a is a constant. You will see later how the value of this constant (PV=25 for the 300K isotherm shown here) is determined.
Figure 9.22 Boyle’s Law graph
A related type of plot with which you should be familiar shows the product PV as a function of the pressure. You should understand why this yields a straight line, and how this set of plots relates to the one immediately above.
Figure 9.23 Boyle’s law graph
Problem Example 1
Figure 9.24 Gas expansion
In an industrial process, a gas confined to a volume of 1 L at a pressure of 20 atm is allowed to flow into a 12-L container by opening the valve that connects the two containers. What will be the final pressure of the gas?
Solution: The final volume of the gas is (1 + 12)L = 13 L. The gas expands in inverse proportion two volumes
P2 = (20 atm) × (1 L ÷ 13 L) = 1.5 atm
Note that there is no need to make explicit use of any “formula” in problems of this kind!
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2 How the temperature affects the volume:
Charles’ law
Figure 9.25 jacques Charles
Figure 9.26 Gay Lussac
All matter expands when heated, but gases are special in that their degree of expansion is independent of their composition. In 1802, the French scientist Joseph Gay-Lussac (right, 1778-1850) found that if the pressure is held constant, the volume of any gas changes by the same fractional amount (1/273 of its value) for each C° change in temperature.
On learning that another scientist, Jacques Charles, had observed the proportionality between the volume and temperature of a gas 15 years earlier but had never published his findings, Gay-Lussac generously acknowledged this fact. Eventually, the “law of Charles and Gay-Lussac” become commonly known simply as “Charles’ law”.
The volume of a gas confined against a constant pressure is directly proportional to the absolute temperature.
A graphical expression of Charles’ law can be seen in these plots of the volume of one mole of an ideal gas as a function of its temperature at various constant pressures.
Figure 9.27: Charles’s law graph
What do these plots show?
The straight-line plots show that the ratio V/T (and thus dV/dT) is a constant at any given pressure. Thus we can express the law algebraically as
V/T = constant or V1/T1 = V2 /T2
(be sure you understand why these two equations are equivalent!)
What is the significance of the extrapolation to zero volume?
If a gas contracts by 1/273 of its volume for each degree of cooling, it should contract to zero volume at a temperature of –273°C. This, of course, is the absolute zero of temperature, and this extrapolation of Charles’ law is the first evidence of the special significance of this temperature.
Why do the plots for different pressures have different slopes?
The lower the pressure, the greater the volume (Boyle’s law), so at low pressures the fraction (V/273) will have a larger value. You might say that the gas must “contract faster” to reach zero volume when its starting volume is larger.
•Try this interactive exploration of the Law of Charles and Gay-Lussac
•Carleton University lecture demonstration video of Charles’ Law (YouTube)
Problem Example 2
The air pressure in a car tire is 30 psi (pounds per square inch) at 10°C. What will be pressure be after driving has raised its temperature to 45°C ? (Assume that the volume remains unchanged.)
Solution:The gas expands in direct proportion to the ratio of the absolute temperatures:
P2 = (30 psi) × (318K ÷ 283K) = 33.7 psi
Historical notes
The fact that a gas expands when it is heated has long been known. In 1702, Guillaume Amontons (1163-1705), who is better known for his early studies of friction, devised a thermometer that related the temperature to the volume of a gas. Robert Boyle had observed this relationship in 1662, but the lack of any uniform temperature scale at the time prevented them from establishing the relationship as we presently understand it.
Jacques Charles discovered the law that is named for him in the 1780s, but did not publish his work. John Dalton published a form of the law in 1801, but the first thorough published presentation was made by Gay-Lussac in 1802, who acknowledged Charles’ earlier studies.
Figure 9.28 Hot-air balloons and Charles’ Law
Ref: commons.wikimedia.org/
The buoyancy that lifts a hot-air balloon into the sky depends on the difference between the density (mass ÷ volume) of the air entrapped within the balloon’s envelope, compared to that of the air surrounding it. When a balloon on the ground is being prepared for flight, it is first partially inflated by an external fan, and possesses no buoyancy at all. Once the propane burners are started, this air begins to expand according to Charles’ law. After the warmed air has completely inflated the balloon, further expansion simply forces excess air out of the balloon, leaving the weight of the diminished mass of air inside the envelope smaller than that of the greater mass of cooler air that the balloon displaces.
Figure 9.29 Hot air-balloon
Figure 9.30 Hot air balloon
Jacques Charles collaborated with the Montgolfier Brothers whose hot-air balloon made the world’s first manned balloon flight in June, 1783 (left). Ten days later, Charles himself co-piloted the first hydrogen-filled balloon. Gay-Lussac (right), who had a special interest in the composition of the atmosphere, also saw the potential of the hot-air balloon, and in 1804 he ascended to a then-record height of 6.4 km. [images: Wikimedia]
Jacques Charles’ understanding of buoyancy led to his early interest in balloons and to the use of hydrogen as an inflating gas. His first flight was witnessed by a crowd of 40,000 which included Benjamin Franklin, at that time ambassador to France. On its landing in the countryside, the balloon was reportedly attacked with axes and pitchforks by terrified peasants who believed it to be a monster from the skies. Charles’ work was mostly in mathematics, but he managed to invent a number of scientific instruments and confirmed experiments in electricity that had been performed earlier by Benjamin Franklin and others.
Gay-Lussac’s balloon flights enabled him to sample the composition of the atmosphere at different altitudes (he found no difference). His Law of Combining Volumes (described below) constituted one of the foundations of modern chemistry. Gay-Lussac’s contributions to chemistry are numerous; his work in electrochemistry enabled him to produce large quantities of sodium and potassium; the availability of these highly active metals led to his co-discovery of the element boron. He was also the the first to recognize iodine as an element. In an entirely different area, he developed a practical method of measuring the alcohol content of beverages, and coined the names “pipette” and “burette” that are known to all chemistry students. A good biography can be viewed at Google Books.
If you ever visit Paris, look for a street and a hotel near the Sorbonne that
bear Gay-Lussac’s name.
Figure 9.31 Gay-Lussac
3 Volume and the number of molecules: Avogadro’s law
Gay-Lussac’s Law of Combining Volumes
For a good tutorial overview of the Law of Combining Volumes, see this ChemPaths page.
In the same 1808 article in which Gay-Lussac published his observations on the thermal expansion of gases, he pointed out that when two gases react, they do so in volume ratios that can always be expressed as small whole numbers. This came to be known as the Law of combining volumes.
These “small whole numbers” are of course the same ones that describe the “combining weights” of elements to form simple compounds, as described in the lesson dealing with simplest formulas from experimental data.
Figure 9.32 Amadeo Avogadro
Ref: commons.wikimedia.org/
The Italian scientist Amedeo Avogadro (1776-1856) drew the crucial conclusion: these volume ratios must be related to the relative numbers of molecules that react, and thus the famous “E.V.E.N principle”:
Equal volumes of gases, measured at the same temperature and pressure, contain equal numbers of molecules
Avogadro’s law thus predicts a directly proportional relation between the number of moles of a gas and its volume.
This relationship, originally known as Avogadro’s Hypothesis, was crucial in establishing the formulas of simple molecules at a time (around 1811) when the distinction between atoms and molecules was not clearly understood. In particular, the existence of diatomic molecules of elements such as H2, O2, and Cl2 was not recognized until the results of combining-volume experiments such as those depicted below could be interpreted in terms of the E.V.E.N. principle.
How the E.V.E.N. principle led to the correct formula of water
Early chemists made the mistake of assuming that the formula of water is HO. This led them to miscalculate the molecular weight of oxygen as 8 (instead of 16). If this were true, the reaction H + O → HO would correspond to the following combining volumes results according to the E.V.E.N principle:
Figure 9.33
But a similar experiment on the formation of hydrogen chloride from hydrogen and chlorine yielded twice the volume of HCl that was predicted by the the assumed reaction H + Cl → HCl. This could be explained only if hydrogen and chlorine were diatomic molecules:
Figure 9.34
This made it necessary to re-visit the question of the formula of water. The experiment immediately confirmed that the correct formula of water is H2O:
Figure 9.35 electrolysis of water
This conclusion was also seen to be consistent with the observation, made a few years earlier by the English chemists Nicholson and Carlisle that the reverse of the above reaction, brought about by the electrolytic decomposition of water, yields hydrogen and oxygen in a 2:1 volume ratio.
A nice overview of these developments can be seen at David Dice’s Chemistry page, from which this illustration is taken.
Figure 9.36 System of gas
https://phet.colorado.edu/en/simulation/legacy/gas-properties
Click on the above simulation and observe the properties of Gas.
Now introduce 50 molecules into the chamber using the handle pump. Slowly decrease the volume of the container using the man symbol. Notice that the pressure monitoring system is giving higher and higher values. This is called Boyle’s law.
Boyle’s law: Volume of a fixed amount of gas is inversely proportional to the pressure applied to the gas if the Temp is kept constant.
Boyle’s law has wide application in various fields like scuba diving, human breahing technique.
Figure 9.38 Application of Boyle’s law
Ref: common.wikimedia.org/
When a person breathe in, the diaphragm contracts and volume of the lungs get bigger and pressure is low inside the body than outside pressure. Since outside pressure is high, air goes inside the body. When we breathe out, the diaphragm relaxes, volume decreases, high pressure gas comes out of the body.
Figure 9.39 Boyle’s law
Ref: common.wikimedia.org/
Mathematically we can write,
at constant T
P1= initial pressure, P2= final pressure, T1= initial temp in Kelvin, T2= final temp in Kelvin
If we plot different Pressure vs. Volume for a gaseous system at a constant temperature, the curve will be parabolic in nature.
Figure 9.40 Boyle’s law graph
Ref: common.wikimedia.org/
Example: A 4.0 L container of Helium gas has a pressure of 15.0 atm. What pressure does the gas exert if the volume is increased to 8.0 L?
Ans: since the volume is doubled, pressure must be half of the initial pressure.
New pressure= 15.0/2= 7.50 atm
Now introduce heat to the activity: Increase the temp from 400K to 1500K by adding heat keeping the pressure at 1:00atm. On the right top corner of the screen, keep pressure as constant parameter. You will notice that volume of the chamber is increasing. This is called Charles’s law.
Charles’s law: Volume of a fixed amount of gas is directly proportional to the kelvin temp if the pressure is kept constant.
This concept is applied in hot air balloon, where the volume of the gas is expanded by applying heat and once it becomes less dense than air, it can float in air.
Mathematically we can write:
at constant P
V1= initial pressure, V2= final pressure, T1= initial temp in Kelvin, T2= final temp in Kelvin
We can understand the absolute zero temperature from Charles’s law. For a sample of gas, when temperature is decreases, volume of the gas molecules decrease. According to the kinetic Molecular theory, the energy of the gas molecules is directly proportional to the kelvin temp. So gas molecules will move slowly as the temperature decreases. Therefore, hypothetically if the absolute temperature of a gaseous system reaches zero i.e. -2730C, all the gas molecules motion will be ceased and the volume of the gas molecule would be zero. In reality, experiments done at lower temp that show the volume decreases steadily but so far, zero volume hasn’t reached.
Figure 9.41 Charle’s Law graph
Ref: common.wikimedia.org/
Example: A volume of 1.00L of gas at 370C is expelled from the lungs to cold outside at temperature -50C. What is the volume of the air at that temperature?
Ans: V2 is unknown.
1.00/(37+ 273) = V2/( -5+ 273) , 1/310=V2/268 or V2= 0.865 L
Avogadro’s law: Increase the number of molecules from 50 to 100 in the simulation system keeping the pressure and Temperature constant. You may notice that volume of the container is increasing. This is Avogadro’s law.
Avogadro’s law: Volume of a of gas is directly proportional to the number of moles of gas if the pressure and Temp are kept constant.
Figure 9.42 Avogadro’s law graph
Ref: common.wikimedia.org/
V1= initial volume P2= final volume, n1= initial moles n2= final moles
Example: The lungs of an average male holds 0.25 mol of air in a volume of 5.5 L. How many mole sof air do the lungs of an average female hold if the volume is 4.5 L?
Ans: n2 is unknown.
5.5/0.25= 4.5/n2, 22= 4.5/n2 or n2= 0.205 mol or 0.20 mols
Gay Lusaac’s law or Amonton’s law: With volume remaining constant, the pressure of a gas molecule is directly proportional to its absolute temperature. According to KMT theory, when temperature increases, the gas molecules possess high kinetic energy. They collide with other molecules and with the walls of the container with high speed and try to expand. If the volume is kept constant, they collide each other more frequently and gas pressure increases.
P1= initial pressure, P2= final pressure, T1= initial temp, T2= final temp
Figure 9.42 Gay-Lusaac’s law
Ref: common.wikimedia.org/
Gay Lusaac’s law can also be observed the above diagram, where application of heat to a gaseous system increases the pressure of the gas.
Example: The tire on a bicycle in a cool garage is stored at 200C and 80. Psi. What is the pressure inside the tire after riding the bike at 430C?
Ans: P2 is unknown, 80/(20+273)= P2/(43+ 273)
0.273= P2/316 or P2= 0.273*316= 86.3 psi or 86 psi.
The following activity has been taken from AACT ( American association of chemical teachers)
https://teachchemistry.org/periodical/issues/november-2015/gas-laws
In this investigation you will examine three gas laws including Boyle’s Law, Charles’ Law and Gay-Lussac’s Law. You will explore how manipulating the variables of volume (L), pressure (atm) and temperature (K) can affect a sample of gas. The formula for each of the gas laws are:
Boyle’s Law: Charles’ Law: Gay-Lussac’s Law:
P1 V1 = P2 V2 V1 = V2 P 1 = P 2
Checking Comprehension
Please create a list of the variable given in each problem and show all your work required to complete the calculation.
- Calculate the temperature of a gas when it is expanded to 5.25L. The gas originally occupies 3.45L of space at 282K.
- The temperature of a gas is increased from 125⁰C to 182⁰C inside of a rigid container. The original pressure of the gas was 1.22atm, what will the pressure of the gas be after the temperature change?
- The volume of gas in a container was originally 3.24L, while at standard pressure, 1.00atm. What will the volume be if the pressure is increased to 1.20atm?
Questions:
- The pressure inside a 1.0 L balloon at 250C was 750 mm of Hg. What is the pressure inside the balloon when it is cooled to -400C and expands to 5.0 L volume?