741
NOTES
F (TI,)
E F (Tld
C-120Dq
+ ~ D +s 45Dt]/20
F (T2,)
F (T2d
[4Ds [BDq
+ 5Dt] ( 15)'12/20
+ 7Dt]/4
p (TI,) Dq = ~ 4 6 Ds ; = a 2 / 7 ; Dt = 4 2 1 ; 012 = qe2(r2)/R3; 014 = qe2(r4)/R5; where qe is the ligand charge, R the ligand-metal distance and ( r n ) the expectation value of the nth power of the d-electron distance from the metal nucleus. The above formulas also describe d4l9 and d3sSconfigurations if the signs of Dq, Ds, and Dt are changed everywhere.
Transference Numbers and Ionic Solvation of Lithium Chloride in Dimethylformamide by Ram Chand Paul, Jai Parkash Singla, and Suraj Prakash Narula Department of Chemistry, Panjah University, Chandigarh-I 4, India (Received August 20, 1 9 6 8 )
In a number of publications, the potentialities of dimethylf~rmamide'-~as a protonic solvent have been highlighted. Sherrington and Prue5 have briefly mentioned the measurement of cation transference number of potassium thiocyanate in DhIF. However, no attempt has been made to calculate the solvation of the ions on the basis of transference data. Gopal and Hussain6 have calculated the solvation number of many alkali ions in different solvents from the available conductance data. In the absence of transference data of various ions, they have claimed only a limited accuracy of their results. Lithium ion being small in size is generally solvated in solutions. Lithium chloride is appreciably soluble in D M F and accurate conductance data for it are available in the l i t e r a t ~ r e . ~It has, therefore, been selected as the electrolyte for the present investigations. The ionic conductance and the solvation number of lithium ion as calculated on the basis of transference data are reported here.
Experimental Section Materials. Lithium chloride (BDH AnalaR) was fused in a platinum crucible under a stream of dry hydrogen chloride, cooled in a desiccator, and lumps of
p (TI,) [.iODq - 12Ds - 15Dt]/lO
+ 5Dt](15)1'2/10 [-7Ds + 5 ( P - F ) ] / 5
C4Ds
the fused salt were powdered and reheated in a weighing bottle to 300" for 3 hr, cooled, and kept in a vacuum desiccator for use. Silver nitrate and potassium thiocyanate (both BDH AnalaR) were used as received. Silver (commercial) was purified by electrolysis in the laboratory and converted into wire for use. Solvent. Dimethylformamide (Baker Analyzed) was purified by keeping it over anhydrous sodium carbonate (BDH AnalaR) for about 48 hr with occasional shaking. It was fractionally distilled. The middle fraction ohm-' cm-l) (bp 148.5-149.5", sp. cond. < 2 X was taken for use. As far as possible all transference of materials was carried out in a drybox and solutions were protected from moisture by silica gel guard tubes. Determination of Transference Numbers. A weighed amount of lithium chloride was dissolved in DMF (250 ml) . A modified Hittorf transference cell with three compartments separated with well-greased stopcocks was used. The experimental technique for the measurement of transference number is exactly the same as described by Amis and cou~orkers.~J There was no evolution of gas at the cathode when a current of 3-10mA was employed. A current stabilizer (Gelman Instrument Co.) was used along with a Richard coulometer to measure the amount of current passed, Each experiment was continued for about 12-24 hr depending upon the concentration of the solution. The time of experiment was increased with the dilute solutions. Because of the solubility of the silver chloride (formed at the anode during electrolysis) in D I I F , the solutions of cathode and middle compartments were analyzed. The chloride ion concentrations of the solutions were estimated by Volhard's method. Two (1) R. C. Paul, P. S. Guraya, and R. R . Sreenathan, Indian J . Chem., 1, 335 (1963). (2) R. 0. Paul, S . Sharda, and B . R . Sreenathan, ihid., 2 , 97 (1964). (3) R. 0. Paul, S. C. Ahluwalia, and 8. R. Pahil, d h i d . , 3 , 300, 306 (1965). (4) R . 0. Paul and B . R. Sreenathan, ihid., 4, 348, 382 (1966). (5) J. E. Prue and P. J. Sherrington, Trans. Faraday Soc., 57, 1795 (1961).
(6) R. (1963).
Gopal and M . M. Hussain, J. Indian Chcm. Soc.,
40, 981
(7) W. Ves Chjlds and E. 13. Amis, J . Inorg. Nucl. Chem., 16, 114 (1960). (8) J. 0. Wear, 0. V. McNully, and E . S. Amis, i h i d . , 18, 48 (1961).
Volume Y9,Number 9 March 1060
NOTEB
742
1
0
I
I
2
I
I 3
I 4
I
fox Figure 1. Plot of transference number us.
I
s V
I
7
4-
-
F
I
8
I P
I
f0
F
4;. The limiting transference number of the lithium ion, t+u = 0.295.
to three sets of experiment for each concentration were performed to check the accuracy of the results. The equation for the calculation of transference number is the same as employed by Amis and c o ~ o r k e r s . ~ , ~
correction factor (rcor/rs)was read from the plot of roor/rBvs. rb for tetraalkylammonium ions in water s o l ~ t i o n . ~The volume of the solvent sheath was obtained from the equationlo
Results and Discussion In view of the very high resistance offered by the cell as well as difficulties of accurate estimation, the investigations regarding transference numbers were carried out in the concentration range 0.08-0.55 N at, 25'. The transference numbers have been found to vary linearly with the square root of the concentration. On extrapolation, the limiting transference number of lithium ion was found to be 0.295 (Figure 1). The equivalent conductance was combined with the limiting transference number to give the individual ionic mobility. The ionic mobility of lithium ion in DMF was calculated to be 23.62 (int ohms)-l cm2mol-l. All the relevant data are recorded in Table I. The radius of the solvated ions ( r s ) was calculated from the Stokes equation ra = F2/6a~X+ON
The crystallographic radius (yo = 0 . 6 0 A ) was obtained from the literature.6 The Robinson-Stokes The Journal of Physical Chemistry
The solvation number was thus obtained by dividing the above volume by the average volume of a DMF molecule (128.4 k3).lo The solvation number of lithium ions in the concentration range discussed was found to be 3.24. The cation transference number of lithium chloride in nonaqueous mixed solvents has been found to decrease with the increase in concentration.8 In formamide, the cation transference numbers of alkali halides decrease with an increase in concentration.lo-12 In water also the cation transference numbers of lithium and sodium ions decrease with increase in their concentration though the transference numbers of K+, Rb+, (9) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," Butterworth and Co. Ltd., London, 1985, p 120. (10) J. M. X'otley and M. Spiro, J. Phys. Chem., 70, 1502 (1966).
(11) L. R . Dawson and C. Berger, J . Amer. Chem. Soc., 7 9 , 4267 (1957). (12) G. P. Johari and P. H. Tewari, J. Phys. Chem., 70, 197 (1966).
NOTES
743
Intensity Contour Maps in Molecular
Table I: Hittorf Transference Numbers for the Ions of Lithium Chloride in Dimethylformamide a t 25' a Concn, niol/l. of solvent
0.0858 0.2233 0.3149 0.5466
WA.=
0.1570 0.3434 0.2975 0.4397
t2
X+O
0.295
23.62
x
F
103
1.45 3.69 2.75 4.05
to 1. ( ~ 1 % )( ~ 1 % )
0.257 0.237 0.235 0.208
0.743 0.763 0.765 0.792
Beam Scattering Experiments1 wsOl, R 45.185 55.935 55.563 45.04
A
re, k
roar. A
solvation no.
0.60
4.37
4.66
3.24
rc,
a WA.=is the weight of silver deposited in the coulometer; F is the Faraday (quantity of electricity) ; t , is the cation transport number; t, is the anion transport number; Wa0iis the weight of the solution in the cathode compartment; tc0 is the transport number of the cation a t infinite dilution; Lois the ionic conductance a t infinite dilution; r , is the crystallographic radius of the cation; rs is the Stokes radius of the solvated cation; and rC,, is the corrected value of the radius of the solvated cation.
Cs+, C1-, Br-, and I- ions remain almost constant with an increase in their concentration.13 This difference in behavior has been explained on the basis of ionsolvent interaction.ld The cation transference number of lithium chloride in DMF also decreases with an increase in concentration and shows a linear relationship. This behavior of lithium chloride is the same as in other solvents including water. DRIF has a high dielectric constant and dipole moment. Thus the ion-dipole interaction between lithium ion and the solvent may result in the solvation of the cation. The ionic mobilities of different alkali ions have been calculated from conductance data in various nonaqueous solvents.6 The ionic conductance of lithium ion (X+O) in DRIF has been reported as 25.0 (int ohms)-l cm2 mol-'. Using this value, Gopal and Hussaina have calculated the solvation number of lithium ion in DMF as 3. However, they have pointed out the limited accuracy of this value particularly because of the absence of transference data of lithium chloride in this solvent. Now the ionic conductance of the Lif ion (A+O) has been obtained from the relation A+' = A, X t+O
and has been found to be 23.62 (int ohms)+ cm2 mol-'. The solvation number of lithium ion calculated on the basis of the above value of ionic conductance has been found to be 3.24. Acknowledgment. The authors thankfully acknowledge the financial assistance from National Bureau of Standards, Washington, D. C. (13) L. G. Longsworth. J. Rmer. Chem. Soc., 54, 2741 (1932). (14) R . Gopal and 0.N. Rhatnagar, J . Phys. Chem., 68, 3892 (1964).
by R. Wolfgang and R. J. Cross, Jr. Chemistry Department, Yale University, New Haven, Connecticut 06520 (Received September l e , 1 9 6 8 )
Molecular beam experiments are now yielding information on the combined velocity and angular distributions of reaction products. The representation of such data in easily interpretable yet unambiguous graphical form has, however, posed unnecessarily vexing problems. We propose here the adoption of a simple convention which, despite its usefulness, does not seem to have been described in the literature. Data are usually presented as relative differential cross sections IL(v, e, @) for a given range of laboratory velocity dv and solid angle dQ = sin eded%. They may be presented as an intensity or flux contour map on a standard Newton diagram3 (see Figure 1). This representation is unanibiguous but its phase space is symmetric only with respect t o the laboratory (LAB) origin, the volume elements varying as v2. A system symmetric with respect to the center of mass is, however, more useful. It enables one to check that the product distribution is symmetric around the collision axis (relative velocity vector) as is required of all randomly oriented systems, and t o ascertain if the forward-backward symmetry identifying a long-lived intermediate is present. The common solution t o this problem has been to transform the LAB cross sections to similar cross sections I C M ( U 8, ,6 ) referred t o an origin at the center of mass (CAI system). The above symmetry considerations can then be readily demonstrated. The transformation relationship is (see Appendix) (u,e,+)
=
lL ( v , e,
(uz/u~)
(1)
(Sote, however, that most published results to date have used the incorrect factor ( u 2 / u 2 ) cos 6, where 6 i r the angle between u and v.)~ Serious difficulties arise with the transformation to the CM system as there are normally velocity and angular spreads in one or both of the colliding beams. Thus there is no unique center of mass. Commonly an (1) Financial support for this work from the National Aeronautics and Space Administration and from the Sational Science Foundation is gratefully acknowledged. (2) IL(v,e.@)dvdfi is the intensity of product molecules between 21 and 2) +do in the solid angle dn divided by ( I A n B ) . where I A iS the beam flux of A (molecules/cmz sec) and n B is the number density of B (molecules/cma), (3) D. R . Herschbach, Advan. Chem. Phys., 10, 319 (1966). (4) The correct transformation has been given by 2. Herman, J. Kerstetter, T. Rose, and R. Wolfgang, Dascussionr Faraday SOC., 44, 123 (1967); W. Miller, 8 . A. Safron, and D. R. Herschbach, i b i d . , 44, 108 (1967). Vnlume 73, Number 3 March 1989
