CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion of the idel gs lw from kinetic theory The idel gs lw ws discovered bsed on observtions of the mcroscopic properties of gses (p, V m, nd T). However, there clerly must be connection between the microscopic (moleculr) behvior of gses nd the stte vribles tht describe the bulk properties of gses. This connection is developed in kinetic theory. The ssumptions of kinetic theory re s follows: 1) Gses re composed of molecules in rndom motion. ) The volume occupied by the molecules is smll compred to the size of the continer holding them. 3) Molecules hve only wek forces of interction. 4) Collisions of molecules with the wlls of the continer re elstic (conserve the totl kinetic energy of the molecule) There re severl wys we cn proceed t this point, depending on the degree of rigor we desire. A simple nd resonbly rigorous derivtion is s follows. Consider single molecule in cubiclly shped continer whose sides re ll length, s indicted below. The pressure exerted by this molecule ginst one prticulr wll of the continer is p = F x/a (per molecule) (1.1) where we hve chosen the wll perpendiculr to the x-xis. The re of wll is A =. As for the force exerted by the molecule F x = m = m (dv x/dt) = d(mv x)/dt = dp x/dt (1.) where m is the mss of the molecule, nd p x = mv x is the momentum of the molecule long the x-xis. Over long time period we my sy dp x/dt p x/t (1.3) Now, when molecule undergoes n elstic collision with the wll of the continer the component of the velocity vector long the x-xis chnges sign, while the components long the y nd z xes re unchnged. Tht is components before collision: v x, v y, v z components fter collision: - v x, v y, v z Therefore p x = mv x - (-mv x) = mv x (1.4)
t represents the time between successive collisions with the wll. It will be equl to the distnce the molecule must trvel long the x-xis between collisions divided by the speed of the molecule long the x-xis. Since the molecule must trvel distnce d = between collision, then it follows tht t = d/v x = /v x (1.5) Putting ll of the bove together, we get F x p x/t = (mv x)/(/v x) = mv x / (1.6) p = F x/a = (mv x /)/ = mv x / 3 = mv x /V (per molecule) (1.7) where we hve used the fct tht V = 3 is the volume of the continer. Now consider the pressure exerted by N molecules in the continer. p totl = N p(per molecule) = Nm<v x >/V (1.8) where <v x > is the verge vlue for v x for the molecules in the continer. However, since there is nothing specil bout motion long the x-xis Since it follows tht <v x > = <v y > = <v z > (1.9) <v > = <v x > + <v y > + <v z > (1.10) <v x > = <v >/3 (1.11) Substitution into eq 1.8 gives p = Nm<v >/3V (1.1) If we multiply nd divide eq 1.1 by N A (Avogdro s number) nd note tht M = N Am nd n = N/N A, we get our finl result p = nm<v >/3V (1.13) If we compre the bove result to tht found from the idel gs lw p = nrt/v (1.14) then we my show tht <v > 1/ = v rms = (3RT/M) 1/ (1.15) is the rms verge speed of molecule in gs. This completes the connection between the microscopic nd mcroscopic behvior of the gs.. Mxwell-Boltzmnn distribution of moleculr speeds Molecules in gs move with rndom velocities (tht is, rndom speeds nd directions of motion). Using the Boltzmnn distribution lw, n expression for the number of molecules in gs or the probbility of gs molecule hving prticulr speed cn be found. There re two useful equtions:
The frction of molecules hving prticulr speed v x long the x xis is f(v x) dv x = (M/RT) 1/ exp( - Mv x /RT) dv x ; - < v x < + (1-dimensionl distribution) (.1) The frction of molecules hving totl speed v (where v = v x +v y + v z ) is f(v) dv = 4 (M/RT) 3/ v exp( - Mv /RT) dv ; 0 v < + (3-dimensionl distribution) (.) Notice tht v x rnges from - to +, since speeds in prticulr direction cn be either positive or negtive. However v, the totl speed (which is the mgnitude of the velocity vector) must be non-negtive. The probbility of the speed being between nd b (or, lterntively, the frction of molecules with speeds between nd b) is given by the expression f( < v < b) = b f(v) dv (.3) The bove integrl does not in generl hve closed form solution, but my be evluted numericlly. Note tht f(0 < v < ) = 0 f(v) dv = 1 (.4) since the speed of molecule must be some vlue between 0 nd. The bove expression for f(v) dv my be used to find three different types of verge speeds men (verge) speed of molecule, v men = <v> = 0 v f(v) dv = (8RT/M) 1/ (.5) rms verge speed of molecule, v rms = <v > 1/ = ( 0 v f(v) dv ) 1/ = (3RT/M) 1/ (.6) most probble speed of molecule, v mp, found by setting df(v)/dv = 0. v mp = (RT/M) 1/ (.7) In finding v men nd v rms, the following definite integrls re used 0 x n exp(-x ) dx = () 1/ (1. 3. 5..... n-1)/() n+1 ; n = 0, 1,,... (.8) 0 x n+1 exp(-x ) dx = (n!/)(1/ n+1 ) ; n = 0, 1,,... (.9) Finlly, to find the frction of molecules with speeds between nd b, we my pproximte the integrl (which hs no closed form solution) using rectngle f( < x < b) = b f(v) dv f(v ve) v (.10) where v ve = ( + b)/ nd v = (b - ). More ccurte pproximtions my be found by dividing the region being integrted into more rectngles of smller width. 3. Men free pth () nd collision frequency (z) The men free pth () of molecule in the gs phse is defined s the verge distnce the molecule trvels between collisions. The collsion frequence (z) of molecule in the gs phse is defined s the verge number of collisions molecule mkes per unit time. For pure gs the expressions for nd z re s follows: = RT (3.1) 1/ N Ap z = 4N Ap (3.) (MRT) 1/
where nd M re the collision cross-section nd moleculr mss of the molecule, N A is Avogdro's number, nd p nd T re the pressure nd temperture of the gs. Note tht z = v men = (8RT/M) 1/ (3.3) where v men is the verge speed of gs molecule. Note tht eq 3.1 nd 3.3 differ from eq 1B.1 nd 1B.13 in Atkins by fctor of 1/. The equtions in Atkins re incorrect, nd eq 3.1 nd 3.3 re correct. In clcultions using eq 3.1 nd 3. it is esiest to get the correct units in your finl nswer by using MKS units for ll terms in the equtions. This mens giving in units of m, p in units of P (1 br = 10 5 P = 10 5 N/m ), nd M in units of kg/mol, nd using R = 8.3145 J/mol K. If this is done, then will hve units of m, nd z will hve units of s -1. 4. Even nd odd functions A function is n even function if f(-x) = f(x) for ll x, nd n odd function if f(-x) = - f(x) for ll x. Exmples of even functions include x, cos(x), nd exp(-x ). Exmples of odd functions include x nd sin(x). Some functions re neither even nor odd, s, for exmple, x + x nd exp(x). A useful property for integrls of even nd odd functions is the following: - f(x) dx = 0 f(x) dx if f(x) is n even function (4.1) - f(x) dx = 0 if f(x) is n odd function (4.) 5. Prtil derivtives nd prtil derivtive reltionships For f(x), function of one vrible, the rte of chnge of the function cn be written s df = (df/dx) dx (5.1) where (df/dx) is the derivtive of the function, found using stndrd techniques from clculus. By nlogy with the bove the rte of chnge of function of two vribles, f(x,y), cn be written s df = (f/x) y dx + (f/y) x dy (5.) where (f/x) y nd (f/y) x re the prtil derivtives of the function f with respect to x nd with respect to y. Note tht we use f/x to indicte prtil derivtive, nd list the independent vribles being held constnt outside of the prentheses enclosing the prtil derivtive. Eqution 5. cn be generlized in strightforwrd mnner to functions of more thn two vribles. The prtil derivtive of function is found by treting ll vribles except the one for which the prtil derivtive is being tken s if they re constnts. Thus, finding the prtil derivtive of function of severl vribles is no more difficult thn finding the norml derivtive of function of one vrible. ----------------------------------------------------------------------------------------------------------------------------------------- EXAMPLE: If f(x,y) = 4x 3 + xy + 7y + 1, then: (f/x) y = 1x + y (f/y) x = 4xy + 7 -----------------------------------------------------------------------------------------------------------------------------------------
The following four reltionships (which cn be derived) re often useful in mnipulting prtil derivtives (note tht x, y, z, nd w re vribles) #1 (y/x) z = 1/(x/y) z (5.3) # (x/y) z (y/z) x (z/x) y = -1 (5.4) #3 (y/x) z = - (z/x) y/(z/y) x (5.5) #4 (z/x) w = (z/x) y + (z/y) x (y/x) w (5.6) The reson we re discussing prtil derivtives is tht thermodynmic functions re generlly functions of severl vribles. A complete discussion of thermodynmics requires working knowledge of the bsics of prtil derivtives. 6. Tylor series expnsion of function Subject to the usul mthemticl restrictions, function f(x) cn be expressed by expnsion bout point. The technicl term for this process is Tylor series expnsion. The expnsion is given by the following reltionship f(x) = f() + (x ) (df/dx) + (x ) (d f/dx ) + (x ) 3 (d 3 f/dx 3 ) + (6.1)! 3! = n=0 (x ) n (d n f/dx n ) (6.) n! A few useful Tylor series expnsions re given below e x = 1 + x + (x /!) + (x 3 /3!) +... ; ll vlues of x (6.3) ln(1 + x) = x - (x /) + (x 3 /3) -... ; -1 < x < 1 (6.4) sin(x) = x - (x 3 /3!) + (x 5 /5!) -... ; ll vlues of x (6.5) cos(x) = 1 - (x /!) + (x 4 /4!) -... ; ll vlues of x (6.6) 1/(1 + x) = 1 - x + x - x 3 +... ; -1 < x < 1 (6.7)