I
J. Olbreats
Determining the Intermolecular Potential Energy in a Gas
II
and J. P. walgra&e Universite Llbre de Bruxelles B-1050 Brussels. Belgium
A physical chemistry experiment
The measurement of the viscosity coefficient of gases is a common experiment in undergraduate physical chemistry laboratories (I). This experiment is performed either with a Rankine or an Ostwald viscosimeter which are both very suitable for measurements a t or near room temperature. It is, however, interesting to obtain the viscosity coefficients over a temperature range as large as possible in order t o analyze the complicated temperature dependence of these coefficients. It is then oossible to determine the oarameters of the intermolecular potential energy V(r) and, from these parameters, to calculate other properties of gases such as virial coefficients. This has been done in our laboratory for a gas such as propane which does not decornoose or react a t temneratures uo to 700°K and which has a 2 d virial coefficientlarge enough to be measured by a simple method (2) available in an undergraduate physical chemistry laboratory. We use an Ostwald viscosimeter (see Fie. 1) for which the ca~illarvis heated in a furnace suitable f i r temperatures up to 700:~. Different temperatures are obtained simply . . by . adiustine the beating power, i.e., the voltage. The evperiment~lprocedure is ver; similar to that described by Shoemaker and Garland (I). I t consists of measuring the time for a volume of gas to flow through a capillary of known dimensions, when one applies .. a iliff~:rence~~I'~~rt.sarrr at theexrremitirs i~frhiscaj~illary. The 5 d ~ m of e gas, delimited IJY the fiducial marks o and b wee FIE. 1) is kept Bman, so that the time required for this volume to flow through the capillary is small (less than 100 s); this permits minimization of errors due to a temperature drift of the furnace during a measurement. The time is measured with a precision of 0.1 s by means of an electronic digital clock which is started and stopped by the signal of the photocells in response to the marks a and b. The diaeram of the electronic circuit i giten in Figure 2. The apparatus ii calibrated with nitrozen at different temDeratures. The values of the viscositv coefficients of nitrogen &e obtained either from reference (3j or from eq. (4).
Table 1. Experimental data
T
.O
299.0
83.0
301.9
84.0
314.0 315.4
85.0
86.5
c n
307.5
E.C.
Figure 1. Diagram of the viscosimeter. a and b, fiducial marks delimiting the volume (about 5-10 cm3)of gas flawing through the capillary c (length: 30 cm; diameter: 0.2 mm): G,gas bottle; F, elechically heated turnace: d, connected to B fumehood: E.C.. electronic clock. Figure 3 shows a typical plot of the temperature dependence of the viscwlty of propane obtained hy a group of students. It can be seen that the 'l" law
v = nrnoLI3 = const(T1l2M'/Z/a2) (1) does not fit the experimental data. Here n is the number of molecules per unit volume, m the mass of a molecule, ~i the mean velocity of molecules, L the mean free path, T the absolute temperature. M the molar mass. and m the collision diameter. Equation (1) is deduced from the element& kinetic A more realistic theow of eases based on a rieid s ~ h e r model. e descripti& is given by the C h a p m a n - ~ n s k oequation ~ v = 26.69 ~ ~ T I O ~ R (2) where the "collision integral" R (noted as R, in ref. (5) and R'2,2'* in ref. ( 6 ) )accounts for the remaining temperature dependence in terms of the potential energy V(r). The coefficient 26.69 in eqn. (2) corresponds to in micropoises and c in A. Equation (2) permits one to calculate values of o2R
Estimation of the e,Ik and o Parameters of the Lennard-Jones Intermolecular Potential e. 'k = 2 7 5 O ~ e . / k = 290°K e , / k = 300'K T* n o2 T* n O' T* n o T*
ea/k = 2 5 0 ' ~
7)
no2
86.6 36.70
T'
n
1.230
1.436
25.56
1.118
1.502 24.43
1.060
1.542
23.80
1.025
1.568
23.41
0.992
~p~
T, t e m p e r a t u r e PKI; 7.ViSCOSitY Coefficient (p); E , and o parameters of t h e LennardJoner potential (c in A); T' = k T / e , , 5 ) ;n o ' , calculated from t h e experimental data using eqn. (2). (A2]; 0,mean values.
602 / Journal of Chemical Education
with
k , constant
Figure 2. Diagram of me circuit of the electronic clock. The output of the cadmium sulfide photoconductive cells, or LDR, is fed into the linear integrated amplifiers TAA 293 cabled as Schmitt's triggers. With this set-up a linear variation at the input voltage results in an abrupt change of the TAA 293 output voltage. Followingme TAA 293 is a series of Nand gates which render the meaoura ments automatic. To give a m e n time count, the coumes must be set to zero at me beginning: this is realized when me two LOR'Sare illuminated. LOR a starts the counter while LDR b stops it. The reference frequence for me counting is the hequence of the local power supply, that is 50 cls. The 50 cis signal is rectified and put into square form by a Schmitt's bigger in order to obtain a Correct rise time. The first SN7490 wuntw divides by 5 a n d w following ones divlde 4.10.The b i w omput of U-e mmers is decoded by me SN7447 circub which command the seven segments LED numeric display.
Figure 3. Viscosity caefficient of popane(q. IP) against square r w t of absolute temprahlre (T).T M sbaight line would correspond to a rigld sphere model (see eqn. (1)).
from the experimental data. The calculation of the collision inteeral itself is verv difficult. However. for a Lennard Jones potential (nonpolar molecules)
-
V ( r )= 4 ~ 0 ( ( 0 l r ) ' ~ (c/r)9 -
(3)
values of Q have been calculated and are tabulated (5,6) as a function of T* = kTleo; to and a in eqn. (3) are characteristic of the denth and the width of the ootential well. (In the case of polar molecules, tables of Q are also available (5, 6 ) for a Stockmaver takine into account the divole moment) " ootential . T o obtain the parameters eolk and a from the viscosity data for nonpolar gases it is necessaryto adjust the curve Ra2 = f(T) obtained from the experimental results to the curve Q = f(T*). This can be done by the following method 1 ) Approrimotiue estimation of d k . The ratios of thevaluesof noZ
obtained from the experimental viscosity data at temperatures T and aT (with a about 1.5-2) for different values of Tare comnnrad t o the ratios of the tabulated values of a for T* and oTX. T h i s yrelds n n2rrespmdenre h o t w r m 7 and T' and hence rstlmates for eo k . Applying this mrrhod t o the ~ a p e r i m e n t adata l of ~~
~
~
~~~~
~
F g.re 4 Second vmai coeff clent of propane against temperature C m e calc.lated from a Lennard Jones polenllal r tn d k = 300% and n = 4 28A ODtamed from tne v m o s q data 0 expermental whes oi McGlasnan an0 Poher (7):A experimental values reinterpretedby Hirschfelderet al. (8)fmm several literature data.
Figure 3, one ohtains values of rdk ranging between 250 and 3 5 0 ' ' ~ far propane. 2 ) Determination of the best set of unlues rolk and o. Assuming a reasonable value for rolk, it is possible to calculate T* corres p o n d i n g ~each of the experimental temperatures T.Using the table of R = f(T*)one obtains values of oZ(02= (ao2)..,/(n)kbl,) which must be constant over all the temperature range. The best value of eolk corresponds to the least scatter in the values of 0%. This is illustrated in Table 1.
~
The values rolk = 300'K
(4)
and
from Viscosity Data of Propane
(5)
c = 4.88A
give a verv satisfactorv descri~tionof the temperature dependence bf the viscosity of propane. This is shown in Table 2 where the viscosity coefficients calculated from eqn. (2) for Table 2. Comparison of Experimental and Calculated Viscosities for Propane
of Boltzman;
n, coliirion integral, values obtained from ref.
(4,
T("K) absolute temperature; vexp and 'l, ,I (LIP] see text: 'lilt (micropoirer) interpolation of data of ref. 51: AI-3) = loo mexp- tica~c)17iexpiA(2-3) = 100 (?lit - 'lcatcl/'liit.
(3,
Volume 53, Number 9, September 1976 / 603
the values of folk and a of eqns. (4) and (5) are compared t o students' experimental values and to literature data. From the values of @/k and a, it is now possible to determine the 2d virial coefficient B(T) which is another parameter characteristic of the interaction between molecules in the gas phase. It is not easy to calculate B(T) from the integral B(T) = 21LNo
J-(1- e-V(')"T)rZdr
(6)
with a Lennard Jones potential V ( r )(see eqn. (3)).However tables or plots are available in several books ( 5 , 6 )on the gas phase, giving a reduced 2d virial coefficient B* = Blbo as a function of T* (bo, equal to WVono3, is usually called the covolume). Values of B(T) for propane calculated at different temperatures using the values of folk and o of eqns. (4) and (5) are compared in Figure 4 to the values of B(T) reported in the literature (7,8). The value calculated for T = 295OK is B(T),j, = -385 cm3 mole-' compared to the best value of the literature (7,8) B(T)iic= -399 cm3mole-' while the values ohtained by the same students using an ap-
604 / Journal of Chemical Education
paratus s i m i i to that described hy Feng and Melzer (2)range between -370 and -430 cm3 mole-'. A possible expansion of the temperature range for theviscosity measurement is now planned by fitting the furnace with a system permitting one to obtain lower temperatures down to 250°K. The experiment presented here permits one to correlate in a very simple way the microsco$c and macroscopic properties of a gas. Acknowledgment
The authors wish to thank Dr. G. R. De Mar6 for critical readings of the manuscript. Literature Cited BmkCa, Inc, New York. 1962. (2) Feng, P. Y., and Melzer, M., J. CHEM EDUC.,49.375 (1972). (31 Handbookof Chemmtrv and Phwics. ChemiealRubber Publishin. Co (41 Dushman. S.. "~ciantific~oundatio"of Vaeuum Technique,'. J& Wiley and Sona, Inc, New York. 1955. (51 Reid. R. C..and Shewmd,T.K.,'ThePmprti~ofGarosandLiquida,"Mffiraw-Hill
"".,.....,.. x,-... ,m"c -- .".%,.""".
rz""l, 7"" """"""
" . b
(6) Hinhfelder, J. O..Curtis,C.F.,andBird.RB.."MalenrlarThso'ydkandLiqui&(I John Wileyand Son% Ine.. NewYork, 1954. (71 Mffilashan, M. L., and Potter. D. J, B.,Pmc.Roy. Sm.,Sen, A 267,178 (19621. (81 Hirsehfdder, J. O., M c C m , F.T., and Weeks, I. W.,J. Chem. Phya., 10,201(1912).
