ProblemsHW1 - Chapter 1 Properties of Cases 10 A linear...

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Unformatted text preview: Chapter 1 Properties of Cases 10. A linear least—squares fit is accomplished by minimizing N . 02 = {MN — 1)1-1 [y — a]; l with respect to the linear coefficients in a series exp a-“SiOn of ‘ independent variable x: - y In term-5 The equationsthat must be solved to obtain the fitting COEfficients a. algebraic equations. ' I Problems Problems that require the use of some type of computation— al device are marked with an asterisk (*). Problems that require some type of plotting routine are indicated with a pound sign (’i). Unless otherwise stated, all gases may be ' assumed to behave ideally. 1.1 One mole of Ar atoms are confined in a vessel whose volume is 1,000 cma. If all of the atoms are moving with identical speeds of 230 m 5'1 in the x, y, and 2 directions, compute the expected pressure, in kPa, inside the vessel. What is the pressure in atm? I A right circular cylinder whose radius is 1 cm con- tains a mixture of Hg (d = 13.596 kg L'l) and H20 (d = 1.000 kg L“), with the mercury on the bottom of the cylinder and the water on the top. It is found that the pressure at the bottom of the cylinder is 1.200 atln. It is also determined that the mass of Hg in the cylinder is eight times the mass of water pre- ' sent. Determine the length of the Hg column, hmand the length of the water column, hm, in the cylinder. (Be careful with the units.) It is found that the lengths of two metal rods, denoted A and B, are linear functions of the absolute temperature T. That is, LA 2a” + “IT, where do is the lengthofrodA atT = OKand a1 isa positive constant, and LB = b, + air, with similar definitions for b, and 111. An investigator now defines two temperature scales based on the lengths of rods A and B, respectively. The tempera- tures t A and in are defined by t _ 100 (LA — L”) A — (L100 _ Lu) K . y = Sal-x}. i=0 are t'_ 10002,; — La) 3' (Lao—Lo) ’ where__L,, and Llooare the lengths of the rodatthe mal freezing and boiling points of water, a u -. - (A) Determine the relationship between t, and ' (B) Show that ti = 1‘}; at all values of T, em 30%boanda15éb1f ' If it is found that the lengths of the rods in ' lem 1.3 are quadratic functions of T; that is, ILA =11“ +H1T +u2T2 and L,5 = b, + blT + by. Determine the relationship that will exist between and T, and show that, in general, t,‘ at: £3. Here is a problem for hot—air balloon enthusiasts. I investigator decides to construct a thermal: using the volume of a balloon filled with Ar measuring device. She defines her temperature by t = 75(V v V25)/(Vm - V25), where V2; and are the volumes of the balloon at 25°C and 1* respectively, when the pressure is 1 atm. The mometer is placed outside on the ground and Si heated with a laser beam. The air temperattm’v 25°C, and the outside pressure is 1 atm. Heam'fi continued until the investigator notes that 1191: mometer has risen off the ground and is flowing? the air. The balloon itself weighs 4 grams and tains'15 grams of Ar gas. What is the tertlptfl'ature the thermometer, t, at the point it 11595 Off ground? Assume that all gases are ideal and 3;; air is 20% 02 and 80% N2 by mass. Igrwte the the balloon’s elasticity on the pressure and V" of the balloon. erms of the of are linear )d at the nor- respectively 1 tA and T. If T, evenif ads 'in Prob— rat is, [‘3- 1thusiasts.An . thermometer with Ar as a perature scale e V5 and Van ’C and 100°C. ttrn. The that nd and slowi)t emperature 15 tu1.I-1eatingi5 that her the“ temperature “f : rises off the :al and that the are the effect “I re and V01!me Lst between t 1 -. 1,5 Suppose the ideal gas laws in a different galaxy far removed from ours are as follows: 1. At constant temperature, pressure is inversely proportional to the square of the volume. 2. At constant pressure, the volume varies direct- ly with the 2/ 3 power of the temperature 3. At 273.15 K and 1 atm pressure, one mole of an ideal gas is found to occupy 22. _414 liters. (A) Under these conditions, show that V‘Pa/T4— — a constant, and obtain the form of the ideal-gas equa- tion of state in this "galaxy. Be certain to give the value of the ideal~gas constant in the galaxy. (B) The coefficient of thermal expansion is defined to be a = V‘1(aV/8T)p. Obtain a: for the ideal-gas equation of state you found-in (A) in terms of the temperature alone. 1.7 The coefficient of thermal expansion is defined to be V1(3V/3T)P- (A) Obtain an expression for o: for an ideal gas. (B) Show that, for a Dieterici gas, a may be written in the form RV + 'a/T PV2 exp{a/VRT} - a where the notation exp{x}means ex- and a is'a para- meter in the Dieterici equation of state, which is given in Table 1. 2. 1.8 25gramsofArand 15 grams ofHClgasaremixedin a I‘D-liter container at 300 K. Determine the partial pressure of each gas and the total pressure inside the container.'You may assume the gases to be ideal. 1.9 A container is divided into two compartments. Compartment A holds ideal gas A at 400 K and 5 atm of pressure. Compartment B is filled with ' ideal gas B at 400 K and 8 atm. The partition between the compartments is removed and the gases are allowed to mix. (It will be shown in later chapters that this mixing produces no change in temperature if the gases are ideal.) The mole frac- tion of A in' the mixture is found to be 25/43 = 0.581395 . The total volume of both compartments is 29 liters. Determine the original volumes of compartments A and B. 1.10 The density of an ideal gas is found to be 1. 8813 g L"1 .' at 298 K and 1 atm pressure. What is the molar mass ' ' Of the gas? 1.11 - A container is known to hold a pure rare gas. A -1—liter sample of the gas at 298 K and 1 atm pressure . - ‘_ is found to weigh 3 .427 grams. What gas is inside the - container? 112 The ponderosity‘ 1n whams of a sample of umbo is directly proportional to the number of frenks' in the PrOblenis ' 53. sample and to its muckle, measured' in fluggas. The constant1 of proportionality is 8.31 whams frenk 1 fluggas‘1 .I-Iow many whams are there in 4 frenks of umbo whose muclde is 120 fluggas? [The author 15 indebted to Fredrick L. Minn, M. D. Ph D, for provid- ing this problem and the associated solution. 1 1.13 Consider the equation Z.— A[y 2 + x2]exp[— xy/a], whereZis afunctionofx andy,while a andAare constants. The notation exp[w} represents e'”. Z has units of joules, while 3:: and y each have units of meters. - (A) What are the units on the constant A? . (B) What are the units on the constant a? (C) Is it possible for Z to be given by the function Z= B[y + x2] exp[— xy/a], where B is another constant? Explain. 1.14 An investigator is told that a closed vessel holds a pure rare gas. She meaSures the density of the gas in the vessel at 298 K and 1 attn pressure and finds it to he 0.8252 g liter—1. (A) What gas does the investigator think is in the vessel? (8) After turning in her report, she is told that the vessel actually contains a mixture of He and Ar. What percentage of the gas in the container is He? _1 .15" The ideal—gas censtant' 15 obtained by measuring P—V -data for a real gas at a fixed temperature. The ratio PV/T is then computed at each measured pressure. The result is fitted'by an appropriate least-squares procedure and extrapolated to zero pressure, at which point the gas will behave ideally. This prob- lem illustrates the procedure. An investigator mea- sures the pressure of 1 mole of a real gas at various volumes at a fixed temperature of 300 K. Her data ' _ are as follows: - .20: -- ”1223046‘ _ _ 2.1._ ..... -_ 11651591 _. . 2.2 1.1125040: 24 1.020238 25 - 0.979605 '.26 w 0.942139 - -- 27" .. '- - 3.0307458 28 0828298 29 ‘ 0.845150 '. 54 Chapter 1 Properties of Gases . 30 .. . -- =ra.817097 . 35 0.700794 .40 0.613473 (A) Compute the apparent value of R at each of the data points. ' ' ' (B) Use the last six data points (at volumes V = 27 liters to V = 40 liters) to execute a least-squares fit ' of the computed value of R to a linear function of the pressure. That is, fit the function R = a, + a1P to the computed values of R at the six lowest pmsures. - . (C) Using the fitted function, obtain the limit of R as P -—'> 0. - _ (D) Plot the fitted function and compare your curve 1.16 ' - ' with the measured data points. A container of fixed volume holds two ideal gases, _ denoted as A and B, such that the mole fraction of 11:! gas A is X A = 3-,. At a given temperature, the pressure in the container, P1, is measured. Two additional moles of one of the gases are now added to the con- tainer at the same temperature. The new pressure, P2, is measured. It is found that the ratio Pz/Pl =_ 11/9. How many moles of A and B were originally present in the container? ' - All those concerned about the temperature of hell will find'this problem interesting. Because the earth’s popu- lation is increasing in a near-exponential fashion, it is reasonable to assume that the number of souls in hell is also increasing exponentially with time. That is, n = number of moles of souls in hell = A eXp{at}, where A and a are positive constants. Since it is like- ly that at t = 0 we had only one soul (the Devil) in hell, we know that A = 1/6.022 X 1023 11101—1 = 1.66 X 10‘24 mol. It is also reasonable to assume that souls entering hell do not leave. Let us further assume that we may treat a collection of souls as an ideal gas. Under these conditions, the temperature of hell will be given by _'PV_ PV T _ E _ AR exp{at}' If the. pressure of hell is constant at 1 atm, as it is on earth (this assumption isreasonable, since there are no suggestions that hell is a place of very high or very low pressure), the temperature will be depen- dent on how fast hell is expanding as souls enter it. (A) There have been approximately 1010 e earth since the Devilentered hell. If w: OP 16°“ that 10% of these people have entered hen 21:18:“ ' the Devil was thrown into hell 5,000 Ye'ntrs that 3. determine the value of the constant a. What 330. 2?: units on a? _ math“ '5 (13) Because at t = 0 there was onl one 5 hell, the initial volume must have 'geen 3:251:35 am of ' The volume of an average house is probabl 5111311 i estimate. If this house had 2,000 ft2 of $1001.22: and 8-ft ceilings, we would have V, = 16.0mm 1.18 1.19 The intermolecular forces between two 835 mole 4.5307 X 105 liters. Given that the number of 501115‘ hell is rising exponentially, the volume must 330$ ._- increasing in exponential fashion; that is ‘- V = V, exp{bt}. Use the foregoing assumptions to obtain the temper-,1. ‘2 tureofhellasafunction oftheparameter-bands,“E (C) Compute the initial temperature of hell at t = D This result shows the origin and the magnum}, oi 'g;‘ the popular slang phrase ”hot as hell." (D) Note that if we have b < a, hell will eventuafly freeze over. Let b = a/ 2, and compute how many years must elapse before hell freezes over (reams the freezing point of water). . (E) Using the same value of b as in (D), detem how long it will take hell to reach a temperature of '- 1 K. How long will it take to reach a temperature of 0.01 K? How long will it take to reach a temperature of 10‘10 K? How long will it take to reach a tempera- ture of 0 K? The answer to this last question shows that T = 0 K is not attainable. We will later show that the second law of thermodynamics prohibits reach— ing absolute zero inhell or anywhere else. It is clear that we have no clue as to the fraction of people whose souls will enter hell. Consequently, let us generalize the results obtained in Problem 1.17m include this fact. Let the fraction of people whose souls enter hell be denoted by f. This fraction does not include the Devil, whom we know went to hell. (A) Use the assumptions contained in Problem 1.17 to obtain the temperature of hell as a function of Ni. and time. _ (B) Show that if the Devil is unsuccessful in capilll": ing any souls (f = 0), the temperature of hell must remain constant or increase. Perhaps that is Why the Devil seeks souls; he has no other way to cool“ place. . cules are'described by the L](12, 6) patenting“? in Eq. 1.55. Show that the potential Mm“: occurs at r = a- and that the well depth, defined .. [VLJ(r = DO) — VLIU = 0)], is equal to 8- . whr (A). 10" ainthetempem. g Eterbandtinm :ofhellatt=u, '- II . will eventually nute how many as over (reaches i? l (D), determine jI temperature at l temperature of g: h a temperature each a tempera 3 question shows 1 later show that '_ prohibits reach- ‘- 2 else. a the fraction of f lonsequentlylef " Problem 1.17m ; .f people whose f: [is fraction does ow went to hell in Problem 1.17 -f 1 frmction of hi ' e‘sshrlincapw' 5 ore of hell 1111152 s that is WW *1“ 2 way to cool the :: e inaccuracy of i If an-lintermolecular'potential of the form given in .-1- 54 is employed, obtain an expression for the vfiglue' of r at the potential minimum as a function of Krcrandn" ‘I . I . I 'm nucleus is separated from an electron by a .121 A}. hlellme 0'55 A (5 X 10‘10m). If both particles are malted as point charges, compute the potential energy betwéen them in units of joules and kcal moi—1. 1.20 132' Another functional form often used to represent intermolecular forces is a Morse function, given by . V... = D[1 — expi-alr * 0}]2, where D, a, and r, are constant parameters that depend upon the nature of the particles inter-acting. (A) Let' D '= 100 k] mol‘l, a = 1.5 A‘1 ' (1 A = 10—10111) and r, = 2.0 A. Make a careful plot of V", versus 1' over the range 1.4 A 5 r s 6.0 A. (B) Show that the minimum for a Morse potential oomrs at r = r,. (C) Show that the well depth, as defined in Pro- blem 1.19, is equal to D. (D) It will later be shown that the vibrational fre- quency of an oscillator is dependent up on the second derivative of the intermolecular potential, evaluated at the minimum of the potential. Show that aZVM/arZIFM = 2a2D. ' 1.23 What mass of N2 gas is present in a 50-Hter container at 400 K under 20 atm of N2 pressure if (A) the gas is ideal and _ (B) the gas obeys the van der Waals equation of state? 1.24 Verify that expansion of the van der Waals equation .of state in virial form, Eq. 1.64, gives a1 = RT, :12 = RTE: - a, and a3 = Rsz. Prove that the van der Waals parameter :1 appears only in the second virial coefficient a2. , 1.25 Using the data in Table 1.1, estimate the second and third virial coefficients for CO2 at 300 K. At what temperature would we expect the second virial coef- ficient for CO; to be zero? 126 A nonideal gas obeys the equation of state PV", = RT + orP, where a is a function of T only. An investigator obtains pressure and volume data for this gas at a fixed temperature. Not knowing the actual equation of state, she fits her data to a virial equation of state. In terms of the quantities appear- mg in the actual equation of state, what will she obtain for the second virial coefficient? L27 Consider a gas that obeys the equation of state 12—51 [-“J VexP VRT' Problems 55 where a isa constant and the notation exp[x} means 2‘. (A) Determine the second and third virial coeffi- cients for this gas as a function of a, R, and T. (B). Determine the residual volume of the gas as a function of a, R, and T. 1.28 A gas obeys the van der Waals equation of state. The critical parameters for this gas are P, = 47.7 atm, Tc = 151.15 K, and V, = 0.0752 liter mol'l. Compute the reduced pressure of the gas at 300 K and a vol- ume of 15 liters. 1.29 A gas is represented by the equation of state 1 a b P—RT[V+ VZ-l'fi], where a and b are constants. By requiring that this equation of state satisfy the three constraints at the critical point, express R, a, and b in terms of the criti- cal variables, and put the equation of state in reduced form. ' 1.39 A nonideal-gas is represented by the equationof state 1 ' a b ‘ c' ' P=RT[V+§E+YZ—3+W], where a, b, and c are constants. By requiring that this equation of state satisfy the three constraints at the critical point, express the parameters a, b, and c in terms of the critical variables. Show that substi- tution of these results into the equation of state does not result in a reduced form that is identical for all gases. 1.31 For a gas represented by a virial equation of state, determine the residual volume in terms of the virial coefficients. _ , 1.32 The equation of state of a nonideal gas is PV,“(1 — aeP) =RT, where a: is a function of tempera- ture only. Obtain an expression for the residual vol- ume ofthis gas as a flmction ofR, T, and a. 1.33‘T'he following set of pressure and volume data is obtained for 1 mole of a nonideal gas at 300 K: 2Looo _. 111.29% ' _ T5538 . 4.313 .2..59,o_ 8.903. _ 6.12.8 ' "3:909. .- . 3.1.79 -_ 7.342 ' - ' 6.718. 3.575 . 3.769 ' 6.246 7.308 - -- , 3293 .- -4359 . 5.435 . 7.897 ' 3.052. : - 4.949 - 4.809 - - 13.437 - "2:844 '7 Properties of Gases - 56. '_ Chapterl 1.407 ' 1.361 1.318 " 51.278. 21505 . 2.362 _= 17.333. 17.923 .- . 13.513: 1 19.103' -_ _-19.692 " ; 124.0, - 20.282"; - 1.204" - '_ .'_2'0.87_2-.-_ '_11'.17o. - .51.":1916212 {1:138 - - _".2.'2..05'1' -' f -"i.108' " 23231 1.052 . _ 523.82:l'_.' 1.026;: --24.410" ' 1.002 9.072. 9.667' ' 10256. ,10345' sass 4.11.136” _. 2.122 _ _' 112.026- p.121”? -. .- _- -_12'.61_5._ . .:-1._9_'26..-1-- ' 11111 1111 ' -' 131.795 1.7611] 14,535 . _.1_".692_}f .' .. 14.914 1.626 15.564. 1565 . - . . 16.154. -_ 1508' _fj.1s.744;:'-s._i1_4.56 - - (A) Using a least-squares method, fit the data to a truncated virial equation of state of the form P = a” + my + a2y2,where y = V‘l, and obtain the best values of an, 111, and a2. Be certain to show the equations that are being solved, and give the values -_ of all required sums. (B) Plot the data and the fit obtained in (A) on the' sauna graph. ', ' (C) Is the expression used in (A) an-appropriate _ equation of state at low pressures and large vol— umes? Explain. ' (D) Set 11,, = 0 and a, = RT in the equation in (A), and use a least—squares procedure to obtain :12. (E) Estimate the residual volume for this nonideal gas. 1.34 The compressibility of a gas is defined to be __'_ 9K fi— V1(3P)r where the subscript T means that the partial deriva- tive is taken holding T constant. (A)1 Show that the compressibility of an ideal gas is P‘ . (B) Derive an expression for ,3 for a van der Waals gas intermsofP, V,a, and b. _ _ (C) Show that in the limit of infinite volume, ,3 for a .van der Waals gas reduces to that for an ideal gas. 1.35 (A) Develop an expression for the compression fac— tor for a gas described by a virial equation of state. (a) Show that Z(T, P)—> 1 as p —10. 1.36 Use a van der Waals equation of state along with the data given in Table 1.1 to compute the compression factor for N; at 273.15 K and 128.09 aim The ' sured value is 0.9829. Calculate the Pmém your computed value. "' ' 1.37 A young scientist has recently broken up Wi a boyfriend. The former boyfriend is an t1l. decides to sabotage her research for MEI? :- seems that the young lady is doing pmfls ume, and temperature measurements o M. at night, her former boyfriend enters her . .. and places a spring inside the cylindeI she is " the experiments. (See accompanying figure.) - Spring Area of piston face = A The force on the piston face produced by the P _= —k [L'1 — L51] + C [L‘2 — Lfl, , where k, C, and L, are constants. At L = Lo, the is at equilibrium. The formerboyfriend Chooses spring such that when 1 mole of ideal gas is p .. the cylinder at 298 K and 1 atm pressure, the . length is precisely L0. The next day, the young scientist enters he! oratory and places 1 mole of an ideal gas in the ratus at 298 K and 1 atm pressure. She then the piston to the right, so as to increase the vol the gas at constant temperature, and measurti resulting pressure. This procedure is repeated she has an extensive set of pressure and v data at 298 K. Being unaware of the presence spring, she believes that her gas is behaViI‘S very nonideal fashion. Therefore, she fits he: sure, volume,...
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