ATMO 551a Fall 08. Diffusion

Transcription

1 Diffusion Diffusion is a net tanspot of olecules o enegy o oentu o fo a egion of highe concentation to one of lowe concentation by ando olecula) otion. We will look at diffusion in gases. Mean fee path Molecules in a gas ove with a kinetic o theal velocity elated to thei tepeatue. Thinking in tes of the billiad ball analogy, olecules ove soe distance befoe colliding with anothe olecule and then the two olecules icochet off in othe diections. The typical distance the olecules ove between collisions is called the ean fee path λ). λ depends on the nube density of the olecules and thei collisional cosssection,. In defining λ, one can think of it in tes of the volue pe olecule, V, which is the invese of the nube density of olecules, N, in units of olecules pe cubic in ks units. V = 1 N 1) This is the sae N we defined in the equation of state fo an ideal gas in icoscopic units ealy in the class P = N k B T ) One can think of this volue pe olecule as a cylinde whose aea is the collisional cossectional aea and the length of the cylinde is the ean fee path. Theefoe this volue pe olecule can also be defined as V = λ 3) So we can cobine 1) and 3) to get an equation fo the ean fee path " = 1 N 4) Bownian otion o ando walk The next step in undestanding diffusion is to undestand the net effect of the ando otion of the olecules that esults fo these collisions. The question we have to answe is What is the typical distance a olecule will ove away fo its initial position afte soe nube of collisions with othe olecules? This is witten as the standad deviation of its position afte soe nube of collisions. Afte one collision it will have oved on aveage λ in soe diection. Afte two collisions, it will have oved the su of two vectos each of ean) length, λ, but the diection of each of the two vectos is ando. So the su of the two vectos ust be descibed in soe pobabilistic way. We add two vectos and ask what is the length of the su of the two vectos. Vecto 1, x 1, is the vecto descibing the position change in the position of the olecule between the 0 th collision and the 1 st collision. It is given as x 1 = "x 1 ˆ x ˆ y ˆ z 5) 1 Kusinski 10/31/08

2 To undestand the net esult of a ando o dunkad s walk which is the su of a seies of ando steps, we begin by consideing the squae of the su of the fist two ando steps witten in Catesian coodinates and then genealizing to an abitay nube of steps X n= X n= X n= " x 1 + x = #x 1 + #x [ ) + #y 1 + #y ) + #z 1 + #z ) ] 1/ X n= = x 1 + x = "x 1 + "x ) + "y 1 + "y ) + "z 1 + "z ) 6) ) + "y 1 "y + "y ) + "z 1 "z + "z ) ) + "x 1 "x "y "z ) + "x + "y + "z ) = "x 1 + "x 1 "x + "x = "x 1 X n= + x ˆ + "x 1 "x "y "z ) 7) Now we need to conside the expected value o ean value of X n= X n= X n= = < X n= >. + x ˆ + "x 1 "x "y "z ) + x ˆ + "x 1 "x "y "z ) 8) The expected value of the length the olecule oves between collisions is the ean fee path, λ. So X n= X n= ) ) 9) = " + " + #x 1 #x + #y 1 #y + #z 1 #z = " + #x 1 #x + #y 1 #y + #z 1 #z Now the question is what to do with the expected value of the coss tes. If the collisions ae eally ando as we think and assue they ae then thee is no pefeed diection and theefoe no coelation between Δx 1 and Δx o Δy 1 and Δy which eans Theefoe <Δx 1 Δx > = 0, <Δy 1 Δy > = 0, <Δz 1 Δz > = 0 10) X n= Genealizing this elation to X afte n collisions gives = " 11) X n = n" 1) The squae oot of this equation gives us the oot of the ean squae distance o s distance a olecule will ove away fo its initial location afte n collisions. X n 1/ = n " 13) Kusinski 10/31/08

3 Tie and velocity Now we ust intoduce tie into the diffusion pocess. In ode to calculate diffusive fluxes and ates of diffusion, we need to know how long it will a olecule take to ove a distance X away fo its initial location. Fo 13) we see that the nube of collisions equied fo the olecule to ove a distance X is given by # n = X & % $ " The typical tie fo a olecule to ove one ean fee path, which is the tie between collisions, is 14) " # = # 15) whee is the theal velocity of the olecule. Theefoe the typical tie fo the olecule to ove a distance, X, is " # = n" # = n # $ = X & ) % # # = X # The aveage velocity of the olecule in oving a distance X is v X = X n" = X# # = v X t X So the fathe the olecule oves fo its initial position, the slowe it oves on aveage. This akes diffusion a vey slow pocess ove lage distances but it is quite fast ove vey shot distances). Diffusive Flux Suppose we have a hoizontal gadient in the density of soe quantity, B, which is ρ B. What is the flux of high ρ B into the low ρ B aea and visa vesa via diffusion? Conside two points sepaated by a distance, Δx. The diffeence in ρ B between the two points is Δρ B = dρ B /dx Δx. The geneal definition of a flux is the density ties the velocity, ρ B v check units). 17) povides the diffusive velocity. The net flux is the diffusive flux in the +x diection inus the diffusive flux in the x diection 16) 17) = " B x # /) v x # /) # " B x + /) v x + /) 18) % = " B # d" B % = " d# B * + ) " % d# B * + ) # % " + d" B B * + ) = " d# B dx * + ) + = " # d$ B dx 19) 3 Kusinski 10/31/08

4 Diffusivity and Equations of Diffusion What is the definition of diffusivity? The diffusion equations wee deived by Adolf Fick in Fick s Fist Law is that the diffusive flux, F, of soe substance, B, is given by F B = "D d# B x dx ˆ 0) o oe geneally F B = "D # $ B 1) whee D is the diffusivity in units of /s. F has units of B-units/ /s. This equation fo is known as downgadient diffusion because the flux is in the opposite diection of the gadient. Copaing 19) and 0), it is clea that D = λ ) Fo copleteness, Fick s Second Law is known as the diffusivity equation which descibes the tie ate of change of the density of soe quantity, B, due to diffusion [ )] 3) = $ D $# B,t If the diffusivity, D, does not vay with position then = D $ # B,t ) 4) Note that 3) and 4) ae siply vesions of the flux divegence equation = $% F B + B 5) whee B epesents a souce te because plugging 0) into 5) with no souce te yields which is the sae as 3). = $% $D%# B ) = % D%# B ) Putting it all togethe D can be thought of as a distance ties a velocity. Given the deivations we just did about the ando otions of olecules in a gas, we can wite D = Xv x = X" X = " = N The theal velocity has at least definitions. The s velocity is "s = 3k B T 6) 7) 4 Kusinski 10/31/08

5 The aveage agnitude of the velocity is Fo the ideal gas law, Cobining these D = N = "ag = N = P k B T a k B T 8k BT # k B T P = a k B T) 3 / P whee a is the constant fo the elevant theal velocity, eithe 3 o 8/π. a 1/ is eithe 1.73 o The biggest uncetainty in using this equation is pobably the collisional coss-section. 8) 9) 30) The Diffusivity of Ai The diffusivity of ai at 300 K and 1000 b, is.16e-5 /s. We can calculate the diffusivity of ai fo 30). The coss-sectional aea of a N olecule is ~.08x10-10 x 1.6x10-10 ~ 4 x10-0. The collisional coss-section of two N olecules colliding should be about Using this value yields a diffusivity of ai of e-4 /s. This is a facto of 10 too high indicating that the coss-sectional aea at 300 K is actually about and the siple coss-sectional aea I used is too low by about a facto of Kusinski 10/31/08