Physics
Ideal Gases
Ideal gases are hypothetical gases that perfectly follow the ideal gas law, which describes their behavior under certain conditions. In an ideal gas, the particles have negligible volume and do not interact with each other. This simplifies calculations and allows for the prediction of gas behavior in various situations.
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eBook - PDFPhysical Chemistry
A Modern Introduction, Second Edition
- William M. Davis(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
23 2 Ideal and Real Gases Examinationofthegaseousstateofmatteroffersanexcellentmeansforunderstanding certainbasicsofthermodynamicsandforseeingconnectionswithatomicandmolecu-larlevelbehavior�Thischapterisconcernedwithexplainingthedifferencesbetweenthe behaviorofrealgasesandthebehaviorofahypotheticalgascalledanidealgasbecause ofitsparticularlysimplebehavior�Theidealgasisamodelthatundercertainconditions canserveasagoodapproximationofrealgasbehavior,andwebeginbyconsideringthe relationshipoftemperaturetootherpropertiesofanidealgas� 2.1 The Ideal Gas Laws Experimentsthatarenowcenturiesoldhaverevealedrelationshipsamongthreeproper-tiesofgases,thevolume, V ,thepressure, P ,andthetemperature, T �Volumeissimplythe three-dimensionalspaceoccupiedbythegassample�Itisfixedbythegeometryofthecon-tainerholdingthegas�Pressureisaforceperunitarea,anditcanbemeasuredbybalancing againstanexternalforceofknownsize�AsshowninFigure2�1,apistonassemblyprovides onemeansofmeasuringpressure�Thegaspressureisexertedagainstthepiston,andthe piston is loaded until its position is unchanging, that is, until it reaches the point of bal-ance�Theforceexertedbythepistonisthegravitationalforceofthemass( m )loadedonit, andthatforceisthegravitationalconstant g times m �Asdiscussedinthepreviouschapter, onefundamentaldefinitionoftemperatureisrelatedtodistributionsamongtheavailable energystates�Inpractice,certainmechanicalchangeshavebeenshowntovarylinearlywith temperature,atleastovercertainranges�Thevolumeofmercuryoverafairlywiderange aroundroomtemperatureisoneofthese�Hence,measuringthevolumeofmercuryinatube (athermometer)servestomeasurethetemperature� Asthetemperatureofasampleoffixedvolumeincreases,thepressureincreases� If the external pressure acting on a gas sample contained in a piston assembly is increased,thevolumewilldiminish�Thesearestatementsofobservedphenomena,
- Armand Berman(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
1 Ideal Gases 1.1 The Ideal Gas Law If an ideal gas, specified by the quantities p { (pressure) and V! (volume) at temperature 7, changes its state to another set of quantities p 2 > ^2 a t ^ 2 » t n e n , ,/ , =p 2 V 2 /T 2 (1.1) where T is the thermodynamic temperature measured in degrees Kelvin (see Appendix, Table A.20). 1.2 Boyle's Law For a given mass of gas, held at T = const. p x V x = p 2 V 2 = const. (1.2) 13 Charles's Law For a given mass of gas, held at p = const. Vi/ T i = V 2 /T 2 = const. (1.3) IA Gay-Lussac'sLaw For a given mass of gas, held at V = const. f>i/ T i = Pi/^i = const. (1.4) 1 2 1 Ideal Gases 1.5 Mole Amount The amount of moles n M in a given mass W t of substance having a molar mass M is n M = W t /M [mol (moles)] (1.5) The molar mass M of a substance (also known as moiar weight) is the mass divided by the amount of substance. The SI base unit is kg mol -1 , and the practical unit is g mol -1 (Compendium of Chemical Technology, 1987, p. 260). 1.6 Dalton's Law In a mixture of gases, each component exerts the pressure that it would exert if it were present alone at the same temperature in the volume occupied by the gas mixture. The total pressure p of a gas mixture is the sum of partial pressures Pv Pv - -· » Pi °f t n e individual components. P = Px + Pi + *·· +ft = ÎPi (1.6) The partial pressure of each component is equal to the total pressure multiplied by its mole fraction c, in the mixture (for c,·, see Eq. 1.7c). Dalton's law holds true for Ideal Gases. At pressures below atmosphere, gas mixtures can be regarded as Ideal Gases.
eBook - PDFChemistry
An Atoms First Approach
- Steven Zumdahl, Susan Zumdahl, Donald J. DeCoste, , Steven Zumdahl, Steven Zumdahl, Susan Zumdahl, Donald J. DeCoste(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Thus ideal gas behavior can best be thought of as the behavior approached by real gases under certain conditions. We have seen that a very simple model, the kinetic molecular theory, by making some rather drastic assumptions (no interparticle interactions and zero volume for the gas particles), successfully explains ideal behavior. However, it is important that we examine real gas behavior to see how it differs from that predicted by the ideal gas law and to determine what modifications are needed in the kinetic molecular theory to explain the observed behavior. Since a model is an approximation and will inevitably fail, we must be ready to learn from such failures. In fact, we often learn more about nature from the failures of our models than from their successes. We will examine the experimentally observed behavior of real gases by measuring the pressure, volume, temperature, and number of moles for a gas and noting how the quantity PVynRT depends on pressure. Plots of PVynRT versus P are shown for sev- eral gases in Fig. 8.25. For an ideal gas, PVynRT equals 1 under all conditions, but notice that for real gases, PVynRT approaches 1 only at very low pressures (typically below 1 atm). To illustrate the effect of temperature, PVynRT is plotted versus P for nitrogen gas at several temperatures in Fig. 8.26. Note that the behavior of the gas ap- pears to become more nearly ideal as the temperature is increased. The most important conclusion to be drawn from these figures is that a real gas typically exhibits behavior that is closest to ideal behavior at low pressures and high temperatures. One of the most important procedures in science is correcting our models as we col- lect more data. We will understand more clearly how gases actually behave if we can figure out how to correct the simple model that explains the ideal gas law so that the new model fits the behavior we actually observe for gases.
eBook - PDF- Arthur Adamson(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
A phenomenological relationship, however, merely reflects some aspect of the behavior of nature, and must therefore be correct (within the limits of the experimental error of the measurement). 4 CHAPTER 1: IDEAL AND NONIdeal Gases Robert Boyle: 1627-1691 As the son of the Earl of Cork, he was born to wealth and nobility. While residing at Oxford, he discovered 'Boyle's law, methyl alco-hol, and phosphoric acid, and noted the darkening of silver salts by light. In The Sceptical Chymist, he at-tacked the alchemical no-tion of the elements, giving an essentially modern defi-nition. A founder of the Royal Society. (From H. M. Smith, Torchbearers of Chemistry, Academic Press, New York, 1949.) Two of Boyle's Experiments FIG. 1-1. On the left : A demonstration that a paddle wheel of feathers fell rapid-ly in a vacuum, and without turning. Boyle was seeing if air had some subtle com-ponent that could not be re-moved. On the right: How Boyle obtained the data of Table 1-1. (From Robert Boyle's Experiments in Pneumatics J. B. Conant, ed., Harvard Univ. Press, 1950.) Mercury column increased by pouring mercury in at Τ ^ Shorter leg with scale Initial level of mercury 1-3 THE IDEAL GAS; THE ABSOLUTE TEMPERATURE SCALE 5 was a function of temperature. At this point the equation of state for all gases was observed to be Pv =f {t). (1-3) Very accurate contemporary measurements add some important refinements. A selection of such results is given in Table 1-2, and we now see that not only does the PV product depend on pressure at constant temperature, but it does so in different ways for different gases. The data can be fitted to the equation PV = A(t) + b(t)P + c(t)P 2 + -· · (1-4) where t in parentheses is a reminder that the coefficients A, b, c, etc., are tempera-ture dependent. The important observation is that while b, c 9 and so forth depend also on the nature of the gas, the constant A does not.
eBook - PDF- Don Shillady(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
For that reason, this author favors the simple formula in terms of percent uncertainties, which can be (subjectively) estimated numerically. NONIDEAL GAS BEHAVIOR While the ideal gas law works well for pressures up to about 10 atm and higher temperatures above 25 8 C, many common processes (air conditioning, refrigeration) involve higher pressures and lower temperatures. If the ideal gas law is truly universal we could de fi ne the ‘‘ compressibility factor ’’ as Z ¼ PV nRT ¼ 1 and expect that if we plot Z against the pressure we should get a fl at line (Figure 1.6). When such graphs are plotted for real data, there are large deviations, particularly at low temperatures and = or high pressures. There are other ways to plot these data to exaggerate the deviations from Z ¼ 1, but on the other hand we can see that over a fairly large range of temperatures and pressures the ideal gas law is approximately correct. What are the reasons for the deviations from the ideal? Let us try to patch the ideal gas law for a more detailed treatment. We start by setting up the basic PV behavior and allow for corrections. ( P þ ? 1 )( V þ ? 2 ) ¼ nRT Consider a correction to the pressure, P . If indeed the pressure we measure is due to molecular impacts with a surface in a manometer or a diaphragm in a pressure gauge, is that the actual pressure within the gas? We are creeping up on a new concept that models a gas as a collection of small Ideal and Real Gas Behavior 13 molecules fl ying around with a lot of space between them (recall Dalton ’ s law). That idea should include collisions of molecules within the volume. Consider a collision of an auto with a fi xed wall compared to a head-on collision with another similar auto.
eBook - PDF- Arther Adamson(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
This type of distinction occurs fairly often in physical chemistry, such as, for example, in the treatment of solutions. 1-4 The Ideal Gas Law and Related Equations Equation (1-10) can be put in various alternative forms, such as Pv = nRT (n = number of moles); (1-11) YYl Pv = RT (M = molecular weight); (1-12) PM = pRT ( P = density). (1-13) Equation (1-13) tells us, for example, that the molecular weight of any gas can be obtained approximately if its pressure and density are known at a given tempera-ture. Furthermore, since the ideal gas law is a limiting law, the limiting value of P/p as pressure approaches zero must give the exact molecular weight of the gas. In effect, by writing Eq. (1-4) in the form P = Pv = RT βΡ y_P> p m Μ Μ Μ 9 U ; one notes that the intercept of Pv/m (or P/p) plotted against Ρ must give RT/M for any gas. Such a plot is illustrated schematically in Fig. 1-2. Example. The density of a certain hydrocarbon gas at 25°C is 12.20 g liter 1 at Ρ = 10 atm and 5.90 g liter 1 for Ρ = 5 atm. Find the molecular weight of the gas and its probable formula. At 10 atm, P/p is 10/12.20 = 0.8197, and at 5 atm, it is 5/5.90 = 0.8475. Linear extrapolation to zero pressure gives P/p = 0.8753. Hence Μ = RT/(P/p) = (0.082057)(298.15)/(0.8753) = 27.95 g m o l e 1 . The probable formula is C 2 H 4 . Example. Convert the data above to SI units and rework the problem. The SI unit of force is the newton, N ; this force gives an acceleration of 1 m s e c 2 to 1 kg. The SI unit of pressure is the pascal, Pa; 1 Pa is 1 Ν per m a . Thus 1 atm = (0.760 m Hg)(13.5981 g c m -8 ) ( 1 0 8 kg g -l ) ( 1 0 e c m 8 m 8 ) (9.80665 m sec 2 ) = 1.01325 x 1 0 5 Pa or Ν m a . Also, 1 g l i t e r -1 = 1 kg m ~ 8 . The problem now reads that the density is 12.20 kg m 8 at Ρ = 1.01325 χ 1 0 e Pa and is 8 CHAPTER 1: IDEAL AND NONIdeal Gases 5.90 kg m -8 at Ρ = 5.06625 χ 10 6 Pa.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The mass per mole (in g/mol) of a substance has the same numerical value as the atomic or molecular mass of one of its particles (in atomic mass units). The mass m particle of a particle (in grams) can be obtained by dividing the mass per mole (in g/mol) by Avogadro’s number, according to Equation 3. 14.2 The Ideal Gas Law The ideal gas law relates the absolute pressure P, the volume V, the number n of moles, and the Kelvin temperature T of an ideal gas, according to Equation 14.1, where R 5 8.31 J/(mol ? K) is the universal gas constant. An alternative form of the ideal gas law is given by Equation 14.2, where N is the number of particles and k 5 R N A is Boltzmann’s constant. A real gas behaves as an ideal gas when its density is low enough that its particles do not interact, except via elastic collisions. A form of the ideal gas law that applies when the number of moles and the temperature are constant is known as Boyle’s law. Using the subscripts “i” and “f” to denote, respectively, initial and final conditions, we can write Boyle’s law as in Equation 14.3. A form of the ideal gas law that applies when the number of moles and the pressure are constant is called Charles’ law and is given by Equation 14.4. 14.3 Kinetic Theory of Gases The distribution of particle speeds in an ideal gas at constant tem- perature is the Maxwell speed distribution (see Figure 14.8). The kinetic theory of gases indicates that the Kelvin temperature T of an ideal gas is related to the average translational kinetic energy KE of a particle, according to Equation 14.6, where v rms is the root-mean-square speed of the particles. The internal energy U of n moles of a monatomic ideal gas is given by Equation 14.7. The internal energy of any type of ideal gas (e.g., monatomic, diatomic) is proportional to its Kelvin temperature.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The resulting equation is called the ideal gas law. IDEAL GAS LAW The absolute pressure P of an ideal gas is directly proportional to the Kelvin temperature T and the number of moles n of the gas and is inversely proportional to the volume V of the gas: P = R(nT/V). In other words, PV = nRT (14.1) where R is the universal gas constant and has the value of 8.31 J/(mol · K). Sometimes, it is convenient to express the ideal gas law in terms of the total number of particles N, instead of the number of moles n. To obtain such an expression, we multiply and divide the right side of Equation 14.1 by Avogadro’s number N A = 6.022 × 10 23 particles/mol* and recognize that the product nN A is equal to the total number N of particles: PV = nRT = nN A ( R N A ) T = N ( R N A ) T The constant term R/N A is referred to as Boltzmann’s constant, in honor of the Austrian physi- cist Ludwig Boltzmann (1844–1906), and is represented by the symbol k: k = R N A = 8.31 J/(mol · K) 6.022 × 10 23 mol −1 = 1.38 × 10 −23 J/K With this substitution, the ideal gas law becomes PV = NkT (14.2) Example 2 presents an application of the ideal gas law. EXAMPLE 2 BIO The Physics of Oxygen in the Lungs In the lungs, a thin respiratory membrane separates tiny sacs of air (ab- solute pressure = 1.00 × 10 5 Pa) from the blood in the capillaries. These sacs are called alveoli, and it is from them that oxygen enters the blood. The average radius of the alveoli is 0.125 mm, and the air inside contains 14% oxygen. Assuming that the air behaves as an ideal gas at body tem- perature (310 K), find the number of oxygen molecules in one of the sacs. Reasoning The pressure and temperature of the air inside an alveolus are known, and its volume can be determined since we know the radius. Thus, the ideal gas law in the form PV = NkT can be used directly to find the number N of air particles inside one of the sacs. The number of oxygen molecules is 14% of the number of air particles.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The three relations just discussed for the absolute pressure of an ideal gas can be expressed as a single proportionality, P ~ nT/V. This proportionality can be written as an equation by inserting a proportionality constant R, called the universal gas con- stant. Experiments have shown that R 5 8.31 J/(mol?K) for any real gas with a density sufficiently low to ensure ideal gas behavior. The resulting equation is called the ideal gas law. Figure 14.4 (a) A partially filled balloon. (b) The air pressure in the partially filled balloon can be increased by decreasing the volume of the balloon as shown. 0 100 200 300 Temperature, K Absolute pressure Gas Figure 14.3 The pressure inside a constant-volume gas thermometer is directly proportional to the Kelvin temperature, a proportionality that is characteristic of an ideal gas. A Andy Washnik Andy Washnik (a) (b) 330 Chapter 14 | The Ideal Gas Law and Kinetic Theory Ideal Gas Law The absolute pressure P of an ideal gas is directly proportional to the Kelvin tempera- ture T and the number of moles n of the gas and is inversely proportional to the volume V of the gas: P 5 R(nT/V). In other words, PV 5 nRT (14.1) where R is the universal gas constant and has the value of 8.31 J/(mol?K). Sometimes, it is convenient to express the ideal gas law in terms of the total number of particles N, instead of the number of moles n.
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