Chemistry
Ideal Gas Law
The Ideal Gas Law describes the behavior of an ideal gas and is represented by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. It shows the relationship between these variables and is used to predict the behavior of gases under different conditions.
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11 Key excerpts on "Ideal Gas Law"
- 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 - PDFIntroductory Chemistry
An Active Learning Approach
- Mark Cracolice, Edward Peters, Mark Cracolice(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
542 Chapter 14 The Ideal Gas Law and Its Applications Goal 1 State how volume and amount of gas are related when pressure and temperature are constant and explain phenomena or make pre- dictions based on that relationship. Avogadro’s Law states that equal volumes of two gases at the same temperature and pressure contain the same number of molecules, V ~ n. Goal 2 Explain how the ideal gas equation can be constructed by combining Charles’s Law (Section 4.5), Boyle’s Law (Section 4.6), and Avogadro’s Law (Section 14.3), and explain how the ideal gas equation can be used to derive each of the three two-variable laws. Since V ~ T, V ~ 1/P, and V ~ n, it follows that V ~ T 3 s1/Pd 3 n. Inserting a proportionality constant R, V 5 RTs1/Pdn, or, rearranging, PV 5 nRT. When pressure and amount are constant, V 5 kT, which is Charles’s Law. When temperature and amount are constant, PV 5 k, which is Boyle’s Law. When pressure and temperature are constant, V 5 kn, which is Avogadro’s Law. Goal 3 Given values for all except one of the variables in the ideal gas equation, calculate the value of the unknown variable. The ideal gas equation is PV 5 nRT. Two values of the gas constant, R, are 0.0821 L ? atm/mol ? K and 62.4 L ? torr/mol ? K. Given all the values in the ideal gas equation except one, the unknown value may be calculated. Substituting m/MM for its equivalent, n, in the ideal gas equation gives PV 5 m MM RT. Goal 4 Calculate the density of a known gas at any specified temperature and pressure. Solving the PV 5 sm/MMdRT form of the ideal gas equation for m/V, which is density, yields D 5 m V 5 sMMdP RT . Goal 5 Given the density of a pure gas at spec- ified temperature and pressure, or information from which the density may be found, calculate the molar mass of that gas. The density of a gas at constant temperature and pressure is directly proportional to its molar mass, D ~ MM. Either molar mass or density can be calculated from the other using the ideal gas equation.
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)
- 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) 14.2 | The Ideal Gas Law 371 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. To obtain such an expression, we multiply and divide the right side of Equation 14.1 by Avogadro’s number N A 5 6.022 3 10 23 particles/ mol* and recognize that the product nN A is equal to the total number N of particles: PV 5 nRT 5 nN A a R N A b T 5 N a R N A b T The constant term R/N A is referred to as Boltzmann’s constant, in honor of the Austrian physicist Ludwig Boltzmann (1844–1906), and is represented by the symbol k: k 5 R N A 5 8.31 J/(mol ? K) 6.022 3 10 23 mol 21 5 1.38 3 10 223 J/K With this substitution, the Ideal Gas Law becomes PV 5 NkT (14.2) Example 2 presents an application of the Ideal Gas Law.
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- 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- 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 - 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,
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 - 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.
eBook - PDF- Robert J. Silbey, Robert A. Alberty, George A. Papadantonakis, Moungi G. Bawendi(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Since different gases give slightly different scales when the pressure is about one bar (1 bar = 10 5 Pascal = 10 5 Pa = 10 5 N m −2 ), it is necessary to use the limit of the P V product as the pressure approaches zero. When this is done, all gases yield the same temperature scale. We speak of gases under this limiting condition as ideal. Thus, the ideal gas temperature T is defined by T = lim P→0 (P V ∕R) (1.4) The proportionality constant is called the gas constant R. The unit of thermodynamic temperature, 1 Kelvin or 1 K, is defined as the fraction 1/273.16 of the temperature of the triple point of water. ∗ Thus, the tem- perature of an equilibrium system consisting of liquid water, ice, and water vapor is 273.16 K. The temper- ature 0 K is called absolute zero. According to the current best measurements, the freezing point of water at 1 atmosphere (101 325 Pa; see below) is 273.15 K, and the boiling point at 1 atmosphere is 373.12 K; however, these are experimental values and may be determined more accurately in the future. The Celsius scale t is formally defined by t∕ ∘ C = T∕K − 273.15 (1.5) The reason for writing the equation in this way is that temperature T on the Kelvin scale has the unit K, and temperature t on the Celsius scale has the unit ∘ C, which need to be divided out before temperatures on the two scales are compared. In Fig. 1.5, the molar volume of an ideal gas is plotted versus the Celsius temperature t at two pressures. FIGURE 1.5 Plots of V versus temperature for a given amount of a real gas at two low pressures P 1 and P 2 , as given by Gay-Lussac’s law. V – P 1 P 2 P 2 > P 1 –273.15 0 t/°C ∗ The triple point of water is the temperature and pressure at which ice, liquid, and vapor are in equilibrium with each other in the absence of air.
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