### The Law of Gasesl

Chemistry Independent Study: Gas Laws

Since the days of Aristotle, all substances have been classified into one of three physical states. A substance having a fixed volume and shape is a solid. A substance, which has a fixed volume but not a fixed shape, is a liquid; liquids assume the shape of their container but do not necessarily fill it. A substance having neither a fixed shape nor a fixed volume is a gas; gases assume both the shape and the volume of their container. The structures of gases, and their behavior, are simpler than the structures and behavior of the two condensed phases, the solids and the liquids

Quantitative measurements on gases were first made in a rational manner by the English chemist Robert Boyle (1627 – 1691). The instruments used by Boyle to measure pressure were two: the manometer, which measures differences in pressure, and the barometer, which measures the total pressure of the atmosphere.

A manometer is simply a bent piece of tubing, preferably glass with one end closed.

When the liquid level in both arms is the same, the pressure of the sample of gas inside the closed end must equal the pressure of the external atmosphere since the downward force on the two columns of liquid is then equal. When the liquid levels are unequal, the pressures must differ. The difference in pressure can be measured in units of length of the vertical column of liquid. The mm Hg, or its modern version the torr, originated in this use of the manometer. Mercury is particularly convenient for use in manometers (and barometers) because at room temperature it has low vapor pressure, does not wet glass, and has a high density. Other liquids such as linseed oil or water have also been used in manometers.

The barometer is a device for measuring the total pressure of the atmosphere. A primitive barometer can easily be constructed by taking a glass tube about a meter long, sealing one end, filling the tube completely with mercury, placing your thumb firmly over the open end, and carefully inverting the tube into an open dish filled with mercury. The mercury will fall to a height independent of the diameter of the tube and a vacuum will be created above it.

The height of the mercury column will be the height which the atmospheric pressure can support. The standard atmospheric pressure, one atmosphere (atm), is 760 mm Hg but the actual atmospheric pressure varies depending upon altitude and local weather conditions. For this reason barometers can be used to help predict the weather. A falling barometer indicates the arrival of a low-pressure air system, which often means stormy weather. A rising barometer indicates the arrival of a high pressure air system, and that often means clear weather.

While mercury is again the most convenient liquid for use in barometers it is by no means the only liquid which can be used. Preparation of a water barometer and many of the early barometers did use water.

With the manometer and barometer used together, the actual pressure of a sample of gas can be measured. Combining the barometer reading of atmospheric pressure with the manometer reading of pressure difference gives the actual pressure. If the manometer is as shown on the left-hand side of the Figure below, then p2 = p (atmospheric) + p1, while if the manometer is as shown on the left-hand side of the Figure below, then p2 = p (atmospheric) – p1. (McQuarrie and Rock, Page 161)

Units of pressure were originally all based on the length of the column of liquid, usually mercury, supported in a manometer or barometer. By far the most common of these units was the mm Hg, although inches of mercury were also used in English-speaking countries. However, the modern SI unit of pressure is derived from the fundamental units of the SI. Pressure is force per unit area, and force is the product of mass times acceleration, so the SI unit of pressure is the kg m s-2/m2 or newton/m2, which is called the pascal (Pa).

All of the older units of pressure have now been redefined in terms of the pascal. One standard atmosphere or atm, the pressure of the atmosphere at sea level, is by definition exactly 101325 Pa. The torr, named in honor of Torricelli, is defined as 1/760 of a standard atmosphere or as 101325/760 Pa. The mm Hg, which is almost but not quite identical to the torr, is defined as (13.5951 x 9.80665) Pa, using a fixed density of mercury and a standard force of terrestrial gravitation. The term bar is used for 100000 Pa, which is slightly below one standard atmosphere. (http://dhswvuds.K12.us/GasLaw/KMT-Gas-Laws.html)

Boyle used the manometer and barometer to study the pressures and volumes of different samples of different gases. The results of his studies can be summarized in a simple statement which has come to be known as the law of Boyle or Boyle’s law:

At any constant temperature, the product of the pressure and the volume of any size sample of any gas is a constant.

For a particular sample of any gas, Boyle’s law can be shown graphically as is done in the Figure below. It is more common to express it mathematically as p1V1 = p2V2 or as

pV = k, where k is a constant which depends upon the particular sample. The pressure and the volume vary inversely; as the pressure of the sample increases the volume of the sample of gas must decrease. (McQuarrie and Rock, Page 163)

The law as formulated by Boyle does not suggest any particular scale of volume or of pressure. The units of volume are simply the cube of any convenient unit of length; the volume is actually measured in a separate experiment in which the tube is filled to the same mark with a liquid.

Temperature and the Law of Charles

The conventional liquid-in-glass thermometer was invented in the seventeenth century. This bulb-and-tube device is still in use.

In these thermometers the diameter of the bulb is much greater than the diameter of the tube so that a small change in the volume of liquid in the bulb will produce a large change in the height of the liquid in the tube. Two things were not clear about the thermometer at this time. The first question was what it was that the thermometer measured. As the temperature or “degree of hotness” apparent to one’s fingers increased, the height of the liquid obviously did also, and this was useful in medicine for checking fevers, but there was no quantitative measurement made, merely the relative degree of hotness between this and that. The second question was whether the degree of hotness of any particular thing was a constant everywhere so that the temperatures of other things could be measured relative to it. Suggested fixed temperatures included that of boiling water, that of melting butter, and the apparently uniform temperature of deep cellars.

Robert Boyle knew of the thermometer, and also was aware that a gas expands when heated. However, since no quantitative temperature scale then existed he could not, and did not, determine the relationship between degree of hotness (temperature) and volume of a gas quantitatively.(Siebring, Richard, Page 32)

Guillaume Amontons (d. 1705) developed the air thermometer, which uses the increase in the volume of a gas with temperature rather than the volume of a liquid. The air thermometer is an excellent demonstration of Charles’ law because the atmosphere maintains a fixed downward pressure above a small mercury plug of constant mass. The volume of a trapped sample of air increases on heating until the pressure of the trapped air equals the pressure of the atmosphere plus the small pressure due to the plug. Nevertheless, Amontons failed to achieve formulation of Charles’ law for the same reason as did Boyle: a quantitative scale of temperature was needed.

A quantitative scale of temperature could only be developed after it was realized that at a fixed pressure any pure substance undergoes a phase change at a single fixed temperature which is characteristic of that substance. The melting point of ice to water was taken as 0oC and the boiling point of water was taken as 100oC to give our common Celsius scale of temperature. The measurements of the French chemists used the very similar Reaumur scale (water freezes at 0oRe and boils at 80oRe) to establish the law of Charles.

The study of the effect of temperature upon the properties of gases took considerably longer to achieve a simple quantitative relation than did study of the effect of pressure, primarily because the development of a quantitative scale of temperature was a difficult process. However, once such a scale was developed, the appropriate measurements were made, primarily by the French chemist Jacques Charles (1746 – 1823).

The experimental data were formulated into a general law which became known as the law of Charles or Charles’ law:

At any constant pressure, the volume of any sample of any gas is directly proportional to the temperature.

Mathematically, the law of Charles can be expressed as

where t represents the temperature on any convenient temperature scale and k’ and k” are constants. However the volume extrapolates to zero at a temperature of -273.15oC. If this temperature were taken as the zero of a temperature scale, the constant k” would be zero and it could be dropped from the equation. Such a temperature scale is now the fundamental scale of temperature in the SI. It is called the absolute scale, the thermodynamic scale, or the Kelvin scale. Temperature on the Kelvin scale, and only on the Kelvin scale, is symbolized by T. The unit of temperature n the Kelvin scale is called the kelvin, and it has the same size as the degree Celsius. The symbol for the unit kelvin is K. (Metcafe H. Clark, Page 273-4)

The law of Charles can be written more simply using the Kelvin scale of temperature as V = k’T, where T represents the absolute temperature. An alternative form, more useful when the volume of one particular sample of gas changes with temperature, is V1/T1 = V2/T2.

Dalton’s studies which led him to the atomic-molecular theory of matter included studies of the behavior of gases. These led him to propose what is now called Dalton’s law of partial pressures:

For a mixture of gases in any container, the total pressure exerted is the sum of the pressures that each gas would exert if it were alone.

This law can be expressed in equation form as:

where p is the total or measured pressure and p1, p2, … are the partial pressures of the individual gases. For air, an appropriate form of Dalton’s law would be:

p(air) = p(N2) + p(O2) + p(CO2) + …

At temperatures near ordinary room temperature, the partial pressures of each of the components of air is directly proportional to the number of moles of that component in any volume of air. When the total pressure of air is 100 kPa or one bar, the partial pressures of each of its components (in kPa) are numerically equal to the mole per cent of that component. Thus the partial pressures of the major components of dry air at 100 kPa are nitrogen, 78 kPa; oxygen, 21 kPa; argon, 0.9 kPa; and carbon dioxide, 0.03 kPa. (Metcafe H. Clark, Page 273-4)

The same substance may be found in different physical states under different conditions. Water, for example, can exist as a solid phase (ice), a liquid phase (water), and a gas phase (steam or water vapor) at different temperatures. The processes by which a substance is converted from one phase to another are called by specific names. The conversion from solid to liquid is melting or fusion and the reverse conversion from liquid to solid is freezing. The conversion from liquid to gas is called boiling or vaporization and the reverse conversion from gas to liquid is called condensation. The conversion from solid to gas, when it occurs directly without going through a liquid state as in the case of iodine and carbon dioxide, is called sublimation; the reverse conversion from gas to solid shares the name of condensation.

The Ideal Gas Law was first written in 1834 by Emil Clapeyron.

This is just one way to derive the Ideal Gas Law:

For a static sample of gas, we can write each of the six gas laws as follows:

Note that the last law is written in reciprocal form. The subscripts on k indicate that six different values would be obtained.

When you multiply them all together, you get:

Let the cube root of k1k2k3k4k5 / k6 be called R. (Wilbraham, Antony C., page 234)

Each unit occurs three times and the cube root yields L-atm / mol-K, the classic units for R when used in a gas law context. (Dickson, T.R., Page 78-9)

PV / nT = R

PV = nRT

R is called the gas constant. Sometimes it is referred to as the universal gas constant. If you wind up taking enough chemistry, you will see it showing up over and over and over.

R’s value can be determined many ways. This is just one way:

Assume we have 1.000 mol of a gas at STP. The volume of this amount of gas under the conditions of STP is known to a high degree of precision. We will use the value of 22.414 L.

By the way, 22.414 L at STP has a name. It is called “molar volume.” It is the volume of ANY ideal gas at standard temperature and pressure. (Siebring, Richard, Page 54)

Let’s plug our numbers into the equation:

(1.000 atm) (22.414 L) = (1.000 mol) (R) (273.15 K)

Notice how atmospheres were used as well as the exact value for standard temperature.

Solving for R gives 0.08206 L atm / mol K, when rounded to four significant figures. This is usually enough. Remember the value. You’ll need it for problem solving.

Notice the weird unit on R: say out loud “liter atmospheres per mole Kelvin.”

This is not the only value of R that can exist. It depends on which units you select. Those of you that take more chemistry than high school level will meet up with 8.3145 Joules per mole Kelvin, but that’s for another time.

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