ENERGY AND BOLTZMANN DISTRIBUTIONS

Transcription

1 MISN--159 NRGY AND BOLTZMANN DISTRIBUTIONS NRGY AND BOLTZMANN DISTRIBUTIONS by J. S. Kovacs and O. McHarris Michigan State University 1. Introduction nergy Distribution Functions a. Genera Distribution Functions b. Therma quiibrium c. The Botzmann Distribution Readings xercises Answers to xercises Acknowedgments Project PHYSNT Physics Bdg. Michigan State University ast Lansing, MI 1

2 ID Sheet: MISN--159 Tite: nergy and Botzmann Distributions Author: J. S. Kovacs and O. McHarris, Michigan State University Version: 2/1/2 vauation: Stage Length: 1 hr; 2 pages Input Skis: 1. Describe an idea gas in terms of (a) the reationship between temperature and the speed of its moecues and (b) the nature of the interactions among its moecues. (MISN--157) Output Skis (Knowedge): K1. Define the partition of a system consisting of N moecues of a gas. K2. Define therma equiibrium starting from the concept of a very arge number of possibe partitions for a given many partice system. Identify the partition for an idea gas in therma equiibrium. Output Skis (Probem Soving): S1. Determine the number of partices that have their energies in a given energy range, given the partition for the many partice system. S2. Given the partition, cacuate the average vaues of kinetic energy, speed and speed squared and the tota energy of a system of N moecues of a gas. xterna Resources (Required): 1. K. W. Ford, assica and Modern Physics, Vo. 2, Xerox (1972). For avaiabiity, see this modue s Loca Guide. 2. M. Aonso and. J. Finn, Physics, Addison-Wesey (197). For avaiabiity, see this modue s Loca Guide. THIS IS A DVLOPMNTAL-STAG PUBLIATION OF PROJT PHYSNT The goa of our project is to assist a network of educators and scientists in transferring physics from one person to another. We support manuscript processing and distribution, aong with communication and information systems. We aso work with empoyers to identify basic scientific skis as we as physics topics that are needed in science and technoogy. A number of our pubications are aimed at assisting users in acquiring such skis. Our pubications are designed: (i) to be updated quicky in response to fied tests and new scientific deveopments; (ii) to be used in both cassroom and professiona settings; (iii) to show the prerequisite dependencies existing among the various chunks of physics knowedge and ski, as a guide both to menta organization and to use of the materias; and (iv) to be adapted quicky to specific user needs ranging from singe-ski instruction to compete custom textbooks. New authors, reviewers and fied testers are wecome. PROJT STAFF Andrew Schnepp ugene Kaes Peter Signe Webmaster Graphics Project Director ADVISORY OMMITT D. Aan Bromey Yae University. Leonard Jossem The Ohio State University A. A. Strassenburg S. U. N. Y., Stony Brook Views expressed in a modue are those of the modue author(s) and are not necessariy those of other project participants. c 21, Peter Signe for Project PHYSNT, Physics-Astronomy Bdg., Mich. State Univ.,. Lansing, MI 48824; (517) For our ibera use poicies see: 3 4

3 MISN NRGY AND BOLTZMANN DISTRIBUTIONS by J. S. Kovacs and O. McHarris Michigan State University 1. Introduction In a gas in therma equiibrium at a given temperature T, the average kinetic energy of the moecues of that gas has a definite vaue. For an idea gas, in fact, the inear reationship K,ave = (3/2)kT defines the temperature of that gas. However, the energy of an individua moecue of that gas varies from instant to instant, changing from coision to coision and at any one instant the individua energies of the moecues are distributed over a broad range of vaues about the average energy. As a consequence of the very arge number of moecues in a gas sampe ( 1 2 moecues), the probabiity that any one moecue wi have any given energy vaue can be known to very high precision when the gas is in therma equiibrium. This probabiity distribution gives us the distribution of partices among the possibe energy vaues and is given by the Botzmann distribution function for an idea gas in therma equiibrium. In this unit distribution functions in genera and the Botzmann function in particuar wi be introduced nergy Distribution Functions 2a. Genera Distribution Functions. Suppose we have a system of N moecues and a set of L avaiabe energies such that any one of the N moecues coud have any of the L energies. Then there are very many possibe ways that the N moecues coud have energies aocated to them. For exampe a of the moecues coud have energy 9, or maybe 1/5 of the moecues coud have energy 7 and the other 4/5 have energy 33, or the energies coud be a equay popuated, meaning the moecues are divided up eveny among them, and so forth. The numbers that te how many of the moecues have each of the energies is caed the partition, or distribution; that is, if n 1 moecues have energy 1, n 2 have energy 2, and so on up to n L moecues with energy L, then the set of numbers MISN n 1, n 2,..., n L is the partition. If one knows the partition, it is easy to cacuate a system s tota energy: and its average energy: tota = n n n L L = ave = L tota N = i=1 n i i. N L n i i, Notice that since N is the tota number of moecues, we aso know: N = n 1 + n n L = L n i. Many different distributions are possibe, we said. However, a manypartice system in therma equiibrium can be shown by both theory and experiment to have ony one distribution, determined by the system s temperature. 2b. Therma quiibrium. At therma equiibrium a system is in its most probabe distribution and thus may aso be said to be in statistica equiibrium. The more partices there are in a system, the more probabe the most probabe distribution is; the very arge number of moecues in even a sma quantity of gas makes the most probabe distribution very stabe and the fuctuations around it very sma. The energy of any one moecue changes drasticay and rapidy as it coides with other moecues in the system. Since, however, the moecues of an idea gas coide easticay, one s oss is iteray another s gain, and since there are so very many of them, for every moecue that attains energy i another oses energy i. At therma equiibrium, the tota number of moecues with any given energy is constant, even though the individua partices trade energies rapidy among themseves. i=1 i=1 1 A concept intimatey reated to the probabiity distribution of the partices of a gas amongst the possibe energy states of the system is entropy. See ntropy (MISN--16). 5 6

4 MISN c. The Botzmann Distribution. The most probabe distribution for an idea gas is a we known function caed the Botzmann distribution (or the Maxwe-Botzmann distribution). It can be derived by statistica mechanics and is given by: = 2πN (πkt ) 3/2 1/2 e /kt. The genera shape of this function is shown in Fig. 1. Notice now we have not written the distribution in the form of a set of numbers n corresponding to specific energies but rather in the form of a continuous function that can te us the number of moecues having energies within a range of energies. The reasons for this are discussed in xercise B. The mode for an idea gas is a system of sma round partices whose ony energy is their kinetic energy. That means that their distribution function coud be in terms of speed instead of energy simpy by substituting = mv 2 /2 and = mv dv. Thus the speed distribution of an idea gas is: ( m ) 3/2 dv = 4πN v 2 e mv2 /2kT. 2πkT The functions / and /dv potted as functions of and v, respectivey, obviousy do not have the same shape. 2 Nevertheess, they do have severa characteristics in common: they are equa to zero at = or v = ; both go to zero exponentiay at high vaues of or v; and both have a maximum at some most probabe vaue of or v, the position and shape of the peak depending on the temperature T. At ow temperatures both / and /dv have high, narrow peaks near = or v =, corresponding to a system with most of its partices moving reativey sowy. At high temperatures, on the other hand, both functions have ow, broad peaks at high vaues of or v, corresponding to a system of partices moving both rapidy and with a reativey wide range of speeds. 3. Readings For access to the foowing readings, see this modue s Loca Guide. Read pages of assica and Modern Physics, Vo. 2, by K. W. Ford. 3 This reading seection deveops the connection between partition probabiity and equiibrium. 2 Notice in particuar that near =, / starts out ike y = x whereas near v =, /dv starts out ike y = x 2. 3 For avaiabiity, see this modue s Loca Guide. MISN Figure 1. The Maxwe-Botzmann distribution. Read Section 13.4 (p. 257) of Physics by M. Aonso and.j. Finn xercises A. onsider the foowing partition shown in the tabe beow (n i is the number of partices with energy i ): with no other energies represented. i n i i (joues) a. What are the average kinetic energy and the tota interna energy for the partices of this system? b. How many moes of substance are there in this gas? c. oud this system, represented by this above partition, be in therma equiibrium? xpain. d. an you assign a temperature to this gas with this partition? xpain. e. When therma equiibrium gets estabished, if this system is isoated so that none of the energy gets away, what can we say about the temperature? What about the energy partition? 4 For avaiabiity, see this modue s Loca Guide. 7 8

5 MISN B. The partition described in xercise A was a discrete partition. That is, 9.8% of the partices had energy equa to exacty joues each, 19.6% of them had joues each, 35.3% had joues, 23.5% had joues, and 11.8% had joues. No other energies were represented. No partice, for exampe, had an energy anywhere in the range between joues and joues, athough about 1 18 of them had exacty either of these two energies. In any physica situation, as a resut of the many coisions that occur in such a many-partice system the energy distribution wi quicky spread to fi in the bank spaces between the discrete n i s so that there is a continuous distribution: you wi find partices with any energy (within the imitations set by the tota energy of the system. Later you see that Quantum Mechanics aso imposes imitations. So how do you describe the energy partition in this case? You cannot write down the partition in tabuar form, as was done in xercise A, uness your tabe incudes every possibe energy in some continuous range. Then you run into what may seem ike a contradiction, because there are an infinite number of possibe energies and ony a finite number of partices the most ikey partition from the discrete tabuation point of view is zero partices at any one specified energy. That is why it makes more sense, when a continuous range of energies is possibe, to describe the partition with a density function (the number of partices per unit energy in the vicinity of specified energy vaue). The Maxwe- Botzmann distribution function is such a density function. (It is the correct density function which gives the energy partition of an idea gas after therma equiibrium has been estabished.) A prototype of a density function / is potted vs, the energy, in Fig. 2. Note that it can aso be expressed and potted as a function of v, the veocity. What does a point on such a curve te you? Or if / was given to you as a function of and you evauated it at a specific vaue of the energy, what woud this number te you? It woud not te you the number of partices that have energy. For one thing, it has the dimensions of number per unit energy, not number. Aso, we concuded above, the number of partices that have precisey the energy is zero for a vaues of. From cacuus, (/) is, the infinitesima MISN ( ) () Figure 2. A prototypica number density (number per unit energy), potted as a function of energy. increment in the variabe n. For finite (but sma) increments,, then: = n. This has the interpretation of being the finite increment in n in the neighborhood where / is evauated. In our case, n is the number of partices in an energy increment around the point on the curve where / is evauated. onsider the curve depicted in Fig. 2. Suppose at energy = joues the vaue of the function / is /,J meaning partices per joue. How many partices are there in an energy range J around this vaue of? How many are there in a arger interva, = joues? or the interva = joues? The answers you got shoud have been 4, 4, and 4 partices, respectivey. The first of these says that 4 partices in the system can be expected to have their energy between joues and joues, and in the ast interva, there are 4 partices with individua energies somewhere between joues and ( ) () Figure 3. As in Fig. 2, reating number density to numbers. } 9 1

6 } MISN MISN n is the shaded region under the curve between and Figure 5. Reating rate-area to number. a. How many partices can be expected to have their energy between a and b? b. If the tota number of partices in the system is N, what is the vaue of? c. What is the tota number of partices with their energy between b and infinity? (xpress your answer in terms of N.) d. Is this system in therma equiibrium? xpain. joues. Let s examine Fig. 3 more cosey. The area of the shaded rectange is just / times : / is the height, the width. xcept for the deviations at the top of the rectange, this is amost the area under the curve in the interva. (see Fig. 4) The smaer is, the coser this is to the area under the curve. In the imit where goes to the infinitesima, we can see that the number of partices in the interva between 1 and 2 is n. (see Fig. 5) 2 1 = n. Because of the reasons discussed above, we need to think from the point of view of continuous distributions rather than discrete ones. With this engthy proogue, examine the distribution iustrated in Fig. 6, where the number of partices per unit energy interva is zero for a energies except between 1 and 2, where it is a constant. onsider the shaded rectanguar energy region of width = b a.. a. Refer to the figure in Probem 4 of the Probem Suppement (/ for a gas satisfying the Maxwe-Botzmann distribution aw). What is the tota number of moecues in the gas? Give a quaitative answer here. b. Again referring to the same figure, what is the expression (in integra form) that gives you the number of partices with energy greater than 2? ompare the resut of this integration at the two temperatures 3 K and 5 K. D. The Botzmann distribution is given by: = 1/2 e β, β = 1 kt, where and β are independent of energy. At very ow, the 1/2 makes this function go to zero, whie at high the exponentia forces it to zero, so there is a maximum somewhere between. 1 a b 2 =b-a Figure 6. Number distribution for Probem B. a. For what vaue of does this maximum occur? The average vaue of some measurabe quantity A is given by: A ave = A 1 + A A N, N where each A i is one of the measured vaues and there are N of them. This can aso be written: A ave = n 1A 1 + n 2 A n p A p, N 11 12

7 MISN where n i is the number of times A i is measured. If, as in the case of the energy of the partices in a gas, there is a continuous range of possibe vaues of A, the discrete sum must go over to the integra: A ave = 1 A N da da. Note that the factor on the right, (/da) da, is the number of partices that have their A vaue between A and A + da. Thus the average energy of a system of partices is given by: ave = 1 N. ave =, the denominator being just N, the tota number. b. For the distribution given in xercise B, what is the average energy? c. For the Botzmann distribution, = 1/2 e β, β = 1 kt, find the vaue of by using the condition that the tota number of partices is N. You need to refer to a tabe of integras. If you don t have one, use: where: x n 1 e x dx = Γ(n), Γ(n + 1) = nγ(n) and Γ(1/2) = π. d. vauate the average energy for a system of partices that have their energies distributed cording to the Botzmann distribution. A. 5. Answers to xercises a. U = joues, ave = joues. MISN b moes c. No. When the therma equiibrium is estabished, the distribution of energy among the moecues is given by the Botzmann distribution, which is the most probabe partition of the many possibe partitions. d. No. Temperature is defined for systems in therma equiibrium. However, in more advanced treatments of thermodynamics, systems not in therma equiibrium may have temperatures defined for them. In fact, our above system by that definition then has a negative temperature! e. This system, with ave = joues, woud have temperature T = 157 K if therma equiibrium were estabished whie ave remained unchanged. The energy partition woud be given by the Botzmann distribution whose T = 157 K and which has N = tota number of partices. B. a.. b. = N 2 1. c. n( > b ) = N 2 b 2 1. d. No, for the same reason as in part (c) of xercise A.. a. The tota area under the curve. b. 2. At 5 K there are more partices with > 2. D. a. The most probabe = KT/2, the point on the curve where the maximum occurs. b. ( )/2. c. N = = 1/2 e β, 13 14

8 MISN MISN Hence: N = β 3/2 (β)1/2 e β d(β). Let β x so: N = β 3/2 x 1/2 e x dx. From the integra tabes this is: Γ(3/2). But Γ(3/2) = (1/2)Γ(1/2) and Γ(1/2) = π. MISN--? cover Graphics Backup So, N = 2β 3/2 π. Hence, = 2πN ( ) 3/2 β = 2πN π (πkt ). 3/2 d. Note that the ast ine beow is as stated in MISN--157: ave = 1 N = N = = Nβ 5/2 = N 3/2 e β = 3 Nβ 5/2 2 x 3/2 e x d(x) = 1/2 e β Nβ 5/2 Γ(5/2) Nβ5/2 1 2πN 1 Γ(1/2) = 2 (πkt ) 3/2 N (kt 3 )5/2 2 (β) 3/2 e β d(β) 1 π 2 Fig 1 1 = 3 2 k T. Acknowedgments Fig 2 Preparation of this modue was supported in part by the Nationa Science Foundation, Division of Science ducation Deveopment and Research, through Grant #SD to Michigan State University. ( ) () 2 Fig 3 15 ( ) } 16

9 MISN--159 LG-1 MISN--159 PS-1 LOAL GUID The readings for this unit are on reserve for you in the Physics-Astronomy Library, Room 23 in the Physics-Astronomy Buiding. Ask for them as The readings for BI Unit 159. Do not ask for them by book tite. PROBLM SUPPLMNT 1. In a diute gas containing very many moecues, such that there is a continuous distribution of energies among the moecues, the Maxwe- Botzmann distribution aw is satisfied: = 2πN (πkt ) 3/2 1/2 e /kt. Identify the terms in this reation, expain what it tes you and pot a rough sketch of this function, abeing axes. [K] What is the partition of this system? [I] How is (/) reated to N? [A] 2. onsider the foowing fictitious system. There are 1 2 partices in the system. Some of these partices have an energy of.6 ev, of them have an energy of.3 ev, whie of them have an energy of.4 ev. a. What is the partition of this system? [] b. What is the tota interna energy of this system in ev? [B] In joues? [L] c. What is the average energy of the partices of this system? [G] 3. How does the partition of the above system differ from the partition of a rea diute gas of 1 2 partices? [] Above is shown the pot of the distribution function (number of moecues per unit energy interva potted versus the energy) for a gas at a temperature T. a. What is the tota number of partices which at any instant have an energy between 1 and 2 (xpress your answer graphicay)? [J] 17 18

10 MISN--159 PS-2 b. Again in terms of the graph, what is the tota number of partices in this gas? [F] 5. Starting from the Maxwe-Botzmann distribution aw, cacuate the average energy of a moecue of an idea gas. [H] 6. Starting from the Maxwe-Botzmann distribution aw, cacuate the RMS vaue of the energy of a moecue of an idea gas. [Find first the average vaue of 2.) [D] Brief Answers: A. N =. B. U = ev.. This is ony one partition, one with extremey ow probabiity. The most probabe partition is the Botzmann distribution for this tota number of partices, with this ave. D. RMS = kt.. The partition is the set of 3 numbers n 1, n 2, n 3. F. The tota area under the curve. G. ave = ev= joues. H. See xercise D, part (d). ave = (3/2) kt. I. (/) is the number of partices with energy between and +. The set of a these numbers for a is the partition. The partition is essentia to the distribution function. J. The area under the curve between 1 and 2. K. See the discussion in this Unit. L..69 joues. 19 2