Chpt 9

1 Pa =

1 N/m^2 N is newton, a unit of force defined as 1kg m/s^2

The Kinetic-Molecular Theory Explains the Behavior of Gases, Part II

According to Graham's law, the molecules of a gas are in rapid motion and the molecules themselves are small. The average distance between the molecules of a gas is large compared to the size of the molecules. As a consequence, gas molecules can move past each other easily and diffuse at relatively fast rates. The rate of effusion of a gas depends directly on the (average) speed of its molecules effusion rate ∝ urms The ratio of rates of effusion is thus derived to be inversely proportional to the ratio of the square roots of their masses. This is the same relation observed experimentally and expressed as Graham's law.

Molecular velocity vs molecular mass

At a given temperature, all gases have the same KEavg for their molecules. Gases composed of lighter molecules have more high-speed particles and a higher urms, with a speed distribution that peaks at relatively higher velocities. Gases consisting of heavier molecules have more low-speed particles, a lower urms, and a speed distribution that peaks at relatively lower velocities. This trend is demonstrated by the data for a series of noble gases shown in Figure 9.34.

Chemical stoichiometry and gases

Chemical stoichiometry describes the quantitative relationships between reactants and products in chemical reactions. We have previously measured qnatities of reactants and products using masses for solids and volumes in conjunction with the molarity of the solutions; now we can also use gas volumes to indicate quantities. If we know hte volume, pressure, and temp of a gas, we can use the ideal gas law equation to calculate how many moles of the gas are present. If we know how many moles of a gas are involved, we can calculate the volume of a gas at any temp and pressure.

How the ideal gas law came to be.

During the seventeenth and especially eighteenth centuries, driven both by a desire to understand nature and a quest to make balloons in which they could fly (Figure 9.9), a number of scientists established the relationships between the macroscopic physical properties of gases, that is, pressure, volume, temperature, and amount of gas. Although their measurements were not precise by today's standards, they were able to determine the mathematical relationships between pairs of these variables (e.g., pressure and temperature, pressure and volume) that hold for an ideal gas—a hypothetical construct that real gases approximate under certain conditions. Eventually, these individual laws were combined into a single equation—the ideal gas law—that relates gas quantities for gases and is quite accurate for low pressures and moderate temperatures. We will consider the key developments in individual relationships (for pedagogical reasons not quite in historical order), then put them together in the ideal gas law.

Amonton's Law of Gay-Lussac's law

Guillaume Amontons was the first to empirically establish the relationship between the pressure and the temperature of a gas (~1700), and Joseph Louis Gay-Lussac determined the relationship more precisely (~1800). Because of this, the P-T relationship for gases is known as either Amontons's law or Gay-Lussac's law. Under either name, it states that the pressure of a given amount of gas is directly proportional to its temperature on the kelvin scale when the volume is held constant. Mathematically, this can be written: P ∝ T or P = constant × T or P = k × T where ∝ means "is proportional to," and k is a proportionality constant that depends on the identity, amount, and volume of the gas. For a confined, constant volume of gas, the ratio TP is therefore constant (i.e., TP = k ). If the gas is initially in "Condition 1" (with P = P1 and T = T1), and then changes to "Condition 2" (with P = P2 and T = T2), we have that P1 = k and P2 = k, which reduces to P1 = P2. This equation is useful for pressure-temperature calculations T1T2 T1T2 for a confined gas at constant volume. Note that temperatures must be on the kelvin scale for any gas law calculations (0 on the kelvin scale and the lowest possible temperature is called absolute zero). (Also note that there are at least three ways we can describe how the pressure of a gas changes as its temperature changes: We can use a table of values, a graph, or a mathematical equation.)

Vapor pressure of water

However there is another factor we must consider when we measure the pressure of the gas b this method. Water evaporates and there is always gaseous water (water vapor) above a sample of liquid water. As a gas is collected over water, it becomes saturated with water vapor and the total pressure of the mixture equals the partial pressure of the gas plus the partial pressure of the water vapor. The pressure of the pure gas is therefore equal to the total pressure minus the pressure minus the water vapor- this is referred to as the "dry" gas pressure, that is, the pressure of the gas only, without water vapor. The vapor pressure of water, which is the pressure exerted by water vapor in equilibrium with liquid water in a closed container, depends on the temp.

Barometer continued...

If the liquid is water, normal atm pressure will supoort a column of water over 10 meters high, which is rather inconvenient for making (and reading) a barometer. Because mercury (Hg) is about 13.6-times denser than water, a mercury barometer only needs to be 1/13.6 as tall as a water barometer- a more suitable size. Standard atmospheric pressure of 1 atm at sea level (101, 325 Pa) corresponds to a column of mercury that is about 760 mm(29.92 in) high.

combined gas law

If the number of moles of an ideal gas are kept constant under two diff sets of conditions, a useful mathematical relationship called the combined gas law is obtained. P1V1/T1=P2V2/T2 Using units of atm, L, and K. Both sets of conditions are equal to the product of n x R (where n = the number of moles of the gas and R is the ideal gas law constant).

Temp and ave KE

If the temperature of a gas increases, its KEavg increases, more molecules have higher speeds and fewer molecules have lower speeds, and the distribution shifts toward higher speeds overall, that is, to the right. If temperature decreases, KEavg decreases, more molecules have lower speeds and fewer molecules have higher speeds, and the distribution shifts toward lower speeds overall, that is, to the left. This behavior is illustrated for nitrogen gas in

V and P...

If we fill a balloon with air and seal it, the balloon contains a specific amount of air at atmospheric pressure, let's say 1 atm. If we put the ballon in a fridge, the gas inside gets cold and the balloon shrinks (although both the amount of gas and its pressure remain constant). If we make the ballon very cold, it will shrink a great deal, and it expands again when it warms up. These examples of the effect of temp on the volume of a given amount of a confined gas at constant pressure are true in general: The volume increases as the temp increases, and decreases as the temp decreases. Volume-temp data for a 1-mole sample of methane gas at 1 atm are listed and graphed in figure 9.12.

Volume and pressure...

If we partially fill an airtight syringe with air, the syringe contains a specific amount of air at constant temp, say 25°C. If we slowly push in the plunger while keeping temp constant, the gas in the syringe is compressed into a smaller volume and the P increases; if we pull out the plunger, the volume increases and the pressure decreases. This example of the effect of V on P of a given amount of a confined gas is true in general. Decreasing the volume of a contained gas will increase its pressure, and increasing its volume will decrease its pressure. In fact, if the volume increases by a certain factor, the pressure decreases by the same factor, and vice versa. Volume-pressure data for an air sample at room temp are graphed in figure... Unlike the P-T and V-T relationships, pressure and volume are not directly proportional to each other. Instead, P and V exhibit inverse proportionality: Increasing the pressure results in a decrease of the volume of the gas. Mathematically this can be written as: Pα1/V orP = k·1/V orP·V = korP1V1 = P2V2 With k being a constant. Graphically, this relationship is shown by the straight line that results when plotting the inverse of the pressure (1/P) versus the volume (V), or the inverse of the volume (1/V) versus the pressure (P). Graphs with curved lines are difficult to read accurately at low or high values of the variables,and they are more difficult to use in fitting theoretical equatiosn and parameters to experimental data. For those reasons, scientists often try to find a way to "linearize" their data. If we plot P versus V, we obtain a parabola. The relationship between the volume and pressure of a given amount of gas at constant pressure was first published by the English natural philosopher Robert Boyle over 300 years ago. It is summarized in the statement now known as Boyle's law: the volume of a given amount of gas held at constant temp is inversely proportional to the pressure under which it is moved.

mean free path

If you've ever been in a room when a piping hot pizza was delivered you have been made aware of the fact that gaseous molecules can quickly spread throughout a room, as evidenced by the pleasant aroma that soon reaches your nose. Although gaseous molecules travel at tremendous speeds (hundreds of meters per second) they collide with other gaseous molecules and travel in many diff directions before reaching the desired target. At room temp, a gaseous molecule will experience billions of collisions per second. The mean free path is the average distance a molecule travels between collisions. The mean free path increases with decreasing pressure; in general, the mean free path for a gaseous molecule will be hundreds of times the diameter of the molecule.

Pressure and temperature: Amontons's law

Imagine filling a rigid container attached to a pressure gauge with gas and then sealing the container so that no gas may escape. If the container is cooled, the gas inside likewise gets colder and its pressure is observed to decrease. Since the container is rigid and tightly sealed, both the volume and number of moles of gas remain constant. If we heat the sphere, the gas inside gets hotter (Figure 9.10) and the pressure increases.

Amonton's Law equation

P/T = constant at constant V and n P1/T1=P2/T2

Boyle's law equation

PV = constant at constant T and n Thus, P1V1=P2V2

As is apparent from Figure 9.35, the ideal gas law does not describe gas behavior well at relatively high pressures. To determine why this is, consider the differences between real gas properties and what is expected of a hypothetical ideal gas.

Particles of a hypothetical ideal gas have no significant volume and do not attract or repel each other. In general, real gases approximate this behavior at relatively low pressures and high temperatures. However, at high pressures, the molecules of a gas are crowded closer together, and the amount of empty space between the molecules is reduced. At these higher pressures, the volume of the gas molecules themselves becomes appreciable relative to the total volume occupied by the gas. The gas therefore becomes less compressible at these high pressures, and although its volume continues to decrease with increasing pressure, this decrease is not proportional as predicted by Boyle's law. At relatively low pressures, gas molecules have practically no attraction for one another because they are (on average) so far apart, and they behave almost like particles of an ideal gas. At higher pressures, however, the force of attraction is also no longer insignificant. This force pulls the molecules a little closer together, slightly decreasing the pressure (if the volume is constant) or decreasing the volume (at constant pressure) (Figure 9.36). This change is more pronounced at low temperatures because the molecules have lower KE relative to the attractive forces, and so they are less effective in overcoming these attractions after colliding with one another. a) Attractions between gas molecules serve to decrease the gas volume at constant pressure compared to an ideal gas whose molecules experience no attractive forces. (b) These attractive forces will decrease the force of collisions between the molecules and container walls, therefore reducing the pressure exerted at constant volume compared to an ideal gas.

SI unit of pressure

Pascal (Pa)

atmosphere (atm)

Pressure can also be measured using the unit atmosphere (atm), which originally represented the average sea level air pressure at the approximate latitude of Paris (45°)

The Kinetic-Molecular Theory Explains the Behavior of Gases, Part I

Recalling that gas pressure is exerted by rapidly moving gas molecules and depends directly on the number of molecules hitting a unit area of the wall per unit of time, we see that the KMT conceptually explains the behavior of a gas as follows: 1. Amontons's law. If the temperature is increased, the average speed and kinetic energy of the gas molecules increase. If the volume is held constant, the increased speed of the gas molecules results in more frequent and more forceful collisions with the walls of the container, therefore increasing the pressure (Figure 9.31). Charles's law. If the temperature of a gas is increased, a constant pressure may be maintained only if the volume occupied by the gas increases. This will result in greater average distances traveled by the molecules to reach the container walls, as well as increased wall surface area. These conditions will decrease the both the frequency of molecule-wall collisions and the number of collisions per unit area, the combined effects of which balance the effect of increased collision forces due to the greater kinetic energy at the higher temperature. Boyle's law. If the gas volume is decreased, the container wall area decreases and the molecule-wall collision frequency increases, both of which increase the pressure exerted by the gas (Figure 9.31). Avogadro's law. At constant pressure and temperature, the frequency and force of molecule-wall collisions are constant. Under such conditions, increasing the number of gaseous molecules will require a proportional increase in the container volume in order to yield a decrease in the number of collisions per unit area to compensate for the increased frequency of collisions (Figure 9.31). Dalton's Law. Because of the large distances between them, the molecules of one gas in a mixture bombard the container walls with the same frequency whether other gases are present or not, and the total pressure of a gas mixture equals the sum of the (partial) pressures of the individual gases.

van der waals equation

There are several different equations that better approximate gas behavior than does the ideal gas law. The first, and simplest, of these was developed by the Dutch scientist Johannes van der Waals in 1879. The van der Waals equation improves upon the ideal gas law by adding two terms: one to account for the volume of the gas molecules and another for the attractive forces between them. The constant a corresponds to the strength of the attraction between molecules of a particular gas, and the constant b corresponds to the size of the molecules of a particular gas. The "correction" to the pressure term in the ideal gas law is n2 a, and the "correction" to the volume is nb. Note that when V is relatively large and n is relatively small, V2 both of these correction terms become negligible, and the van der Waals equation reduces to the ideal gas law, PV = nRT. Such a condition corresponds to a gas in which a relatively low number of molecules is occupying a relatively large volume, that is, a gas at a relatively low pressure. Experimental values for the van der Waals constants of some common gases are given in Table 9.3. At low pressures, the correction for intermolecular attraction, a, is more important than the one for molecular volume, b. At high pressures and small volumes, the correction for the volume of the molecules becomes important because the molecules themselves are incompressible and constitute an appreciable fraction of the total volume. At some intermediate pressure, the two corrections have opposing influences and the gas appears to follow the relationship given by PV = nRT over a small range of pressures. This behavior is reflected by the "dips" in several of the compressibility curves shown in Figure 9.35. The attractive force between molecules initially makes the gas more compressible than an ideal gas, as pressure is raised (Z decreases with increasing P). At very high pressures, the gas becomes less compressible (Z increases with P), as the gas molecules begin to occupy an increasingly significant fraction of the total gas volume. Strictly speaking, the ideal gas equation functions well when intermolecular attractions between gas molecules are negligible and the gas molecules themselves do not occupy an appreciable part of the whole volume. These criteria are satisfied under conditions of low pressure and high temperature. Under such conditions, the gas is said to behave ideally, and deviations from the gas laws are small enough that they may be disregarded—this is, however, very often not the case.

relationship between T and P under constant V...

This relationship between temperature and pressure is observed for any sample of gas confined to a constant volume. An example of experimental pressure-temperature data is shown for a sample of air under these conditions in Figure 9.11. We find that temperature and pressure are linearly related, and if the temperature is on the kelvin scale, then P and T are directly proportional (again, when volume and moles of gas are held constant); if the temperature on the kelvin scale increases by a certain factor, the gas pressure increases by the same factor.

Charles law equation

V/T = constant at constant P and n V1/T1=V2/T2

Avogadro's law equation

V/n = constant at constant P and T V1/n1 = V2/n2

rate of diffusion

We are often interested in the rate of diffusion, the amount of gas passing thru some area per unit time. rate of diffusion = (amount of gas passing thru an area)/(unit of time) the diffusion rate depends on several factors: the concentration gradient, amount of surface area available for diffusion, and the distance gas particles must travel. Time required for diffusion to occur is inversely proportional to rate of diffusion.

barometer

We can measure atmospheric pressure, the force exerted by the atmosphere on the earth's surface, with a barometer. A barometer is a glass tube that is closed at one end, filled with a nonvolatile liquid such as mercury, and then inverted and immersed in a container of that liquid. The atmosphere exerts pressure on the liquid outside the tube, the column of the liquid exerts pressure inside the tube, and the pressure at the liquid surface is the same inside and outside the tube. The height of the liquid in the tube is therefore proportional to the pressure exerted by the atmosphere.

Standard Temp and Pressure (STP)

We have seen that the volume of a given quantity of gas and the number of molecules (moles) in a given volume of gas vary with changes in pressure and temperature. Chemists sometimes make comparisons against a standard temperature and pressure (STP) for reporting properties of gases: 273.15 K and 1 atom (101.325 kPa). At STP one mole of an ideal gas has a volume of about 22.4 L- this is referred to as the standard molar volume.

manometer

a mamometer is a device similar to a barometer that can be used to measure the pressure of a gas trapped in a container. A closed-end manometer is a U-shaped tube with one closed arm, one arm that connects to the gas to be measured, and a nonvolatile liquid (usually mercury) in between. As with a barometer, the distance between the liquid levels in the two arms of the tube (h in the diagram) is proportional to the pressure of the gas in the container. An open-end manometer is the same as a closed-end manometer, but one of its arms is open to the atmosphere. In this case, the distance between the liquid levels corresponds to the difference in pressure between the gas in the container and the atmosphere.

effusion

a process involving movement of gaseous similar to diffusion is effusion, the escape of gas molecules thru a tiny hole such as a pinhole in a balloon into a vacuum. Although diffusion and effusion rates both depend on the MM of teh gas involved, their rates are not equal; however, the ratio of their rates are the same. Diffusion involves the unrestricted dispersal of molecules throughout space due to their random motion. When this process is restricted to passage of molecules through very small openings in a physical barrier, the process is called effusion. This means that if two gases A and B are at the same temperature and pressure, the ratio of their effusion rates is inversely proportional to the ratio of the square roots of the masses of their particles:

Collection of gases over water

a simple way to collect gases that do not react with water is to capture them in a bottle that has been filled wiht water and inverted into a dish filled wiht water. The pressure of the gas inside the bottle can be made equal to the air pressure outside by raising or lowering the bottle. When the water level is the same both inside and outside hte bottle the pressure of the gas is equal to atmospheric pressure, which can be measured with a barometer.

One pascal is...

a small pressure. Thus, in many cases, it is more convenient to use units of kilopascal or bar (1 bar = 100,000 Pa).

How can pressure be decreased?

by decreasing the force or increasing the area

In the US, pressure is often measured in...

pounds of force on an area of one square inch- pounds per square inch (psi)- for example, in car tires.

Heated gas expand...

which can make a hot air balloon rise, or cause a blowout in a a bicycle left in the sun on a hot day

Avogadro's law revisited

Sometimes we can take advantage of a simplifying feature of the stoichiometry of gases that solids and solutions do not exhibit. All gases that show ideal behavior contain the same number of molecules in the same volume (at same temp and pressure). thus, the ratios of volumes of gases involved in a chemical reaction are given by the coefficients in the equation for the reaction, provided that the gas volumes are measured at the same temp and pressure. We can extend Avogadro's law (that the volume of a gas is directly proportional to the number of moles of the gas) to chemical reactions with gases: Gases combine, or react, in definite and simple proportions by volume, provided that all gas volumes are measured at the same temp and pressure. For example, since nitrogen and hydrogen gases react to produce ammonia gas according to N2 (g) + 3H2 (g) -> 2NH3 (g), a given volume of nitrogen gas reacts with three times that volume of hydrogen gas to produce two times that volume of ammonia gas, if pressure and temp remain constant. The explanation for this is illustrated in figure 9.23. According to Avogadro's law, equal volumes of gaseous N2, H2, and NH3, at the same temp and pressure, contain the same number of molecules. Because one molecule of H2, reacts with three molecules of H2 to produce two molecules of NH3, the volume of H2 required is three times the volume of N2 and the volume of NH3. produced is two times te volume of N2.

Gas pressure

The earth's atmosphere exerts a pressure as does any gas. Although we do not normally notice atmospheric pressure, we are sensitive to pressure changes. For example, when your ears "pop" during take-off and landing while flying, or when you dive underwater. Gas pressure is caused by the force exerted by gas molecules colliding with the surfaces of objects. Although the force of each collision is very small, any surface of appreciable area experiences a large number of collisions in a short time, which can result in a high pressure. In fact, normal air pressure is strong enough to crush a metal container when not balanced by equal pressure from inside the container. Atmospheric pressure is caused by the weight of the column of air molecules in the atmosphere above an object, such as the tanker car. At sea level, this pressure is roughly the same as that exerted by a full grown African elephant standing on a doormat, or a typical bowling ball resting on your thumbnail. These may seem like huge amounts, and they are, but life on earth has evolved under such atmospheric pressure. If you actually perch a bowling ball on your thumbnail, the pressure experienced is twice the usual pressure, and the sensation is unpleasant.

Kinetic Molecular Theory

The gas laws that we have seen to this point, as well as the ideal gas equation, are empirical, that is, they have been derived from experimental observations. The mathematical forms of these laws closely describe the macroscopic behavior of most gases at pressures less than about 1 or 2 atm. Although the gas laws describe relationships that have been verified by many experiments, they do not tell us why gases follow these relationships. The kinetic molecular theory (KMT) is a simple microscopic model that effectively explains the gas laws described in previous modules of this chapter. This theory is based on the following five postulates described here. (Note: The term "molecule" will be used to refer to the individual chemical species that compose the gas, although some gases are composed of atomic species, for example, the noble gases.) 1. Gases are composed of molecules that are in continuous motion, travelling in straight lines and changing direction only when they collide with other molecules or with the walls of a container. 2. The molecules composing the gas are negligibly small compared to the distances between them. 3. The pressure exerted by a gas in a container results from collisions between the gas molecules and the container walls. 4. Gas molecules exert no attractive or repulsive forces on each other or the container walls; therefore, their collisions are elastic (do not involve a loss of energy). 5. The average kinetic energy of the gas molecules is proportional to the kelvin temperature of the gas. The test of the KMT and its postulates is its ability to explain and describe the behavior of a gas. The various gas laws can be derived from the assumptions of the KMT, which have led chemists to believe that the assumptions of the theory accurately represent the properties of gas molecules. We will first look at the individual gas laws (Boyle's, Charles's, Amontons's, Avogadro's, and Dalton's laws) conceptually to see how the KMT explains them. Then, we will more carefully consider the relationships between molecular masses, speeds, and kinetic energies with temperature, and explain Graham's law.

gas density and molar mass

The ideal gas law described previously in this chapter relates the properties of pressure P, volume V, temp T, and molar amount n. This law is universal, relating these properties in identical fashion regardless of the chemical identity of the gas. PV = nRT The density d of the gas, on the other hand, is determined by its identity. As described in another chapter of this text, the density of a substance is a characteristic property that may be used to identify the substance. d = m/V Rearranging the ideal gas equation to isolate V and substituting into the density equation yields .. look at photo... d = (MM)P/RT This relation may be used for calculating densities of gases of known identities at specified values of pressure and temp. When the identity of a gas is unknown, measurements of the mass, pressure, volume and temp of a sample can be used to calculate the MM of the gas (a useful property for identification purposes) combining the ideal gas law equation PV = nRT and the def of molarity MM = m/n Equation MM = mRT/PV

Avogadro's law

The italian scientist Amedeo Avogadro advanced a hypothesis in 1811 to account for the behavior of gases, stating that the equal volumes of all gases, measured under the same conditions of temp and pressure, contain the same number of molecules. Over time, this relationship was supported by many experimental observations as expressed by Avogadro's law: For a confined gas, the volume (V) and number of moles (n) are directly proportional if the pressure and temp both remain constant. In equation form, this is written as: V ∝ n or V = k × n or V1 = V2 Mathematical relationships can also be determined for the other variable pairs, such as P versus n, and n versus T.

Hydrostatic pressure, p.

The pressure exerted by fluid due to gravity is known as hydrostatic pressure, p. p = hdg Where h is the height of the fluid, d is the density of the fluid, and g is acceleration due to gravity.

Non-ideal gas behavoir

By the end of this section you will be able to: Describe the physical factors that lead to deviations from ideal gas behavior Explain how these factors are represented in the van der Waals equation Define compressibility (Z) and describe how its variation with pressure reflects non-ideal behavior Quantify non-ideal behavior by comparing computations of gas properties using the ideal gas law and the van der Waals equation Thus far, the ideal gas law, PV = nRT, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and molecular formula problems to determining the density and molar mass of a gas. As mentioned in the previous modules of this chapter, however, the behavior of a gas is often non-ideal, meaning that the observed relationships between its pressure, volume, and temperature are not accurately described by the gas laws. In this section, the reasons for these deviations from ideal gas behavior are considered. One way in which the accuracy of PV = nRT can be judged is by comparing the actual volume of 1 mole of gas (its molar volume, Vm) to the molar volume of an ideal gas at the same temperature and pressure. This ratio is called the compressibility factor (Z) with: Z = (molar volume of gas at same T and P)/(molar volume of ideal gas at same T and P) = (PVm/RT)measured Ideal gas behavior is therefore indicated when this ratio is equal to 1, and any deviation from 1 is an indication of non-ideal behavior. Figure 9.35 shows plots of Z over a large pressure range for several common gases.

ideal gas law

Combining these four laws yields the ideal gas law, a relationship between pressure, volume, temp, and number of moles of a gas PV = nRT Where P is the pressure of the gas, V is its volume, n is the number of moles of the gas, T is its temp on the kelvin scale, and R is a constant called the ideal gas constant or the universal gas constant. The units used to express P, V, and T will determine the proper form of the gas constant as required by dimensional analysis, the most commonly encountered values being 0.08206 L atm mol^-1 K^-1 and 8.314 kPa L mol^-1 K^-1. Gases whose properties of P, V, and T are accurately described by the ideal gas law (or other gas laws) are said to exhibit ideal behavior or to approximate the traits of an ideal gas. An ideal gas is a hypothetical construct that may be used along with kinetic molecular theory to effectively explain the gas laws as will be described in a later module of this chapter. Although all the calculations presented in this module assume ideal behavior, this assumption is only reasonable for gases under conditions of relatively low pressure and high temp. In the final module of this chapter, a modified gas law will be introduced that accounts for the non-ideal behavior observed for many gases at relatively high pressures and low temps. The ideal gas equaton contains five terms, the gas constant R and the variable properties, P, V, n, and T. Specifying any four of these terms will permit use of the ideal gas law to calculate the fifth term as demonstrated in the following example exercises.

Molecular velocities and kinetic energy

The previous discussion showed that the KMT qualitatively explains the behaviors described by the various gas laws. The postulates of this theory may be applied in a more quantitative fashion to derive these individual laws. To do this, we must first look at velocities and kinetic energies of gas molecules, and the temperature of a gas sample. In a gas sample, individual molecules have widely varying speeds; however, because of the vast number of molecules and collisions involved, the molecular speed distribution and average speed are constant. This molecular speed distribution is known as a Maxwell-Boltzmann distribution, and it depicts the relative numbers of molecules in a bulk sample of gas that possesses a given speed (Figure 9.32). The kinetic energy (KE) of a particle of mass (m) and speed (u) is given by: KE= 12mu^2 Expressing mass in kilograms and speed in meters per second will yield energy values in units of joules (J = kg m2 s-2). To deal with a large number of gas molecules, we use averages for both speed and kinetic energy. In the KMT, the root mean square velocity of a particle, urms, is defined as the square root of the average of the squares of the velocities with n = the number of particles: The average kinetic energy for a mole of particles, KEavg, is then equal to: KE = 1Mu2rms where M is the molar mass expressed in units of kg/mol. The KEavg of a mole of gas molecules is also directly proportional to the temperature of the gas and may be described by the equation: KEavg = 32 RT where R is the gas constant and T is the kelvin temperature. When used in this equation, the appropriate form of the gas constant is 8.314 J/mol⋅K (8.314 kg m2s-2mol-1K-1). These two separate equations for KEavg may be combined and rearranged to yield a relation between molecular speed and temperature:

Charle's law

The relationship between volume and temp of a given amount of gas at constant pressure is known as Charles' law in recognition of the French scientist and balloon flight pioneer.. Charles' law states that the volume of a given amount of gas is directly proportional to its temp on the kelvin scale when the pressure is held constant. Mathematically, this can be written as: V αT orV = constant·T orV = k·T orV1/T1 = V2/T2 with k being a proportionality constant that depends on the amount and pressure of the gas. For a confined, constant pressure gas sample, VT is constant (i.e., the ratio = k), and as seen with the P-T relationship, this leads to another form of Charles's law: V1 = V2.

Torr

The torr was originally intended to be a unit equal to one millimeter of mercury, but it no longer corresponds exactly.

pressure

in general, pressure is defined as the force exerted on a given area: P = F/A. Note that pressure is directly proportional to force and inversely proportional to area. Thus, pressure can be increased either by increasing the amount of force and inversely proportional to area. Thus, pressure can be increased either by increasing the amount of force or by decreasing the area over which it is applied; pressure can be decreased by the force or increasing the area.

Stoichiometry of gaseous substances, mixtures, and reactions

by the end of this section you will be able to: -Use the ideal gas law to compute gas densities and molar masses -Perform stoichiometric calculations involving gaseous substances -State Dalton's law of partial pressures and use it in calculations involving gaseous mixtures The study of chemical behavior of gases was part of the basis of perhaps the most fundamental chemical revolution in history. French nobleman Antoine Lavoisier, widely regarded as the "father of modern chemistry" changed chemistry from a qualitative to a quantitative science through his work with gases. He discovered the law of conservation of matter, discovered the role of oxygen in combustion reactions, determined the composition of air, explained respiration in terms of chemical reactions, and more. As described in an earlier chpt of this text, we can turn to chemical stoichiometry for answers to many of the questions that ask "How much?" The essential property involved in such use of stoichiometry is the amount of a substance, typically measured in moles (n). For gases, molar amount can be derived from convenient experimental measurements of pressure, temp, and volume. Therefore, these measurements are useful in assessing the stoichiometry of pure gases, gas mixtures, and chemical reactions involving gases. This section will not introduce any new material or ideas, but will provide examples of applications and ways to integrate concepts already discussed.

diffusion

in general we know that when a sample of gas is introduced to one part of a closed container, its molecules very quickly disperse throughout the container; this process by which molecules disperse in space in response to differences in concentration is called diffusion. The gaseous atoms or molecules are, of course, unaware of any concentration gradient, they simply move randomly- regions of higher concentration have more particles than regions of lower concentrations, and so a net movement of a species from high to low concentration areas takes place. In a closed environment, diffusion will ultimately result in equal concentrations of gas throughout as depicted... the gaseous atoms and molecules continue to move, but since their concentrations are the same in both bulbs, their concentrations are the same in both bulbs, the rates of transfer between the bulbs are equal (no net transfer of molecules occurs).