NOTES
Jan., 1958 Acknowledgment.-This investigation was supported by the Columbian Carbon Corporation. The infrared spectroscopy was done in the labora-
117
tory of Dr. Foil A. Miller. The rubber measurements and the butadiene adsorption was done in the laboratories of the Columbian Carbon Corporation.
NOTES T H E 0-H BOND LENGTH AS A FUNCTION OF T H E 0 . .. . .O DISTANCE IN HYDROGEN BONDS BY HANSFEILCHENFELD Petrochemistry Laboratory of the Research Council of Israel and the Department 01Physzcal Chemibtry, Hebrsiu University, Jerusalem, Israel Received Auoust 13, 1967
Recently it has been proposedl that the energy of the different C-C bonds is related t o their length by the equation
0-H distance is about 2% and the distance enters equation 2 in the third power. For large 0. . .O distances (E' -t 0) the actual error is even smaller than it might seem. F o r instance, in the casea of Crt(0H)z La.0 = 3.333 A. and LO-= = 0.984 A. The second term of equation 2 becomes therefore 7 kcal./ mole. The error in taking E' as 6 kcal./mole instead of 0 is automatically compensated for by the fact that even though no hydrogen bonding occurs, the distance of a hydrogen atom to the nearest oxygen atom of an adjacent molecule is finite.
Inserting the above numerical values for E
which holds also for the C-H bond though with different k . It would be interesting to investigate whether the 0-H bond length is not connected to the 0-H bond energy in a similar way. In a normal 0-H bond E
=
4-
E' and IC, equation 2 becomes 1
1 + (LO..O
- LO-H)a
=
1.20
'4.-
(3)
T o test the correctness of the equation, LO-H is plotted against LO. .O in Fig. 1. The experimental
110 kcal./mole 1.20 -
and LO-= e 0.958 A. which leads t o k = 96.7
A.8
1.15
kcal./mole.
Since the 0-H bond length is practically constant except in hydrogen bonded molecules, equation 1 can only be tested by application t o the latter case. Fortunately a number of measurements of 0-H. .O distances have been made recently for linear hydrogen bonds. Using equation 1 both for the 0-H bond and the H. .O hydrogen bond leads t o the equation
-
'\ 'I
I
od1.10 d
d
bJ
1.05
1.0
where IC is given above and E and E' are the 0-H and H . . 0 bond energies, respectively. Though both E and E' vary with the 0. . . .O distance, the sum E E' remains virtually unchanged. Actually what is usually termed hydrogen bond energy is the energy increment of the hydrogen bonded over the not hydrogen bonded structure. This increment amounts in the case of the 0-H. .O system on the average to 6 kcal./mole12 i.e.
+
+
.
E E' = 116 kcal./mole The error introduced by this simplification will in general not be greater than 3 kcal./mole. This is bearable since anyhow the error in the experimental determination of the (1) H. Feilohenfeld, THISJOURNAL, 61, 1133 (1957). (2) C. A. Coulson, Research, 10, 149 (1957).
'
0.95 2.25
I
2.50
3.0
3.5
L o . . . .ol A. Fig. 1.-Plot of the 0-H bond length against the 0. .O distance: 0,experimental data$ -.-.-., curve due to Welsh;' - - -, curve due to Lippincott and Schroeder.6
..
-
results are shown by circles (for original references see 4). The fully drawn curve shows the plot calculated according to equation 3. For comparison's sake some other curves recently proposed have been included. Lippinmtt and Schroeder6 base their curve on a n expression derived from (8) (4) (6)
W. R. Busing and H. A. Levy, J . Chcrn. Phyr.,
26, 563 (1957).
H. K. Welsh, ibid., 86, 710 (1957). E. R. Lippinoott and R. Scbroeder, ibid., 23, lOQQ (1955).
NOTES
118
theory which can, however, only be solved by successive approximations. Welsh4 uses Lippincott and Schroeder's expression though with a different parameter. Due to the incertitude of the experimental results it is difficult to favor one curve over the other. The curve calculated according t o equation 3 has, however, the following advantages. (1) It fits the results without using any free parameter obtained from hydrogen bond data. ( 2 ) It is based on a simple mathematical expression. (3) It is a smooth curve and does not show the discontinuity of Lippincott's and of Welsh's curves a t LO-H = '/zLo. -0. Conversely the good fit of the curve with the experimental results shows that the inverse cube law (1) holds also for the 0-H and H. .O bonds. The above treatment can, of course, be extended to other straight hydrogen bonds. For the C-H. .O system, for instance E +E"
kW k = L C - E a + ( L C . . O - LC-E)*
(4)
where E and k are given above, and the energy of the C-H bond E" = 3 kcal./mole2 and according t o b" = 128.2 A.Skcal./mole. Unfortunately there are no data t o check expression (4) with experiment.
Vol. 62
tion solution a time r before the regular reaction mixture is initiated, one has the equation for this solution Lt+r
- m~[A]o=
BYDAVID M. GRANTAND RANDALL E. HAMIM Department of Chemiatry, University of Utah, Salt Lake City, Utah Received J u l g 23, 1967
The specific conductivity a t time t of a solution in which the reaction A -+ B is occurring is given by the equation Lt = LA
+ LB = m ~ [ A l tf m ~ [ B ] t
(1)
where Lt is the specific conductance at time t, LA and LB are the specific conductance of the individual species A and B, mA and mB are the molar conductances of A and B each divided by 1,000, and [A], and [Bit are the molar concentrations at time t. If the rate law for this change is d[A]/dt = -klA], use of the conservation of mass expression, [Bit = [AI0 - [AIt, yields an equation which expresses the dependence of L on t Lt
- rn~[A]o= [AID(
m ~ mB)e-kr
(2)
Thus, a plot of the natpral logarithm of the quantity on the left side of the equation versus time will give a straight line with a slope of - k . This method requires a knowledge of the quantities [AIoand mB. I n a modification of the method suggested by Guggenheiml, King2 showed that first-order rate constants may be obtained directly on the spectrophotometer. In a similar manner a conductometric modification, of the above method yields first-order rate constants without a knowledge of either [Ala or mB. By starting an equivalent reacE. A. Guggenheim, Phil. Mag.. 171 8, 538 (1926). (2) E. L. King, J . Am. Chem. Soo., 74, 563 (1952). (1)
"?Zg)e-k(t+T)
(3)
Lt - Lt+r = [A]o(m~- m ~ ) ( 1 e+")e+' (4) A plot of logarithm (Lt - LL+,)vs. t will now yield a straight line of slope -k . The circuit proposed as a means of obtaining the quantity (Lt - Lt+.) directly is a conductance bridge which has been altered by placing a potentiometer, Rg, between the ratio arms Rs and R4, with the variable point on the potentiometer connected t o the detector. For the purpose of the following discussion Rgohms of Rsplus Ra is the left ratio arm and Rsohms of R8 plus Rq is the right ratio arm. The other two arms of the bridge are two conductance cells of identical construction, having cell constants of n1 and n2 for the left and right cells, respectively. By means of a three position single throw switch, a variable resistance, R7,is in position t o be connected to shunt either cell 1 or cell 2, or be left unconnected. As an initial step the earlier-mixed solution is introduced into both cells 1 and 2 making it possible to write
L1
CONDUCTOMETRIC DETERMINATIONS O F THE RATE CONSTANT O F FIRSTORDER REACTIONS
[A]o(m~-
Subtracting equation 3 from equation 2, equation 4 is obtained
= Lz = Lt+.
(5)
from which follows n1 ( 1 / R A = nz (1/Rz)
(6)
where Li,l/Ri, and ni are the specific conductance, actual conductance, and cell constant, respectively, for the ith cell. With the switch in position SO that R7 is not in the circuit, the bridge is balanced with the variable resistance Rg. The circuit has now been adjusted t o compensate for any differences between the two cell constants, and R8 is then left unchanged as long as the other circuit components are not altered. The equation for circuit balance is
+ R's)/(Re+ R4) = Ri/R2 which upon combining with equation 6 yields (R5 + &)/(Re + R I ) = ndna (Rs
(7) (8)
Equation 8 is now valid even though different solutions are placed in cells I and 2. Cell 1 is then emptied and filled with the second reaction mixture started r seconds after the solution used to sthndardize the bridge. As LI and Lz will no longer be equal, R7 must be introduced into one of the lower branches of the bridge depending upon the relative values of mA and mB. If mB is greater than mA then the specific conductance of the first reaction mixture would be greater than that of the second requiring that R7 be placed across cell 1. This is necessary to compensate for the proportionately higher resistance noted for cell 1 as compared t o cell 2. For mA greater than mB, the case where the specific conductance of the solution decreases as A goes to B, R7is placed in parallel with cell 2. For the argument which follows, the latter case is assumed t o be valid for the postulated reaction. One can now write the following expression as the condition for a balanced bridge where R, is in parallel with cell 2
5
