CHEM-UA 65: Thermodynmics nd Kinetics Notes for Lecture I. WHAT IS PHYSICAL CHEMISTRY? Physicl chemistry is truly interdisciplinry brnch of chemistry tht exists t the boundry between chemistry nd physics. In prticulr, the im of physicl chemistry is to explin chemicl processes nd the properties of chemicl systems using the principles of physics. Becuse physicl chemistry ttempts to put chemistry on firm physicl foundtion, it is ble to free chemistry from being lrgely empiricl science. Once this is possible, new, testble predictions for new phenomen cn be mde tht cn led to design nd engineering of new mterils for energy or electronic pplictions or the development of new nd novel drugs, to list just few exmples. This is why physicl chemistry is one of the key core disciplines in chemistry. Physicl chemistry is somewht unique in the extent to which it integrtes theory nd experiment. Experimenttion in chemistry is essentil, but without theory, it leds only to ctlog of dt nd results of disprte mesurements with no underlying rtionle. Theory in chemistry plys the role of providing tht rtionliztion by constructing mthemticl nd physicl models t different levels of resolution nd ccurcy. In ddition to providing n underlying rtionliztion of experimentl results, theory must lso be ble to mke testble predictions for new experiments. Theory often strts from bsic physics nd physicl chemistry, nd so physicl chemistry relly embodies the tightest synergy between experiment nd theory. The two min brnches of physicl chemistry nicely exemplify this interply between theory nd experiment. Quntum theory ws invented to explin certin experiments from the lte 9th nd erly 0th centuries tht could not be explined using clssicl physics. These included blckbody rdition, electron diffrction, nd the photoelectric effect. As the theory developed nd reshped our view of the mechnicl universe, its ppliction led to importnt developments technology, biology, engineering, nd medicine, such s modern computers, enzyme engineering, drug design, mgnetic resonnce imging, nd solr energy, not to mention its influence in tomic physics, nucler physics, prticle physics, nd cosmology. Thermodynmics gives us every-dy concepts such s engines nd refrigertion. However, when plced on firm physicl, microscopic bsis using the theory of sttisticl mechnics nd/or combined with kinetics, we re led to novel developments such s piezoelectronics, fuel cell nd bttery technologies, crystl engineering, nd mechnochemistry. Becuse physicl chemistry employs physics, nd physics uses the lnguge of mthemtics (it is often considered to be pplied mthemtics ), physicl chemistry is, by its very nture, mthemticl subject. In fct, n entire hierrchy of fields of study cn be trced bck to mthemtics, s the crtoon below illustrtes: The focus of this semester will be on thermodynmics nd kinetics, which re concerned with bulk mtter, or wht is often clled condensed mtter, including gses, liquids, solids, solutions, etc. Thermodynmics focuses on sttic, equilibrium properties of such systems, while kinetics focuses on rtes of condensed-phse processes nd mechnisms underlying these processes. II. THERMODYNAMICS: A BRIEF OVERVIEW Thermodynmics is the study of reltionships between the mcroscopic properties tht chrcterize bulk system. These include quntities such s totl energy, pressure, temperture, volume, number of moles, density, chemicl potentil, het cpcity, isotherml compressibility, work performed, nd equilibrium constnts. For properties such s these, thermodynmics mkes no reference to the underlying chemicl composition of system, nd in this respect, it is n elegntly self-consistent theory, lbeit purely phenomenologicl one. At the sme time, we know tht there is n underlying microscopic theory of mtter tht dels with individul prticle motions, microscopic energy levels, wve functions, nd so forth. It must, therefore, be possible to upscle this underlying microscopic theory to predict the mcroscopic properties of mtter treted in thermodynmics. The bridging theory tht ccomplishes
FIG. : A rnking of fields by purity. this is clled sttisticl mechnics, nd it is the theory of sttisticl mechnics tht plces thermodynmics on firm footing. This semester s mteril will include n introduction to sttisticl mechnics, nd we will see exctly how this upscling is chieved. Anyone interested in lerning more bout sttisticl mechnics nd its implementtion on modern computers should refer to the book Sttisticl Mechnics: Theory nd Moleculr Simultion by M. E. Tuckermn (not tht I m trying to sell you nything)! Thermodynmics cn be extended to incorporte trnsport properties such s the diffusion constnt, therml conductivity, nd electricl conductivity. While thermodynmics cn generlly tell us if process will occur spontneously or not, it cnnot tell us how long we might hve to wit for the process to occur. For this, we need kinetics, which will be treted tken up towrd the end of the semester. The following is preliminry list of concepts tht we will need to strt off our study of thermodynmics: i. A thermodynmic system is mcroscopic system. Thermodynmics lwys divides the universe into the system nd its surroundings. A thermodynmic system is sid to be isolted if no het or mteril is exchnged between the system nd its surroundings nd if the surroundings produce no other chnge in the thermodynmic stte of the system. ii. A system is in thermodynmic equilibrium if its thermodynmic stte does not chnge in time. iii. The fundmentl thermodynmic prmeters tht define thermodynmic stte, such s the pressure P, volume V, the temperture T, nd the totl mss M or number of moles n re mesurble quntities ssumed to be provided experimentlly. A thermodynmic stte is specified by providing vlues of ll thermodynmic prmeters necessry for complete description of system. iv. The eqution of stte of system is reltionship mong the thermodynmic prmeters prescribing how these prmeters vry from one equilibrium stte to nother. Thus, if P, V, T, nd n re the fundmentl thermodynmic prmeters of system, the eqution of stte tkes the generl form g(n,p,v,t) 0. () As consequence of eqn. (), there re in fct only three independent thermodynmic prmeters in n equilibrium stte. When the number of moles remins fixed, the number of independent prmeters is reduced to two. An exmple of n eqution of stte is tht of n idel gs, which is defined (thermodynmiclly) s system whose eqution of stte is PV nrt 0, () where R 8.35 J mol K is the gs constnt. The idel gs represents the limiting behvior of ll rel gses t sufficiently low density ρ n/v. Note tht if we divide the idel gs lw by the volume, we obtin the eqution of stte in the form P ρrt (3)
More discussion of idel gses will follow briefly. However, extrpolting this ide, we see tht generl equtions of stte for thermodynmic systems cn lso be expressed s g(p,ρ,t) 0 (4) v. A thermodynmic trnsformtion is chnge in the thermodynmic stte of system. In equilibrium, thermodynmic trnsformtion is effected by chnge in the externl conditions of the system. Thermodynmic trnsformtions cn be crried out either reversibly or irreversibly. In reversible trnsformtion, the chnge is crried out slowly enough tht the system hs time to djust to ech new externl condition imposed long prescribed thermodynmic pth, so tht the system cn retrce its history long the sme pth between the endpoints of the trnsformtion. If this is not possible, then the trnsformtion is irreversible. vi. A stte function is ny function f(n,p,v,t) whose chnge under ny thermodynmic trnsformtion depends only on the initil nd finl sttes of the trnsformtion nd not on the prticulr thermodynmic pth tken between these sttes (see illustrtion below). 3 P V T FIG. : The thermodynmic stte spce defined by the vribles P, V, nd T with two pths (solid nd dshed lines) between the stte points nd. The chnge in stte function f(n,p,v,t) is independent of the pth tken between ny two such stte points. An exmple of stte function is the totl internl energy E of the system, which cn be expressed s function of the thermodynmic stte, i.e., E(n,P,V,T). Contributions to this energy include the kinetic energy of motion of the toms in the system nd their potentil energy of interction. Since E is the internl energy of the system, it does not include energy from overll motion of the system of energy of ny externl fields. III. PROPERTIES OF GASES: THE IDEAL GAS Thermodynmiclly, the idel gs is defined s thermodynmic system whose eqution of stte is PV nrt 0 or P ρrt 0 nd whose totl thermodynmic energy E is E 3 nrt (5) However, this does not give us physicl picture of wht such gs is. At microscopic level, n idel gs is defined to be system of N prticles (which could be toms or molecules) in which no interctions exist between the prticles,
i.e., there re no interprticle forces. Obviously, such system cnnot relly exist in nture, however, t sufficiently low density, interprticle interctions do become negligible, nd s this low-density limit is pproched, ll gses pproch idel-gs behvior. For this reson, the ideliztion of no interprticle interctions is interesting to consider, nd the simplifiction fforded by removing these interctions mkes the problem of predicting the properties of n idel gs trctble. In fct, we will show how to strt with set of microscopic quntum mechnicl energy levels for system of N non-intercting prticles nd derive both the eqution of stte nd the energy reltion in Eq. (5). For now, we must simply tke them s thermodynmic definitions of n idel gs. We will describe the idel gs using the lws of clssicl mechnics. In clssicl mechnics, the microscopic stte of system of N prticles in contining volume V is fully specified by giving the vector positions r,...,r N of ech prticle t ny point in time s well s the velocity vector v,...,v N of ech prticle. The position vectors re restricted to lie within the sptil domin defined by the contining volume. Moreover, since the N prticles re in constnt stte of motion, the positions nd velocities re functions of time r (t),...,r N (t), v (t),...,v N (t), nd if we know how the positions depend on time, the velocities cn be esily computed by differentition of the positions with respect to time: v i (t) dr i(t) (6) dt Finlly, we need to specify wht hppens to the prticles when they rech the boundries of the continer. In n idel gs, it is ssumed tht the prticles undergo perfectly elstic collisions (no chnge in the kinetic energy) with the wlls. This mens tht in n idel gs, the prticle speeds, defined to be the vector mgnitude of the velocities, v i (t) v i (t), re constnt in time, i.e., v i (t) v i ( constnt). The velocities still chnge with time s result of collisions with the wlls of the continer, however, the chnge is only in the direction of the velocity, not its mgnitude. Since there re no interprticle interctions in n idel gs, the totl energy E of the gs is purely kinetic (no potentil energy), nd is, therefore, constnt in time. Let us ssume tht the gs prticles re ll of the sme time nd hve mss m. Then, the totl mechnicl energy of the gs is simply E N i mv i (7) If we now equte the bove microscopic energy expression with the thermodynmic expression in Eq. (5), we obtin simple connection between the mcroscopic temperture T nd the microscopic kinetic energies of the prticles: 3 N nrt mv i i T 3nR N i mv i (8) It is not prticulrly convenient to hve n expression tht involves both the number of moles n nd the number of prticles N, so let s try to clen things up bit. The product nr cn be written s (nn 0 )(R/N 0 ), where N 0 is Avogdro s number. The product nn 0 is just the number of prticles, while the constnt R/N 0 is known s Boltzmnn s constnt, denoted k B, where k B.3806 0 3 J/K. Hende, we cn write T 3k B ( N N i mv i which shows tht the temperture if relted to the verge kinetic energy of the system. If we hve system contining severl moles of idel-gs prticles, there is simply no wy we cn know the exct velocity of ech prticle in the system. After ll, this is something like 0 4 or trillion trillion prticles, which, in computer terms, would require trillions of terbytes(lso known s yottbytes)! Thus, we need to dopt probbilitistic description of the system. ) 4 (9) The energy of ny prticle in the system ε is simply given by its kinetic energy: ε mv m( vx ) +v y +v z (0)
where v is the mgnitude of its velocity, nd v x, v y, nd v z re the components of the velocity vector v. Since we do not know ny of these velocity components, we ppel to the fct tht we hve very lrge number of prticles nd simply ssume tht the probbility of ny component, sy v x, hving vlue in smll intervl dv x centered on vlue v x is given by Gussin probbility distribution P(v x )dv x π e v x dvx () 5 where is constnt. Note tht P(v x )dv x () s is required for probbility. Tht is, we ssume tht v x is Gussin rndom vrible, nd the sme for v y, nd v z. The overll probbility distribution of the three components v x, v y, nd v z hving vlues in the intervls dv x, dv y, nd dv z, respectively, is just product of the three Gussins P(v)dv x dv y dv z P(v x )P(v y )P(v z )dv x dv y dv z ( )( )( ) π e v x π e v y π e v z dv x dv y dv z ( π ) 3/e (v x +v y +v z) (3) The function P(v) P(v x )P(v y )P(v z ) is n exmple of probbility distribution function. We cn ctully motivte this bit better nd derive the constnt from the Boltzmnn distribution. According to theboltzmnndistribution, theprobbilitythtprticleinsystemttemperturet hsenergyεisp exp( βε), where β /k B T, where k B is Boltzmnn s constnt k B.3806 0 3 J/K. Substituting in ε mv /, we obtin probbility distribution of the form P(v) e βmv / e βmv x / e βmv y / e βmv z / (4) From this, we see tht the constnt βm/, which gives the probbility distribution s P(v) ( ) 3/ mβ e βmv / e βmv x / e βmv y / e βmv z / (5) This distribution is known s the Mxwell-Boltzmnn velocity distribution. If our ssumption tht velocities re just Gussin rndom vribles is correct, then we should be ble to derive, from this ssumption, the thermodynmic reltion between energy nd temperture of n idel gs, E 3nRT/. We interpret the thermodynmic energy E s just n verge of the mechnicl energy E: E E N m vi (6) Since the gs prticles re indistinguishble in ll wys, there is no reson to expect tht the verge of ech individul velocity would be different from the verge of ny other velocity in the system. Hence, we cn write i E N m v (7) where v is the verge of the mgnitude of ny velocity in the system, which we ssume is the result we would obtin for ny of the N prticles in the system.
6 There re two wys we cn compute this verge. We will exmine both. The first wy involves writing v v x +v y +v z v x + v y + v z (8) nd recognizing tht if the gs is sptilly isotropic, mening tht it looks the sme from ll viewing ngles, then there is no reson to expect tht verges of the squred velocity components in the different sptil directions would be different. Thus, if then v x v y v z (9) v 3 v x (0) Now, in order to compute the verge v x, we simply use the probbility distribution P(v x ): vx P(v x )dv x ( ) / mβ vx e βmv x / dv x () How do we compute n integrl like this? Well, we cn do lborious nd tedious integrtion by prts, nd tht will certinly work, but there s n esier wy. Let us first recll the identity e x dx π () Now consider tking the derivtive of both sides with respect to the prmeter : d d e x dx d π d dx e x π π x e x x e x π (3) But this is exctly the integrl tht we need. We just need to set βm/: vxe βmv x / dv x βm βm (4) nd vx mβ βm βm βm (5) Thus, v 3 βm (6) nd the verge energy is simply E Nm 3 βm 3N β 3 Nk BT (7)
7 We now multiply nd divide on the right side by Avogdro s number N 0 6.0 0 3 : E 3 N N 0 k B T (8) N 0 The rtio N/N 0 n the number of moles of gs, nd the product N 0 k B R, the gs constnt. This gives the correct thermodynmic result: E 3 nrt (9) A second wy to compute the verge uses sphericl polr coordintes. If we recognize tht v v x +v y +v z is the squred length of the vector v, then this is nlogous to the rdius r x + y + z in ordinry sphericl polr coordintes. Thus, denoting v v, let us trnsform from the integrtion vribles v x,v y,v z to v,θ,φ ccording to the stndrd reltions with v x vsinθcosφ v y vsinθsinφ v z vcosθ (30) dv x dv y dv z v sinθdvdθdφ (3) Note tht v [0, ), θ [0,π], φ [0,]. Now, if we substituting the sphericl coordinte trnsformtion into the expression for v, we obtin v ( ) 3/ mβ ( dv x dv y dv z v x +v ) y +v z e βm(v x +v y +v z )/ ( ) 3/ mβ π dφ sinθ dθ v 4 e βmv / dv 0 0 0 ( ) 3/ π) mβ ( cosθ v 4 e βmv / dv 0 0 ( ) 3/ mβ ( ( )) v 4 e βmv / dv 0 ( ) 3/ mβ 4π v 4 e βmv / dv (3) 0 In order to perform the lst integrl, we cn use the sme procedure s ws done bove, but this time we strt with x e x π π/ 3/ (33) Now tke the derivtive of both sides with respect to : d d x e x d d π/ 3/ x 4 e x 3 4 π/ 5/ 3 π 4 x 4 e x 3 4 π/ 5/ 3 π 4 (34)
8 This is lmost the integrl we hve, but ours rnges from 0 to. Fortuntely, x 4 exp( x ) is n even function bout x 0, so we cn simply chnge the integrl bove from one tht rnges from 0 to nd multiply the right side by /: 0 x 4 e x 3 8 π/ 5/ 3 π 8 (35) Now this is the integrl we need. Setting mβ/, we obtin v 4π ( ) 3/ mβ 3 m β βm 3 βm (36) which gress with the result obtined previously nd will give the sme expression for the verge energy. Note tht ( ) 3/ mβ e βm(v x +v y +v z )/ dv x dv y dv z ( ) 3/ mβ e βmv / v sinθdvdθdφ (37) If we integrte over the ngles, we end up with distribution s function of the mgnitude of v, which is the speed of the prticle: F(v) 4π ( ) 3/ mβ v e βmv / (38) is known s the Mxwell-Boltzmnn speed distribution. Let us look t the difference between the velocity distribution P(v x ) nd the speed distribution P(v). These re shown in the following two figures for different tempertures (note tht wht ppers s u x or u in these figures is the equivlent of our v x nd v). FIG. 3: The probbility distribution of the x-component of the velocity of nitrogen molecule t 300 K nd 000 K. Wht is denoted u x in the figure is wht we hve been referring to s v x, component of the velocity vector. Note tht velocity component cn be positive or negtive, ccording to Fig. 3, while the speed must lwys be positive, s one would expect for the mgnitude of vector.
FIG. 4: The probbility distribution of the speed of nitrogen molecule t 300 K nd 000 K. Wht is denoted u in the figure is wht we hve been referring to s v, the mgnitude of the velocity vector. 9