6.1 THE KINETIC MOLECULAR THEORY
Scientists are always searching for fundamental reasons to explain the phenomena which they observe. One of their objectives is to relate the properties of large samples of matter to the nature and behavior of the individual molecules or atoms in the sample.
All elements can exist as individual molecules in the gaseous phase. The temperature at which different elements change into gases varies over a wide range. Helium with a boiling point of -269^{o}C and tungsten with a boiling point of 5900^{o}C represents the extremes. As a result of thousands of observations and experiments, scientists have developed the KINETIC MOLECULAR THEORY, one of the most successful of all scientific theories. This theory enables us to interpret observed properties of matter in terms of the nature and behavior of individual molecules.
By applying mathematical principles to the qualitative aspects of the theory, scientists have developed general laws expressed in mathematical terms which accurately describe the behavior of IDEAL GASES. We will find that most REAL GASES obey these laws fairly well at normal temperatures and pressures. The deviation in behavior of real gases from that of ideal gases provides valuable clues as to the sizes of molecules and the interactions between them.
The main assumptions of the KINETIC MOLECULAR THEORY are:
I. Gases consist of small particles (molecules or atoms) that are far apart in comparison to their own size. These particles are considered to be tiny points occupying a negligible volume compared to that of their container. |
II. Molecules are in rapid and random straight-line motion. This motion can be described by well-defined and established laws of motion. |
III. The collisions of molecules with the walls of a container or with other molecules are perfectly elastic. That is, no loss of energy occurs. |
IV. There are no attractive forces between molecules or between molecules and the walls with which they collide. |
V. At any particular instant, the molecules in a given sample of gas do not all possess the same amount of energy. |
It should be noted that the above assumptions are made only for a "perfect" or ideal gas. They cannot be rigorously applied (mathematically) to real gases but can be used to explain their observed behavior qualitatively.
6.2 PRESSURE
It was not until the seventeenth century that experiments seem to have been undertaken to prove that the earth was surrounded by a "sea of air" that exerts pressure. Most likely an attempt to prove that the air exerts pressure led Evangelista Torricelli (1608-1647), a young pupil of Galileo Galilei (1564-1642), to invent the barometer in 1643. Torricelli's original barometer was basically the same instrument that we still use for measuring air pressure today. This device, which first made it possible to measure the pressure of gases, opened a new era in the study of gases.
The pressure exerted by the atmosphere (a mixture of gases) may be measured by a simple Torricellian-type barometer. Such an instrument can be prepared by filling a glass tube, closed at one end and about 80 cm long (800 mm), with mercury and inverting it in a dish of mercury. The mercury falls in the tube until the pressure exerted by the column of mercury is exactly equal to the pressure exerted by the air upon the surface of the mercury in the dish. (PRESSURE MEANS THE FORCE EXERTED UPON A UNIT AREA OF SURFACE.) The pressure of the atmosphere varies with the location on the surface of the earth and with climatic changes. The average pressure of the atmosphere at sea level at a latitude of 45^{o} will support a column of mercury 76 cm (760 mm) in height. THE FORCE PER UNIT AREA EXERTED BY A COLUMN OF MERCURY 760 MM IN HEIGHT IS KNOWN AS ONE ATMOSPHERE OF PRESSURE when the mercury is at 0^{o}C.
At 0^{o}C, mercury has a density of 13.59 g/ml. A column of mercury 760 mm high and 1 cm in cross-sectional area has a volume of 76 ml. The mass of mercury in the column is, therefore,
13.59 g/ml X 76 ml = 1033 g.
In chemistry, gas pressures have traditionally been expressed in terms of mm of mercury or in atmospheres (atm.). A pressure unit, the TORR, named in honor of TORRICELLI, has been used by many chemists. The TORR is a pressure unit equal to the pressure exerted by 1 mm of mercury at 0^{o}C. In other words, one standard atmosphere (760 mm of Hg) is equal to 760 Torrs.
Because of gravity, our atmosphere exerts a downward force and consequently a pressure on Earth's surface. The force F exerted by any object is the product of its mass, m, times its acceleration, a: F = ma. Earth's gravity exerts an acceleration of 9.8 m/s^{2}. A column of air 1 m^{2} in cross section extending through the atmosphere has a mass of roughly 10,000 kg. The force exerted by this column is
F = (10,000 kg)(9.8 m/s^{2}) = 1 X 10^{5} kg-m/s^{2} = 1 X 10^{5} N
The SI unit of force is kg-m/s^{2} and is called the newton (N):
1 N = 1 kg-m/s^{2}.
The pressure exerted by the column is the force divided by its cross-sectional area, A:
P = F/A = 1 X 10^{5} N/1m^{2} = 1 X 10^{5} N/m^{2} = 1 X 10^{5} Pa = 1 X 10^{2} kPa
The SI unit of pressure is currently N/m^{2}. It is given the name pascal (Pa) after Blaise Pascal (1623-1662), a French mathematician and sicentist:
1 Pa = 1 N/m^{2}
As previously indicated standard atmospheric pressure, which corresponds to the typical pressure at sea level, is the pressure sufficient to support a column of mercury 760 mm in height. In SI units, this pressure equals 1.01325 X 10^{5} Pa. Thus,
1 atm = 760 mm Hg = 760 torr = 1.01325 X 10^{5} Pa = 101.325 kPa = 14.7 PSI
6.3 BOYLE'S LAW
It has been observed from experiment that the volume of a given mass of gas, at constant temperature, is reduced to half when the pressure on the gas is doubled. Conversely, the volume is doubled by a decrease in pressure to one-half. Observations of this sort were summarized by Robert Boyle (1627-1691) in 1660 in form now known as BOYLE'S LAW: THE VOLUME OF A FIXED MASS OF GAS VARIES INVERSELY WITH THE PRESSURE AT A CONSTANT TEMPERATURE. This law may be expressed mathematically as
P_{1}V_{1}= P_{2}V_{2}
where V_{1} is the initial volume, V_{2} the final volume, P_{1} the initial pressure, and P_{2} the final pressure.
EXAMPLE: A student collects 20 ml of gas at 720 Torrs. What volume would this gas occupy at 800 Torrs?
P_{1}V_{1}= P_{2}V_{2}
(20 ml)(720 Torrs) = (800 Torrs)V_{2}
V_{2} = (20 ml)(720 Torrs)/(800 Torrs)
V_{2} = 18 ml
6.4 KELVIN OR ABSOLUTE TEMPERATURE
By experimentation it has been found that when 273 ml of gas at 0^{o} is
warmed to 1^{o}, its volume increases by 1 ml to 274 ml; at 20^{o} its
volume increases to 293 ml; at 273^{o} its volume increases to 546 ml (double
that at 0^{o}), and so on, provided that in each case the pressure remains
constant. Note that the volume of the gas at 0^{o} increases 1/273 of its volume
for each increase of 1^{o} on the Celsius scale. The volume of a gas decreases in
the same proportion when the temperature falls. If the temperature of 273 ml of a gas
could be lowered from 0^{o} to -273^{o}, then the gas should have no
volume at -273^{o} because its volume should decrease at the rate of 1/273 of its
volume at 0^{o} for each degree of fall in temperature. Before the temperature of
-273^{o} is reached all gases become liquids, to which this rate of change in
volume does not apply. The temperature of -273^{o} (or more exactly - 273.16^{o}C)
is called ABSOLUTE ZERO. Sir William Thompson, Lord Kelvin (1824-1907), a British scientist, used the idea of absolute zero to develop a temperature scale which we call by his name today. Each degree on the Kelvin scale (^{o}K), which is also known as the absolute temperature scale (^{o}A), is equal to one degree Celsius. But since 0^{o}K is equal to -273^{o}C, there are no negative temperatures on the Kelvin, or absolute, scale. To convert ^{o}C to ^{o}K, simply add 273^{o} to the number of degrees on the Celsius scale. ^{o}K = ^{o}C + 273 To convert ^{o}K to ^{o}C, subtract 273^{o} from the number on the Kelvin scale. ^{o}C = ^{o}K - 273 |
6.5 CHARLES' LAW
Section 6.4, were first measured by the French scientist Jacques Charles (1746-1823) between 1787 and 1801. These studies led to the generalization known as CHARLES' LAW: THE VOLUME OF A FIXED MASS OF GAS VARIES DIRECTLY WITH THE ABSOLUTE TEMPERATURE (on the Kelvin scale) AT A CONSTANT PRESSURE The changes that occur in the volume of a given mass of gas, as described in:
V_{1}/T_{1}= V_{2}/T_{2}
where V_{1} is the initial volume, V_{2} the final volume, T_{1} the initial temperature, and T_{2} the final temperature.
EXAMPLE: A sample of gas occupies 200 ml at 10^{o}C and 750 Torrs. What volume will the gas have at 20^{o}C and 750 Torrs?
The Celsius temperatures must first be converted to Kelvin temperatures.
^{o}K = ^{o}C + 273
K = 10^{o} + 273
K = 283^{o}
^{o}K = ^{o}C + 273
K = 20^{o} + 273
K = 293^{o}
V_{1}/T_{1}= V_{2}/T_{2}
(200 ml)/(283^{o}K) = V_{2}/(293^{o}K)
V_{2} = (200 ml)(293^{o}K)/(283^{o}K)
V_{2} = 207 ml
6.6 STANDARD CONDITIONS OF TEMPERATURE AND PRESSURE (S.T.P.)
From previous considerations of the variation of the volume of a given mass of gas with changes in pressure and temperature, it should be clear that the volume and the density (mass per unit volume) of a gas vary with these conditions. Thus, to be able to compare different gases with regard to their densities or to fix a definite density for any gas, it is desirable to adopt a set of STANDARD CONDITIONS OF TEMPERATURE AND PRESSURE (S.T.P.) for all measurements of gases. Accordingly, 0^{o}C (273^{o}K) and 760 Torrs (1 atmosphere) are universally used as standard conditions.
6.7 DALTON'S LAW OF PARTIAL PRESSURES
THE TOTAL PRESSURE IN A MIXTURE OF NONREACTING GASES IS EQUAL TO THE SUM OF THE PARTIAL PRESSURES OF THE INDIVIDUAL GASES. The gas laws can be applied to gaseous systems containing several different gases as well as to a one-component system. Dalton's law of partial pressures, formulated by the English scientist John Dalton (1766-1844), summarizes the behavior of a system consisting of a mixture of gases. This law states that THE TOTAL PRESSURE EXERTED BY A GASEOUS MIXTURE AT CONSTANT TEMPERATURE IS EQUAL TO THE SUM OF THE PRESSURES THAT EACH OF THE GASES COMPOSING THE MIXTURE WOULD EXERT IF EACH OCCUPIED THE SAME VOLUME ALONE.
6.8 GRAHAM'S LAW OF DIFFUSION
One of the properties of all gases is that they rapidly spread out to fill their containers. This tendency to move about rapidly and to intermingle readily as a result of molecular motion is known as DIFFUSION. A gas consisting of very light molecules, such as hydrogen, diffuses much more rapidly than a gas consisting of heavier molecules. The Scottish chemist Thomas Graham (1805-1869) first formulated a mathematical expression of this principle (1829). It is known as GRAHAM'S LAW OF DIFFUSION, which states that UNDER IDENTICAL CONDITIONS, THE RATES OF DIFFUSION OF GASES ARE INVERSELY PROPORTIONAL TO THE SQUARE ROOTS OF THEIR MOLECULAR MASSES (or densities).
The tendency of lighter gases to diffuse more rapidly than heavier gases can be used to separate atoms which have the same chemical properties but different masses. When such a mixture is passed through a series of porous barriers, the lighter molecules emerge first. After many repetitions of this process, the lighter atoms will be separated from the heavier ones. This process is called GASEOUS DIFFUSION. The average kinetic energy (energy due to their motion) of the molecules of different gases is the same at the same temperature, regardless of differences in mass. The kinetic energy of gaseous molecules increases with a rise in temperature, and decreases as the temperature falls. The Kinetic Energy of a molecule is equal to ½ mv^{2}, where m is mass a v is velocity. It follows that molecules of small mass, like hydrogen, must move with higher velocities than molecules of larger mass, like oxygen, if they have the same kinetic energy at the same temperature.
6.9 DEVIATIONS FROM THE GAS LAWS
When gases under ordinary conditions of temperature and pressure are compressed, the volume is reduced by crowding the molecules closer together. This reduction in volume is really a reduction in the amount of empty space between the molecules. At high pressures, the molecules are crowded so closely together that the volume which they occupy is a large fraction of the entire volume of the gas. Because the volume of the molecules themselves is not compressed, only a small fraction of the entire volume can be affected by an increase in pressure. Thus at high pressures the whole volume is not inversely proportional to the pressure as predicted by Boyle's Law.
The molecules in a gas at relatively low pressures and high temperatures have practically no attraction for one another because they are far apart. As the molecules are crowded close together at low temperatures and high pressures, the force of attraction between the molecules increases. This attraction has the same effect as an increase in external pressure; it causes a slightly greater decrease in volume than corresponds to the increase in external pressure. The compression caused by intermolecular attraction is more pronounced at lower temperatures because the molecules move more slowly and have a smaller tendency to fly apart after collision with one another. Strictly speaking, then, the gas laws apply exactly only to gases whose molecules do not attract one another and which occupy no part of the whole volume. Because there are no gases which have these properties, we can speak of them only as IDEAL or PERFECT GASES. Under ordinary conditions the deviations from the gas laws are so slight they may be neglected.
6.10 THE LAW OF CHARLES AND GAY-LUSSAC
Charles' Law (Section 6.5) holds true only as long as the pressure remains constant. When a gas is put under pressure, as when air is compressed by a tire pump, its temperature increases. If the pressure on a gas is released, as when gas escapes from a cylinder of compressed propane, the temperature of the gas falls. Moreover, the pressure exerted by a confined gas becomes greater when the temperature is raised, and becomes less when the temperature cools. A law describing this relationship between the pressure exerted by a confined gas and its temperature was proposed from the combined efforts of Charles and Gay-Lussac of France. This law states that THE PRESSURE OF A FIXED MASS OF GAS VARIES DIRECTLY WITH THE ABSOLUTE TEMPERATURE (Kelvin) AT A CONSTANT VOLUME. This law may be expressed mathematically as
P_{1}/T_{1}= P_{2}/T_{2}
where P_{1} is the initial pressure, P_{2} the final pressure, T_{1} the initial temperature, and T_{2} the final temperature.
EXAMPLE: A gas occupies 85 ml at 0^{o}C and 710 Torrs. What final temperature would be required to increase the pressure to 760 Torrs?
P_{1}/T_{1}= P_{2}/T_{2}
(710 Torrs)/(273^{o}K) = (760 Torrs)/T_{2}
T_{2} = (760 Torrs)(273^{o}K)/(710 Torrs)
T_{2} = 292^{o}K or 19^{o}C
6.11 GAY-LUSSAC'S LAW
From studies of the volumetric relations in which gaseous sub-stances interact, the early chemists came to the conclusion that gases combine or react in definite and simple proportions by volume. It has been determined by experiment that one volume of nitrogen will combine with three volumes of hydrogen to give two volumes of ammonia gas, provided that the volumes of the reactants and product are measured under the same conditions of temperature and pressure.
N_{2} + 3 H_{2} 2 NH_{3}
The term "volume" is used here in a general sense, and if the volume of nitrogen is measured in liters, then the volumes of hydrogen and ammonia must also be measured in liters. Such observations from experiment were summarized by Joseph Louis Gay-Lussac (1778-1850) in the LAW OF COMBINING VOLUMES: THE VOLUMES OF GASES INVOLVED IN A REACTION, AT CONSTANT TEMPERATURE AND PRESSURE, CAN BE EXPRESSED AS A RATIO OF SMALL WHOLE NUMBERS. It is important to remember that this law applies only to substances in the gaseous state measured at constant temperature and pressure, and that the volumes of any solids or liquids involved in the reactions are not considered. When liquids or solids undergo chemical reaction, no generalizations can be made concerning the volumes of the reactants and products involved in the reaction. Additional examples illustrating Gay-Lussac's Law are as follows:
2 H_{2} + O_{2} 2 H_{2}O
H_{2} + Cl_{2} 2 HCl
C + O_{2} CO_{2}
4 H_{2}O + 3 Fe 4 H_{2} + Fe_{3}O_{4}
The simplest ratio for the combining reacting gaseous volumes is always used.
6.12 AVOGADRO'S LAW
The law of combining volumes of gases can be satisfactorily explained in terms of the molecular structure of gases if we assume that EQUAL VOLUMES OF ALL GASES, MEASURED UNDER THE SAME CONDITIONS OF TEMPERATURE AND PRESSURE, CONTAIN THE SAME NUMBER OF MOLECULES (Sections 2.7 & 3.5). Amedeo Avogadro (1776-1856) enunciated this hypothesis in 1811 to account for the behavior of gases. His hypothesis, which has since been experimentally proven, is now accepted as fact and is known as AVOGADRO'S LAW.
Consider the union of hydrogen and chlorine to produce hydrogen chloride, the volumes of the reactants and product being measured under the same conditions of temperature and pressure. According to Avogadro's Law, equal volumes of hydrogen, chlorine, and hydrogen chloride contain the same number of molecules. During the reaction two molecules of hydrogen chloride are formed from one molecule each of hydrogen and chlorine. Now each molecule of hydrogen chloride must contain at least one atom of hydrogen and one atom of chlorine; hence, two molecules of hydrogen chloride will contain at least two atoms of hydrogen and two atoms of chlorine, which must have been present in one molecule of hydrogen and one molecule of chlorine. No gaseous reaction has been found in which one molecule of hydrogen (or chlorine) contains enough of the element to form more than two molecules of product. We can assume, therefore, that each molecule of hydrogen (and chlorine) contains two and only two atoms. If it were possible to cause the diatomic molecules in a sample of hydrogen all to dissociate to monoatomic molecules, we would observe that the volume would double for the same conditions of temperature and pressure. Each of the original hydrogen molecules would have dissociated into two new molecules. It can be shown that elemental oxygen, nitrogen, fluorine, chlorine, bromine, and iodine also consist of diatomic molecules.
Dalton had considered and rejected the idea that equal volumes of gases contain the same number of atoms; it did not occur to him that elements might exist as polyatomic molecules (O_{2}, N_{2}). Gay-Lussac's Law can be used to determine the volumes of gases involved in a reaction.
EXAMPLE: Calculate the number of liters of hydrogen that will combine with 20 liters of nitrogen to form ammonia.
N_{2} + 3 H_{2} 2 NH_{3}
From the equation we see that one volume of nitrogen will combine with three volumes of hydrogen. Therefore, 20 liters of nitrogen will combine with:
(1 volume of nitrogen) = (20 liters)
X = (20 liters)(3 volumes of hydrogen)
X = 60 liters of hydrogen
6.13 RELATIVE DENSITIES OF GASES
Inasmuch as molecules of different substances have different masses and equal volumes of different gases under the same conditions of temperature and pressure contain the same number of molecules, it follows that the densities of different gases will not be the same. The density of oxygen at S.T.P. is 1.429 grams per liter (g/l), whereas that of hydrogen is 0.0899 g/l. Because equal volumes of different gases contain the same number of molecules, the ratio of the densities of two gases is the same as the ratio of their molecular weights.
Density of O_{2} = 1.429 g/l / Density of H_{2} = 0.0899 g/l
This will equal
molecular wt. of O_{2} = 32.00 / molecular wt. of H_{2} = 2.016
Suppose we calculate the molecular weight of carbon dioxide, the density of which is 1.977 g/l at S.T.P. Noting that the density of carbon dioxide is greater than that of oxygen (1.429 g/l), it follows that its molecular weight should be greater than that of oxygen (32.00). We may obtain the molecular weight of carbon dioxide by multiplying that of oxygen by the ratio of the two densities.
Density of O_{2} = 1.429 g/l / Density of CO_{2} = 1.977 g/l
This will equal
molecular weight of O_{2} = 32.00 / molecular wt. of CO_{2} = X
X = 44.00
It should be obvious to the observer that if the molecular weights of two gases are known and if the density of one of them is known, the density of the other may be easily calculated.
6.14 GRAM-MOLECULAR VOLUME (G.M.V.) OF GASES
We can calculate the volume that one gram-molecular weight of a gas will occupy under standard conditions if we know its density at S.T.P. and its molecular weight. Oxygen has a density of 1.429 g/l at S.T.P. and its molecular weight is 32.00, therefore
(1.429 grams)/(1 liter) = (32.00 grams)/X
X = (32 grams)(1 liter)/(1.429 grams)
X = 22.4 liters
THE VOLUME OCCUPIED BY ONE GRAM-MOLECULAR WEIGHT OF A SUBSTANCE (in the gaseous state) IS KNOWN AS ITS GRAM-MOLECULAR VOLUME (G.M.V.). A gram-molecular weight of any substance contains 6.023 X 10^{-23} molecules (Avogadro's Number) of that substance, and the same number of molecules of different gases occupy equal volumes under the same conditions of temperature and pressure. The highly important and interesting fact follows, then, that ONE GRAM-MOLECULAR VOLUME (one gram-molecular weight or mole) OF ANY GAS OCCUPIES 22.4 LITERS AT STANDARD TEMPERATURE (273^{o}K or 0^{o}C) AND PRESSURE (760 Torrs or 1 Atmosphere).
6.15 COMBINED GAS LAW EQUATION
It is often necessary to correct gas volumes for both temperature and pressure changes. This can be done in a single operation by combining Boyle's Law, Charles' Law and the Law of Charles and Gay-Lussac.
(P_{1})(V_{1})/(T_{1}) = (P_{2})(V_{2})/(T_{2})
The COMBINED GAS LAW EQUATION may be stated as follows: THE VOLUME OF A FIXED MASS OF GAS VARIES DIRECTLY WITH THE ABSOLUTE TEMPERATURE AND INVERSELY WITH THE PRESSURE.
EXAMPLE: A student collects 300 ml. of nitrogen gas measured at 754 Torrs and 21^{o}C. What volume will it occupy at S.T.P.?
(P_{1})(V_{1})/(T_{1}) = (P_{2})(V_{2})/(T_{2})
(754 Torrs)(300 ml.)/(294^{o}K) = (760 Torrs)(V_{2})/(273^{o}K)
V_{2} = (754 Torrs)(300 ml.)(273^{o}K)/(294^{o}K)(760 Torrs)
V_{2} = 276 ml.
6.16 MOLECULAR WEIGHTS OF GASES
The molecular weight of a gas can be found readily by determining experimentally the weight of a known volume of the gas under laboratory conditions of temperature and pressure and, using these data, calculate the weight of 22.4 liters of the gas corrected to standard conditions.
EXAMPLE: A sample of gas occupies 400 ml. at 20^{o}C and 740 Torrs and weighs 0.842 grams. What is the molecular weight (m.w.) of the gas?
(P_{1})(V_{1})/(T_{1}) = (P_{2})(V_{2})/(T_{2})
(740 Torrs)(400 ml.)/(293^{o}K) = (760 Torrs)(V_{2})/ (273^{o}K)
V_{2} = (740 Torrs)(400 ml.)(273^{o}K)/(293^{o}K)(760 Torrs)
V_{2} = 362.9 ml.
By utilization of the COMBINED GAS LAW EQUATION it is possible to find the volume of 0.842 grams of the unknown gas at standard temperature and pressure. Therefore, the weight of 22,400 ml. (22.4 l) of the gas at S.T.P. can be found by the following procedure:
One gram-molecular weight occupies 22.4 l at S.T.P. (Section 6.14) Therefore 0.842 grams occupies .3629 l.
X/(22.4 l) = (0.842 grams)/(.3629 l)
X = (0.842 grams)(22.4 l)/(.3629 l)
X = 51.97 grams
This method of determining molecular weights is applicable to gases and to substances that can be vaporized without decomposition. Many substances decompose before they are completely changed to the vapor state, and many others vaporize at such high temperatures that it is either impossible or impractical to determine their molecular weights by this method.
6.17 VOLUMES OF GASES
The volumes of gases involved in a chemical reaction can be determined from a chemical equation due to the fact that one gram-molecular weight of a gas occupies 22.4 liters at standard temperature and pressure.
EXAMPLE: What volume of oxygen, measured at 0^{o}C and 760 Torrs, would be formed by the complete decomposition of 21.66 grams of mercury (II) oxide?
2 HgO 2 Hg + O_{2}
A 21.66 gram sample of HgO is equivalent to 0.10 moles of HgO (one mole equals 216.6 grams). The equation shows that two moles of HgO will liberate one mole of oxygen (22.4 liters at S.T.P.). Therefore, 0.10 mole of HgO will liberate 0.05 moles of oxygen.
If one mole of oxygen gas occupies 22.4 liters at S.T.P., then 0.05 moles of oxygen will occupy 1.12 liters.
(1 mole)/(22.4 l) = (0.05 moles)/X
X = (0.05 moles)(22.4 l)/(1 mole)
X = 1.12 liters
6.18 THE IDEAL GAS LAW EQUATION
The four variables, pressure (P), volume (V), absolute temperature (T), and number of moles (n), which determine the state of a gaseous system are related by a general equation which may be derived from the individual gas laws or from the Kinetic Molecular Theory.
PV = nRT
The equation PV = nRT is called the IDEAL GAS LAW EQUATION. The value of the new constant, R, can be obtained by substituting into the equation a set of known values of n, P, V, and T. We know that under ideal conditions (S.T.P.) one mole of any gas occupies 22.4 liters. In calculating for our constant, R, pressure (P) is 1 atmosphere, molar volume (V) is 22.4 liters, standard temperature (T) is 273^{o}K, and the number of moles (n) is 1 mole.
PV = nRT
R = PV/nT
R = (1 atm.)(22.4 l)/(1 mole)(273^{o}K)
R = 0.0821 l-atm/mole-^{o}K
This value for R is constant, and can always be used if the units of the other quantities are not changed. We can determine the number of moles of a quantity of a substance by dividing the mass (in grams) of the substance being used by the gram-molecular weight of the substance.
no. of moles = grams of gas/(G.M.W.)
EXAMPLE: What is the molecular weight of a gas if 372 ml. weigh 0.8 grams at 100^{o}C and 800 Torrs?
372 ml. = .372 l
0.8 grams = (.8 grams)/(G.M.W.) moles = (.8)/(M.W.) moles
100^{o}C = 373^{o}K
PV = nRT
(800 atm.)(.372 l.)/( 760 ) = (0.8 moles)(0.82 l-atm)(373^{o}K)/( M.W. )( mole-^{o}K )
M.W. = (760)(0.8 moles)(0.082 l-atm)(373^{o}K)/(800 atm)(.372 l)(mole-^{o}K)
M.W. = 62.5
Any problem that can be solved by this equation can also be solved by direct application of the gas laws.
6.19 CRITICAL TEMPERATURE AND PRESSURE
Critical temperature and pressure are important properties of gases. When great pressure is applied to any gas at a low enough temperature, the molecules are forced closely together, and the gas condenses to a liquid. Each gas has a certain temperature above which it cannot be liquefied at any pressure. This temperature is called its CRITICAL TEMPERATURE. The critical temperature may be thought of as the maximum possible temperature at which a particular substance may exist as a liquid. The critical temperature of helium is -267.9^{o}C. Above this temperature, helium cannot be liquefied even if thousands of atmospheres of pressure are applied. The CRITICAL PRESSURE is the minimum pressure required to liquefy a gas at its critical temperature. The critical pressure of helium is 2.26 atm. This means that a pressure of 2.26 atm. or greater will liquefy a sample of helium gas if the temperature of the gas is no higher than -267.9^{o}C. Substances with low critical temperatures behave more ideally than those with higher critical temperatures. Thus, the behavior of helium approaches that of an ideal gas.
6.20 VAN DER WAALS FORCES
The attraction of gas molecules for each other under certain conditions accounts for the fact that Boyle's Law must be modified to describe the behavior of gases under very high pressure. For a similar reason, Charles' Law does not apply to gases at such low temperatures that they are near the point of liquefying. Unlike liquids and solids, gases always fill the container in which they are placed because their molecules can move freely. But compressing or cooling the gas beyond a certain point "pushes" the molecules of the gas so close together that the attractive forces between them become effective. These intermolecular forces resulting from THE ATTRACTION OF THE POSITIVE NUCLEI OF ONE MOLECULE FOR THE ELECTRON CLOUD OF A NEARBY MOLECULE are known as VAN DER WAALS FORCES. They were named after the Dutch scientist Johannes Diderils van der Waals (1837-1923), who was the first to recognize their importance. Although van der Waals forces are present in all matter, they come into play in gases only when the gases are under high pressure or at temperatures near the point of liquefication. At high pressures or at reduced temperatures, the gas molecules possess less kinetic energy to counteract the attraction of the van der Waals forces. Moreover, the greater the number of electrons in a molecule, usually the greater is the force of attraction between the molecules.
Summary - Summary of the Gas Laws
Boyle's Law: The volume of a fixed mass of gas varies inversely with the pressure at a constant temperature.
Charles' Law: The volume of a fixed mass of gas varies directly with the absolute temperature at a constant pressure.
Law of Charles & Gay-Lussac: The pressure of a fixed mass of gas varies directly with the absolute temperature at a constant volume.
Dalton's Law of Partial Pressures: The total pressure in a mixture of nonreacting gases is equal to the sum of the partial pressures of the individual gases.
Graham's Law of Diffusion: Under identical conditions of temperature and pressure, the rates of diffusion of gases are inversely proportional to the square roots of their molecular masses (or densities).
Gay-Lussac's Law (Law of Combining Volumes): The volumes of gases involved in a reaction, at constant temperature and pressure, can be expressed as a ratio of small whole numbers.
Avogadro's Law: Equal volumes of all gases, measured under the same conditions of temperature and pressure, contain the same number of molecules.
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