Physics
Ideal Gas Model
The ideal gas model is a theoretical concept used to describe the behavior of gases under certain conditions. It assumes that gas particles have negligible volume and do not interact with each other, and that their kinetic energy is directly proportional to the temperature. This model is often used to simplify calculations and understand the basic properties of gases.
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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- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
We move from the Ideal Gas Model to a more widely applicable approximation, called the Van der Waals model. To understand gases even better, we must also look at them on the microscopic scale of molecules. In gases, the molecules interact weakly, so the microscopic behavior of gases is relatively simple, and they serve as a good introduction to systems of many molecules. The molecular model of gases is called the kinetic theory of gases and is one of the classic examples of a molecular model that explains everyday behavior. Chapter 2 | The Kinetic Theory of Gases 67 2.1 | Molecular Model of an Ideal Gas Learning Objectives By the end of this section, you will be able to: • Apply the ideal gas law to situations involving the pressure, volume, temperature, and the number of molecules of a gas • Use the unit of moles in relation to numbers of molecules, and molecular and macroscopic masses • Explain the ideal gas law in terms of moles rather than numbers of molecules • Apply the van der Waals gas law to situations where the ideal gas law is inadequate In this section, we explore the thermal behavior of gases. Our word “gas” comes from the Flemish word meaning “chaos,” first used for vapors by the seventeenth-century chemist J. B. van Helmont. The term was more appropriate than he knew, because gases consist of molecules moving and colliding with each other at random. This randomness makes the connection between the microscopic and macroscopic domains simpler for gases than for liquids or solids. How do gases differ from solids and liquids? Under ordinary conditions, such as those of the air around us, the difference is that the molecules of gases are much farther apart than those of solids and liquids. Because the typical distances between molecules are large compared to the size of a molecule, as illustrated in Figure 2.2, the forces between them are considered negligible, except when they come into contact with each other during collisions.
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 - PDF- Arthur Adamson(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
We have taken the pheno-menological observation of Eq. (1-5), noticed the approximate validity of Eq. (1-6), and then defined our temperature scale so as to make Eq. (1-9) exact. In effect, the procedure provides an operational, that is, an unambiguous experimental, 1-4 THE IDEAL GAS LAW AND RELATED EQUATIONS 7 definition of temperature. At no point has it been in the least necessary to under-stand or to explain why gases should behave this way or what the fundamental meaning of temperature is. To summarize, Eq. (1-9) is an equation obeyed (we assume) by all gases in the limit of zero pressure. As Boyle and Charles observed, it is also an equation of state which is approximately obeyed by many gases over a considerable range of temperature and pressure. At this point it is convenient to introduce the concept of a hypothetical gas which obeys the equation PV = RT (1-10) under all conditions. Such a gas we call an ideal gas. It is important to keep in mind the distinction between Eq. (1-9) as an exact limiting law for all gases and Eq. (1-10) as the equation for an ideal gas or as an approximate equation for gases generally. 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) Pv = — RT (M = molecular weight); (1-12) PM = P RT (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 β_Ρ , 7 ^ ! • ... Π-14) p ~ m ~ M M M ' 1 4 J one notes that the intercept of Pv/m (or P/p) plotted against Ρ must give RTjM for any gas.
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)
Problem-Solving Insight In the ideal gas law, the temperature T must be expressed on the Kelvin scale. The Celsius and Fahrenheit scales cannot be used. 372 Chapter 14 | The Ideal Gas Law and Kinetic Theory Historically, the work of several investigators led to the formulation of the ideal gas law. The Irish scientist Robert Boyle (1627–1691) discovered that at a constant tempera- ture, the absolute pressure of a fixed mass (fixed number of moles) of a low-density gas is inversely proportional to its volume (P ~ 1/V). This fact is often called Boyle’s law and can be derived from the ideal gas law by noting that P 5 nRT/V 5 constant/V when n and T are constants. Alternatively, if an ideal gas changes from an initial pressure and volume (P i , V i ) to a final pressure and volume (P f , V f ), it is possible to write P i V i 5 nRT and P f V f 5 nRT. Since the right sides of these equations are equal, we may equate the left sides to give the following concise way of expressing Boyle’s law: Constant T, constant n P i V i 5 P f V f (14.3) Figure 14.6 illustrates how pressure and volume change according to Boyle’s law for a fixed number of moles of an ideal gas at a constant temperature of 100 K. The gas begins with an initial pressure and volume of P i and V i and is compressed. The pressure increases as the volume decreases, according to P 5 nRT/V, until the final pressure and volume of P f and V f are reached. The curve that passes through the initial and final points is called an isotherm, meaning “same temperature.” If the temperature had CONCEPTUAL EXAMPLE 3 | The Physics of Rising Beer Bubbles If you look carefully at the bubbles rising in a glass of beer (see Figure 14.5), you’ll see them grow in size as they move upward, often doubling in volume by the time they reach the surface. Beer bubbles contain mostly carbon dioxide (CO 2 ), a gas that is dissolved in the beer because of the fermentation process.
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.
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 - PDFGeneral Chemistry I as a Second Language
Mastering the Fundamental Skills
- David R. Klein(Author)
- 2015(Publication Date)
- Wiley(Publisher)
CHAPTER 4 THE IDEAL GAS LAW In this chapter, we will learn tips and techniques for solving the most common types of problems that focus on the Ideal Gas Law (PV nRT). Some of these problems can appear quite complex, even though they are actually very simple. You just need to train your eye to see the patterns, and you have to know what to look for. Before we get started, we should recognize that there are two broad groups of problems: those describing a gas under a specific set of conditions, and those de- scribing a gas undergoing a change of conditions. To illustrate the difference, imag- ine that you have a balloon filled with helium gas. We can make calculations on the gas inside the balloon, while there is no change taking place. Or, we can mea- sure how the volume will expand if you heat the balloon. In the second case, our calculations revolve around a change in conditions. In order to quickly solve any Ideal Gas Law problem, you must train your eye to detect the difference between these two scenarios. You should always ask yourself: “Is there a change of condi- tions, or not?” You should be on the lookout for key phrases like these: • the gas expands to a volume of . . . • the temperature of the gas is raised to . . . • the pressure on the gas is decreased to . . . Whenever you read a problem with these (or similar) phrases, you should im- mediately recognize that the problem is referring to a change of conditions. The first five sections of this chapter will focus on problems that pertain to a gas under a specific set of conditions (a gas that is not undergoing a change of con- ditions). The last three sections of the chapter are devoted to problems that pertain to a gas that is changing its conditions. 4.1 UNIT CONVERSIONS FOR THE IDEAL GAS LAW In this chapter, we will see many different types of problems. In every problem, you must always make sure that your units are consistent before you do any calcula- tions.
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