Langmuir 1994,10, 1882-1887
1882
Surface Adsorption and Volume Behavior of Local Anesthetics Hitoshi Matsuki,' Shinichiro Hashimoto, and Shoji Kaneshina Department of Biological Science and Technology, Faculty of Engineering, T h e University of Tokushima, Minamijosanjima, Tokushima 770, J a p a n
Michio Yamanaka College of General Education, Kyushu University, Ropponmatsu, Fukuoka 810, J a p a n Received September 27, 1993. I n Final Farm: January 3, 1994@
The surface tension and densities of the aqueous solutions of six hydrochloride salts of local anesthetics in clinical use, tetracaine (TCeHCl), procaine (PC-HCl), dibucaine (DC-HCl), bupivacaine (BC-HCl), mepivacaine (MC-HCl), and lidocaine (LC-HCl), were measured as a function of the molality at 298.15 K under atmospheric pressure. It was observed that the surface tension vs molality and density vs molality curves of DC-HC1and TC.HC1 have distinct break points at concentrations corresponding to the critical micelle concentration (cmc) and the surface tension vs molality curves of DC-HCl, TC-HC1, and BC-HC1 have other break points at lower concentrations than the cmc. The surface densities of anesthetics were evaluated numerically by applying the thermodynamic equations to the surface tension data. They decreased in the order of DCBHC1, TC-HC1,BC-HC1, MC.HC1, LC-HC1, and PC-HC1in accordance with anesthetic potency. Further, the adsorbed films of DC-HC1, TCeHCl, and BCeHCl were turned out to occur phase transition from a gaseous state to an expanded state by examining the surface pressure vs area per adsorbed molecule curves. On the other hand, by analyzing the density data, volume properties of anesthetics in aqueous solution were evaluated and the volume parameters of micelle formation of DCeHCl and TC.HC1 were also determined. The surface adsorption and volume behavior of these anesthetics were revealed on the basis of the thermodynamic quantities.
Introduction Anesthesia is caused by the interaction between anesthetic molecules and lipid molecules constituting a biological membrane a t the membrane interface. The action of anesthetics is also known to be reversed by high pressure, and this pressure reversal of anesthesia is closely related to the volume change of membranes. Thus, the surface and volume properties of anesthetics are supposed to play an important part in the mechanism of anesthesia. The molecular mechanism of anesthesia has been extensively studied by many researchers. However, the main interest of most studies is only directed toward the influence of anesthetics on the model membranes; the fundamental information about the properties of anesthetic molecules themselves seems to be lacking though it is necessary to understand the mechanism of anesthesia better. Therefore, we investigate the surface and volume properties of local anesthetics in this study. Local anesthetics are basically amphiphile molecules of tertiary amines, and some of them have colloidal properties in aqueous solution. They are classified into the ester type and amide type by the difference in the chain which binds the hydrophobic group and hydrophilic group in a molecule. The anesthetic potency of these drugs is significantly dependent on the hydrophobicity of the molecules. Many reports have been made on their clinical properties as drug compounds, whereas there have been only a few studies on the physicochemical properties such as the surface adsorption and volume behavior of
* To whom correspondence @
should be addressed. Abstract published in Aduance ACS Abstracts, May 1, 1994.
local anesthetic^^-^ though some reports concerning their micelle formation have been published.a14 In the present study, we employ six hydrochloride salts of local anesthetics in clinical use. The surface tension and densities of the aqueous solutions of these anesthetics are measured as a function of concentration at constant temperature under atmospheric pressure, and the surface adsorption of anesthetics and their volume behavior in water are discussed in terms of the thermodynamic quantities evaluated. Experimental Section Materials. The hydrochloride salts of six local anesthetics, tetracaine (TCSHCl), procaine (PC-HCl),dibucaine (DC-HCl), bupivacaine (BC-HCl),mepivacaine (MGHCl), and lidocaine (LC-HCl),were purchased from Sigma Chemical Co. except for MC-HC1which was obtained from Fujisawa PharmaceuticalCo., Ltd. (Osaka, Japan). The molecular structures of these anesthetics are summarized in Figure 1. The anesthetics were recrystallized several times from ethanol except for dibucaine (1) Skou, J. C. Acta Pharmacol. Toxicol. 1954, 10, 317. (2) Sekera, A.; Urba, C. J . Am. Pharm. Assoc. 1960, 49, 394. (3)Eckert, Th.; Kilb, E.; Hoffmann, H. Arch. Pharm. 1964, 297, 31. (4) Lin, H.-C.;Ueda, I.; Lin, S.H.; Shieh, D. D.; Kamaya, H.; Eyring, H. Biochim. Biophys. Acta 1980,598, 51. (5) Iqbal, M.; Verrall, R. E. Can. J . Chem. 1989, 67, 727. (6) Tomoaia-Cotisel, M.; Cadenhead, D. A. Langmuir 1991, 7, 964. (7) Miller, K. J., 11; Goodwin, S. R.; Westermann-Clark,G. B.; Shah, D. 0. Langmuir 1993,9, 105. (8)Jaenicke, R. Kolloid-2. 1966, 212, 36. (9)Johnson, E. H.;Ludlum, D. B. Biochem. Pharmacol. 1969,18,2675. (10) Fernandez, M. S.Biochim. Biophys. Acta 1980, 597, 83. (11) Srivastava, R. C.; Sharma, R. K.; Srinivasan, R.; Bhise, S. B. J. Colloid Interface Sci. 1983, 94, 456. (12) Attwood, D.;Florence, A. T. Surfactant Systems; Chapman and Hall: London, 1983; Chapter 4, pp 153-155. (13) Attwood, D.; Natarajan, R. J . Pharm. Pharmacol. 1983,35,317. (14) Attwood, D.;Fletcher, P. J. Pharm. Pharmacol. 1986, 38,494.
0743-746319412410-1882$O4.5O/O 0 1994 American Chemical Society
Adsorption and Volume Behavior of Local Anesthetics
Langmuir, Vol. 10, No. 6, 1994 1883
m
Tetracaine*HCI(TC-HCI)
Procaine-HCI(PC-HCI) r
'E 60 -
z
E
. cn,
Dibucaine-HCI (DC*HCI)
G
N
H
C
H*clF
~
cn,
Mepivacaine*HCI(MC-HCI)
.
&
Bupivacaine-HCI(BC-HCI)
50 -
~ N H C 0 C H Z% , -"5 N ~ ~ '
40 CHI
1
200
100
0
Figure 1. Molecular structures of the local anesthetics.
hydrochloride which was recrystallized from an ethanol and carbon tetrachloride mixture. Water was distilled three times from dilute alkaline permanganate solution for measurements of surface tension and twice for measurements of density. Methods. The surface tension of the aqueous anesthetic solutions was measured by means of the drop volume technique describedpreviously?~Theresulta of density measurements were used to calculate the surface tension of the anesthetic solutions. The error estimated for the value of the surface tension was less than 0.05 mN m-1. The measurement was carried out at 298.15 K, kept constant within 0.01 K by using a thermostat under atmospheric pressure. The densities of the aqueous anesthetic solutions were determined with a vibrating tube density meter (Anton Paar DMA60/602) under atmospheric pressure. The apparatus was calibrated by using density values of water16and dry air1' before measurements of the solutions. The temperature of the vibrating tube was maintained at 298.15 f 0.001 K by circulating water thermostated by a PID temperature controller (Yamashita Giken Co. Ltd. (Tokushima, Japan), and monitored with a digital thermometer (Techno1Seven Co. Ltd. D632 thermal referencer). All the solutions were degassed by ultrasonic irradiation before their use and allowed to stand for at least 30 min in order to achievethermal equilibrium in the tube. The experimental error for the value of the density was within 0.002 kg m4 (2 ppm).
Results The surface tension y of the aqueous anesthetic solutions was measured as a function of the molality ml a t 298.15 K under atmospheric pressure. The y vs ml curves of the anesthetics are illustrated in Figure 2, where the measurementa of BC.HC1 solution a t concentrations above 150 mmol k g l could not be made because of the low solubility of BC-HC1. It is seen that the y values of DCQHC1,TC.HC1, and BC.HC1 decrease steeply with increasing ml and the y vs ml curves of DC.HC1 and TC.HC1 have distinct break points because the anesthetics begin t o form micelles in the solution. The concentration is referred to as the critical micelle concentration (cmc). The y values of MC.HC1 and LC.HC1 are somewhat similar and are observed to decrease monotonously with increasing ml while that of PC.HC1 is observed to decrease slightly and linearly with ml. Further, it is noted that the y vs ml curves of DCeHC1, TC.HC1, and BC-HCl have other break points in the lower concentration region. In Figure 3 are magnified the y vs ml curves of these anesthetics in the neighborhood of the break points a t low concentrations in Figure 2. Such behavior has been observed in the cases of various (15) Motomura,K.; Iwanaga, S.;Hayami, Y.; Uryu, S.; Matuura, R. J. Colloid Interface Sci. 1981,80, 32. (16) Chen, C. T.; Millero, F. J. Nature (London) 1977,266, 707. (17)Kagaku Benran Kisohen, 3rd ed.; The Chemical Society of Japan: Maruzen, Tokyo, 1984; Vol. 2, p 3.
300
ml/ mmol kg-'
Lidocaine-HCI (LC-HCI)
Figure 2. Surface tension vs molality curves of the local anesthetics: (1)DC-HC1, (2) TC-HC1, (3) BC-HCl, (4) MC-HC1, (5) LC-HC1, (6) PC-HC1. 75
70
-'E
2.65 r-
60
55
10
0
20
30
40
ml/ mmol kg"
Figure 3. Surface tension vs molality curves of the local anes-
thetics: (1)DCmHC1, (2) TC.HC1, (3) BCeHCl.
surfactants and has been proved to be attributed to the phase transition in their adsorbed films a t the water/air interface.'* Therefore, we may say that the phase transition occurs in the adsorbed films of these anesthetics. The densities p of the aqueous anesthetic solutions were measured under the same conditions as the surface tension measurements. Figure 4 shows the variation in p with ml. The p values of the anesthetic solutions increase linearly with increasing ml, and p vs ml curves of DC-HC1 and TC.HC1 break a t concentrations corresponding to the cmc. Their cmc values are in good agreement with those obtained from the surface tension measurements given in Figure 2. The apparent molar volume of the anesthetic 41 was evaluated from the p value given in Figure 4 by the following equation: 41 = (UP - l/pw)/mi + M/P
(1)
where pw and Mare the density of water (kg m-3) and the molar mass of the anesthetic (kg mol-'), respectively. The 41values are plotted against ml in Figure 5. A linear relation between 41 and ml is found to hold for anesthetics which do not form micelles over the whole concentration range and for DC-HC1 and TCaHCl in the concentration ~~~
~
~
~
~
(18) Aratono, M.; Uryu, S.; Hayami, Y.; Motomura, K.; Matuura, R. J . Colloid Interface Sci. 1984, 98,33.
Matsuki et al.
1884 Langmuir, Vol. 10, No. 6, 1994 1012
1008 ?
E
1000
996
'
0
100
200
300
I
0
100
ml/ mmol kg-'
Figure 4. Density vs molality curves of the local anesthetics: (1)DC.HC1, (2) TC.HC1, (3) BCaHCl, (4) MCaHCl, ( 5 ) LC.HC1, (6) PC.HC1.
322 1 , 3
290
5 . g
300
Figure 6. Surface density vs molality curves of the local anesthetics: (1)DCeHCl, (2) TC-HCl, (3) BC.HC1, (4) MC-HCl,( 5 ) LC.HC1, (6) PC.HC1, ( 0 )surface density at the cmc.
dissociation of the anesthetic cation is negligibly small. So we assume that the local anesthetic is a uni-univalent electrolyte (for more details, see the Appendix). Let us first investigate the adsorption behavior of the anesthetics at the waterlair interface. The surface density of the anesthetic I'p can be calculated by applying the following equation:2'
I
.
292
200 m, / mmol kg-'
259 242
4
,
I
240 241
,
,
5
1
239 7
1
to the y vs ml curve shown in Figure 2. The results are depicted in the form of I'p vs ml curves in Figure 6. It is observed that the I'F value of the anesthetic increases with increasing m land decreases in the order of DC-HC1, TCeHC1, BC.HC1, MC.HC1, LC.HC1, and PC-HC1. We can see that the increase of I'y is large for DCqHC1, TC-HCl, and BC-HC1, while considerably small for PC.HC1. This suggests that the surface activity of the anesthetics is greatly dependent on the difference in the structures of their hydrophobic groups shown in Figure 1. DC-HC1, TCBHCl, and BC-HCl adsorb remarkably at the waterlair interface because they have a large hydrophobic group consisting of an aromatic ring with a butyl chain, and then, DC-HC1 and TC-HC1 form micelles in solution. The low surface activity of PC.HC1 may be explained by the strong hydrophilicity of the amino group of the aromatic ring in the molecule as compared with that of TCeHC1. MC-HCl and LC-HC1have surface activities due to some extent to the somewhat hydrophobic dimethylphenyl group in their molecules. It is further found that the order of the surface activities of the anesthetics is consistent with that of their partition coefficients between octanol and buffer solution20 and that of their solubilization into sodium dodecyl sulfate (SDS) micelles.22 This finding indicates that the surface activity of the anesthetic is proportional to the facility of partitioning the anesthetic into membranes: its magnitude is in accord with the potency of the anesthetic a ~ t i o n . ~ ~ ~ ~ ~ On the other hand, we notice that the values of DC-HC1,TC-HC1, and BC-HC1change discontinuously at concentrations corresponding to the break points in Figure 3. This observation confirms that the first-order phase
226b.-. 6 224
0
100
200
300
ml / mmol kg"
Figure 5. Apparent molar volume vs molality curves of the local
anesthetics: (1)DCSHC1, (2) TC-HCl,(3) BC-HCl, (4) MC-HC1, ( 5 ) LC-HCl, (6) PC-HCl. range below the cmc though the $1 values decrease slightly with ml. On the other hand, the $1 values of DC-HCl and TC-HC1increase with increasing ml and approach certain values in the concentration range above the cmc.
Discussion The local anesthetic cation partially dissociates into its uncharged form and proton in the aqueous solution as follows: AH+ + H 2 0 2 A + H30f
(2)
where AH+ and A represent the charged and uncharged forms of the anesthetic, respectively. Thus, it is necessary to take into account the above reaction in estimating the thermodynamic quantities for the anesthetics. However, because local anesthetics used in this study have relatively high pK, values (7.5-9.0),19v20and the pH values of the solutions were observed to have values of between 5.0 and 5.5 in the concentration range measured though each anesthetic has a slightly different value, the partial (19) Kamaya, H.; Hayes, JJ., Jr.; Ueda, I. Anesth. Analg. 1983, 62, 1025. (20) Strichartz, G.R.;Sanchez, V.; Arthur, G. R.; Chafetz, R.; Martin, D.Anesth. Analg. 1990, 71, 158.
(21) Motomura, K. J. Colloid Interface Sci. 1978, 64, 348. (22) Kaneshina, S.;Miyata, T.; Matsuki, H.; Satake, H.; Kuroki, M. Langmuir, to be submitted for publication. (23) Clark, E. R.; Hughes, I. E. Br. J . Pharmacol. Chemother. 1966, 28,105. (24) Kitagawa, N.; Kaminoh, Y.; Takasaki, M.; Ueda, I. J . Pharm. Sci. 1990, 79, 344.
Adsorption and Volume Behavior of Local Anesthetics 12
Langmuir, Vol. 10, No. 6, 1994 1885 and the number of molecules of anesthetic in one micelle particle with reference to the spherical dividing surface, respectively.2s The d(m14l)ldml vs ml curves for various anesthetics obtained from the 41vs ml curves shown in Figure 5 are demonstrated in Figure 8. The V y vs ml curves for various anestheticsare found to behave similarly to the 41 vs ml curves in Figure 5. The values of the partial molar volumes of the anesthetics a t infinite dilution are given in Table 2. The V F o value of PC-HCl is in good agreement with that reported in the literature.6 It is further seen that the d(ml41)ldml vs ml curves of DC-HC1and TC-HC1 change discontinuously a t the cmc because of their micelle formation. The volume of micelle V: from anesthetic monomers is defined by formation A
I
Po
I 0
2
4
6
A /nm2
Figure 7. Surfacepressure vs area per adsorbed molecule curves of the local anesthetics: (1)DC-HC1, (2) TC.HC1, (3) BCaHCl. transition does occur in their adsorbed films. In order to make the state of adsorbed films attended by the phase transition clearer, it is advantageous to examine the surface pressure vs area per adsorbed molecule (?r vs A) curve. Here ?r and A are defined by ?r
= yo-y
(4)
and
A = l/NAl?f respectively, where yois the surface tension of pure water and NA is Avogadro's number. The ?r vs A curves of DCOHC1, TC-HC1, and BC.HC1, which are obtained by making use of Figures 3 and 6, are shown in Figure 7. It is seen clearly that the ?r vs A curve shows a discontinuous change in A a t the equilibrium surface pressure of the phase transition point. Comparing the values of the surface pressure and area at the phase transition point with those of the surfactants,18 we can conclude that the phase transition occurs from a gaseous stateto an expanded state. It is worthwhile to note that the expanded film of BC.HC1 has a larger area than those of DC-HCland TCeHCl a t a given surface pressure. This result may be attributable to the bulky hydrophobic group of BC-HC1in comparison with those of DC-HC1and TC-HC1. The thermodynamic quantities inherent in the phase transition for these anesthetics are summarized in Table 1. Next, we consider the volume behavior of the anesthetic solutions. According to the thermodynamic treatment of the volume behavior of surfactants by Yamanaka et al.,26 in the case where the anesthetic solution is free from micelles, the derivative of the quantity mldl with respect to ml a t constant T and p is related to the partial molar volume of the monomeric anesthetic V F by the equation
[d"dl)/dm,lT,p = vy
m1< c
A V:
v"/e
Acknowledgment. This study was supported in part by a Grant-in-Aid for Scientific Research (B) (No. 05453059)from the Ministry of Education, Science and Culture of the Japanese Government. Appendix Surface Density of the Local Anesthetic. Let us consider the system composed of air, water, and the hydrochloride salt of a local anesthetic. The anesthetic completely dissociates into anesthetic cation and chloride ion while the anesthetic cation and water partially dissociate into the uncharged form of the anesthetic and corresponding ions, respectively,by the followingreactions: AHCl e AH'
where
VM
ml>> C
(7)
and NF are the molar volume of the micelle
(25) Yamanaka, M.; Kaneshina, 5.J. Solution Chem. 1990, 19, 729.
+ C1-
(AI)
AH+SA+H+
(A21
H,O e H++ OH-
(A3)
and
The surface tension y is expressed as a function of the ~~
[d(m141)/dm11T,p =P/NF
(8)
and evaluated by use of eqs 6 and 7. The estimated volume parameters of micelle formation of DC.HC1 and TC-HCl are listed in Table 3. Here the d(m14l)ldml values a t 300 mmol k g l were taken as the v " / q values for their micelles. Taking into account the molecular structures and cmc values of DC-HC1 and TC-HCl and comparing ! values with those of hydrocarbon-chain surtheir VA factants, the VA ! values of DC-HCl and TC-HC1 are comparable those of decyl- and nonyltrimethylammo! values of decyltrinium chloride as judged by the VA methylammonium chloride and the chain-length depenV !, respe~tively.~~*~~-~~ dence on the A Local anesthetic molecules incorporated into the lipid membrane are well-known to cause membrane expansion. This membrane expansion contains contributions from both anesthetic molecules and lipid molecules. The present results are useful to estimate the contribution from anesthetic molecules themselves when considering the mechanism of anesthesia from the thermodynamic point of view.
(6)
where C denotes the cmc. On the other hand, if the anesthetic forms micelles in aqueous solution, this derivative is related to the molar volume of the micelle per in the sufficiently high anesthetic molecule concentration range above the cmc by the equation
= P / N Y - VY
~
(26) Motomura, K.; Yamanaka, M.; Aratono, M. Colloid Polym. Sci. 1984,262,948. (27) Corkill, J. M.; Goodman, J. F.; Walker, T. Trans. Faraday SOC. 1967,63,768. (28) Yamanaka,M.; Kaneshima, S. J. Solution Chem. 1991,20,1159. (29) Satake, H.; Matauki, H.; Kaneshina, S. Colloids Surf.A: Phys. Eng.Aspects 1993, 71, 135.
1886 Langmuir, Vol. 10, No. 6, 1994
Matsuki et al.
Table 1. Thermodynamic Quantities Inherent in the Phase Transition at 298.15 K and 0.1 MPa. (mdw (mmol kg')
DCqHCl TC*HCl BGHC1
-yq(mN m-l)
fl
69.17 68.49 69.50
3.95 9.39 14.11
(mN m-1) 2.79 3.47 2.46
l'?
l'y(pmol m-2)
(pmol m-2) 0.56 0.70 0.50
1.32 1.07 0.64
A (nmz) 2.94 2.37 3.35
A e (nm2) 1.26 1.56 2.60
+,
Notation: ( m P , concentration of the phase transition point; yq, surface tension of the phase transition point; surface pressure of the phase transition point; I'ya, surface density of the gaseous film; surface density of the expanded film; A g, mean area per adsorbed molecule of the gaseous film; A 0 , mean are per adsorbed molecule of the expanded film.
l'y,
330
I
320
I
292
that the dissociation of water is negligibly small, the following relation holds: mA = mH+ >> mOH-
3
Assuming the anesthetic solution to be ideal, the total differential of the chemical potentials of species i at constant T and p is written by use of ml as
r
'0 290
+
P
(A61
261
dPi = (RT/mi)(ami/dml)T,pdml
259 242
(A71
Substituting eq A7 into eq A4 and using eqs A5 and A6 and the surface densities of species i defined by
4
E 240 240 5
230 226
I
6
224 I 0
100
200
300
and
I
ml i mmol kg"
Figure 8. Partial derivative of m1& with respect to molality vs molality curves of the local anesthetics: (1)DC-HCl, (2) TC-HC1, (3) BCSHCl, (4)MCaHCl, (5)LC-HC1, (6)PC.HC1. Table 2. Partial Molar Volume of the Local Anesthetics at Infinite Dilution at 298.15 K and 0.1 MPa DC-HCl BC*HCl TC*HCl MCSHCl LC-HCl PCeHCl vy 323.6 291.6 261.0 241.7 240.5 225.5 (cma mol-') Table 3. Volume Parameters of Micelle Formation of the Local Anesthetics at 298.15 K and 0.1 MPa.
DC-HCl TCSHCl
cmc (mmol k 1 78.7 127.8
vy
)
(cm3 mol-') 322.4 260.8
V "/Ny (cma mol-') 327.3 263.0
V: A (cm3 mol-') 4.9 2.2
Vyc,
Notation: partial molar volume of the monomeric local anesthetics at the cmc; V "I*, molar volume of the micelle per : , volume of micelle local anesthetic molecule at 300 mmol kg'; AV formation.
chemical potential pi of the species i at constant temperature T and pressure p:30 dr
-rfH+dPAH+- rfdPA - r:+dPH+ - r:l-dPcl- r&.i-dPoH- (A4)
where I'r is the surface excess number of moles per unit area of species i with respect to the two dividing planes.21 It is appropriate for the present system to choose as a variable the stoichiometric concentration of the anesthetic ml defined by m, = mm+ + mA = mcl-
(A5)
Here mi is the molality of species i. Taking into account (30) Aratono, M.; Uryu, S.;Hayami, Y.;Motomura, K.; Matuura, R. Mem.Fac. Sci., Kyushu Univ. 1983,C14, 19.
we have the equation dy = -RZTy(4/[(Kn
+ 4m1) - (K: + 4Kaml)1/21+ l/ml)dml (A101
where I'y is the surface density of the anesthetic defined by
ry = rL++ rf and Kn is the acid dissociation constant of AH+ defined by Ka = mAmH+fmm+
(A121
Thus, the value of Fy can be evaluated experimentally by use of the equation
ry = -[(2
-4/(4 -
(A131
Here we have introduced the degree of dissociation of AH+ a given by a = mA/(mm++ mA)=
-[Ka - (K:
+ 4K,ml)'/21/2ml
(A14)
Since the PKa values of anesthetic cations used in this study are relatively high, 7.5-9.0,19*20the value of a is found to be negligibly small. So eq A13 can be approximated as follows:
rr = -(m,/2RT)(dy/aml)T,p
(A151
It is noted that this equation corresponds to that of a uni-univalent electrolyte.21 Partial Molar Volume of the Local Anesthetic. The volume V of the anesthetic solution containing 1 kg of water at constant T and p is expressed as
Langmuir, Vol. 10, No. 6,1994 1887 then eq A17 can be approximated as follows:
where Vw and MW are the partial molar volume and the molar mass (kg mol-') of water, respectively, and Vi is the partial molar volume of species i. By use of eqs A5, A6, A12, and A14, eq A16 is rewritten in the form
where AVAH+ is the volume change for eq A2 given by
V = Vw/Mw+ mlVl
where V Iis the partial molar volume of anesthetic in the solution defined by
Further, the apparent molar volume of anesthetic in the solution 41 is related to V and the molar volume of water by the equation
vow
Since the value of a nearly equals zero as mentioned above,
(A191
