424 CHAPTER 10 Gases THE KINETIC-MOLECULAR THEORY OF GASES (SECTION 10.7) The knetc-molecular theory of gases accounts for the propertes of an deal gas n terms of a set of statements about the nature of gases. Brefly, these statements are as follows: Molecules are n contnuous chaotc moton. The volume of gas molecules s neglgble compared to the volume of ther contaner. The gas molecules nether attract nor repel each other. The average knetc energy of the gas molecules s proportonal to the absolute temperature and does not change f the temperature remans constant. The ndvdual molecules of a gas do not all have the same knetc energy at a gven nstant. Ther speeds are dstrbuted over a wde range; the dstrbuton vares wth the molar mass of the gas and wth temperature. The root-mean-square (rms) speed, Unns, vares n proporton to the square root of the absolute temperature and nversely wth the square root of the molar mass: u,ms = V3RT/ M. The most probable speed of a gas molecule s gven by Ump = VZRT / M. MOLECULAR EFFUSION AND DIFFUSION (SECTION 10.8) It follows from knetc-molecular theory that the rate at whch a gas undergoes effuson (escapes through a tny hole) s nversely proportonal to ufl square root of ts molar mass (Graham's law). The dffuson of one gl through the space occuped by a second gas s another phenomenon {J\ lated to the speeds at whch molecules move. Because movng molecu(j;,. undergo frequent collsons wth one another, the mean free path-tl,. mean dstance traveled between collsons-s short. Collsons betwe molecules lmt the rate at whch a gas molecule can dffuse. REAL GASES: DEVIATIONS FROM IDEAL BEHAVIOR (SECTION 10 Departures from deal behavor ncrease n magntude as presst ncreases and as temperature decreases. Real gases depart from ~ behavor because (1) the molecules possess fnte volume and (2) t molecules experence attractve forces for one another. These two fects make the volumes of real gases larger and ther pressures smaj than those of an deal gas. The van der Waals equaton s an equano. state for gases, whch modfes the deal-gas equaton to account f ntrnsc molecular volume and ntermolecular forces. Learnng Outcomes After studyng ths chapter, you should be able to: Calculate pressure and convert between pressure unts wth an emphass on torr and atmospheres. (Secton 10.2) Related Exercses: 10.15, 10.17, 10.19, 10.21 Calculate P, V, n, or Tusng the deal-gas equaton. (Secton 10.4) Related Exercses: 10.33, 10.39, 10.43 Explan how the gas laws relate to the deal-gas equaton and apply the gas laws n calculatons. (Sectons 10.3 and 10._4) Related Exercses: 10.25, 10.39, 10:41 Calculate the densty or molecular weght of a gas. (Secton 10.5) Related Exercses: 10.47, 10.51, 10.53 Calculate the volume of gas consumed or formed n a chemcal reacton. (Secton 10.5) Related Exercses: 10.55, 10.57 Calculate the total pressure of a gas mxture gven ts part pressures or gven nformaton for calculatng partal pressur (Secton 10.6) Related Exercses: 10.7, 10.61, 10.63 Descrbe the knetc-molecular theory of gases and ho explans the pressure and temperature of a gas, the gas la and the rates of effuson and dffuson. (Sectons 10.7 and 10. Related Exercses: 10.9, 10.75, 10.79, 10.85 ' Explan why ntermolecular attractons and molecular velum cause real gases to devate from deal behavor at hgh pressu or low temperature. (Secton 10. 9) Related Exercses: 10.89, 10.91, 10.114 Key Equatons PV = nrt P V1 P2V2 T T d = PM RT P=P+P2+P3+ (10.5] (10.8] (10.10] (10.12] Ideal-gas equaton / The combned gas law, showng how P, V, and Tare related for a constant n Densty or molar mass of a gas Relatng the total pressure of a gas mxture to the partal pressures of; components (Dalton's law of partal pressures) / P1 =(::)Pr= X1P (10.16] Relatng partal pressure to mole fracton Urms = ~ (10.20] Defnton of the root-mean-square (rms) speed of gas molecules (,a) P + v2 (V - nb) = nrt Exercses (10.22] [10.25] Relatng the relatve rates of effuson of two gases to ther molar masses The van der Waals equaton Vsualzng Concepts IO.I Mars has an average atmospherc pressure of 0.007 atm. Would t be easer or harder to drnk from a straw on Mars than on Earth'? Explan. [Secton 10.2] 10.2 You have a sample of gas n a contaner wth a movabl'e_ ton, such as the one n the drawng. (a) Redraw the cont,u to show what t mght look lke f the temperature of tl s ncreased from 300 to 500 K whle the pressure s kep\ stant. (b) Redraw the contaner to show what t mght I_
Exercses 425 )l, Jl<e f the external pressure on the pston s ncreased from 1.0 atm to 2.0 atm whle the temperature s kept constant. (c) Reel raw the contaner to show what t mght look lke f 'the temperature of the gas decreases from 300 to 200 K whle. 'the pressure s kept constant (assume the gas does not lquefy). [Secton 10.3] 10.7 The accompanyng drawng represents a mxture of three dfferent gases. (a) Rank the three components n order of ncreasng partal pressure. (b) If the total pressure of the mxture s 1.40 atrn, calculate the partal pressure of each gas. [Secton 10.6] 3' Consder the sample of gas depcted here. What would the drawng look lke f the volume and temperature remaned constant whle you removed enough of the gas to decrease the pressure by a factor of 2? [Secton 10.3] 1 (a} It would contan the same number of molecules., (b} It would contan half as many molecules. (c) It would contan twce as many molecules...,. (d) There s nsuffcent data to say. / 10.8 On a sngle plot, qualtatvely sketch the dstrbuton of molecular speeds for (a) Kr(g) at -50 C, (b} Kr(g) at 0 C, (c) Ar(g) at O 0 C. [Secton 10. 7] 10.9 Consder the followng graph. (a) If curves A and B refer to two dfferent gases, He and 0 2, at the same temperature, whch curve corresponds to He? (b} If A and B refer to the same gas at two dfferent temperatures, whch represents the hgher temperature? (c) For each curve, whch speed s hghest: the most probable speed, the root-mean-square speed, or the average speed? [Secton 10. 7] <Jl Q) '"5 u Q) 0 s Imagne that the reacton 2 CO(g) + 0 2 (g) -----> 2 C0 2 (g) occurs n a contaner that has a pston that moves to mantan., a constant pressure when the reacton occurs at constant tern ' perature. Whch of the followng statements descrbes how the volume of the contaner changes due to the reacton: (a) the 1 volume ncreases by 50%, (b} the volume ncreases by 33%, (c) the volume remans constant, (d) the volume decreases by 33%, (c) the volume decreases by 50%. [Sectons 10.3 and 10.4] Suppose you have a fxed amount of an deal gas at a constant volume. If the pressure of the gas s doubled whle the volume s held constant, what happens to ts temperature? [Secton 10.4] The apparatus shown here has two gas-flled contaners and one empty contaner, all attached to a hollow horzontal ' tube. When the valves are opened and the gases are allowed to mx at constant temperature, what s the dstrbuton of atoms n each contaner? Assume that the contaners are of equal volume and gnore the volume of the connectng tube. Whch gas has the greater partal pressure after the valves are opened? (Secton 10.6] Molecular speed IO.IO Consder the followng samples of gases: () () () O=He =N2 If the three samples are all at the same temperature, rank them wth respect to (a) total pressure, (b) partal pressure of helum, (c) densty, (d) average knetc energy of partcles. [Sectons 10.6 and 10.7] 10.11 A thn glass tube 1 m long s flled wth Ar gas at 1 atm, and the ends are stoppered wth cotton plugs as shown below. HCl gas s ntroduced at one encl of the tube, and smultaneously NH 3 gas s ntroduced at the other encl. When the two gases dffuse through the cotton plugs clown the tube and meet, a whte rng appears clue to the formaton of NI-l 4 Cl(s). At whch locaton-a, b, or c-clo you expect the rng to form? [Secton 10.8] HCI
426 CHAPTER 10 Gases 10.12 The graph below shows the change n pressure as the temperature ncreases for a 1-mol sample of a gas confned to a 1-L contaner. The four plots correspond to an deal gas and three real gases: CO 2, N 2, and Cl 2. (a) At room temperature, all three real gases have a pressure Jess than the dea! gas. Whch van der Waals constant, a orb, accounts for the nfluence ntermolecular forces have n lowerng the pressure of a real gas? (b) Use the van der Waals constants n Table 10.3 to match the labels n the plot (A, B, and C) wth the respectve gases (CO 2, N 2, and C1 2 ). [Secton 10.9] 45,---.,-----,----..,.----,----,----, 40 ~ 30 ;:J (/)... (/) <l/ o... 25 20 ' I -----t- - ----r--------- 1 15'----'----'----'----1--- -...J J 250 300 350 400 450 500 550 Temperature (K) s 1.00 g/cm 3 = 1.00 x 10 3 kg/m 3. The gravtatonal co stant s 9.81 m/s 2, and 1 Pa = 1 kg/m-s 2. 10.18 (a) The compound 1-odododecane s a nonvolatle l ud wth a densty of 1.20 g/ml. The densty of mercur/ 13.6 g/ml. What do you predct for the heght of a bar 0111 ter column based on 1-oclocloclecane, when the atmosphe/ pressure s 749 torr? (b) What s the pressure, n atm~ spheres, on the body of a dver f he s 21 ft below the surfac o_f_the water when the atmospherc pressure s 742 torr? 10.19 The typcal atmospherc pressure on top of Mount Ever (29,028 ft) s about 265 torr. Convert ths pressure to (a) at (b) mm Hg, (c) pascals, (d) bars, (c) ps. 10.20 Perform the followng conversons: (a) 0.912 atm to tbr (b) 0.685 bar to klopascals, (c) 655 rnm Hg to atmospher ("-cl-'- ) 1.323 x 10 5 Pa to atmospheres, (e) 2.50 atm to ps. 10.21 In the Unted States, barometrc pressures are generally report n nches of mercury (n. Hg). On a beautful summer clay Chcago, the barometrc pressure s 30.45 n. Hg. (a) Con~e ths pressure to torr. (b) Convert ths pressure to atrn. 10.22 Hurrcane Wlma of 2005 s the most ntense hurrcane 0 record.n the Atlantc basn, wth a low-pressure readng 882 mbar (mllbars). Convert ths readng nto (a) atrn, s""p_heres, (b) ton, and (c) nches of Hg. 10.23 If the atmospherc pressure s 0. 995 atm, what s the pressu of the enclosed gas n each of.the three cases depcted n tl drawng? Assume that the gray lqud s mercury. Open ='-. Gas Characterstcs; Pressure (Sectons 10.1 and 10.2) 10.13 Whch of the followng statements s false? (a) Gases are far less dense than lquds. (b) Gases are far more compressble than lquds. (c) Because lqud water and lqud carbon tetrachlorde do not mx, nether do ther vapors. (d) The volume occuped by a gas s determned by the volume of ts contaner. 10.14 (a) Are you more lkely to see the densty of a gas reported n g/ml, g/l, or kg/cm 3? (b) Whch unts are approprate for expressng atmospherc pressures, N, Pa, atm, kg/m 2? (c) Whch s most lkely to be a gas at room temperature and o_r_dnary atmospherc pressure, F 2, Br 2, K 2 0. IO.IS Suppose that a woman weghng 130 lb and wearng hghheeled shoes momentarly places all her weght on the heel of one foot. If the area of the heel s 0.50 n. 2, calculate the pressure exerted on the underlyng surface n (a) pounds per square nch, (b) klopascals, and (c) atmospheres. 10.16 A set of bookshelves restson a hard floor surface on four legs, each havng a cross-sectonal dmenson of 3.0 x 4.1 cm n contact wth the floor. The total mass of the shelves plus the books stacked on them s 262 kg. Calculate the pressure n pascals exerted by the shelf footngs on the surface. ----- 10.17 (a) How hgh n meters must a column of glycerol be to exert a pressure equal to that of a 760-mm column of mercury? The densty of glycerol s 1.26 g/rnl., whereas that of mercury s 13.6 g/ml. (b) What pressure, n atmospheres, s exerted on the body of a dver f she s 15 ft below the surface of the water when the atmospherc pressure s 750 torr? Assume that the densty of the water h =;152 cm () () () 10.24 An open-encl manometer contanng mercury s conned' to a contaner of gas, as depcted n Sample Exercse 10.. What s the pressure of the enclosed gas-n torr n each of,tl followng stuatons? (a) The mercury n the arm attacl), to the gas s 15.4 mm hgher than n the one open to the' mosphere; atmospherc pressure s 0.985 atm. (b) Them cury.n the arm attached to the gas s 12.3 mm lower than the one open to the atmosphere; atmospherc pressur. 0.99 atm. The Gas Laws (Secton 10.3) 10.25 You have a gas at 25 c confned to a cylnder wth am. able pston. Whch of the followng actons would cjoull the gas pressure? (a) Lftng up on the pston to doublet, volume whle keepng the temperature constant; (b) He, ng the gas so that ts temperature rses from 25 c to 50 whle keepng the volume constant; (c) Pushng clown the pston to halve the volume whle keepng the tempe ture constant. 10.26 A fxed quantty of gas at 21 c exhbts a pressure of 7 torr and occupes a volume of 5.12 L. (a) Calculate tle'( ume the gas wll occupy f the pressure s ncreased to 1
Exercses 427 atrn whle the temperature s held constant. (b) Calculate the volume the gas wll occupy f the temperature s ncreased to 175 C whle the pressure s held constant. (a) Amon ton's law expresses the relatonshp between pressure and temperature. Use Charles's law and Boyle's law to derve the proportonalty relatonshp between P and T. (b) If a car tre s flled to a pressure of 32.0 lb/n. 2 (ps) measured at 75 F, what wll be the tre pressure f the tres heat up to 120 P durng drvng? Ntrogen and hydrogen gases react to form ammona gas as follows: N 2 (g) + 3 H 2 (g) ---> 2 NH3(g) At a certan temperature and pressure, 1.2 L of N 2 reacts wth 3.6 L of H 2. If all the N 2 and H 2 are consumed, what volume of NH 3, at the same temperature and pressure, wll be produced? :' Ideal-Gas Equaton (Secton 10.4) (a) What condtons are represented by the abbrevaton STP? (b) What s the molar volumeof an deal gas at STP? (c) Room temperature s often assumed to be 25 C. Calculate the molar volume of an deal gas at 25 C and 1 atm pressure. (d) If you measure pressure n bars nstead of atmospheres, calculate the correspondng value of R n L-bar/mol-K. To derve the deal-gas equaton, we assume that the volume of the gas atoms/molecules can be neglected. Gven the atomc radus of neon, 0.69 A, and knowng that a sphere has a volume of 41Tl 3 /3, calculate the fracton of space that Ne atoms occupy n a sample of neon at STP. Suppose you are gven two 1-L flasks and told that one contans a gas of molar mass 30, the other a.gas of molar mass 60, both at the same temperature. The pressure n flask A s x atm, and the mass of gas n the flask s 1.2 g. The pressure n flask B s 0.5x atm, and the mass of gas n that flask s 1.2 g. Whch flask contans the gas of molar mass 30, and whch contans the gas of molar mass 60? Suppose you are gven two flasks at the same temperature, one of volume 2 Land the other of volume 3 L. The 2-L flask contans 4.8 g of gas, and the gas pressure s x atm. The 3-L flask contans 0.36 g of gas, and the gas pressure s O.lx. Do the two gases have the same molar mass? If not, whch contans the gas of hgher molar mass? Complete the followng table for an deal gas: p V n T P, I I,~ I '1, 'HI> ',) I\ '!j( ;l\ '! 2.00atfll_,,, _;LpqF, p.509,rp.o,l,,.. 0.300 atm 0.250 L? mo!?atm 585 ml 0.2501110!?'K '. 27 c 350K 295 K Calculate each of the followng quanttes for an deal gas: (a) the volume of the gas, n lters, f 1.50 mo! has a pressure of 1.25 atm at a temperature of -6 C; (b) the absolute temperature of the gas at whch 3.33 x 10-3 mol occupes 478 ml at 750 torr; (c) the pressure, n atmospheres, f 0.00245 mol occupes 413 ml at 138 C; (d) the quantty of gas, n moles, f 126.5 Lat 54 c has a pressure of 11.25 kpa. The Goodyear blmps, whch frequently fly over sportng events, hold approxmately 175,000 ft 3 of helum. If the gas s at 23 c and 1.0 atm, what mass of helum s n a blmp? 10.36 A neon sgn s made of glass tubng whose nsde dameter s 2.5 cm and whose length s 5.5 m. If the sgn contans neon at a pressure of 1. 78 torr at 35 C, how many grams of neon are n the sgn? (The volume of a cylnder s 1rr 2 /-!.) ---- 10.37 (a) Calculate the number of molecules n a deep breath of ar whose volume s 2.25 Lat body temperature, 37 C, and a pressure of 735 torr. (b) The adult blue whale has a lung ca-. pacty of 5.0 x 10 3 L. Calculate the mass of ar (assume an average molar mass of 28.98 g/mol) contaned n an adult blue whale's lungs at 0.0 C and 1.00 atm, assumng the ar behaves deally. 10.38 (a) If the pressure exerted by ozone, 0 3, n the stratosphere s 3.0 x 10-3 atm and the temperature s 250 K, how many ozone molecules are n a lter? (b) Carbon doxde makes up approxmately 0.04% of Earth's atmosphere. If you collect a 2.0-L sample from the atmosphere at sea level (1.00 atm) on a warm d_a~y (27 C), how many CO 2 molecules are n your sample? 10.39 A scuba dver's tank contans 0.. 29 kg of 0 2 compressed nto a volume of 2.3 L. (a) Calculate the gas pressure nsde the tank at 9 C. (b) What volume would ths oxygen occupy at 26 C and 0.95 atm? 10.40 An aerosol spray can wth a volume of 250 ml contans 2.30 g of propane gas ( C 3 H 8 ) as a propellant. (a) If the can s at 23 C, what s the pressure n the can? (b) What volume would the propane occupy at STP? (c) The can's label says that exposure to temperatures above 130 P may cause the can to burst. W_hat s the pressure n the can at ths temperature? 10.41 A 35.1 g sample of sold CO 2 (dry ce) s added to a contaner at a temperature of 100 K wth a volume of 4.0 L. If the contaner s evacuated (all of the gas removed), sealed and then allowed to warm to room temperature (T = 298 K) so that all of the sold CO 2 s converted to a gas, what s the pressure nsde the contaner? 10.42 A 334-mL cylnder for use n chemstry lectures contans 5.225 gofhelum at 23 C. How many grams of helum must be released to reduce the pressure to 75 atm assumng deal ---~g:-a_s behavor? 10.43 Chlorne s wdely used to purfy muncpal water supples and to treat swmmng pool waters. Suppose that the volume of a partcular sample of Cl 2 gas s 8. 70 Lat 895 torr and 24 C. (a) How many grams of Cl 2 are n the sample? (b) What volume wll the Cl 2 occupy at STP? (c) At what temperature wll the volume be 15.00 L f the pressure s 8.76 x 10 2 torr? (d) At what pressure wll the volume equal 5.00 L f the temperature s 58 C? 10.44 Many gases are shpped n hgh-pressure contaners. Consder a steel tank whose volume s 55.0 gallons that contans 0 2 gas at a pressure of 16,500 kpa at 23 C. (a) What mass of 0 2 does the tank contan? (b) What volume would the gas occupy at STP? (c) At what temperature would the pressure n the tank equal 150.0 atm? (d) What would be the pressure of the gas, n kpa, f t were transferred to a contaner at 24 C whose volume s 55.0 L? ---- 10.45 In an experment reported n the scentfc lterature, male cockroaches were made to run at dfferent speeds on a mnature treadmll whle ther oxygen consumpton was measured. In 1 hr the average cockroach runnng at 0.08 km/hr consumed 0.8 ml of 0 2 at 1 atm pressure and 24 C per gram of nsect mass. (a) How many moles of 0 2 would be consumed n 1 hr by a 5.2-g cockroach movng at ths speed? (b) Ths same cockroach s caught by a chld and placed n a 1-qt frut jar wth a tght ld. Assumng the same level of contnuous actvty as n the research, wll the cockroach consume more than 20% of the avalable 0 2 n a 48-hr perod'? (Ar s 21 mol % 0 2.)
428 CHAPTER 10 Gases 10.46 The physcal ftness of athletes s measured by "Vo, max," whch s the maxmum volume of oxygen consumed by an ndvdual durng ncremental exercse (for example, on a treadmll). An average male has a V 02 max of 45 ml 02/kg body mass/mn, but a world-class male athlete can have a V 02 max readng of 88.0 ml 0 2 /kg body mass/mn. (a) Calculate the volume of oxygen, n ml, consumed n 1 hr by an average man who weghs 185 lbs and has a V 02 max readng of 47.5 ml Oz/kg body mass/mn. (b) If ths man lost 20.lb, exercsed, and ncreased hs V 02 max to 65.0 ml 02/kg body mass/mn, how many ml of oxygen would he consume n 1 hr? Dumas bulb flled wth vaporzed unknown substance Further Applcatons of the Ideal-Gas Equaton (Secton 10.5) 10.47 Rank the followng gases from least dense to most dense at 1.00 atm and 298 K: CO, N20, Cl 2, HE 10.48 Rank the followng gases from least dense to most dense at l._00 atm and,298 K: S02, HBr, CO2. 10.49 Whch of the followng statements best explans why a closed balloon flled wth helum gas rses n ar? (a) Helum s a monatomc gas, whereas nearly all the molecules that make up ar, such as ntrogen and oxygen, are datomc. (b) The average speed of helum atoms s greater than the average speed of ar molecules, and the greater speed of collsons wth the balloon walls propels the balloon upward. (c) Because the helum atoms are of lower mass than the average ar molecule, the helum gas s less dense than ar. The mass of the balloon s thus less than the mass of the ar dsplaced by ts volume. (cl) Because helum has a lower molar mass than the average ar molecule, the helum atoms are n faster moton. Ths means that the temperature of the helum s greater than the ar temperature. Hot gases tend to rse. IO.SO Whch of the followng statements best explans why ntrogen gas at STP s less dense than Xe gas at STP? (a) Because Xe s a noble gas, there s less tendency for the Xe atoms to repel one another, so they pack more densely n the gaseous state. (b) Xe atoms have a hgher mass than N2 molecules. Because both gases at STP have the same number of molecules per unt volume, the Xe gas must be denser. (c) The Xe atoms are larger than N2 molecules and thus take up a larger fracton of the space occuped by the gas. (cl) Because the Xe atoms are much more massve than the N2 molecules, they move more slowly and thus exert less upward force on the gas contaner and make the gas appear denser. IO.SI (a) Calculate the densty of N02 gas at 0.970 atm and 35 C. (b) Calculate the molar mass of a gas f 2.50 g occupes 0.875 Lat 685 torr and 35 C. 10.S2 (a) Calculate the densty of sulfur hexafluorde gas at 707 torr and 21 C. (b) Calculate the molar mass of a vapor that has a d_e_nsty of 7.135 g/l at 12 C and 743 torr. 10.S3 In the Dumas-bulb technque for determnng the molar mass of an unknown lqud, you vaporze the sample of a lqud that bols below 100 Cna bolng-water bath and determne the mass of vapor requred to fll the bulb. From the followng data, calculate the molar mass of the unknown lqud: mass of unknown vapor, 1.012 g; volume of bulb, 354 cm 3 ; pressure, 742 torr; temperature, 99 C. 10.S4 The molar mass of a volatle substance was determned the Dumas-bulb method descrbed n Exercse 10.53. The u known vapor had a mass of 0.846 g; the volume of the bll was 354 cm 3, pressure 752 torr, and temperature 100 C. Ca c_u_late the molar mass of the unknown vapor. 10.SS Magnesum can be used as a "getter" n evacuated enclosu to react wth the last traces of oxygen. (The magnesum usually heated by passng an electrc current through a 0. or rbbon of the metal.) If an enclosure of 0.452 Lhasa par pressure of 02 of 3.5 x 10-6 torr at 27 C, what mass ofm nesum wll react accordng to the followng equaton? :< 2 Mg(s) + 0 2 (g) ---'> 2 MgO(s).I 10.56 Calcum hydrde, CaH2, reacts wth water to form hydro gas: CaH2(s) + 2 H20(1) - Ca(OH)z(aq) + 2 H2(g) " Ths reacton s sometmes used to nflate lfe rafts, wea balloons, and the lke, when a smple, compact mean generatng H 2 s desred. How many grams of CaH,2 needed to generate 145 L of H2 gas f the pressure of I 825 torr at 21 C? ----,. I 10.57 Th 7 metabolc oxdaton of glucose, C 6H120 6, r our bod produces CO2, whch s expelled from our lungs as a gas:. C6H1206(aq) + 6 02(g) --> 6 COz(g) + 6 H20().'.. (a) Calculate the volume of dry CO2 produced at b temperature (37 C) and 0.970 atm when 24.5 g of glu s consumed n ths reacton. (b) Calculate the volum. oxygen you would need, at 1.00 atm and 298 K, to cornple oxdze 50.0 g of glucose. 10.58 Both Jacques Charles and Joseph Lous Guy-Lussac were~ balloonsts. In hs orgnal flght n 1783, Jacques Cha. used a balloon that contaned approxmately 31,150 L of He generated the H2 usng the reacton between ron and drochlorc acd: Fe(s) + 2HCl(aq) --> FeC1 2 (aq) + H 2 (g) How many klograms of ron were needed to produce v_o_lume of H2 f the temperature was 22 C? 10.59 Hydrogen gas s produced when znc reacts wth sulfurc a Zn(s) + H2S0 4 (aq) --> ZnS0 4 (aq) + H2(g) If 159 ml of wet H2 s collected over water at 24 C arr barometrc pressure of 738 torr, how many grams of Zn I been consumed? (The vapor pressure of water s tabulate Appendx 13.) 10.60 Acetylene gas, C2J-I2(g), can be prepared by the reacto calcum carbde wth water: CaC2(s) + 2 I-120(1) --> Ca(OH)z(aq) + C 2 Hz(g) \
\ Exercses 429 Calculate the volume of C 2 H 2 that s collected over water at 23 c by reacton of 1.524 g of CaC 2 f the total pressure of the gas s 753 torr. (The vapor pressure of water s tabulated n Appendx!3.), Consder the apparatus shown n the followng drawng. ; (a) When the valve between the two contaners s opened and the gases are allowed to mx, how does the volume occuped by the N 2 gas change? What s the partal pressure of N 2 after mxng? (b) How does the volume of the 0 2 gas change when the gases mx? What s the partal pressure of 0 2 n the mxture? (c) What s the total pressure n the contaner after the gases mx? 2.0 L l.oatm 25 C 3.0 L 2.0 atm 25 C, 2 Consder a mxture of two gases, A and!3, confned n a closed vessel. A quantty of a thrd gas, C, ts added to the :1. same vessel at the same temperature, How does the addton r, of gas C affect the followng: (a) the partal pressure of gas A,, (b) the total pressure n the vessel, (c) the mole fracton of.gas!3? '.' A mxture contanng 0. 765 mol He(g), 0.330 mol Ne(g), :;' and 0.110 mol Ar(g) s confned n a 10.00-L vessel at 25 C. ' (a) Calculate the partal pressure of each of the gases n the 1 mxture. (b) Calculate the total pressure of the mxture. A deep-sea dver uses a gas cylnder wth a volume of 10.0 L ' and a content of 51.2 g of 0 2 and 32.6 g of He. Calculate the ', partal pressure of each gas and the total pressure f the tern., perature of the gas s 19 C. s;,'.'he atmospherc concentraton of CO 2 gas s presently, 407 ppm (parts per mllon, by volume; that s, 407 L of every 10 6 L of the atmosphere are CO 2 ). What s the mole fracton of CO 2 n the atmosphere? {!)l, f A plasma-screen TV contans thousands of tny cells flled,;_ wth a mxture of Xe, Ne, and He gases that emts lght of ;, specfc wavelengths when a voltage s appled. A partcular 1- plasma cell, 0.900 mm x 0.300 mm X 10.0 mm, contans 4% Xe n a 1:1 Ne.He.mxture at a total pressure of 500 torr. Calculate the number of Xe, Ne, and He atoms n the cell and state the assumptons you need to make n your calculaton. A pece of dry ce (sold carbon doxde) wth a mass of 5.50 g ll. s placed n a 10.0-L vessel that already contans ar at :, 705 torr and 24 c. After the carbon doxde has totally, sublmed, what s the partal pressure of the resultant CO 2., gas, and the total pressure n the contaner at 24 C? 8 A sample of 5.00 ml of d e t h y l e t h e r (C 2 H 5 0C 2 H 5, densty = O. 7134 g/ml) s ntroduced nto a 6.00-L vessel that already contans a mxture of N 2 and 0 2, whose partal pressures are PN, = 0. 751 atm and P 02 = 0.208 atrn. The temperature s held at 35.0 c, and the dethylether totally evapr orates. (a) Calculate the partal pressure of the dethylether. (b) Calculate the total pressure n the contaner. A rgd vessel contanng a 3: 1 rnol rato of carbon doxde and ;, Water vapor s held at 200 C where t has a total pressure of 2.00 atm. If the vessel s cooled to 10 C so that all of the water vapor condenses, what s the pressure of carbon doxde? Neglect the volume of the lqud water that forms on coolng. 10.70 If 5.15 gof Ag 2 0 s sealed n a 75.0-mL tube flled wth 760 torr of N 2 gas at 32 C, and.the tube s heated to 320 C, the Ag 2 0 decomposes to form oxygen and slver. What s the total pressure nsde the tube assumng the volume of the tube remans constant? ----- 10.71 At an underwater depth of 250 ft, the pressure s 8.38 atm. What should the mole percent of oxygen be n the dvng gas for the partal pressure of oxygen n the mxture to be0.21 atm, the same as n ar at 1 atrn? 10.72 (a) What are the mole fractons of each component n a mxture of 15.08 g of 02, 8.17 g of N 2, and 2.64 g of H 2? (b) What s the partal pressure n atrn of each component of ths mxture f t s held n a 15.50-L vessel at 15 C? ----- 10.73 A quantty of N 2 gas orgnally held at 5.25 atm pressure n a 1.00-L contaner at 26 C s transferred to a 12.5-L contaner at 20 C. A quantty of 0 2 gas orgnally at 5.25 atm and.26 C.n a 5.00-L contaner s transferred to ths same contaner. What s the total pressure n the new contaner? 10.74 A sample of 3.00 g of S0 2 (g) orgnally n a 5.00-L vessel at 21 C s transferred to a 10.0-L vessel at 26 C. A sample of 2.35 g of N 2 (g) orgnally n a 2.50-L vessel at 20 C s transferred to ths same 10.0-L vessel. (a) What s the partal pressure of S0 2 (g) n the larger contaner? (b) What s the partal pressure of N 2 (g) n ths vessel? (c) What s the total pressure n the vessel? Knetc-Molecular Theory of Gases; Effuson and Dffuson (Sectons 10.7 and 10.8) 10.75 Determne whether each of the followng changes wll ncrease, decrease, or not affect the rate wth whch gas molecules collde wth the walls of ther contaner: (a) ncreasng the volume of the contaner, (b) ncreasng the temperature, (c) ncreasng the molar mass of the gas. 10.76 Indcate whch of the followng statements regardng the knetc-molecular theory of gases are correct. (a) The average knetc energy of a collecton of gas molecules at a gven temperature s proportonal to m 1 1 2 (b) The gas molecules are assumed to exert no forces on each other. (c) All the molecules of a gas at a gven temperature have the same knetc energy. (d) The volume of the gas molecules s neglgble n comparson to the total volume n whch the gas s contaned. (e) All gas molecules move wth the same speed f they are at the same te_mperature. 10. 77 WF 6 s one of the heavest known gases. How much slower s the root-mean-square speed of WF 6 than He at 300 K? [10.78] You have an evacuated contaner of fxed volume and known mass and ntroduce a known mass of a gas sample. Measurng the pressure at constant temperature over tme, you are surprsed to see t slowly droppng. You measure the mass of the gas-flled contaner and fnd that the mass s what t should be-gas plus contaner-and the mass does not change over tme, so you do not have a leak. Suggest an e_xplanaton for your observatons. 10.79 The temperature of a 5.00-L contaner of N 2 gas s ncreased from 20 c to 250 C. If the volume s held constant, predct qualtatvely how ths change affects the followng: (a) the average knetc energy of the molecules; (b) the rootmean-square speed of the molecules; (c) the strength of the mpact of an average molecule wth the contaner walls; (d) the total number of collsons of molecules wth walls per second.
430, CHAPTER 10 Gases unknown gas. (Remember that the faster the rate of ef( son, the shorter the tme requred for effuson of 1.0 L other words, rate s the amount that dffuses over the tj t takes to dffuse.) 10.80 Suppose you have two 1-L flasks, one contanng N 2 at STP, the other contanng CH 4 at STP. How do these systems compare wth respect to (a) number of molecules, (b) densty, (c) average knetc energy of the molecules, (d) rate of effu- s_on through a pnhole leak? 10.81 (a) Place the followng gases n order of ncreasng average Nondeal-Gas Behavor (Secton 10.9) molecular speed at 25 C: Ne, J-113r, S0 2, NF 3, CO. (b) Calculate therms speed of NF 3 molecules at 25 C. (c) Calculate the most probable speed of an ozone molecule n the stratosphere, where the temperature s 270 K. 10.82 (a) Place the followng gases n order of ncreasng average molecular speed at 300K:CO,SF 6,J-1 2 S,Cl 2,J-113r. (b) Calculate therms speeds of CO and Cl 2 molecules at 300 K. (c) Calculate the most probable speeds of CO and Cl 2 molecules at 300 K. 10.83 Whch one or more of the followng statements are true? 10.91 (a) 0 2 wll effuse faster than Cl 2. (b) Effuson and dffuson are dfferent names for the same process. (c) Perfume molecules travel to your nose by the process of effuson. (d) The hgher the densty of a gas, the shorter the mean free path. [10.84] At constant pressure, the mean free path (A) of a gas mol- 10.92 ecule s drectly proportonal to temperature. At constant temperature, A 's nversely proportonal to pressure. If you compare two dfferent gas molecules at the same tempera- 10.93 ture and pressure, A s nversely proportonal to the square of the dameter of the gas molecules. Put these facts together to create a formula for the mean free path of a gas molecule wth a proportonalty constant (call t Rm1p, lke the deal-gas constant) and defne unts for Rmrp 10.85 Hydrogen has two naturally occurrng sotopes, 1 H and 2 J-I. Chlorne also has two naturally occurrng sotopes, 35 Cl and 37 Cl. Thus, hydrogen chlorde gas conssts of four dstnct types of molecules: 1 H 35 Cl, 1 H 37 Cl, 2 H 35 CJ, and 2 H 37 Cl. Place these four molecules n order of ncreasng rate of 10.94 effuson. 10.86 As dscussed n the "Chemstry Put to Work" box n Secton 10.8, enrched uranum can be produced by effuson of gaseous UF 6 across a porous membrane. Suppose a process were developed to allow effuson of gaseous uranum atoms, U(g). Calculate the rato of effuson rates for 235 U and 238 u, and compare t to the rato for Ul~ gven n the essay. [10.95] 10.87 Arsenc(lII) sulfde sublmes readly, even below ts meltng pont of 320 C. The molecules of the vapor phase are found to effuse through a tny hole at 0.28 tmes the rate of effuson of Ar atoms under the same condtons of temperature and pressure. What s the molecular formula of arsenc(iii) sulfde n the gas phase? [10.96] 10.88 A gas of unknown molecular mass was allowed to effuse through a small openng under constant-pressure condtons. It requred 105 s for 1.0 L of the gas to effuse. Under dentcal expermental condtons t requred 31 s for 1.0 L of 0 2 gas to effuse. Calculate the molar mass of the 10.89 (a) Lst two expermental condtons under whch gases vate from deal behavor. (b) Lst two reasons why the ga devate from deal behavor. 10.90 The planet Jupter has a surface temperature of 140 K anct mass 318 tmes that of Earth. Mercury (the planet) has a su face temperature between 600 Kand 700 Kand a mass o. tmes that of Earth. On whch planet s the atmosph n_1ore lkely to obey the deal-gas law? Explan. Whch statement concernng the van der Waals constan and bs true? (a) The magntude of a relates to molecular volume, 1 whereas b relates to attractonsbetween molecules. (b) The magntude of a relates to attractons between molecules, whereas b relates to molecular volume., (c) The magntudes of a and b depend on pressure.,l (d) The magntudes of a and b depend on temperature, Based on ther respectve van der Waals corsf (Table 10.3), s Ar or CO 2 expected to behave more ne lke an deal gas at hgh pressures?, -----.. vl In Sample Exercse 10.16, we found that one mole of confned to 22.41 Lat O C devated slghtly from behavor. Calculate the pressure exerted by 1.00 mo! confned to a smaller volume, 5.00 L, at 25 c. (a) Frst the deal-gas equaton and (b) then use the van der W equaton for your calculaton. (Values for the van der constants are gven n Table 10.3.) (c) Why s the clff~{ between the result for an deal gas and that ~alculate ng the van der Waals equaton greater when the gas s 1 fned to 5.00 L compared to 22.4 L?..,...,~ Cal 7 ulate the-pressure that CC1 4 wll exert at 809 1.00 mo! occupes 33.3 L, assumng that (a) CC1 4 obeys deal-gas equaton; (b) CC1 4.obeys the van der Waals e ton. (Values for the van der Waals constants are gve Table 10.3.) (c) Whch would you expect to devate from deal behavor under thesecondtttons, Cl 2 or C' Explan. Table 10.3 shows that the van der Waals b parameter unts of L/mol. Ths mples that we can calculate the sz atoms or molecules from b. Usng the value of b for Xe;' culate the radus of a Xe atom and compare t to the va 0 " found n Fgure 7. 7, that s, 1.40 A. Recall that the vo]um aspheres(4/3)1t1' 3. 1 Table 10.3 shows that the van der Waals b parameter unts of L/mol. Ths means that we can calculate the ;t s of atoms or molecules from the b parameter. Refer bac, the dscusson n Secton 7.3. Is the van der Waals radu calculate from the b parameter of Table 10.3 more cl. assocated wth the bondng or nonbondng atomc ra o dscussed there? Explan. Addtonal Exercses 10.97 Torrcell, who nvented the barometer, used mercury n ts constructon because mercury has a very hgh densty, whch makes t possble to make a more compact barometer than one based on a less dense flud. Calculate the densty of mercury usng the observaton that the column of mercury s 760 mm hgh when the atmospherc pressur 1.01 x 10 5 Pa. Assume the tube contanng the mercur a cylnder wth a constant cross-sectonal area. 10.98 A gas bubble wth a volume of l.omm 3 orgnates at bottom of a lake where the pressure s 3.0 atm. Calcll
Addtonal Exercses 431 ts volume when the bubble reaches the surface of the lake where the pressure s 730 torr, assumng that the temperature doesn't change. Q.99 A 15.0-L tank s flled wth helum gas at a pressure of 1.00 x 10 2 atm. How many balloons (each 2.00 L) can be nflated to a pressure of 1.00 atm, assumng that the temperature remans constant and that the tank cannot be empted below 1.00 atm? /100 To mnmze the rate of evaporaton of the tungsten flament, 1.4 x 10-s mol of argon s placed n a 600-cm 3 lghtbulb. What s the pressure of argon n the lghtbulb at 23 C? 101 Carbon doxde, whch s recognzed as the major contrbutor to global warmng as a "greenhouse gas," s formed when fossl fuels are combusted, as n electrcal power plants fueled by coal, ol, or natural gas. One potental way to reduce the amount of CO 2 added to the atmosphere s to store t as a compressed gas n underground formatons. Consder a 1000-megawatt coal-fred power plant that produces about 6 x 10 6 tons of CO 2 per year. (a) Assumng deal-gas behavor, 1.00 atm, and 27 C, calculate the volume of CO 2 produced by ths power plant. (b) If the CO 2 s stored underground as a lqud at 10 C and 120 atm and a densty of 1.2 g/cm3, what volume does t possess? (c) If t s stored underground as a gas at 30 C and 70 atm, what volume does t occupy? 02 Propane, C 3 H 8, lquefes under modest pressure, allowng a large amount to be stored n a contaner. (a) Calculate the number of noles of propane gas n a 110"-L contaner at 3.00 atm and 27 C. (b) Calculate the number of moles of lqud propane that can be stored n the same volume f the densty of the lqud s 0.590 g/ml. (c) Calculate the rato of the number of moles of lqud to moles of gas. Dscuss ths rato n lght of the knetc-molecular theory of gases. 03] Nckel carbonyl, N(C0) 4, s one of the most toxc substances known. The present maxmum allowable concentraton n laboratory ar durng an 8-hr workday s 1 ppb (parts per bllon) by volume, whch means that there s one mole of N(C0) 4 for every 10 9 moles of gas. Assume 24 c and 1.00 atm pressure. What mass of N(C0) 4 s allowable n a laboratory room that s 12 ft x 20 ft x 9 ft? ';04 When a large evacuated flask s flled wth argon gas, ts mass ncreases by 3.224 g. When the same flask s agan evacuated and then flled wth a gas of unknown molar mass, the mass ncrease s 8.102 g. (a) Based on the molar mass of argon, estmate the molar mass of the unknown gas. (b) What assumptons dd you make n arrvng at your answer? 105 Consder the arrangement of bulbs shown n the drawng. Each of the bulbs contans a gas at the pressure shown. What s the pressure of the system when all the stopcocks are opened, assumng that the temperature remans constant? (We can neglect the volume of the capllary tubng connectng the bulbs.) 1.0 L 265 torr 1.0 L 800 torr 0.5 L 532 torr 106 Assume that a sngle cylnder of an automoble engne has a volume of 524 cm 3. (a) [f the cylnder s full of ar at 74 c and 0.980 atm, how many moles of 0 2 are present? (The mole fracton of 0 2 n dry ar s 0.2095.) (b) How many grams of C 8 H 18 could be combusted by ths quantty of 0 2, assumng complete combuston wth formaton of CO 2 and H 2 0? 10.107 Assume that an exhaled breath of ar conssts of 74.8% N 2, 15.3% 0 2, 3.7% CO 2, and 6.2% water vapor. (a) If the total pressure of the gases s 0.985 atm, calculate the partal pressure of each component of the mxture. (b) If the volume of the exhaled gas s 455 ml and ts temperature s 3 7 C, calculate the number of moles of CO 2 exhaled. (c) How many grams of glucose (C 6 I-I 12 0 6 ) would need to be metabolzed to produce ths quantty of CO 2? (The chemcal reacton s the same as that for combuston of C 6 I-I 12 0 6. See Secton 3.2 and Problem 10.57.) 10.108 A 1.42-g sample of helum and an unknown mass of 0 2 are mxed n a flask at room temperature. The partal pressure of the helum s 42.5 torr, and that of the oxygen s 158 torr. What s the mass of the oxygen? [10.109] An deal gas at a pressure of 1.50 atm s contaned n a bulb of unknown volume. A stopcock s used to connect ths bulb wth a prevously evacuated bulb that has a volume of 0.800 Las shown here. When the stopcock s opened, the gas expands nto the empty bulb. If the temperature s held constant durng ths process and the fnal pressure s 695 torr, what s the volume of the bulb that was orgnally flled wth gas? 10.110 The densty of a gas of unknown molar mass was measured as a functon of pressure at O C, as n the table that follows. (a) Determne a precse molar mass for the gas. [Hnt: Graphd/P versus P.] (b) Why s d/p not a constant as a functon of pressure? Pressure (atm) 1.00 0.666 0.500 0.333 0.250,; P~n#~'f(g){Y: " 1 ~:L:fo?s4),\T~z~t ):l o_''.'tfz~7\': o.5660 [10.111] A glass vessel ftted wth a stopcock valve has a mass of 337.428 g when evacuated. When flled wth Ar, t has a mass of 339.854 g. When evacuated and reflled wth a mxture of Ne and Ar, under the same condtons of temperature and pressure, t has a mass of 339.076 g. What s the mole percent of Ne n the gas mxture? 10.112 You have a sample of gas at -33 C. You wsh to ncrease therms speed by a factor of 2. To what temperature should the gas be heated? 10.113 Consder the followng gases, all at STP: Ne, SF 6, N 2, CH 4. (a) Whch gas s most lkely to depart from the assumpton of the knetc-molecular theory that says there are no attractve or repulsve forces between molecules? (b) Whch one s closest to an deal gas n ts behavor? (c) Whch one has the hghest root-mean-square molecular speed at a gven temperature? (cl) Whch one has the hghest total molecular volume relatve to the space occuped by the gas? (c) Whch has the hghest average knetc-molecular energy? (f) Whch one would effuse more rapdly than N 2? (g) Whch one would have the largest van cler Waals h parameter?