C. Relaxation Time ( T1) Measurements

The DD relaxation mechanism is combined of both intramolecular and

intermolecular processes. A detailed account of the derivation of T1 from the

various intermolecular and intramolecular dipole–dipole interactions has been

described by Bloembergen et al. (30). The intramolecular relaxation process is

governed by the angular reorientation of the vector connecting the spin one-half

1

( 2 ) nuclei—in this case either 1 H or 19 F in benzene or perfluorobenzene. The

relaxation rate (1/T1 ) is

1/T1 (DD − intra) = 9/10γH 4



2 −6



r



(H−H) τc



(2)



here γ is the magnetogyric ratio for the proton (or 19 F), is Planck’s constant

over 2π, r(H−H) is the proton–proton distance in benzene (or fluorine–fluorine

distance in perfluorobenzene), and τc is the rotational correlation time of the

molecule. All of these molecules have a C6 symmetry axis for in plane rotational

orientation and a C2 symmetry axis of tumbling about the molecular plane. It

is impossible from this investigation to determine the difference between these

two types of molecular motion. The intermolecular relaxation process has a

complex dependence on angular position and spatial reorientation. This has been

simplified by expressing the dependence of the relaxation rate in terms of the

self-diffusion coefficient (D):

1/T1 (DD − inter) = (3π/10)No γH 4



2



/aD



(3)



here No is the number density and a is the distance of closest approach of the

nuclei.

The spin-rotation relaxation process becomes important in gases or supercritical fluids at low densities and high temperatures (16). This relaxation rate

is expressed as

1/T1 (SR) = (2/3)kT



−2



I(2c⊥ 2 + c 2 )τJ



(4)



where I is the moment of inertia for the molecule, k is Boltzmann’s constant,

T is temperature, c⊥ and c are the spin-rotation coupling constants, and τJ

is the angular momentum correlation time for the molecule. The spin-rotation

relaxation rate is greater for 19 F as compared to 1 H because the moment of

inertia and the coupling constants are larger for perfluorobenzene (31). It should

be noted that τJ and τc have opposite dependence on temperature, i.e., as the gas

temperature increases τc decreases whereas τJ increases. The density dependence

is opposite also, i.e., at high density/low temperatures τc is long and τJ is short,

whereas for high temperatures/low density, τc is short and τJ is long. In fact,

τc , such that spin rotation does not play a role in relaxation.

for liquids τJ

The quadrupolar relaxation mechanism is the dominant process for nuclei

1

with spins greater than 2 . Quadrupolar relaxation efficiency is determined by



Copyright 2002 by Marcel Dekker. All Rights Reserved.



the magnitude of the nuclear quadrupole and the electric field gradient at the

nucleus. This interaction is modulated by molecular rotation in a similar manner

as for dipole–dipole relaxation. The quadrupolar relaxation rate is

1/T1 (Q) = (3π2 /10)(2I + 3/(I 2 (2I − 1)))(1 + ηs 2 /3)χ2 τc



(5)



where ηs is the measure of asymmetry of the quadrupolar nuclei, I is the spin

quantum number, and χ is the nuclear quadrupole coupling constant (which is

the product of the electric field gradient and the nuclear quadrupole moment).

The other relaxation processes, CSA and SC, are assumed to play a negligible

role in molecular relaxation for the CO2 solution (32).

The effect of density on the relaxation time for benzene and perfluorobenzene is shown in Figure 2 for the two temperature extremes. At high

densities/low temperatures the relaxation times for both C6 F6 and C6 H6 are

similar. At density values ≥ 0.8 g/cm3 the relaxation times for C6 F6 over the

temperature range studied (25–150◦ C) were very similar. Benzene has a large

variation in T1 as a function of temperature and density. The T1 values for C6 F6

and C6 H6 in CO2 are similar to those of the pure liquids for the lower temperature (31,33). The relaxation time for C6 D6 was much faster than the other

two solute molecules due to quadrupolar relaxation. Using C6 D6 allows one

to separate the intermolecular from the intramolecular dipole–dipole relaxation

contributions for this series of solute molecules. As the molecular reorientation

correlation time in Eq. (5) is the same as the molecular reorientation correlation

time in Eq. (2). For benzene the dipole–dipole intramolecular relaxation time



Figure 2 Plot of relaxation time for C6 H6 and C6 F6 vs. CO2 density; C6 H6 :

(᭺, 30◦ C), (᭹, 150◦ C), and C6 F6 : (ᮀ, 25◦ C), (᭿, 150◦ C).



Copyright 2002 by Marcel Dekker. All Rights Reserved.



calculated from Eq. (2) is dominant throughout most of the density range. At

high density, intermolecular dipole–dipole relaxation [determined from Eq. (3)]

begins to play a role in relaxation as the diffusion coefficient decreases. This

is similar to the T1 results reported for methanol as a function of pressure and

temperature (16). For C6 F6 at low temperatures the relaxation mechanism is

similar to that determined for benzene. On the other hand, for high temperatures, the spin-rotation relaxation mechanism [determined from Eq. (4)] effects

nuclear relaxation. This relaxation process is related to the number of molecular

collisions in solution. At high temperature/low density spin rotation becomes

a major factor in molecular relaxation as reported for benzene near its critical

temperature (31) and methanol (16). The difference between the T1 values for

C6 H6 and C6 H6 at 150◦ C with decreasing density seen in Figure 2 is due to

spin-rotation relaxation. 19 F is affected to a much greater degree than 1 H since

C6 H6 has a larger moment of inertia then benzene (IC6 F6 /IC6 H6 = 5.6) and the

spin-rotation coupling constants are larger for 19 F than 1 H and appear squared

in Eq. (4).

As apparent in these measurements of the relaxation times for C6 H6 and

C6 H6 in CO2 over similar pressures and temperatures, there is no experimental

manifestation of a specific intermolecular interaction between CO2 and fluorine.

These interactions, if prevalent, would be expected to be seen in a change in

relaxation rate or mechanism at high densities where the intermolecular distance between the CO2 molecule and the fluorine group would be the smallest

and their potential specific interaction the greatest. It appears that at high densities, solution viscosity dominates the relaxation process, and the relaxation

mechanism for both 19 F and 1 H are similar. Therefore, there is no experimental evidence for a specific CO2 -F interaction that impacts on the relaxation of

these two molecules, which supports the calculations of Diep et al. (34) and the

experimental efforts of Yee et al. (35).

D. Vapor Liquid Equilibrium Measurements

NMR can be used to investigate the phase behavior of complex, multicomponent solvent systems as a function of pressure and temperature, with the molar

composition of the different phases being determined simultaneously, in situ, using a high-pressure capillary NMR cell (36). In a similar manner, the hydrogen

bonding behavior of the polar modifier can be determined and provides important physicochemical information regarding solvent interactions occurring in

both the liquid and vapor phase. Typically, the vapor–liquid equilibrium (VLE)

for a solution is determined using variable-volume high-pressure view cells,

with remote sampling and off-line analysis to determine phase composition. In

this way, the phase behavior of the system with regard to pressure, temperature, and composition can be determined. However, these techniques are labor



Copyright 2002 by Marcel Dekker. All Rights Reserved.



and equipment intensive and are not commonly available in most typical laboratories. For most supercritical fluid solvent systems of interest the solvent

molecules are hydrocarbons (a notable exception is supercritical CO2 ). This

presents an opportunity for the use of high-pressure NMR to determine the

pressure–temperature–composition behavior of binary hydrocarbon solvents because of the ease of proton detection on the solvent molecules in both the vapor

and liquid phase simultaneously. In practice, the application of high-pressure

NMR for the determination of VLE phase behavior could be extended to any

1

spin one-half ( 2 ) nuclei of adequate sensitivity. A hydrocarbon-containing solvent system, ethylene/methanol, was investigated demonstrating the advantages

and limitations of high-pressure NMR for VLE determinations.

The VLE experimental data for the ethylene/methanol binary solvent system at 140◦ C is shown in Figure 3. The initial mole fraction of methanol at

the starting conditions of the NMR experiment was 0.54. This was determined

from the peak areas in the single-phase liquid region of the VLE phase diagram.

Liquid phase equilibrium data determined by McHugh et al. (37) is shown for

comparison in Figure 3. At pressures above the two-phase region only a single liquid phase was detected in the capillary cell. As pressure decreased the

two-phase region was entered, resulting in an NMR spectrum containing both

liquid and vapor phase. Figure 4A and 4B shows the two-phase and singlephase NMR spectra for the ethylene/methanol binary system at the pressures

of 130.0 and 269.5 bar, respectively. The vapor and liquid phases are readily

distinguished due to the differences in the chemical shifts between the two sepa-



Figure 3 Plot of the experimental phase behavior for ethylene/methanol at 140◦ C;

vapor phase (᭹), liquid phase (᭺), and liquid phase data (᭿) reported from McHugh

et al. (From Ref. 37.)



Copyright 2002 by Marcel Dekker. All Rights Reserved.