The pressure and composition dependence of mutual diffusion in the system helium-nitrogen at 300 K

.Volume 32/numberS

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ANn C0 0SMOND VE

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a t 1, 3 , 5 ~ u d 7

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atmo~heres.

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The re~

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show

- pressure being monitored by a13ourd0rt gaUge(Texas )

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" " Inarc~ntarfiete, Carsonetal..[l]describeda > : . .'arid me V a c u u m s y s t e m w e m c o u t ~ e e t e d tO the cC]~ ..ra~.tl~nrl for thenreeiSe-i~easurement 0f-diefi~ion ~0- " ..witl~ ~tainh~s steelflexible tubing, all ioints being " ~'f~c[~nts-/0r'id~a~}~asT~tures~The present~aper-i_~, made_pressute~fightwith SwageIo~t~doubte~ferrm~ fit .... i shoihdbe read in conjunction with the above article.. ::: tingS (Crawford Fitting Co,, Cleveland, Ohio). A p a i r . i -.Where"they: pr~selited"resUlts for comP0siti0n det~n-, r" "~.! of matched thermi~t0zs (.type G!.!2P, Feawal E!ec: . -:. dence: at 1 atinospliere forhelium~-afgtm and hydro- ". ,..- .tronics, Eramingham, Massachusetts) was mounted at n-nitro en mixtures w e resent resuits for the"sys~. : " i p0sitions(t76) and (sl/6) aioagthe ceil. t~e len~a~, ~, : !_t~e: h~r0mg~-nitr0gen:as!a~etion ofpres~ure and. :. '.- ::.- of tla¢ Cellw~ 45.019cm, 60tapered with I02.00 cm these experimentS:was totest tlie'appHCability to a ' -- .... T h e thermistors_were.connected through shielded " ...-. r e ~ System;" of Thome's generalisafibn (ref. [2], eh. i 6 ,~:..:. Cables .to opp 0~te anus of ai~.Wheatstone b ridge, to :. : .- p. 3i z) or me::Enskog.m~o~ of-a e ~ ~ e ~ : ~el re--?.: : ~i.:-wriieh:a Corts~tant ~oltage o f 3 ;5 volt was appIied i ' . . . ;/. 7 s ~ g 0 i , ~ e a iti.-thR~eell de .viatCO~y ~tigtit!Y fibmi .-..! i"::{ii::At-any time, th e C o m p o s ~ n ~ eiffel.nee in..the :.: : the,Thome-Enskog predicti0n.-.i f {17!:_ ~: ::i:::-:.:::. :..i,7,mix,~_a~:at po~tiOn~ 116 :arid 5!16 eOUia be m 0 m t 0 ~ .i'. -:: . -."...~"~i! ,~ ::- -": i-: =: : i--::'-' i":.~-.:i :~ : i.::: "~i::;:::::~' :::.:~indirecfly from:thedifferent rates:of dissipation o f . . . ZT-

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:. :-":.~"~." v.:L_-:" -": ' .:~ "- .... " :~ <:..-: _.:.. " : " - : ~ n t ~ l - ~ 7 ~ i l i ~ w ! t:tt~_ t t t = t t t t t a ~ l : n o

~ttt~,~.t~t~. u L v z t t ~ , ~ _

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:..-:i:~:"::2. E X p e r ~ e n ~ aiidie.~tb (-], 3 !::.i:-:;7::.::. ~ i::-::(.:i,.(:7{i-:.•i::~-~_-i,i! (:.iwere, !emperature sensitiv~,:thediff4r¢.nt ratesofhea t :i(.i..".-: :: .::.-~-~-::' "-!-.-i!::.-}-.i-.:- :;:::. }-.:.:-::-:-.i:: .dis~ipation:were-reflected b y a difference in resistance: :~ :.:.:-7. ::::.The_ diffu~ioffeeII:{vaS!ad0sed!smnl_esS' ~tdel(type:,-!(:.i:-e.:.b.etweeln the.thermi~mis:whicIS,ideallytendedt-owards. ?i':.:':~:~:Sl6I c~;I/ndrieal IVeSS~Iw ifli ~ / i t e ~ifilets. ~it.:~ae-fi,-:.:.,: :!i:.-iZero a~ the gas mixti/re :became uniform. Instead .of ..

::-.:~i{d::"Gas flo~~0ti~:ea~li=~t:-W~"d~.hlf0aed;bY~ila (.:.).:-"measudng ~-.-t~:an.ce:bY"nUlling'{-t1~e bridge,:a.~.-._ ::(~.,,:;bell6~!vaive.~uprd:,c4:,-~4~i~a~: :Okio)~-th~:~ia~~".7-':.i "i:.!pairof voltag~gr~latedto tlie,resistancedifferencewas

•Vo*ume SZ, humanS.::

:":" :

: : : : : : - c H E M m . ~ L v-aVsms r~ta-'r~lzs :

:: :!. : : . . . . ' . . . .

'.. " . ' : :

,M.~Y 197S ::

:: x-,aeasure:d 0n' the"badg a. with. a digital.voltmeter]:.: ..7-' : .~ ~ .7 ~. :) - ~ . . ! . : . . ' . ' During"a run,-this pair. oi'iVoitages:Was'scanned.and rea-: .. '..3 .i ' . 1. ~ ~ -L --/T /3. .. . 7. ' corded automatie~-lly at giventime intervals:by.adata. :. : . : ' " ,6 -~ ' ~ - ' ~ ~- - ~ + ~ 7 . : :~:perimentallypower during*he coursethat this bridge:.bfawas raft, esseritiallyatconstant in contrast tO t h e . : : : ' ::. : ::: i ". hulled this:difference . . , 75 i 17:..:, {::.4. : di~cussddat some length inan article bY Yabsley a n d . '. . • Dunl0p:[a]:U~uallyi.100 " " .. . . . . . . . . . pointswere . . . . . . . . . . . . . . takenl startin= ..... . .when

t h e bridge.-, w a s - r o u g i t ! y., . 1 0 0 I'~. 0 u e ° t b a l a n c e . . . .

•

. ,: . .

-- ,.r:, , i . . : I'O . Z . r . . £i0 ..

When the mixture was completely uniform, a further : .:-'m . . enes of 20 readings.were taken and averaged.Due to . the nonPerfect.matchingof.the thermistors [i] a fi-: :~,.,~ nite resistance, AR(~) was Obtained. The relm,~ati0n: 7 . time, r; 0fthe runwas calculatedbyfitting the equa:: : ..- .: ~, tion. : : . ..... " " ..... " S

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•

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~(,)=~_,,.(~)

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a e.~p(-t?r) : :

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.... :

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least.squares:.A

run Was generallycortsideied

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method

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t 0 t h e S e t Ot: r e ~ i s t ~ i n e e s , b y a n o n - l i n e a r

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iftheexpefimentalAR(~=) agreed With the Value cal- . . : o.0 0 . 0., ' o s . 0 ~ " ,_a "culated by themethod ofleastsquares io witla_in t h e " . .MOLE rRACrION.0rN TROGrN : " experimentaluneertainty 6f 0.02 ft. By treating eq. (1)-' i Fig. L Presstire and cOmPOSitiondependence of n co 1=: • : b o t h asa two-parameter and a three-paramet:er p r o b - ' 0 " ° : , ca."(4); , a - a : , eq. (s);-~--=-, eq. (6); v, 7 arm; •

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a~ae'ed:to within.O.:~,e~ed~ome~a~V~a~ue'~of~the : : - . " ~lowly varyingquantity " •

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n t"~i2, given by theequation

=(p/za:r)~2i~2r, ' :

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by. theec_uation D , 5] . . . . . .

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+, 9 a t m " ,:o l, l . a .t m '. v ; .1 2 5. a r m ~ T.h e b i a c k s p o. t °fasymb°t-den0tcsa, duplicat.¢rcSUlt"

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inthe middle .

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" could be calculated. In eq. (2), n is the overalindmbe[ hOOt2 2 0 1 "1".1 +.l.3855X2/~0"0582x2 ~ " (4) density,~12 is the mutual diffUsiOn coefficient, P, k " : . • " : . . •: . . .. " : and ghave.lheir usualsignifiCance and z isthe Corn= " w i t h a stamdard deviation 0f 0.O5%. At ~ atmospheres,' •pressibflity Of themixture, given by .: " " *he"best'-' curve was . " " •. • \ .

~=,:~:~e+:... ....... , • . . . . .:.- .' • .. -

+ :0.°407x= • ~, (S) " •where B is the second pressure virial coemcient for t h e " I + 0..793ax2). : : :mixturel .... ...". ::: ..i i i...... : w i i h a s t i ~ n d a r d d e v i a t i o r i o f , 0 1 0 3 % : " . : . : The results ofthe:pressure~0mp0siti0n dependence i : . . The d~itgat 5 atmospheres wasbest represented b y i stUdy.0t:heliumZnitrogen are listed chronologica]Jy.in: the straight line..:.. " 'i i : " : . , •tablel.Themolefractionofnitrogenisden0:ted b y i : n e a i 7 2 3 I+:0 " i " . ! . • ( 6 ~. 2-referst0 the heavier Compo! ,7. . . . . ' 7 ....2!'. : . : ." neut. The m01e fractions were.calculated from the p~r- : Becauseof thelargescatter; it Was not conSidered . fial and finalpressures read tO 0.1% fr0m.the BoUrdon : :.. worthwhile t o perform leaSt:squaxescalculations o n ::. :•g a. u g e : . -T. h e . p

. . . ....

~

- :.

. .,

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.

. o. . ). ~ % Z : = : ~ 1 7 = : ( ~

g e n. w e. r e .

:.better: than 99,99%. T h e values of n ~ ~ . i n the last :

...

t h e C..... o m ops i } . i o n

de p endences

a t h i . g h e r .p 1 4 s w a .r e s.:.h. .o.w - :

"

:: .. ever a visualstraight line.of best fit was drawn through

::,c01umn"We~e calciJlated from ~the iizregparameter

'~ : " th-e.7 atrno-sphe/e p6intsJ :: "2 :::.~i.! ' : . .: .i .:, , .._ , Ab.0ve ~as: densiti~S.0~"appi0~matelyOlQO2 g Cm-:3 ( . .resultzat i atm0sphere:could belrepresented it wasifot:rkd that the voltage readings :cOuld not be fit:

:i{:5621

:. : i:.:!i:7:! : . " . .

:i::::.:7i.(.ii:

i.i .

i::I: i.:::.

7i :!(:.i:!i ::i::.... ::

: i~:

ii"::

Volume ?2, kumbei 3

at 300K

x2 ..

1, 2.'. .;;,

O.OS& ; q.9497 0.0851 ;

,:

P

n

Wm.)

x lo

'5.049 5.002. 5.004 ~ 5.089

1.232. 1.226

Exp.

.caic. ’ aRi-) fR1

Q 12 x :wZL (in+-L) ‘.

-35.19 -69.09

-35.17 -69.10

1.726 1.756

1.221

-35.30

-35.28

1.725'

-53.75

-53.76

I.756 -.

-82.36

-*“(lT?)

5..

0.9331

5.101

6

0.9499:

5.008

1.227

7 '., 0.7044 .8 :. '.0.3172 .0.172g 9 .lO: .' 0.7991 .ll .' 0.9403 12 13 14

3.004

1.000

0.9604 0.9601

15 '..

0.080~

16. 17,

0.96gq 0.0746

18 19 .20 21 22 23 24

.'

27 32 33

34 35 36 37. 38 43

yJg"o 1 o.*oo: 0.9301 0.199cJ

,: ,. 0.9400

.:.++

.O.lOO,

45 .: 46 ,. : ” 41 48 .,I,49 -_ so : ‘_ 51 :

:

‘.

0.9702 0.07og O.6668 0.3731 0.7398,. O.SOlg 0.2994 0.0989 0.3771 0.6530 0.9301) O.SOD()

s;

7.004 9.007

0.7998 : 0.2000

osoo* 0.6500 0.29830.6503

0.376a

,:

9.000

..

11.001:. : 10.996

:

1.001

,.’

-1.000

1.000 1.000 1.000

1.000 1.000 1.000

3.000. 3.OOl3 ,. 3.000 .’ 3.000 3.000 ,’ 3.000 2.993 ,' 1.000

I.?00 .-

-68.82 -6824 -69.11

,-

., .. tedFaithacceptable residuafs to eq. (i). It is beEe+id, ‘. that the deviations fro.ni eq. (1 j could $e due to two

-’

1.759 L.760 1.760

-68.02

-68.07,

I.763

-35.18 -66.46 -35.03

-35.16 -66.72 -3f.01

2.719

-64.37 -48.47 '-39.55 .; -31.32 -71.92 -35.67 -72.09. A:_;;; : -35:27 -48.44

.. -55.85 "

1.757

L.758 I.749 1.733 i.729 1.752

-68.22 -69.16

-49.3s -53.48 : -72.00

0.2647: 0.2446. 0.2444 0.2446 0.2447 .. 0.2445 0.7345 0.7327 -.

-74.77

-82.36 -61.70 -46.27 -39.30 -63.31

.. -68.82

-66.64. -34.78' -s4.01 -49.19 -57.70 -46.26 -45.49 -35.95

02446 : 0.2446. ,.. 0.2447

: 0.244+

:

-61.72

:0.7355 ,0.2446 1.717 2.210 2.191 2.702 2.675

0.7339 6.7328 0.7329 ‘, 0.7333 0.7310 -0.2445

.,,.

-46.29 -39.33 '.' -63.31

10.998 .. 2.701 3.041 12.511 8.998 _. ,.2.196 8.998 2.191 ,7.002 1.711 1.770 7.258 1.706 7.003 .0.2442. 0.999: 1.000

..

-74.78

1.238 1.228 1.202

4.91.q

0.9302

‘,

AR(-) (J-0

1.202.

4.919', 5.076 5.032

.‘.

”

1,246. 1.249

0.9079

. .

‘.

.

iable 1 Rq_~Its for helium-nit&en

Expi:

LMay 1975

CHEhlICXL PHYSICS,LZTTERS

., -39.24 -55.92

-66.73 ,' -34.76 -54.03 -49.18 -57.69 -46.27 -45.48 -35.94

1.765 1.72L

I.765 l-720 1.743 1.733 f-747. 1.737 1.731. 1.729

-49.34 -53.46 -72.02

1.745 1.754 1.76L

-64.37 48.49 -39.54 -31.32 -71% -35.66

I.759 I.749 1.743 1.728 1.76L L.735

L72.10

I.758

-31.26 -63.91 -35.26 -48.43 -55.83 -39.22 -55.92

I.725 L.755 1.533' I.746 : 1.750

: ,.42;5&' -. .,'.-42.58 : .:.’[7] [email protected]’a ~emkmpiricai

t&heat

:

.

1.738 I.753

..

,-

I.745

equation which relates

tbkifgr goeffrcient tp the gas density, p, tie

1 temperature difierenti; hT, between the thermistor effects. .Firstly, heat genqrated by the &king gases and +3idldly~ f+ convkction from the.spheri+l flie&i+ ” : kid the Sutiounding fluid, &XI other parameters which : tbr be8.ds.Kie~eibach [6] has d&jibed how theikisto~. : ._ usu,$ly cannot be kied. fie heat &an&r coefficient. ‘[email protected] re1atbd.k such conv&$n. R&z and I$f&shali ,which’hes&ibes the heat dissipated by corrductioh tid : ,__.

:

..,.:

., -, .- ‘, ..I _. _.\

,:.1 ,p. :’ :.‘.:.,. ,: ” . .. . ::,, .’ ,,, : ., ,.,. _‘. .:- ..__: : :_ ‘.. ., ., ” .; ..,I : .. ,, . . I.,_.

;. ,:,_... :. ,:.; : 1. .. . ‘_ ., .‘,,. ..-:, ,’ _ : . ‘,( I._ . “.. - .,_ .. .. .._, . . ._ ‘,, ,. ., .: _ ‘-.. ... .. -, .___;;

563.

_. ‘--‘.

‘... ..l:

Wnnc-32;

n”mber 3 .‘.

.,. .’ ‘,

:

:

.’ ;-

&EM+.

.’

PHYSICS LETlTRi

.-

:

1 May 1975

..

,.‘.,. ‘,j&&i&~on d&aids linearly oi-p1~2’+bd~(Al$4i / . coefficient, iDl$ [S]; BE is the excess second pressure ... virial cocfficicnt d$insd’by~ . .’ .” : ; ‘. Giv,en the constant heat dissipation used ex&&n. : ..’ ‘. :,. .,. tally, one r;lds that as the Gen,sity.is increased by in-, : ..BE+2--;(Q f&).’ I-,,:.. :, ,: .(% crca&ng P‘or ‘the molecular wei’&t .of one compone’nt

.of the gas mixture; the fit becomes worse, presumably .because convectior-~ contributes’s larger.@rcentage of the heat lost by thermistors.. Consequently the ther-.mistors wee shielded with.silver gauze, as su&ested b~~Kie&l~ach, tg’reduce this effk& &d aho possibly heat-generated by.&ng. The fit ras then, much im-. proved at hi,@rer densities while,the low densityres&..were~ unch~an&d.’ At higher pressures the effect

To test~rq(7)~agairist the experimental data it is neces; theform.: 1 .; ,. : .,’

:. ” sary td rev+jte,jt.in .’

,ieappearedand the dif&sion coefficients became’ slightly dependent on the wattage dissipated by the, thermistors.

:,

.theoj of dilute gases is extended to include molecular ‘size, since it is not negligible with respect to the mean .. intermolecular distance. Since the mclecules have finite size the collision frequency is altered by a factor Ywhich has two contributions(i),the increased prob- ‘duk to the increasbd

;

(10)

wh&e.K,= [nCDD,,]& [MCD,,] 1, [- 11_refers to the val... y’e at 1 iltmosphere,‘since no experiments were dqne below

1 atmosphere

pressure.’

:

In fig. 2 the function nQIz/[nO&l is plotted against number density n at mole fractions x2 = 0.3, 0.5 and 0.7. The corresponding experimental ratios

effciems

.’ ‘In me fin&q deory of &se gases(ref. [2], ch., 16), derived-for rigid spheres, the Chapman-Enskog

of a collisioti

I.= K(l -2xrX,EEp)lY,2

are calculated from tig. i by interpolation and represented by the points on the graph,The 7 atmosphere point at xi =.0.7 differs-from the Theme-Enskog line by 0.3% The estimafed precision of .@e diffusion co_

:

3. Discuss-ion

ability

,ncg,;;[;m,i]

up to 7 atmospheres is 0.2%, hence the ep

ror in the ratio is roughly 0.4%. Atthough runs were carried iout above 7 ,atmospheres, they were ,only ex.plorato!y, therefore nq composition dependence was studied. There are two possible reasons why the experimental points alllie consistently above the ThomeEnskog Line. Firstly (favouring the e+&imental results) the rigid spheres model may not be realisticenough

“excluded

j&me”

and (ii) the reduced probability ot‘ a collision due to $rieldmg by other molecules. The functionI depends.only-on the number,density and represents .’ the-factor. by which the number of collisions per unit time is increased. ..

For dense, bmary gas, sysfems the diffusion :cient is related to the,di!ute gas value by nQ12

= _:,

II -. r.aoo-

i P-1

almoiphere

i A

*

n;=o3

coefti-

[n’di2]0il-2X1XZBEp)/Y12, -

ihere Y,i is,the generalisation of 9 to a rigd sphere gases derived by Thorne [2],

where,‘b;.ke the rigidgphere diameters of the mole,-:. ; .._ 0 -, ,,. .- .me factor(l-2rlx2BEP)converts &orneTs diffu_‘:.‘.- &.‘~~‘&np&son of experiment with thecry: G, : .: :.sion @efficient.(Q& to.$e experimental diffusion ..’ -Theme--Ensko,o line; p, inferpb~ted experime~taLvalu6. :,_ .’ .I.,, _y -’,_‘Y .t, ,:-.,, ‘-.. : .‘. .. :. : .; _._ .’ SM.,;, ‘.I’, ‘Y,, ,,; ,,.‘,,.; ,‘_..:.;’ .‘,. ... ,:’ ;.. -,. ,;. . .,, ., ‘. ,. ‘; .J ; .’ : . .,,. .,_. ,._.‘.; .,, ..-. ;. .” ,;: ,’ ,,-”

$es:-

,-

__. ,~olnme32,number3: ‘.:

‘.

CHE&AL

PRY&S

Referenees‘

‘f& &i helium--nit;o~e!en.system. s&&y, t&z, accu‘. racy qf the expefmenta! results remams to be.proveil, A new hi& iressurecell is presently tinder construe-

boundary in a :’ cell described by : - [email protected] al. [9]. This method of forming the bounda$ is beli&ed to’be,superior to that Esed in the pres- .‘.

.

.’

ii\ckno,~ledg~me&

‘.

‘, : .,

..

[l] P.J,.Carson; PJ. Durzlopand I’&. Bell, I. Chem. PhyC .. 56 (1972),531.: (21 S. Chapman, and T.G. C!owIing,The mathemati& theory of ‘non-uniform gases (Cambridge Univ. Precu, London, 197ol ch 16 ,b 311

tion.which forms the initial diff~ion

..’

., :

.:-

,’similar.mbnner to.&&, 1 attiosphke

etit cell_

1 May 1975

LETTERS

[3] hi,X.‘,Y&e; &d I?.;. Dbdop, io be published. ‘, . . [4] I; Amdur and T.F. Schatzki; I. ‘3-m. E’hys 29 (1958, 1. . [Sl T-R. hiarrer’o’and E.& Mason, 5. Phys. Chem. Ref. Data ,1’(1972) 3. ‘, :. [6] R. Ki-~b=h,

..

And

C!hem. 32 [!%o)

1749.

I.71 W.E. Ranz and W.R. Siiarshti, C&m. Eng. Prog. 48 (1952) ‘_ 141,173; [S] M.K. l’imn and R,E. Gubbins, 1. Chem. Phys. 55 (197!) 268. [9i G.R. Staker, M.A. Yabsley, I.M. Symons and P.J. Dunlop, J. aem. Sot Faiiday Trnns. t 70 (1974) 825. ‘-

.This w&k was supported h part by.a g&t f&m the Australitin Re&rch@znts Committee. The au-’ thors wish to thank Professor CF. Curtiss fqr helpful dis~u~s~onsdu~~ the course of this work. _

: ..

565