Section1.515TheMaxwellDistributionofSpeedsMass (amu)8002040601003000500ET= 300 K2400400LLLLL1800300(s/u) poods(sru) podsLLLLI- (t)Crm1200200L600F100OLJO1015200525Mass (amu)Figure1.6Various average speeds as a function of mass for T =300 K.atoms having speeds in excess of ue,while minuscule (about10-31), is still 10475 times larger than the fraction of oxygenmolecules having speeds in excess of u!As a consequence, thecomposition of the atmosphere is changing; much of the heliumreleased during the lifetime of the planet has already escaped intospace. A plot of various speeds as a function of mass for T = 300Kis shown inFigure1.6.1.5.5Experimental Measurement of theMaxwellDistribution of SpeedsExperimental verification of the Maxwell-Boltzmann speed distribution can bemade by direct measurement using the apparatus of Figure 1.7.Two versions of themeasurement are shown,In Figure1.7a, slits (S) define a beam of molecules mov-ing in a particular direction after effusing from an oven (O). Those that reach thedetector (D)must successfully havetraversed a slotted,multiwheel chopperby trav-eling a distanced whilethechopperrotated throughan angle.Ineffect, thechop-per selects a small slice from the velocity distribution and passes it to the detector.The speed distribution is then measured by recording the integrated detector signalfor each cycle of the chopper as a function of the angular speed of the chopper.A somewhat more modern technique, illustrated in Figure 1.7b, clocks the timeittakesformolecules totravel afixeddistance.Avery shortpulse ofmoleculesleavesthe chopper at time t = 0. Because these molecules have a distribution of speeds, theyspread out in space as they traveltoward thedetector,whichrecords as afunction oftime the signal due to molecules arriving a distance L from the chopper
Section 1.5 The Maxwell Distribution of Speeds Mass (amu), Mass (amu) II Figure 1.6 Various average speeds as a function of mass for T = 300 K. atoms having speeds in excess of v, while minuscule (about 1OP3l), is still times larger than the fraction of oxygen molecules having speeds in excess of v,! As a consequence, the composition of the atmosphere is changing; much of the helium released during the lifetime of the planet has already escaped into space. A plot of various speeds as a function of mass for T = 300 K is shown in Figure 1.6. 1.5.5 Experimental Measurement of the Maxwell Distribution of Speeds Experimental verification of the Maxwell-Boltzmann speed distribution can be made by direct measurement using the apparatus of Figure 1.7. Two versions of the measurement are shown. In Figure 1.7a, slits (S) define a beam of molecules moving in a particular direction after effusing from an oven (0). Those that reach the detector (D) must successfully have traversed a slotted, multiwheel chopper by traveling a distance d while the chopper rotated through an angle 4. In effect, the chopper selects a small slice from the velocity distribution and passes it to the detector. The speed distribution is then measured by recording the integrated detector signal for each cycle of the chopper as a function of the angular speed of the chopper. A somewhat more modern technique, illustrated in Figure 1.7b, clocks the time it takes for molecules to travel a fixed distance. A very short pulse of molecules leaves the chopper at time t = 0. Because these molecules have a distribution of speeds, they spread out in space as they travel toward the detector, which records as a function of time the signal due to molecules arriving a distance L from the chopper
16ChapterKineticTheoryof Gases(A)V(B)Figure 1.7Two methodsfor measuring the Maxwell-Boltzmann speeddistribution.Analysis of the detector signal from this.second experiment is instructive, sinceit introduces the concept of flux. Recall that the distribution F(u) du gives thefractionof molecules with speeds intherange from uto +du;itis dimensionless.If thenumber density of molecules is n, then n'F(u) du will be the number of moleculesper unit volume with speeds in the specified range. The flux of molecules is definedas the number of molecules crossing a unit area per unit time. It is equal to the den-sity of molecules times their velocity: flux (number/m?/s) = density (number/m3) ×velocity (m/s).Thus,thefluxJof molecules with speeds between u and u+ du isJdu=un'F(u)du.(1.36)Wewill consider theflux inmore detail in Section 4.3.2andmakeextensive useofit in Chapter4.We now return to the speed measurement. Most detectors actually measure thenumber of molecules in a particular volume during a particular time duration,Forexample, the detector mightmeasure current after ionizing those molecules thatenter a volume defined by a cross-sectional area of A and a length . Because mol-ecules with high velocity traverse the distance in less time than molecules withlowvelocity,thedetectionsensitivityisproportional to1/u.Thedetectorsignal S(t)is thus proportional to JAe dv/u,or to n'AF(u)du, where n"is the number densityof molecules in theoven.Assuming that a very narrow pulse of molecules is emittedfrom the chopper, the speed measured at a particular time t is simply u = Llt. Wemust now transform the velocity distribution from a speed distribution to a timedistribution. Note that du = d(Ut) =-L dtlt2, and recall from equation 1.31 thatF(v)duαvexp(-βu2)du(1/t)exp(-βL2it)(Lt).WethusfindthatS(t)αt-4exp(-βL?it). Figure 1.8 displays an arrival time distribution of helium measuredgStrictly speaking, the flux, J, is a vector, since the magnitude of the flux may be different in differentdirections. Here, since the direction of the flux is clear, we will use just its magnitude, J
Chapter 1 Kinetic Theory of Gases II Figure 1.7 Two methods for measuring the Maxwell-Boltzmann speed distribution. Analysis of the detector signal from this second experiment is instructive, since it introduces the concept offlux. Recall that the distribution F(v) dv gives the fraction of molecules with speeds in the range from v to v + dv; it is dimensionless. If the number density of molecules is n", then n*F(v) dv will be the number of molecules per unit volume with speeds in the specified range. The flux of molecules is defined as the number of molecules crossing a unit area per unit time. It is equal to the density of molecules times their velocity: flux (number/m2/s) = density (number/m3) X velocity (m/s).g Thus, the flux J of molecules with speeds between v and u + dv is J dv = un*~(u) du. (1.36) We will consider the flux in more detail in Section 4.3.2 and make extensive use of it in Chapter 4. We now return to the speed measurement. Most detectors actually measure the number of molecules in a particular volume during a particular time duration. For example, the detector might measure current after ionizing those molecules that enter a volume defined by a cross-sectional area of A and a length t. Because molecules with high velocity traverse the distance t in less time than molecules with low velocity, the detection sensitivity is proportional to llv. The detector signal S(t) is thus proportional to JAC dvlv, or to n*AtF(v) dv, where n* is the number density of molecules in the oven. Assuming that a very narrow pulse of molecules is emitted from the chopper, the speed measured at a particular time t is simply v = Llt. We must now transform the velocity distribution from a speed distribution to a time distribution. Note that dv = d(L/t) = -L dt/t2, and recall from equation 1.31 that F(v) dv cc v2exp(-pv2) dv = (llt2)exp(-pL2/t2)(L/t2). We thus find that S(t) tP4 exp(-PL2/t2). Figure 1.8 displays an arrival time distribution of helium measured gstrictly speaking, the flux, J, is a vector, since the magnitude of the flux may be different in different directions. Here, since the direction of the flux is clear, we will use just its magnitude, J
Section 1.6 EnergyDistributions17(ares Keniie aisae2004000600Flight time (μs)Figure1.8Time-of-flight measurements: intensity as a function of flight timeFrom J.F.C.Wang and H.Y.Wachman,as illustrated in F.O.Goodman and HY.Wachman,Dynamics of GasSurface Scattering (Academic Press,New York,1976).Figurefrom"Molecular Beams"in DYNAMICS OFGAS-SURFACE SCATTERING by F. O. Goodman and H Y. Wachmann, copyright 1976 by AcademicPress, reproduced by permission of the publisher.All rights or reproduction in any form reserved.using this“"time-of-flight"technique.The open circles are the detector signal, whilethe smooth line is a fit to the data of a function of the form expected for S(t). Thebestfitparametergivesatemperatureof300K1.6ENERGYDISTRIBUTIONSIt is sometimes interesting to know the distribution of molecular energies ratherthan velocities. Of course, these two distributions must be related since the molec-ular translational energy e is equal to mu?. Noting that this factor occurs in theexponent of equation 1.31 and that de=mu du= (2me)I/2du,we can convertvelocities to energies in equation 1.31 toobtaindeG(e)de=4mV2me(1.37)2TVeexp(TkTThe function G(e) de tells us the fraction of molecules which have energies in therangebetween and +de.Plots of G(e)are shown in Figure1.9.The distribution function G(e) can be used to calculate the average of any function of e using the relationship of equation1.16.In particular,it can be shown asexpectedthat<e>=3kT/2(seeProblem1.9)Let us pause here to make a connection with thermodynamics. In the case ofan ideal monatomicgas,thereare no contributionsto theenergyof thegas frominternal degrees offreedom such as rotation or vibration,and there is normally very
Section 1.6 Energy Distributions 0 200 400 600 Flight time (ps) II Figure 1.8 Time-of-flight measurements: intensity as a function of flight time. From J. F. C. Wang and H. Y. Wachman, as illustrated in F. 0. Goodman and H. Y. Wachman, Dynamics of GasSurjiace Scattering (Academic Press, New York, 1976). Figure from "Molecular Beams" in DYNAMICS OF GAS-SURFACE SCArnRING by F. 0. Goodman and H. Y. Wachmann, copyright O 1976 by Academic Press, reproduced by permission of the publisher. All rights or reproduction in any form reserved. using this "time-of-flight" technique. The open circles are the detector signal, while the smooth line is a fit to the data of a function of the form expected for S(t). The best fit parameter gives a temperature of 300 K. 1.6 ENERGY DISTRIBUTIONS It is sometimes interesting to know the distribution of molecular energies rather than velocities. Of course, these two distributions must be related since the molecular translational energy E is equal to ;mu2. Noting that this factor occurs in the exponent of equation 1.31 and that d~ = mv dv = (2me)lJ2 dv, we can convert velocities to energies in equation 1.31 to obtain The function G(E) d~ tells us the fraction of molecules which have energies in the range between E and E + d~. Plots of G(E) are shown in Figure 1.9. The distribution function G(E) can be used to calculate the average of any function of E using the relationship of equation 1.16. In particular, it can be shown as expected that <E> = 3kTl2 (see Problem 1.9). Let us pause here to make a connection with thermodynamics. In the case of an ideal monatomic gas, there are no contributions to the energy of the gas from internal degrees of freedom such as rotation or vibration, and there is normally very
18Chapter1KineticTheoryof Gases0.120.10T=300K0.08H0.04T = 1000 K0.020.00201030400e(1021))Figure1.9Energy distributions for two different temperatures.Thefraction of molecules for the 300Kdistribution having energy in excess of e' is shown in the shaded region.littlecontribution tothe energyfromexcitation of electronic degrees of freedomConsequently, the average energy U of n moles of a monatomic gas is simply nNAtimes the average energy of one molecule of the gas, or3U=nNA号kT=号nRT.(1.38)2Note that theheat capacity at constant volume is defined as Cy= (aU/aT)y,so thatforan ideal monatomicgaswefindthat3(1.39)Cv ==nR.2This result is an example of the equipartition principle, which states that each termin the expression of the molecular energy that is quadratic in a particular coordinatecontributes kT to the average kinetic energy and R to the molar heat capacity.Since there are three quadratic terms in the three-dimensional translational energyexpression,themolarheatcapacityofamonatomicgasshouldbe3R/2.It is sometimes useful to know what fraction of molecules has an energy greaterthan or equal to a certain value e. In principle, the energy distribution G(e) shouldbeableto providethis information,since thefraction of moleculeshaving energy inthe desired range is simply the integral of G(e) de from e* to infinity, as shown bythe hatched region in Figure 1.9.In practice, the mathematics are somewhat cumbersome,but the result is reasonable.Letf(e)be the fraction of molecules withkinetic energy equal to or greater than e.This fraction is given by the integralf(e) = 2元(Veexp(de(1.40)TkT
Chapter 1 Kinetic Theory of Gases Figure 1.9 Energy distributions for two different temperatures. The fraction of molecules for the 300 K distribution having energy in excess of E* is shown in the shaded region. little contribution to the energy from excitation of electronic degrees of freedom. Consequently, the average energy U of n moles of a monatomic gas is simply nN, times the average energy of one molecule of the gas, or Note that the heat capacity at constant volume is defined as C, = (dUIdT), so that for an ideal monatomic gas we find that This result is an example of the equipartition principle, which states that each term in the expression of the molecular energy that is quadratic in a particular coordinate contributes i kT to the average kinetic energy and i R to the molar heat capacity. Since there are three quadratic terms in the three-dimensional translational energy expression, the molar heat capacity of a monatomic gas should be 3Rl2. It is sometimes useful to know what fraction of molecules has an energy greater than or equal to a certain value E". In principle, the energy distribution G(E) should be able to provide this information, since the fraction of molecules having energy in the desired range is simply the integral of G(E) d~ from E* to infinity, as shown by the hatched region in Figure 1.9. In practice, the mathematics are somewhat cumbersome, but the result is reasonable. Let A€*) be the fraction of molecules with kinetic energy equal to or greater than E*. This fraction is given by the integral
Section1.7Collisions:MeanFreePathandCollisionNumber191.00.8f(e)0.6(a)f0.40.2(2/V)aexp(-a2)0.0230145Q2=e/kTFigure1.10The fraction of molecules having energy in excess of e as a function of e'/kT.Problem 1.10 shows thatthis integral is given by2ae" + erfe(a),f(e) =(1.41)VTwhere a = (e*/kT)/2 and erfc(a) is the co-error function defined in Appendix 1.3. Aplot of f(e) as a function of e'/kT is shown in Figure 1.10. Note that fore*>3kTthe functionf(e)is nearly equal to the firstterm in equation1.4l,2V(e/kT)exp(-e/kT), shown by the dashed line in thefigure. Thus, the frac-tion of molecules with energygreater than efalls off asVe exp(-e/kT),providedthate*> 3kT1.7COLLISIONS:MEANFREEPATHANDCOLLISIONNUMBEROne of thegoals of this chapter is to derive an expression for the number of colli-sions that molecules of type 1 make with molecules of type 2 in a given time. Wewill argue later that this collision rate provides an upper limit to the reaction rate,since the two species must have a close encounter to react.Theprincipal properties of the collision rate can beeasily appreciated by any-one who has ice skated at a local rink.Imagine two groups of skaters, some rathersedate adults and some rambunctious 13-year-old kids.If there is only onekid andone adult in the rink, then the likelihood that they will collide is small, but as the num-ber of either adults or kids in the rink increases, so does the rate at which collisions
Section 1.7 Collisions: Mean Free Path and Collision Number II Figure 1.10 The fraction of molecules having energy in excess of E* as a function of ~*lkl: Problem 1.10 shows that this integral is given by where a = (~*lkT)"~ and erfc(a) is the co-error function defined in Appendix 1.3. A plot off(€*) as a function of ~*lkT isshown in Figure 1.10. Note that for E* > 3kT the function fie*) is nearly equal to the first term in equation 1.41, 2mexp(-e*/k~), shown by the dashed line in the figure. Thus, the fraction of molecules with energy greater than E* falls off as V? exp(-e*/k~), provided that E* > 3kT. 1.7 COLLISIONS: MEAN FREE PATH AND COLLISION NUMBER One of the goals of this chapter is to derive an expression for the number of collisions that molecules of type 1 make with molecules of type 2 in a given time. We will argue later that this collision rate provides an upper limit to the reaction rate, since the two species must have a close encounter to react. The principal properties of the collision rate can be easily appreciated by anyone who has ice skated at a local rink. Imagine two groups of skaters, some rather sedate adults and some rambunctious 13-year-old kids. If there is only one kid and one adult in the rink, then the likelihood that they will collide is small, but as the number of either adults or kids in the rink increases, so does the rate at which collisions