NOTES
2459
Table 111: Dissociation Constant and Associated Parameters of Trityl Perchlorate in lJ2-Dichloroethsne Temp, "C Ao,cm$/(ohm equiv)b Kd x lo4, mol/l." AGO, kcal/mol
AS", cal/(deg mol) AH" , kcal/mol
20 70.2 2.85 6.5 4.75 -25 -2.6
(21 , 5)a (66.5) (12.58)
(8.7) (3.91) (-20) (f2)
25 70.7 2.52 6.5 4.88 -25 -2.6
30 78.4 2.41 6.5 5.02 -25 -2.6
a Values in brackets are those of Longworth and Mason,6 being included here for comparison. within 12%.
we calculate AH" and AS" to be -2.4 kcal/mol and -24 cal/(deg mol), respectively, in very good agreement with our results. For the systems trityl hexachloroantimonate and trityl hydroxypentachloroantimonate in methylene chloride, Kalfoglou and Szwarcs also find that AH" is negative. It has been suggested* that the large Kd values obtained by Longworth and Mason were due to the presence of moisture which thus increased the measured conductance. However, we have deliberately added water to our system and found a large decrease of conductance, which contradicts this suggestion. The effect of water is probably the hydrolysis of trityl perchlorate to triphenyl carbinol and perchloric acid which almost certainly has a lower conductance than trityl perchlorate; Gandini and Plesch'o have determined the specific conductance of perchloric acid (4 X M ) in methylene dichloride at 20" to be 9 X loMsohm-l cm-' compared with our value of 7.4 X lod4 ohm-' cm-' for trityl perchlorate a t a comparable concentration (3.5 X M ) in a solvent of similar dielectric constant (1,2-dichloroethane) at 25". I n fact, reexamination of Longworth and NIason's data reveals that they probably placed too much reliance on values of the conductance a t high concentrations in making the extrapolation to obtain a trial value of no. Such reliance is hazardous, since at higher concentrations than M, higher order aggregated ions may be formed. Deviation from linear behavior in the conductance plot is generally regarded as an indication of the formation of triple ions in solvents of low dielectric constant.l' If a value of 39.2 is taken to be the ionic conductance of C104- in 1,2-dichloroethane at 25",12 the ionic conductance of trityl carbonium ion would be 31.0. This rather low value of Ao+ is attributed to its bulkiness only. Apparently the trityl carbonium ion is not solvated as evidenced by the low -AS" of dissociation. (10) A. Gandini and P. H. Plesch, J . Chem. SOC.,B , 7 (1966); Eur. Polyn. J . , 4, 55 (1968). (11) The original paper is that of R. M. Fuoss and C. A. Kraus, J . Amer. Chem. Soc., 55,2387 (1933). A recent example is given by D. N. Bhattacharyya, C. L. Lee, J. Smid, and M. Szwarc, J . Phys. Chem., 69, 608 (1965). (12) L. F. Gleysteen and C. A. Kraus, J . Amer. Chem. Soc., 69,451 (1947).
40 90.1 2.14 6.6 5.25 -25 -2.6
Reliable within 3%.
' Reliable
Acknowledgment. We wish to acknowledge the Melbourne University Research Grant (to W. Y. L.) for support of this investigation.
On the Tait Equation of Compressibility for Solids
by Y. E(. Huang Research Laboratory, Watervliet Arsenal, Watervliet, New York (Receieed January 20, 1969)
It is of interest to consider the well-known Tait compressibility equation from an analytical point of view. The Tait equation'v2 may be written as K/v
=
-(bp/dv).
=
(voC*)-'[p
+ B*(T)I
(1)
where K-' is the isothermal compressibility, v is the volume, p is the pressure, T is the temperature, and v0, C*, and B* (2') are three parameters to be specified. As pointed out by earlier investigator^,^-^ this equation is applicable not only to liquids but also to solids. I n this note we are concerned with solids a t high pressures. We find a useful connection between eq 1 and the Slater formula6 for the Gruneisen parameter y. On the basis of this link, we can explicitly determine C* and B*(T) in terms of known solid properties. Thus a new model of Tait-Gruneisen solids is formulated. I n what follows, it will be seen that this model is analytically simpler than the Gruneisen-Debye model.' Let vo be the volume of solid under standard conditions of po = 1 atm and TO= 293°K. It will become (1) A. T. J. Hayward, Brit. J . A p p l . Phys., 18, 965 (1967). (2) G. A. Neem and D. R. Squire, J . Phys. Chem., 7 2 , 128 (1968). (3) H. A. Bethe, "The Theory of Shock Waves for an Arbitrary Equation of State," OSRD Report No. 545 (1942). (4) M. A. Cook and L. A. Rogers, J . A p p l . Phys., 34, 2330 (1963). ( 5 ) R. Ginell and T. J. Quigley, J . Phy.9. Chem. Solids, 26, 1157 (1965). (6) J. C. Slater, "Introduction to Chemical Physics,'' McGraw-Hill Book, Co., Inc., New York, N. Y., 1939, p 239. (7) Y. K. Huang, J . Chem. Phys., 45, 1979 (1966). Volume 78, Number 7 July 1969
2460
NOTES
clear that C* is nearly a constant by virtue of the Gruneisen law, while B* ( T ) is temperature dependent only. Differentiating eq 1 once again, we get
c* =
- (l/vo)(b2p/bv2)T--I(bp/bv)T
Since the Gruneisen parameter formula
y
(2)
is given by the Slater
-
= -(v/2)(bp/bv)T-'(b2p/bv2)T
"3
(3)
from eq 2 and 3 follows
c* = ' / Z ( V / V O )
(7
+
2/8)
-'
Co
=
3/(6yo
=
f(v> + Tdv)
which is similar to the Griineisen-Debye approximation.' Let v = vT at any arbitrary temperature T and p = 0. Neglecting the effect of low pressure on thermal expansion, we may write vT = vo[l ao(T - To)]. Now integrating eq 1 gives
+
p = &(T)Iexp(vT - v)/v~C*- 11 (5) which is the desired form for small temperature rises with UT ~ 0 .
@a>
which may be considered approximately equal to the constant Cowith v = vo. Thus we may put
C*
P
+ 4)
(2b)
Temperature Measurement in Fluorine Magnetic Resonance Spectroscopy1
From eq 1 and 2 we can solve for B*(T) in the form by Norbert 14uller and Timothy W. Johnson
B * ( T ) = (vo/v)KC* - p =
-p
(b2P/~V2)T-'(~P/~V)Tz
(4)
Department of Chemistry, Purdue University, Lafayette, Indiana 47007 (Received January 24, 1969)
It will be convenient to put B*(To)' = KoCo - 1
Bo
CoKo
(44
in which KO is related to yo by the Griineisen constant yo = crovoKo/c, with specific heat c, and thermal expansivity 010 nearly constant. But eq 4 is not the final form we seek. Since Gruneisen parameter is otherwise given by Y = (v/cu)(bp/Wo
(34
eliminating y from eq 2a and 3a gives ( b p / W t J= (C,/2VOC*) - (W3V)
Differentiating this equation once again, we get d2p/bvbT = 2c,/3v2 but differentiation of eq 1 gives
b2p/bTdv =
- (l/VoC*) [(bp/bT),+ dB*/dT] - (l/voCu)(yc,/v
+ dB*/dT)
From these we obtain dB*/dT = - ( c , ~ / u ) - (2c,voC*/3v2)
- (c,/v) [Y
=
+ (37 + 2) -'I
(4b)
which is nearly a constant for solids obeying the Gruneisen law. Also, eq 4b shows that B*(T) is a decreasing function as pointed in ref 2. From eq 4a and 4b we now get
B*(T) = CoKo
(co/v)[r
+ (37 + 2)-'1(T
- To)
(44
Once the parameters vo, C*, and B*(T) are determined, eq 1 can serve to establish the thermic equation of state for the solid. We find that by the use of eq 1, 2a, and 4c this equation of state is in the form The Journal of Physical Chemistry
The study of high-resolution nmr spectra as a function of temperature is greatly facilitated if sample temperatures can be determined indirectly with an "nmr thermometer.'' Ideally such a thermometer consists of a compound or mixture containing nuclei of the same isotopic species as those under investigation and giving rise to a spectrum consisting of two sharp peaks with a known, strongly temperature-dependent separation. When such a mixture is placed in a capillary within the working sample, both the desired temperature-dependent data and the signals which are used for the temperature determination can be recorded in a single sweep over the spectrum. Several types of hydrogen nmr thermometer are in common US^,^-^ and it was reported6 while this work was in progress that signals from CFC1, and CFzCClz in a mixture which also contained CFC12CFC12 served as a fluorine nmr thermometer between -10 and - 110". Other temperature-dependent fluorine nmr spectra are known6 which could provide a basis for thermometry, but no convenient method has been described for determining temperatures in the range -40 to +loo" using the fluorine spectra of readily available compounds. I n studying the spectra of fluorinated solutes in aqueous solutions we have found (1) Supported by the National Science Foundation through Grant GP-8370 and by the National Institutes of Health through a predoctoral fellowship awarded to T. W. J. (2) .Publication No. 87-202-006 B168, Varian Associates, Palo Alto,
Calif. (3) R. Duerst and A. Merbach, Rev. Sei. Instrum., 36, 1896 (1965). (4) F. Conti, (bid., 38, 128 (1967). (5) R. A. Newmark and R. E. Graves, J . Phys. Chem., 72, 4299 (1968). (6) See for example J. Jonas, L. Borowski, and H. 5. Gutowsky, J. Chem. Phys., 47, 2441 (1967).
