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 , i s t h e cation transport number; t, is the anion transport number; Wa0iis t h e weight of t h e solution in t h e cathode compartment; tc0 is the transport number of the cation a t infinite dilution; Lois the ionic conductance a t infinite dilution; r , is t h e crystallographic radius of the cation; rs is the Stokes radius of t h e solvated cation; and rC,, is the corrected value of t h e radius of t h e 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 t h e 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
NOTES
744 Tra nsfor m o t ion
thermore, since Cartesian volume elements are constant in size, the “hole” around the center of mass is absent.
VB
’
vn
LAB
a‘
Cartesian
Acknowledgment. A proposal made by us to adopt the Cartesian coordinate system was accepted by an informal group attending the Gordon Conference on Molecular Collisions, June 1968, and we are grateful to these colleagues. Some of them, particularly R. B. Bernstein, B. H. Mahan, and D. R . Herschbach, have already used this system privately. Earlier discussions with Z. Herman and J. Ross following the Fifth International Conference on Electronic and Atomic Collisions, Leningrad, July 1967, were most stimulating to us in developing this convention.
Appendix : Derivation of the LAB-Cartesian-CM Transformations To prove (1) it is convenient to transform lust from the intensity per solid laboratory angle as measured by a detector of constant area, I L ( u ,0, @) to the relative probability of finding a product molecule between 9) and v dv, 8 and 0 d e , and 4, and 4, d@, PL(v,8,@)dvded@. This relation is
+
vn
CM
Figure 1. The Newton diagrams, coordinates, and volume elements are shown for the three-coordinate systems used. The transformations from IL to Pc to I c M are given at the left. Note t h a t the Cartesian volume element is unaffected by rotation or translation.
+
PI. = I L sin 0
Like I L it is unambiguous. As the choice o€ origin is arbitrary, it is independent of any assumption as to the center of mass. The asymmetry due to unequal size of volume elements in the LAB system has been removed, though obviously that introduced by the initial beam distributions remains. Hence Pc possesses all the symmetry properties of I C Mabout any chosen center of mass. It follows that the tests for axial and forwardbackward symmetry described above can be made as well with the Cartesian as with the CAI system. FurThe Journal of Physical Chemistry
(AI)
We may write a Fimilar transformation between the analogous intensity as measured by a detector in the Ci\I system IcaI (u, 0, 4 ) and Pcu ( 7 4 0, 4 )
P C M= ICV sin 0 approximate transformation is made, assuming some sort of average or most probable center of mass. The represent ation of the data has thus been adulterated by an arbitrary assumption. The result of this is particularly unrealistic near the center of mass, where there will be a “hole” in the intensity distribution due to the factor (uz/vz)becoming zero. An exact treatment actually requires prior knowledge of I c b ~ ,the goal of the experiment. A complicated recursive procedure is therefore required to unfold the averaging due to the range of initial condition^.^ These difficulties may be largely overcome by using a probability function in Cartesian coordinates
+
(A2)
NOW
P,,(v, 8 , @)dvded@= PcIr(u, 0, 4)dzLd0d+ (h3) where PI, slid Pch1are related by the ,Jacobian of the LAB-CM transformation, J = a(?’, 8, @ ) / a ( u ,8, 4 ) “ (ie., the ratio of volunie elements). This Jacobian is the product of the Jacobians for three transformations: first, transform PI, to a I A R Cartesian probability, Pc (u,, vu, u,) ; next, transform Pc to the analogous C Y probability P’c(ur, uv, u,); finally, transform P’c l o PcaI. These are, respectively (u2 sin 1, (u2sin 0) Thus J = (u2sin 6 ) / ( v 2 sin 0) (h4)
.’
Expressing 11,and Ichl in terms of PL and PCMgives (1). Similarly Pc = I,/v2 = P I C = I C M / W 2 (A5) (5) E. A. Entemann, Ph.D. Thesis, Harvard University, 1968; R . K. B. Helbing, J. Chem. Phys., 48, 472 (1968): P. T. Warnock and R. B. Bernstein, WIS-TCI-283 (University of Wisconsin, Jan 1968). (6) For a discussion of Jacobians, see H. Margenau and G. M. Murphy, “Mathematics of Physics and Chemistry,” D. Van Nostrand Co. Inc., New York, h-. Y., 1956, p 18; W. Kaplan, “Advanced Calculus,” Addison-Wesley, Reading, RIass., 1959, p 90. (7) The flrst and third Jacobians are simply those for the transformation between Cartesian and spherical coordinate systems, and the second is for the translation and rotation of a Cartesian coordinate system.
