Internal Pressure of Liquids and Solutions - Chemical Reviews (ACS

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Internal Pressure of Liquids and Solutions Yizhak Marcus Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel quantity), V is the molar volume, and (∂U∂V)T is called the internal pressure, Pint. This expression can be rewritten as Pint = (∂U /∂V )T = T (∂P /∂T )V − P

(2)

The isochoric thermal pressure coefficient, γV = (∂P/∂T)V, is measurable, using large pressures. Suryanarayana2 presented reservations regarding application of eq 1 to liquids (rather than to gases), but these have not been heeded by others, and T(∂P/ ∂T)V − P may be considered as just a definition of the quantity Pint. Finally, noting that (∂P/∂T)V = −(∂V/∂T)P/(∂V/∂P)T and that the isobaric expansibility is αP = (∂V/∂T)P/V and the isothermal compressibility is κT = −(∂V/∂P)V/V, the internal pressure can be expressed as

CONTENTS 1. Introduction 2. Internal Pressure of Pure Liquids 2.1. Internal Pressure of Liquids at Ambient Conditions 2.1.1. Internal Pressures of Room-Temperature Ionic Liquids 2.1.1. Internal Pressures of Liquid Polymers 2.2. Internal Pressures of Liquids at Elevated Temperatures 2.3. Temperature and Pressure Effects on the Internal Pressure 2.4. Correlations and Derived Quantities 3. Internal Pressure of Liquid Mixtures 4. Internal Pressures of Dilute Solutions 4.1. Internal Pressures of Solutions of Nonelectrolytes 4.2. Internal Pressure of Aqueous Electrolyte Solutions 4.3. Internal Pressures of Electrolytes in Nonaqueous Solvents 5. Reactions in Solution and the Internal Pressure of the Solvent 6. Discussion and Conclusions Author Information Notes Biography References

Pint = TαP /κT − P

A B

The magnitude of Pint is on the order of >100 MPa, so that at ambient temperatures and pressures (1 atm = 0.101325 MPa) and saturation vapor pressures the last term, −P, in eq 3 can generally be neglected. In earlier years the isochoric thermal pressure coefficient γV was measured directly in a piezometer. This was filled to a known volume by the liquid or solution, temperature was raised, causing expansion, and pressure was then applied to restore the contents to their original volume. The procedure was repeated stepwise to obtain the required data. It was found that the pressure was substantially linear with temperature, so that the slope of its dependence (ΔP/ΔT at constant volume) represented γV.3−10 This method determines the internal pressure Pint = TγV − P directly and can be carried out with high accuracy, provided corrections are applied for compression of the containing vessel, amounting to 1−2% of the observed value.3 The alternative mode of obtaining Pint, namely, by use of eq 3, was used by Hildebrand11 prior to devising the required instrumentation employing eq 2. It could be as accurate if measurements of the isobaric expansibility and isothermal compressibility are carried out by the same investigators at the same time. The latter quantity, κT, has been obtained12−14 by use of the empirical Tait equation

B B C C E G H J J J L L M N N N N

[V (P) − V (1)]/V (1) = A log[(B + P)/(B + 1)]

(1)

where (∂P/∂T)V is the isochoric thermal pressure coefficient, U is the molar internal energy of the substance (a negative © XXXX American Chemical Society

(4)

where A and B are temperature-dependent constants specific for each substance. The value of κT is the slope of a plot of the left-hand side of eq 4 against P. Other authors15 obtained κT from the adiabatic compressibility, κS, which is obtained from the density, ρ, and sound velocity, u: κS = 1/ρu2. The molar volume V and the isobaric expansibility αP as well as the isobaric molar heat capacity, CP, are then used to convert the adiabatic compressibility to the isothermal one

1. INTRODUCTION The internal pressure of a liquid or a solution is a well-defined thermodynamic quantity. It derives from the thermodynamic equation of state of the liquid or solution1 P = T (∂P /∂T )V − (∂U /∂V )T

(3)

Received: November 5, 2012

A

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κT = κS + TVαP 2/CP = 1/ρu 2 + TVαP 2/CP

have small internal pressures, whereas the third has a very large value of Pint. The internal pressures Pint of the liquids at ambient temperatures are compared in Table 2 with their cohesive energy densities, ced, namely, in two manners: as the absolute difference (ΔHV − RT)/V − Pint and as the ratio

(5)

On the whole, κT is about 10% larger than κS. Values of the isothermal compressibility and the isobaric expansibility, κT and αP, are available in extensive compilations for ambient conditions.16,17 However, different sources have generally been employed to provide these data needed for obtaining by means of eq 3 the internal pressures of pure liquids listed in Pint tables.15,16 The drawback of this path is that such input data are liable to lead to possible inaccuracies. Dack18 compared the values of Pint for 25 °C obtained by the two methods, namely, eqs 2 and 3, for several pure liquids, Table 1. Appreciable discrepancies are noted (the entries in Table 1 need not be the most accurate values available today).

n = Pint /[(ΔHV − RT )/V ]

The absolute differences are negative for some liquids (they have n > 1) and positive for others (with n < 1), but values of the difference that are absolutely less than 50 MPa and values 0.95 ≤ n ≤ 1.05 are hardly significant (in view of the uncertainties in Pint) and designate such liquids as being van der Waals liquids according to Hildebrand.22 For associated liquids ced is considerably larger than the internal pressure, and such liquids can be termed “stiff” or “tight” and have n ≤ 0.80. Examples are water, lower alkanols, ethanolamine, formamide, trimethyl phosphate, and other highly polar liquids. On the contrary, liquids where the difference is more negative than −50 MPa (n ≥ 1.20) can be termed as “loose”, examples being mesitylene, diisopropyl ether, piperidine, triethylamine, tetramethyl urea, and mercury. Internal pressure values for many other liquids are readily obtained from tables of compressibilities and expansibilities. Some more recently reported values are, e.g., Pint/MPa = 327 for 1,8-cineole,25 322 for squalane, and 330 for pentaerythritol tetra(2-ethylhexanoate) (read from figures),26 495 for 1-butyl formate27 (all these at ambient conditions), and 375 for pentaerythritol tetrapentanoate (at 25 °C and 20 MPa, read from a figure).28 Allen et al.15 reported extensively on the Pint of hydrocarbons at 20 °C but less on other kinds of liquids. They discussed the values of n in greater detail than above but concluded that their absolute values (accurate to ±0.05) should be accepted as “experimental facts of somewhat obscure significance”. The main insight they provided was that the ced depends on complete breakage of the attractive interactions between molecules, be they dipole−dipole ones for dipolar molecules or hydrogen bonds for protic liquids. The internal pressure describes the change in the energy on incremental isothermal expansion where the intermolecular distances increase slightly without major disruption of the attractive forces between the molecules. This view, then, accounts for the very small n values for the highly hydrogen-bonded water and hydrogen fluoride and the small n values for alkanols and other protic liquids as well as aprotic dipolar liquids with large dipole moments. It does not account for 1.0 ≤ n ≤ 1.2 values for hydrocarbons and nonpolar liquids. Fluorocarbons that have small intermolecular attractive forces but large repulsive ones do have large n values, ≥1.4. 2.1.1. Internal Pressures of Room-Temperature Ionic Liquids. In recent years densities of several room-temperature ionic liquids (RTILs) have been reported over a range of temperatures and pressures, and their internal pressures can be obtained from these data. In some of these studies numerical ρ(P,T) data were presented from which αP and κT could be calculated for 25 °C and 0.1 MPa. In other studies αP(P,T) and κT(P,T) or their ratios γV = αP(P,T)/κT(P,T) were reported directly for use with eq 3, and in still others Pint(P,T) values were reported. Such data could be interpolated for the desired ambient conditions (25 °C and 0.1 MPa), and the resulting Pint values are shown in Table 3. These RTILs have Pint values similar to those of common organic solvents under such

Table 1. Internal Pressure Pint/MPa for 25 °C and Ambient Pressure of Some Pure Liquids Obtained by Different Methods18 and from the Reference Equation of State138

a

liquid

Pint = T(∂P/ ∂T)V

n-hexane tetradecanea benzene toluene water methanol ethanol 1,2-ethanediol acetone tetrachloromethane acetonitrile formamide carbon disulfide dimethyl sulfoxide

243 295 370 346 172 297 282 536 333 337 402 548 372 518

Pint = TαP/ κT

Pint from REFPROP138 236.8

305 379 151 285

375.4 350.0 169.4 281.4 278.2

502 337 345 389 544 377 502

Allen et al.15 at 20 °C.

Frank19 suggested that the internal energy U is inversely proportional to Vn. If the value of n were exactly unity, the internal pressure of liquids, Pint = (∂U/∂V)T, would equal their cohesive energy density ced = −U /V = (ΔV H − RT )/V

(7)

(6)

where ΔVH is the molar enthalpy of vaporization. Trying to obtain the former, Pint, from the latter, ced, by equating them, however, fails since n ≠ 1 as is shown further below.

2. INTERNAL PRESSURE OF PURE LIQUIDS 2.1. Internal Pressure of Liquids at Ambient Conditions

At ambient conditions, 25 °C and 0.1 MPa (∼1 atm), the extensive tables15−17 of αP and κT provide values of Pint for a large number of liquids by use of eq 3. A limited listing of representative liquids is shown in Table 2. The range of the values is not large; most of them are between 200 and 500 MPa; noteworthy liquids with low Pint values (≤200 MPa) are water, hydrogen fluoride, and perfluorohexane. Pint values tend to increase in a homologous series with the number of carbon atoms and aromatic liquids generally having larger Pint values than aliphatic ones with the same number of carbon atoms. No appreciable effect of the polarity of the molecules of the liquid is manifested. Included in Table 2 are three inorganic liquids: water, hydrogen fluoride, and metallic mercury. The former two B

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Table 2. Internal Pressures of Representative Liquids at Ambient Conditions (25 °C and 0.1 MPa), Differences between the Cohesive Energy Density and the Internal Pressure, and Inverse of Their Ratios liquid

Pint/MPa

(ced − Pint)/MPa

n

liquid

Pint/MPa

(ced − Pint)/MPa

n

n-pentane n-hexane c-hexane n-octane n-dodecane benzene toluene p-xylene mesitylene water methanol ethanol n-propanol i-propanol n-butanol tert-butanol n-hexanol trifluoroethanol 1,2-ethanediol glycerol phenol (undercooled) o-cresol diethyl ether diisopropyl ether tetrahydrofuran 1,4-dioxane anisole benzaldehyde acetone 2-butanone c-hexanone acetophenone formic acid acetic acid trifluoroacetic acid ethyl acetate butyl acetate diethyl carbonate ethylene carbonate (undercooled) propylene carbonate 4-butyrolactone perfluoro-n-heptane

216 243 322 272 286 381 346 347 401 172 284 282 294 242 245 380 310 291 472 594 440 472 244 274 404 444 417 324 327 331 428 447 470 348 351 341 333 328 416 480 420 214

−9 −18 −40 −32 −36 −28 7 −19 −73 2287 574 394 301 320 248 87 165 280 578 542 190 8 −7 −61 −43 −56 −29 41 161 18 −40 −15 −12 10 86 −10 −23 −11 255 86 230 −63

1.04 1.08 1.14 1.13 1.14 1.08 0.98 1.06 1.21 0.069 0.33 0.41 0.49 0.43 0.50 0.81 0.65 0.51 0.45 0.52 0.71 0.98 1.03 1.29 1.12 1.14 1.08 0.89 0.67 0.95 1.10 1.04 1.03 0.97 0.80 1.03 1.07 1.04 0.62 0.85 0.65 1.42

perfluoromethyl-c-hexane chlorobenzene dichloromethane 1,1-dichloroethane 1,2-dichloroethane o-dichlorobenzene chloroform tetrachloromethane bromoform n-butylamine diethylamine pyrrole pyrrolidine piperidine morpholine triethylamine aniline pyridine ethanolamine diethanolamine triethanolamine acetonitrile propionitrile benzonitrile nitromethane nitroethane nitrobenzene formamide N-methylformamide N,N-dimethylformamide N-methylacetamide N,N-dimethylacetamide N-methylpyrrolidinone-2 tetramethylurea carbon disulfide dimethyl sulfoxide sulfolane (undercooled) trimethyl phosphate tri-n-butyl phosphate hexamethyl phosphoric triamide hydrogen fluoride (0 °C) mercury (20 °C)

226 383 392 345 420 415 372 345 494 352 307 398 414 439 461 281 538 425 546 667 398 385 356 398 430 376 499 560 469 464 416 464 317 462 383 518 430 235 171a 325 150 1323b

−65 9 16 −11 −20 5 8 −35 −15 −36 −38 217 6 −122 15 −50 42 46 465 36 69 496 119 117 85 86 −10 1008 499 116 460 79 240 −105 29 190 321 210 63 40 939 −461

1.40 0.98 0.96 1.03 1.05 0.99 0.98 1.11 1.15 1.11 1.14 0.65 0.99 1.39 0.97 1.22 0.93 0.90 0.54 0.95 0.85 0.44 0.75 0.77 0.83 0.81 1.02 0.36 0.48 0.80 0.47 0.85 0.57 1.29 0.93 0.73 0.57 0.53 0.73 0.89 0.138 1.53

The low value depends on an estimated20 large value of κT = 1.62 GPa−1 that is not compatible with an appreciably smaller value for κS = 0.65 GPa−1,21 which would lead to Pint ≈ 390 MPa, expected from the value for trimethyl phosphate. bValues of ρ, αP, and κT are from Lide,23 and that of ΔVH is from Boudala et al.24

a

pressures can be calculated. Allen et al.43 measured the thermal pressure coefficient of low molar mass polymers that are liquid at 10−50 °C and obtained their internal pressures. Cho et al.44 calculated from an equation of state the ced and n values (from which Pint = n·ced is obtainable) of such polymers at 20 °C, not all being the same as those studied by Allen et al.43 Molten polymers having high molar mass, therefore melting at higher temperatures, were studied by Sauer and Dee.45 These authors also reported their ced and n values, from which the resulting Pint = n·ced are shown in Table 4.

conditions, but their spread is even more restricted, 300−500 MPa. Within this range, for a given RTIL, the differences between Pint values from several authors are large, so trends with the cation size or nature of the anion cannot be readily discerned. The cohesive energy densities of the RTILs (Table 3) are generally 30−50% larger than their internal pressures, or conversely, the ratios n are between 0.60 and 0.85, characteristic of associated nonionic liquids (Table 2). The dependence of the internal pressure of RTILs on the temperature and the pressure is discussed in section 2.3.38 2.1.1. Internal Pressures of Liquid Polymers. Liquid and molten polymers have been subjected to density measurements at various temperatures and pressures, from which the internal

2.2. Internal Pressures of Liquids at Elevated Temperatures

A category of liquids for which the internal pressure could be determined at much higher temperatures than ambient is C

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Table 3. Internal Pressures, Pint/MPa, at Ambient Conditions (25 °C and 0.1 MPa) of Room-Temperature Ionic Liquids (RTILs), Their Initial Temperature Gradients (∂Pint/∂T)P/MPa·K−1 (at P = 5 MPa), Their Initial Pressure Gradients 103(∂Pint /∂P)T (at 25 °C), and Their Cohesive Energy Densities, ced/MPa RTILa Etmim (CF3SO3)2NH Prmim (CF3SO3)2NH Bumim BF4 Bumim Bumim Bumim Bumim

BF4 BF4 PF6 PF6

Bumim PF6 Bumim CF3SO3 Bumim MeOSO3 Bumim OcOSO3 Bumim (CF3SO3)2NH Pemim (CF3SO3)2NH Hxmim BF4 Hxmim PF6 Hxmim (CF3SO3)2NH Ocmim Cl Ocmim BF4 Ocmim PF6 Ocmim PF6 Ocmim PF6 Bummim PF6 Bu3MePy BF4 Bu3MePy BF4 Bu4MePy BF4 Hx3C14H29P Cl Hx3C14H29P CH3CO2 Hx3C14H29P (C2F5)3PF3 Hx3C14H29P (C2F5)3PF3

Pint

dPint/dT

103dPint /dP

29

392

14.61

−0.340

30, ρ(P,T)

430

31, ρ, αP, κT 32, γV 34, ρ(P,T) 32, γV 31, ρ, αP, κT 33, ρ, αP, κT 34, ρ(P,T) 35, ρ, αP, κT 35, ρ, αP, κT 36, γV

458

ref and data

447 487 433 450

cedc 511

Table 4. Internal Pressures and Cohesive Energy Densities of Liquid Low Molar Mass (Mn < 1000) Polymers at 20 °C43,44 and Molten Polymers at Larger Temperatures45

764

0.857

0.783

743

431 462 439

700

374

548

360

2.53e

0.778

451

−3.37

−0.278

20.88

1.967

659 654 419

30, ρ(P,T)

405

37, ρ(P,T) 34, ρ(P,T) 36, γV

432 504 361

35, ρ, αP, κT 34, ρ(P,T) 31, ρ, αP, κT 32, ρ, αP, κT 34, ρ(P,T) 34, ρ(P,T) 39, Pint(P,T) 32, ρ, αP, κT 39, Pint(P,T) 30, γV 30, γV

444

521d

484 400

605 604

407 457 539b 412

−1.352

341 344

30, γV

362

40, Pint(P,T)

310

−1.263

t/°C

Pint/ MPa

hexadecane polyethylene poly(trifluorochloroethylene) polyethylene (C150) polypropylene polyisobutylene polyisobutylene polystyrene polystyrene polystyrene (Mn = 60 000) polyethylene glycol (Mn = 1500) poly(ethylene oxide) poly(ethylene oxide) (Mn = 4000) poly(propylene oxide) poly(ethylene oxide dimethyl ether) poly(propylene oxide) (Mn = 4000) poly(propylene oxide dimethyl ether) poly(vinyl acetate) poly(ethyl acrylate) poly(methyl methacrylate) poly(n-butyl methacrylate) poly(ethylene terephthalate) poly(propyleneterephthalate) poly(butyleneterephthalate) PEKK (polyaryl ether ketone ketone) polycaprolactam (Mn = 20000) poly(4-vinylpyridine) (Mn = 25 000) poly(2-vinylpyridine) (Mn = 20 000) polyamide ester copolymer nylon 66 polyamide MPMD poly(dimethylsiloxane) poly(dimethylsiloxane) poly(methyl phenylsiloxane)

150 20 20 200 20 20 20 20 20 200 70 20 200 20 20 200 20 20 20 20 20 300 260 260 340 200 250 250 200 290 290 20 20 20

238 340 337a 266 329 369 313a 392 431a 311 465b 397 238 326 370a 238 351a 497 407a 464 404 450 420 399 387 398 540 309 494 441 335 341 234a 218a

a

0.025

428 424

polymer

1.311

2.94

−0.806

ced/ MPa

1.21 1.31

197 259

1.21 1.38 1.40 1.13

220 238 243 243 360

1.35

230

1.25 1.25 1.32

356 190 259

1.25

190

1.35

339

1.27 1.25 0.95 1.05 0.95 0.88 1.17 0.90 1.15 1.05 1.05 1.05 1.43

331 331 474 400 420 440 340 600 269 470 420 319 184

From Allen et al.43 bFrom Allen and Sims.137

CsNO3) and at 350 °C by Ejima and Yamamura48 (1180 for NaNO3, 1210 for KNO3, and 1060 for CsNO3). Values of Pint/ MPa presumably at Tm (not directly stated) reported by Sirousse-Zia et al.49 are 910 for LiNO3, 1240 for NaNO3, 1220 for KNO3, 1100 for RbNO3, and 970 for CsNO3. Cerisier and El-Hazime50 reported values of Pint calculated according to eq 3 at Tm, and Marcus51 reported such values for T = 1.1Tm (considered a “corresponding temperature” at which values for different molten salts ought to be compared) shown in Table 5. Bockris and Richard52 reported κT and αP values at 800 °C for some alkali metal halides and at 400 °C for the nitrates. Bockris et al.53 reported κT and αP values for 800 °C for some alkaline earth halides, and from these the Pint values at these temperatures were calculated, shown in Table 6. Also shown there are Pint values for some 2:1 molten salts reported by Denielou et al.54 On the other hand, Murgulescu55 reported the internal pressures of alkali metal chlorides, bromides, and iodides at their melting points, Tm, obtained according to three different formulations. These agree fairly well among themselves, but the resulting Pint values are some 50% larger than those shown in

−0.074

−16.10

n

a

mim = 3-methylimidazolium, mmim = 2,3-dimethylimidazolium, Bu3MePy = 1-butyl-3-methylpyridinium, Bu4MePy = 1-butyl-4methylpyridinium. bAt 40 °C. cFrom ref 41. dFrom ref 42. eAt 10 rather than 5 MPa.

molten salts. Cleaver et al.46 obtained Pint values for molten sodium nitrate by direct γV = (∂P/∂T)V measurements between 340 and 400 °C. However, the resulting Pint/MPa values, 1734 at 340 °C and 1890 at 400 °C, are considerably larger than the Pint/MPa values obtained according to eq 3. These values at 400 °C were reported by Barton et al.47 (929 for LiNO3, 1209 for NaNO3, 1115 for KNO3, 1192 for RbNO3, and 1322 for D

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from surface tension and reduced volume data and cannot be correct. It is noteworthy that the Pint values for the molten salts are generally 4−20 times larger than those for liquids at ambient temperatures, and this depends only partly on the higher temperatures involved. The strong Coulombic forces in molten salts are responsible for this increase in the internal pressure. On the other hand, whereas for liquids at ambient temperatures the Pint values are commensurate with those of ced, the cohesive energy density, with generally 0.5 ≤ n ≤ 1.5 (n = Pint/ced), this is not the case for molten salts. For these the cohesive energy density is generally >10 times the internal pressure.51 Another class of high-temperature liquids for which internal pressures can be obtained is liquid metals. Amoros et al.57 reported values for mercury, cadmium, and indium, and Azad and Sreedharan58 reported values for the liquid alkali metals as functions of temperatures; values at the melting point, Tm, are shown in Table 7. Values for other metals can be calculated

Table 5. Internal Pressure of 1:1 Molten Salts Tm/K (Tm/ °C)

salt LiF LiCl LiBr NaF NaCl NaBr NaI KF KCl KBr KI RbCl RbBr RbI CsF CsCl CsBr CsI LiNO3 NaNO3 NaBF4 NaCH3CO2 KNO3 KCH3CO2 RbNO3 CsNO3 AgNO3 a

1121 887 823 1265 1073 1023 935 1129 1043 1008 958 988 953 913 955 918 909 894 525 603 567 679 611 582 583 687 476

(848) (614) (550) (992) (800) (750) (662) (856) (770) (735) (685) (715) (680) (640) (682) (645) (636) (621) (252) (330) (294) (406) (338) (309) (310) (414) (203)

Pint/MPa at Tm50 2940 1320 1000 2430 1320 1100 870 1730 1080 950 770 1180 1050 860

Pint/MPa at 800 °C52 1300

1350 990 860 1090 930 730

850 740 890 1190

920 1180

1110

1150

820 950

Pint/MPa at 1.1 (Tm/K)51 3240 1460 1100 2670 1240 1130 960 1900 1230a 960 750 1300 1160 950 1290 800 670 580 980 1310 960 740 1220 590 900 1050 1230

Table 7. Internal Pressure of Liquid Metals at Their Melting Temperatures

The value 1630 in ref 51 is a misprint.

Table 6. Internal Pressure of 1:253 and 2:154 Molten Salts salt

t/°C

Pint/GPa

MgCl2 MgBr2 MgI2 CaCl2 CaBr2 CaI2 SrCl2 SrBr2 SrI2 BaCl2 BaBr2 BaI2 ZnCl2 ZnBr2 ZnI2 Li2SO4 Na2SO4 K2SO4 Rb2SO4 Cs2SO4 Ag2SO4 Li2CrO4 Na2CrO4

800 800 800 800 800 800 900 800 800 1000 900 800 600 600 600 857 882 1068 1066 1001 664 482 797

2.5 (3.66100) 2.1 1.8 15.3 9.5 8.4 17.1 14.1 8.8 18.7 (18.65100) 12.9 11.8 3.7 3.6 4.0 2.2 1.9 1.5 1.4 1.2 2.0 1.5 1.7

metal

Tm/K

Pint/MPa

ref

Li Na K Rb Cs Cu Ag Au Cd Hg Al In Pb Fe Co Ni

454 371 337 313 302 1357 1235 1337 594 234 933 430 601 1811 1773 1727

757 447 229 175 127 867 815 646 1659 1165 2674 2863 202 2235 1717 1916

58 58 58 58 58 59, 60 59 59 57 57 59, 60 57 59, 61 59 59 59

using eq 3 from κT = (CP/CV)/ρu2, i.e., the ratio of the heat capacities, density, and ultrasound velocity reported by Blairs59 and the temperature dependence of the density reported by Egry and Brillo60 and Sobolev.61 Resulting calculated values at Tm are also shown in Table 7. 2.3. Temperature and Pressure Effects on the Internal Pressure

The directly measured γV = (∂P/∂T)V values decrease with increasing temperatures near ambient (in a range of ΔT ≈ 15 K), with Pint being proportional to (1 − αPΔT)2. This was established for such diverse liquids as n-heptane, acetone, benzene, carbon disulfide, ethylene chloride and bromide, and bromoform,3 as well as tetrahalides of carbon, silicon, titanium, and tin,4 and several fluoro- and fluorochlorocarbons.62 A quadratic expression of the dependence of Pint on the temperature pertains to dimethyl sulfoxide for 15−37 °C Pint /MPa = 517[1 − 1.446 × 10−3(t − 25°C) + 16.1 × 10−6(t − 25°C)2 ]

Table 5. The same is the case for the Pint values for alkali metal halides reported by Sanguri and Singh56 at 800 °C. These values depend on indirect evaluation of the κT and αP values

(8)

(but note that the authors erroneously reported a linear dependence).8 A similar quadratic expressions was obtained for E

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carbon tetrachloride,6 studied over a wider temperature range, from −7 to 70 °C

Table 9. Internal Pressure and Cohesive Energy Density of Water

Pint /MPa = 339[1 − 2.97 × 10−3(t − 25°C) − 12.6 × 10−6(t − 25°C)2 ]

(9)

(Note that the factor 10−6 in the last term in brackets was missing in the original paper.) Such a dependence appears to characterize also Pint for other liquids, over a similar temperature range, including some room-temperature ionic liquids studied from 10−70 °C31,35 and tetramethylsilane from −48 to 0 °C.63 For molten sodium nitrate the data by Cleaver et al.46 between 340 and 400 °C are described by Pint /MPa = 3.035(T /K) − 3.36 × 10−4(T /K)2

(10)

However, for a variety of liquids, including liquefied permanent gases (e.g., inert gases, nitrogen), liquid metals, and nonpolar organic liquids, linear dependences of Pint on the temperature were obtained by Amoros et al.,57 shown in Table 8. This is the

t/°C

Pint/MPa

ced/MPa

(ced − Pint)/MPa

n

0 25 50 75 100 125 150 175 200 225 250 275 300 325 350

172 334 479 593 672 719 733 721 685 635 578 525 488 481

2371 2287 2199 2104 2002 1891 1771 1640 1498 1344 1177 995 798 584 353

2129 1865 1625 1409 1219 1052 907 777 659 542 417 273 96 −128

0.069 0.152 0.228 0.246 0.355 0.406 0.447 0.481 0.510 0.540 0.581 0.656 0.836 1.363

Table 8. Temperature Dependence of the Internal Pressure, Pint = a − b(T − Tm), of Some Liquefied Permanent Gases and Other Liquids57 and Molten Salts52 liquid

T/K range

Tm/K

a/MPa

b/MPa·K−1

Ne Ar Kr Xe N2 O2 CH4 Hg Cd In H2O CCl4 C6H6 CF3-c-C6F5 C6F14 C6H14 C8H18 C12H26 LiCl LiBr NaCl NaBr KCl KBr CsCl CsBr

25−40 84−140 116−180 165−255 64−120 60−140 91−180 274−473 600−900 500−927 273−647a 250−343 279−473 287−341 273−328 293−422 294−422 293−393 973−1273 973−1273 973−1273 973−1273 973−1273 973−1273 973−1273 973−1273

24.55 83.81 115.78 181.86 63.3 54.4 90.7 234.28 594.0 429.76 273.15 250 278.7 228.4 186.0 178.15 216.4 267.7 887 823 1073 1023 1043 1008 918 909

84.6 179.7 212.0 207.9 271.8 255.8 317.7 1150 2826 1637 136.8 381.4 393.4 287.7 296.3 345.1 319.5 385.4 1332 1001 1350 1012 1106 964 881 788

2.426 1.768 0.538 0.422 1.779 1.616 1.254 −2.52 −2.03 −2.11 −0.352 0.903 1.178 0.949 1.019 0.892 0.735 0.694 0.17 0.27 0.47 0.43 0.53 0.52 0.20 0.29

Figure 1. Internal pressure (circles) and cohesive energy density (triangles) of water.

on the other hand, does diminish with increasing temperatures (it vanishes at the critical temperature), so that the difference ced − Pint diminishes and finally changes sign at 333 °C. Similarly, the ratio n = Pint/ced increases from its very small value, 0.069 at room temperature, reaches values characteristic of the lower alkanols, 0.40 at 150 °C, and increases beyond unity at 333 °C. This behavior is explained by the gradual destruction of the three-dimensional hydrogen-bonded network of water as the thermal agitation of the molecules increases with increasing temperatures.13 Note that since water has a maximal density at 3.98 °C at ambient pressure, so that αP < 0 at lower temperatures, it would have according to eq 3 negative internal pressures below ca. 5 °C, so that no entry for Pint at 0 °C is shown in Table 9. The pressure dependence of Pint was studied some 70 years ago by Gibson and Loeffler,12,13 but was reported in conjunction with the temperature dependence in plots against the molar volume, showing isotherms and isobars. Plots for the nonpolar CCl4 and C6H6 showed Pint to increase, but for the dipolar C 6H 5Cl, C6H 5Br, C6 H5NO2, and C6H 5NH2 Pint diminishes with increasing pressures at a given temperature. There is a clear temperature dependence, Pint diminishes with increasing temperatures,12,13 which, however, is not seen in similar plots for n-C6H14 and n-C8H18 studied by Eduljee et al.14

a This range is incorrect, because, as Table 8 shows, Pint of water reaches a maximum within the range, hence its linearity with the temperature exists over a much narrower range.

case also for molten salts according to Bockris and Richards,52 also listed in Table 8. Linear plots of Pint vs T are also shown for NaCl, NaBr, KCl, and KBr between 750 and 950 °C by Cerisier and El-Hazine.50 Water, contrary to most liquids (except liquid metals, Table 8), has Pint values that increase with increasing temperatures up to a maximal value near 173 °C, Table 9 and Figure 1.64 Its ced, F

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Goharshadi and Moosavi65 calculated the internal pressure of fluoroethers at several temperatures as functions of the external pressure P in the range from 0.1 to 3.0 MPa. For pentafluoroethylmethyl ether the values of (∂Pint/∂P)T increase from 6.9 × 10−4 at 7 °C to 9.6 × 10−4 at 37 °C and to 16.2 × 10−4 at 67 °C, but the values of Pint at a given pressure, say 1 MPa, diminish from 220 to 170 to 130 MPa at these temperatures. The thermal pressure coefficient of diethyl ether was studied as a function of its density along the saturation line up to high pressures and at the near-critical region and above by Polikhronidis et al.66,67 These authors68 also pointed out that under such conditions the isochoric temperature dependence of the internal pressure, (∂Pint/∂T)V, is zero (Pint has an extremum) at the same conditions that the thermodynamically equal isothermal volume dependence of the constant volume heat capacity, (∂CV/∂V)T, is zero. Davila et al.69 reported the internal pressures of 1-alkanols CnH2n+1OH (2 ≤ n ≤ 11) over the temperature range from 5 to 85 °C and pressure range from 0.1 to 60 MPa. The pressure dependence at 25 °C was small (slightly positive for n = 2, 3, and 4, hardly noticeable for n = 4, 5, and 6, and slightly negative at P > 10 MPa for the higher alkanols). Temperature dependence was roughly inversely proportional to the number of carbon atoms in the alkanol: (∂Pint/∂T)P ≈ 0.86/n MPa·K−1. Several authors reported Pint (or γV = Pint/T) values of roomtemperature ionic liquids as functions of the temperature and pressure. Guerrero et al.38 showed three-dimensional plots of Pint of 1-butyl-n-methylpyridinium tetrafluoroborate (n = 2, 3, and 4) against the temperature and pressure but did not provide explicit numerical data or regression expressions for the values. Data of other authors could be expressed by quadratic equations in the temperature at a constant pressure (5 MPa was the pressure common to all the studies available) and in the pressure at 25 °C (selected as the common temperature for comparison). The initial gradients, (∂Pint/∂T)P and (∂Pint/∂P)T, are shown in Table 3. Both positive and negative dependencies are seen, but no regularities in terms of the natures of the cation and anion of the RTILs for which the data are available can be discerned. According to Goldmann and Tödheide,70 the internal pressure of molten KCl at 772 °C remained at 860 MPa as the external pressure was varied from 67 to 282 MPa, but at 1033 °C it decreased from 820 to 770 MPa as the pressure increased from 144 to 371 MPa. Pandey et al.71 found at 400 °C that Pint remained at 810 MPa for LiNO3 as the pressure was increased from 0.1 to 100 MPa but that it decreased from1221 to 1188 MPa for NaNO3 and increased from 1169 to 1912 MPa for KNO3 over this pressure interval, a behavior that is not very plausible.

Figure 2. Internal pressure of liquids at ambient conditions plotted against their surface tension. Outlier (black triangle) is water.

exists a relationship between the surface tension and the compressibility of liquids. According to Mayer73 κTσ = a(1 − 3y + y 3 )/4(1 + 2y)2

(12)

where a is the rigid sphere diameter of the molecules and y = (πNA/6)a3/V is the packing fraction of the liquid. Since κTPint ≈ TαP, the right-hand side of eq 12 should play the role of TαP for a perfect correlation between Pint and σ to exist. Castellanos et al.72 discussed the relationship between the surface tension and the internal pressure of liquid drops of radius r. The resulting relationship can be written as Pint = p(r =∞) exp(2σV /RT ) + 2σ /r

(13)

where p(r=∞) is the vapor pressure of the liquid with a flat surface. Gibson and Loeffler12 represented the internal pressure of liquids as the sum of an attractive and a repulsive term: Pint = Pint A + Pint R, the latter being negative. They then equated Pint R with −B, where B is the parameter of the Tait eq 4, so that the attractive internal pressure is Pint A = Pint − B and can be calculated. From their pressure dependence studies on Pint they concluded that for the dipolar benzene derivatives Pint A = a′V−2.75. Earlier, Hildebrand and co-workers3,4 found TV2(∂P/ ∂T) V to be constant for nonpolar liquids over short temperature intervals. They related the internal pressure, obtained from the thermodynamic equation of state, to the quantity a′, representing the molecular interactions in the van der Waals equation of state: Pint ≈ a′V−2. When hydrogen bonding is present, then a further negative term, Pint H, should be added to Pint = Pint A + Pint R according to Gibson and Loeffler.13 This being the case for water and ethylene glycol explains their small Pint values. This aspect of the internal pressure was later reviewed by Barton.9 It was taken up by Bagley et al.,74 who related the internal pressure of dipolar and hydrogen-bonded solvents to their three-dimensional solubility parameters, according to the Hansen formulation.75 They74 showed that

2.4. Correlations and Derived Quantities

Several authors11,18,72 have pointed out that the internal pressures of liquids correlate with their surface tensions, σ. This is indeed the case to a certain extent; the correlation expression for the 80 liquids listed in Table 2 is Pint /MPa = (140 ± 18) + (7.77 ± 0.55)(σ /mN·m−1) (11)

ced − Pint = δp2 + δ h 2

as shown in Figure 2. However, the correlation coefficient squared is only 0.7165, and the standard error of the fit is 50 MPa. Water is an obvious outlier from this correlation with its very large surface tension and small internal pressure. There

(14)

where δp and δh are the Hansen polarity- and hydrogenbonding-related solubility parameters and applied this relationship to acetone and the lower alkanols. G

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The concept of ‘free volume’ is related to the internal pressure. Bockris and Richard52 and Cerisier et al.50 related the free volume of molten salts to their internal pressures, but the expressions are applicable to liquids in general. One expression, according to Kincaid and Eyring,76 provides an experimental method to measure the free volume Vf = (ug /ul)3 V = V −2[2RT /(P + Pint)]3

of the internal pressure. A small fraction only of this work is surveyed here. Westwater et al.,3 who were among the first to obtain internal pressures of liquids from direct measurements of the isochoric thermal pressure coefficient, applied this method also to some equimolar mixtures of n-heptane, benzene, and carbon disulfide with some other liquids, both polar and nonpolar. They reported γV = (∂P/∂T)V values at 20, 25, and 35 °C, and the values at 25 °C were smaller than those estimated from the mean values of the pure components, labeled (1) and (2): 0.5γV(1) + 0.5γV(2), corresponding to a negative excess Pint. A better prediction was obtained when 0.5V(1)γ V (1) + 0.5V(2)γV(2) was used. Macdonald and Hyne10 measured γV values for aqueous mixtures with dimethyl sulfoxide (DMSO), methanol (MeOH), and tert-butanol (t-BuOH) and reported values for 25 °C. γV have maxima at xDMSO ≈ 0.35, xMeOH ≈ 0.35, and xt‑BuOH ≈ 0.1. The positive excess Pint in these systems is attributed to a minimum in the free volume, Vf, and intermolecular distances in these mixtures. Dack,18 who reviewed internal pressures of liquids published to 1975, did not mention other studies of liquid mixtures up to this date. However, Srivastava and Berkowitz81 compared results for equimolar mixtures calculated according to their own theoretical approach with the experimental results of Westwater et al.,3 finding for both pure components and mixtures ∼10% too low calculated values of Pint. Some further developments regarding the internal pressure of liquid mixtures are due to Pandey and co-workers,82−85 among others, who estimated Pint according to Flory’s theory. In the earlier paper,82 the calculated Pint values were compared with the experimental ones for benzene + p-xylene, benzene + 1,4-dioxane, and acetone + iodomethane with poor agreement. In the later publications the theory was modified and much better agreement was obtained, e.g., for c-hexane + c-hexanol and for acetone + iodomethane.83,84 The modification consisted of using the modified isobaric expansibility, αP(F), and isothermal compressibility, κT(F), calculated according to Flory’s statistical method to determine the internal pressure: Pint(F) = TαP(F)/κT(F). In a later paper Pandey and Sanguri85 used also empirical relations

(15)

where ug and ul are the speeds of ultrasound in the gaseous and liquid forms of a substance at a given temperature and low pressure. Other formulations that involve the internal pressure, traceable to Buehler et al.,77 are Vf = V [1 − (1 − RT /PintV )3 ] ≈ (πNA /6)a3(RT /PintV )3 (16)

where a is the hard-sphere diameter of the molecules. For the nonpolar globular CCl4 the temperature dependence of the free volume, calculated from the collision diameter of the Lennard− Jones potential energy, does not agree with the experimental internal pressure function, although agreement ought to be expected.6,9 Dack18 surveyed several further relationships of the internal pressure to other quantities. He quoted Cammarata and Yau78 but did not note that they equated Pint with ced = −U/V so that their expression for 50 liquids of low polarity pertains to the (Hildebrand) solubility parameter δH = ced1/2. Dack18 did show rough relationships between Pint and the polarizability α of solvent molecules and of ced − Pint with their dipole moments. Srivastava7 reported a relationship of the internal pressures of liquids to their normal boiling points Pint = 4.5[24.5(Tb/K) − 1400]/V

(17)

Suryanarayana and Pillai79 related the vapor pressure p(T) of 30 liquids, both dipolar and nonpolar, to their internal pressures log p = a − bPint

(18)

where a and b are substance-specific coefficients that are independent of the temperature. Water and 1-propanol were outliers from this linear dependence. Suryanarayana2 later related the internal pressure of liquids to their density (ρ in g·cm−3), viscosity (η in poise), molar mass (M in g·mol−1), and speed of ultrasound (u in cm·s−1) in them Pint = 1.165 × 105RT (η /u)1/2 ρ2/3 M −7/6

αPemp/K−1 = 75.6 × 10−3T −1/9u−1/2ρ−1/3

(20)

κTemp/Pa = 1.71 × 10−2T −4/9u−2ρ−4/3

(21) −1

and measured values of the ultrasound velocity u (in m·s ) and density ρ (in g·cm−3) of the mixtures to obtain the internal pressures of mixtures employing eq 3. Pandey et al.85 also used an alternative approach, namely, hole theory, to calculate αPhole and κThole values for various liquid mixtures, although the results were not as satisfactory. Dey et al.84 argued that the excess internal pressure of liquid mixtures, PintE, ought to be calculated by subtraction of the ideal Pintid involving volume-fraction-prorated αP and κT

(19)

The quasi-lattice dynamics of liquids can be expressed by their Grüneisen parameters, and Sharma80 presented expressions relating them to the internal pressure, reporting calculated values for benzene, n-octane, c-hexane, and carbon tetrachloride. The relationship between the internal pressure of solvents and the activation volume for chemical reactions is discussed in section 5.

Pint id = T Σiφα /Σiφκ i Pi i Ti

3. INTERNAL PRESSURE OF LIQUID MIXTURES The internal pressure of a very large number of binary as well as ternary liquid mixtures has been studied, and only a few of these systems can be reviewed here. Use of ultrasound speed to measure adiabatic compressibilities of liquids appears to be a specialty of Indian investigators, and these compressibilities in conjunction with other data, e.g., eq 19, then led them to values

(22)

When this is done the Pint = Pint − Pint is positive for mixtures of dimethyl sulfoxide with 1-butanol, 1-hexanol, and 1-octanol, as expected from other properties of these mixtures that indicate preferred self over mutual interactions of the components. This manner of treating PintE contrasts with earlier approaches, such as that of Srivastava and Tripathi,86 who used mole fraction prorating of the Pint values of the E

H

id

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Table 10. Internal Pressure Increment, ΔPint = Pint(1 M) − Pint(W), in MPa for 1 M Aqueous Solutions at 25 °C

a

solute

Dack101

Conti104

methanol ethanol 1-propanol 1-butanol tert-butanol 1-pentanol 1,2-ethanediol 1,3-propanediol PEG-200a tetrahydrofuran 1,4-dioxane acetone piperidine ethylenediamine acetonitrile formamide N-methylformamide dimethylformamide N-methylacetamide urea dimethyl sulfoxide ascorbic acid sucrose

18

4 2 23 30

Zaichikov102,103

Dhondge90,91c

2 3 15 34 11 47

51 45 25 15

62 48 74

Dhondge92

141 56 76

40 56

53 41

40 38 61 46b

44 41 67 100

87 59 66

86

Polyethylene glycol with average molar mass of 200 g·mol−1. bAt 35 °C. cAt 6 °C.

Zorebski93 argued that since Pint is not an extensive property, its deviations ΔPint from the values for ideal mixtures, Pintid (eq 22), ought not formally to be called the excess internal pressure, PintE. Nevertheless, he showed (positive) values of PintE for mixtures of 1-butanol and 1,3-butanediol at 25 °C, signifying preferred self-interactions of the components. Even larger PintE values were obtained94 for mixtures of 1-butanol with 1,4-butanediol at 25−45 °C. On the other hand, mixtures of acetonitrile with 2-methoxyethanol, dimethylacetamide, and propylene carbonate at 25 °C show linear dependencies of Pint on the mole fraction compositions.95 Small deviations only of Pint from ideal are also exhibited by mixtures of 2-alkanones with dialkylamines.96 A large number of other binary systems have been studied over the years by noting their internal pressures as functions of the compositions, those presented above being just representative examples. Ternary mixtures of organic liquids have also been studied in a proliferate manner, and again only a few very recent examples can be mentioned here. Aprotic mixtures of dimethylformamide with c-hexane and benzene, or chlorobenzene, or nitrobenzene were studied by Thirumaran and Sudha at 30 °C.97 Mixtures with chlorobenzene were subsequently studied at 15, 25, 35, and 45 °C by Ku et al.98 Protic mixtures involving carbon tetrachloride with c-hexanone + 2-propanol or + tert-butanol were studied by Uvarani and Sivapragasam.99 A problem with some of these studies as well as many others is that the internal pressure of the mixtures was estimated by means of an empirical expression, such as eq 19, rather than from properly measured expansibilities and compressibilities, and this casts doubt on the results. A few studies were devoted to mixtures of molten salts. Sternberg and Vasilescu100 studied mixtures of molten MgCl2 + KCl and of MgCl2 + BaCl2 but were concerned mainly with the excess volumes, expansibilities, and compressibilities. However, they mentioned that if the molten salt mixtures conformed to

components to calculate PintE. Resulting values were negative for the four aqueous mixtures studied: water +1-propanol, water + 1-butanol, water + ethylene glycol, and water + glycerol. For the former two mixtures self-interaction of the water outweighs the mutual interactions, but the opposite is the case for the latter two mixtures,87 so that different signs of PintE are expected. Similarly, Acevedo et al.88 calculated PintE in terms of mole fraction prorating of the Pint values of the components and arrived at negative PintE for mixtures of n-pentane with dichloromethane and methyl acetate, 2-propanol with methyl acetate, and 1-butylamine with 1,4-dioxane. Results were interpreted in terms of appreciable mutual interactions, but these do not appear to take place in these systems (except, perhaps, for 2-propanol + methyl acetate). For mixtures of nheptane and n-octane, as expected for ideal mixtures, Pint values vary linearly with the mole fraction composition at temperatures of 25, 50, and 75 °C and pressures of 0.1−40 MPa.89 On the other hand, aqueous mixtures of dimethyl sulfoxide and ethylenediamine at 6 °C studied by Dhondge et al.90 do show positive PintE = Pint − Pintid, where Pintid was calculated in terms of mole fraction prorating. Results were interpreted as indicating stronger self than mutual interaction and in the case of aqueous dimethyl sulfoxide are in agreement with earlier results of Macdonald and Hyne9 at 25 °C. Dilute aqueous polyethylene glycol PEG 200 has negligible PintE values, but PEG 400 had positive ones at 6 °C. Dhondge and Ramesh91 reported positive PintE values for aqueous dimethylformamide, 1,4-dioxane, acetonitrile, and tetrahydrofuran at 6 °C, again in line with self-interaction of water in such mixtures (except, perhaps, for dimethylformamide to a small extent).87 However, in a recent paper Dhondge et al.92 arrived at erroneous conclusions concerning dilute solutions of alkanols in water based on erroneous values of the isobaric expansibilities they used in the application of eq 3 for obtaining the internal pressures of these solutions. I

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A further aspect of dilute solutions of nonelectrolytes occurs when molecular complexes are transferred from the gas phase into a liquid solvent and exposed there to its internal pressure. Trotter105,106 investigated the spectral changes accompanying this transfer. He distinguished between the thermodynamic internal pressure Pint and the microscopic mechanical pressure PM. The latter arises from the work required to form the cavity for accommodating the solute according to the scaled particle theory

regular solution behavior, as indeed many do, then the enthalpy of mixing can be expressed in terms of the excess internal pressure −PintE (on a mole fraction Pintid basis). Ejima and Yamamura48 studied mixtures of molten NaNO3 + KNO3 at 350 °C and found very small (negative) PintE values (again on a mole fraction Pintid basis). Sanguri and Singh56 applied Flory theory to mixtures of molten alkali halides and obtained values for their Pint. As mentioned in section 2.2 for their values for the individual salts indirect evaluation of the expansibility and compressibility did not lead to correct values.

PM = (kT /πb2){[6y/(1 − y)](a + 2b)

4. INTERNAL PRESSURES OF DILUTE SOLUTIONS

+ [18y 2 /(1 − y)2 ]b}

(23)

4.1. Internal Pressures of Solutions of Nonelectrolytes

In this expression a is the molecular diameter of the solvent, b is that of the solute, and y is the packing fraction defined for eq 12, and the macroscopic pressure is neglected in view of the magnitude of PM of hundreds of MPa. The mechanical pressure on a solute by a solvent is smaller than the internal pressure; for a solute of diameter of b = 1.1 nm the PM/MPa and Pint/MPa values at 25 °C for several solvents are for n-heptane (a = 0.67 nm) 234 and 254, for carbon tetrachloride (a = 0.59 nm) 336 and 369, for benzene (a = 0.53 nm) 157 and 410, and for carbon disulfide (a = 0.44 nm) 149 and 547.105 Relatively weak molecular complexes that exhibit charge transfer spectra (such as iodine-aromatic ones) transferring from the vapor phase into nonpolar, inert solvents are then expected to show considerable red shifts (of 1000−4000 cm−1) and intensity enhancement of their spectra. This is due to the complexes undergoing compression by the PM of the solvent as they do on application of large external pressures. Hydrogen-bonded molecular complexes are stronger, and their donor−acceptor bond length is less sensitive to compressive pressures than the molecular complexes; they show smaller red shifts on transfer from the vapor to the solution phases.106 Duran-Zenteno et al.107 recently reported the internal pressures of dilute solutions of an alkaloid, boldine, in mixtures of ethanol and 1-propanol over extensive temperature and pressure ranges (40−90 °C and up to 20 MPa).

Special consideration may be accorded to the internal pressures of dilute aqueous solutions of nonelectrolytes, because according to Dack101 these may play a role in understanding their water structure-making and -breaking properties. He obtained Pint from the measured thermal pressure coefficient, γV, for dilute aqueous solutions at 25 °C of eight organic substances, added the results from Macdonald and Hyne10 for another three, and interpolated the data to obtain values for 1 M solutions. Additional Pint values are available from Zaichikov,102,103 three amides at 25 °C and N-methylacetamide at 35 °C, that could also be interpolated to 1 M concentration. Dhondge et al.92 provided data dPint/dm for dilute aqueous solutions of alkanols at 25 °C. If the linear dependence of Pint on the molality m persists beyond the measurement limit of 0.25 m, then dPint/dm could represent the values of ΔPint = Pint(1 m) − Pint(W) at 1 M with good approximation. Dhondge and co-workers90,91 supplied data, albeit at 6 °C, for another seven organic substances that again may be interpolated to 1 M concentration. Conti and Matteoli104 reported ΔPint = Pint(1 m) − Pint(W) values for mono- and dihydric alkanols and cyclic mono- and diethers at 25 °C. They obtained the Pint(1 m) from the apparent molar isobaric expansibilities and isothermal compressibilities of the solutes to yield with the values for 1 kg of water the values for the 1 m solutions (they should be practically the same as for 1 M). PintE data for aqueous 1propanol, tert-butanol, ethylene glycol, and glycerol presented by Srivastava and Tripathy86 are negative, contrary to results by other workers. They were obtained by a theoretical expression that cannot be valid as mentioned above (section 2.3) and cannot be used. Other than the latter, the published values for dilute aqueous solutions of organic solutes are shown in Table 10, where it is seen that agreement among data reported by different authors is generally poor. Dack101 argued that the increase of ΔPint with molar volumes of the solutes is related to transfer of water molecules from bulky to compact domains (leaning on the two-structure model of water), i.e., to water structure breaking. However, he arbitrarily selected five solutes (urea, formamide, acetonitrile, 1,4-dioxane, and piperidine) as representing ‘noninteracting’ solutes and proposed that solutes having ΔPint values lying below a line defined by these five solutes are water-structure makers and those lying above it are structure breakers. Strong hydrogen-bonding protic solutes, such as the amides and alkanols, indeed have ΔPint values below the line defined by Dack.101 It is certainly desirable, in view of the poor agreement among the data shown in Table 10, to have measurements (at 1 M concentration) of the internal pressures of more nonelectrolyte aqueous solutes for better (and less arbitrary) establishment of the water structural effects.

4.2. Internal Pressure of Aqueous Electrolyte Solutions

Values of the internal pressures Pint of aqueous electrolyte solutions are rather scarce, but available data indicate that they are linear with the concentration in dilute solutions. Dack101 reported values for 14 salts at 1 M concentration and 25 °C obtained by isochoral thermal pressure coefficient measurements. Leyendekkers108 added estimates of Pint for another 8 salts at 1 m concentration and 25 °C, obtained from several expressions. Available values are shown in Table 11. Kumar109 provided values for 2 M aqueous salts (at an unspecified temperature), from which values for 1 M concentration are obtained by linear interpolation and included in Table 11. Srivastava and Berkowitz81 showed plots of Pint for 7 rather concentrated aqueous salts against their osmotic pressures. Pint values were obtained from their relationship to the speed of ultrasound and the molar refraction. Although the plots were linear beyond some low concentration, they could not be used to calculate Pint values from the osmotic pressures at as low as 1 M or 1 m concentrations. Electrostriction of aqueous electrolyte solutions has been related by several authors to the effective pressure, Peff, that the electrolyte induces in solution, but some authors chose to call this quantity the internal pressure.77,110 The idea can be traced J

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Table 11. Internal Pressure Increments ΔPint = Pint(solution) − Pint(W) of Electrolyte Solutions at 25 °C electrolyte LiCl LiI Li2SO4 NaCl NaBr NaI NaOH NaSCN NaBF4 NaClO4 NaPF6 NaAsF6 Na2CO3 Na2SO4 KCl KBr KSCN KNO3 (NH4)2SO4 Et4NBr GuClc GuBrc GuBF4c GuClO4c MgCl2 MgSO4 BaCl2 CuSO4 CdCl2

ΔPint/MPa at 1 M101

ΔPint/MPa at 1 M108

19 33 74 60 72

27 36 83 57

Pint ‐nih/MPa = Pint(W) + 22.0 + 0.518(Vnih /cm 3·mol−1) (27)

ΔPint/MPa at 1 M109

Thus, Peff could be obtained, and its concentration derivative dPeff/dcE was used in eq 25. Resulting electrostriction volumes Velst∞ agree fairly well with Mukerjee’s values, notable exceptions being lithium and magnesium sulfates. The notion of the effective pressure was used by McDevitt and Long115 to express the salting-out effect of electrolytes on nonelectrolytes. The Setchenow constant is given by

112 108

49

kE,N = lim(cE , c N → 0)(s N0 /s N)

80 50 58

143 138 62

= lim(cE→ 0)(dPeff /dc E)VN * × a /(a + b)

35 −2 −6 −10

where sN0 and sN are the solubilities of the nonelectrolyte (subscript N) in the absence and presence of cE molar electrolyte. On the right-hand side of eq 28 the molar volume of the pure solute VN* is modified as VN*a/(a + b), where a is the mean diameter of the ions of the electrolyte and b is the diameter of the molecules of the solute. This should take care of the noncontinuous volume of the solute and the distance of approach of its molecules to the ions. Deno and Spink116 (equating conceptually Peff with Pint), setting arbitrarily a/(a + b) = 0.3, extended the calculation to several nonelectrolytes, both polar and nonpolar, other than the benzene treated by McDevitt and Long.115 A further modification is due to Xie et al.,117 who set

131 125 60 64 58 55 76

78 −7 −8 −16 −21 80 113 98 89

a = [ν+(r+ + 0.085) + ν−(r − + 0.010)]/(ν+ + ν−)

57 80 102

Assuming Pint(W) = 167.8 MPa and taking the Pint values from Table 3 in ref 108 or the mean of the entries of the second and third columns of Table 7 there. bInterpolated linearly from the reported values for 2 M solutions using Pint(W) = 158 MPa. cGu = guanidinium.

to Tamman111 and Gibson112,113 in the following manner. The Tait equation, eq 4, should be modified for solutions112 to read [V (P) − V (1)]/V (1) = A log[(B + P + Peff )/(B + Peff )] (24)

In dilute aqueous solutions the effective pressure is linear with the product cW·cE of the molar concentrations of water cW and the electrolytes cE, but the limiting slopes should represent dPeff/dcE, because at infinite dilution cW is a constant.113 Mukerjee110 derived the molar electrostriction caused by the electrolyte at infinite dilution as

Pint = Pint ‐nih + Peff + Plr

(25)

(30)

Here Pint‑nih is given by eq 27, Peff is obtained from the Tamman−Tait−Gibson (TTG) expression (eq 24) as

where V∞ is the standard partial molar volume of the electrolyte and Vitr its intrinsic volume and taking B in eq 24 to be independent of the electrolyte concentration. Dack114 proposed that the effective pressure Peff in an electrolyte solution is the difference between its internal pressure and the internal pressure due to a noninteracting homomorph Pint‑nih of the same size (molar volume) as that of the electrolyte Peff = Pint − Pint ‐nih

(29)

Here ν are the stoichiometric coefficients and r the ionic radii of cation and anion (addends were due to Latimer). Xie et al.117 also set b = [0.7402 (3/4πNA)VN*]1/3, where 0.7402 is the packing fraction. By this device good agreement was obtained between the calculated (by eq 28) and experimental values of the Setchenow constant kN,E for several aromatic hydrocarbons and some two dozen electrolytes. Application of eqs 25 and 28 relates the Setchenow constant to electrostriction. Returning to eqs 26 and 27, Dack101 suggested that the electrolytes studied by him (Table 11) have Peff > 0 and disrupt the structure of the water, except for LiCl, LiI, and Et4NBr that have Peff < 0 and are water structure promoters. He could not explain why, e.g., Na2SO4, BaCl2, and Na2CO3 should be deemed as structure breakers, whereas other phenomena, such as positive viscosity B coefficients, do classify them as structure makers. Leyendekkers108 described the internal pressure of aqueous electrolyte solutions by means of several expressions, but the results converge to a form arising from inversion of eq 26 with an extra term

a

Velst∞ = V ∞ − Vitr = κT × lim(c → 0)(dPeff /dc E)

(28)

Peff = (B + 1)[exp(m · f(m)) − 1]

(31)

where B is the temperature-dependent Tait parameter, m is the molality of the electrolyte f (m) = a0 + a1m1/2 + a 2m

(26)

(32)

and the ai parameters are obtained from the apparent molar volume and the limiting compressibility of the aqueous electrolyte. The long-range charge effect, Plr, is estimated from the temperature dependence of the internal pressure, the

The latter quantity was obtained from the molar volume dependence of Pint of the five arbitrarily selected nonelectrolytes mentioned above101 K

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4.3. Internal Pressures of Electrolytes in Nonaqueous Solvents

near-infrared band wavelength, and the dielectric relaxation of pure water in a manner not further elaborated. A further development of this theme by Leyendekkers118 ascribed individual ionic values to the ΔPint = Pint(1 m) − Pint(W) of electrolyte solutions. The TTG effective ionic pressure is given by Peff (ion)/MPa = 2.190m[− (V ∞ − Vitr) − SV m1/2]

A few values have been reported for the internal pressures of metal perchlorates in nonaqueous solvents in relation to their effects on reaction rates of certain systems. Kumar124 reported data for the concentration dependence of Pint, from which values can be calculated for 1 M concentration, for comparison with the values in aqueous solutions (Table 11). The resulting ΔPintr/MPa values for lithium perchlorate at 1M, read from a plot for 3 M solutions assuming linearity with the concentration, are 60 in dimethylformamide, 150 in diethyl carbonate, 167 in propylene carbonate, and 233 in diethyl ether. However, he also reported ΔPintr/MPa of only 171 in diethyl ether in a table, where additionally the values 209 in ethyl acetate, 259 in tetrahydrofuran (note that the table lists ethyl acetate twice, the second entry should be tetrahydrofuran), and 197 in acetone are given. For acetone solutions of other 1 M perchlorates ΔPintr/MPa is 107 for the sodium salt, 407 for the magnesium salt, and 201 for the barium salt.

(33)

where the first term in the square bracket is the negative of the electrostriction volume, eq 25, and SV is the slope of the Masson expression for the apparent molar volumes φV φ

V = φV ∞ + SV m1/2

(34)

The volume−pressure term, eq 27, was modified to read Pint ‐nih(ion)/MPa = 1.173Vitr

(35)

However, no explicit independent expression for obtaining the ionic long-range term, Plr(ion)/MPa, was given, and this term was estimated, ranging from 3.3 for Cl− (arbitrarily fixed) to 122.7 for Al3+ (Cs+ with 2.8 and F− with 2.7 had lower values than that for Cl−). Volumes in the above expressions are in cm3·mol−1, and Vitr is the TTG intrinsic molar volume, 2522(r + Δ)3, r is the ionic radius (in nm), and Δ is an addend that cannot be evaluated independently (i.e., it is derived from the TTG value of Vitr).119 A table was provided118 of additive ionic Δpi (not defined in the paper) values at 25 °C and 1 m that ought to represent Pint(1 m) − Pint(W) when the cation and anion values are added, but it does so only roughly. Calculated ionic values are included for ions (Rb+, Cs+, Ca2+, Sr2+, Al3+, F−, and CrO42−) for which no experimental or estimated values for salts involving them were reported (see Table 11). Internal pressures were obtained for aqueous solutions of electrolytes in the presence of nonelectrolytes in several publications, some of which were very recent. Macdonald et al.120 evaluated the internal pressures of solution of alkali halides in 0.4 M aqueous mannitol at 25−40 °C from their free volumes, Vf. The latter were obtained from the viscosity and ultrasound speeds, and the internal pressure was calculated from Pint = 2RTVm−2/3Vf −1/3

5. REACTIONS IN SOLUTION AND THE INTERNAL PRESSURE OF THE SOLVENT Reactions proceeding in solution are affected by many factors, and the authors have searched the most effective ones in order to promote the favorable outcome of the reactions in terms of their rates or the nature of the products. In some cases authors have ascribed the observed effect to the internal pressure of the medium. In fact, Ouellette and Williams125 stated that interpretation of observed experimental facts presents a challenge and proposed that the internal pressure of the solvent, which previously escaped exploitation in this regard, ought to be considered as responsible for certain results. This has been done, e.g., by Turgis et al.97 in the case of conversion of the nitrous oxime precursor to ε-caprolactam catalyzed by the room-temperature ionic liquid 1-methyl-3-(4sulfonylbutyl)imidazolium hydrogensulfate. They suggested that the large internal pressure of the medium caused effective catalysis but did not prove this point. Ouellette and Williams125 considered using the internal pressure of the medium for affecting conformation equilibria. They chose four solvents that should present negligible specific solvent effects, namely, perfluoroheptane, carbon tetrachloride, benzene, and carbon disulfide, with Pint of 214, 345, 370, and 383 MPa, and obtained equilibrium constants for conformational equilibria of 2,3-dimethyl-2-silabutane: 4.18, 5.10, 5.62, and 6.10, respectively, showing a good correlation and enhancement of the gauche form with increasing Pint. However, the readers may note that other solvent properties can play a role: solvent polarizabilities, for instance, diminish in the same order as Pint increase, so that it might be concluded that the gauche form is disfavored by high solvent polarizability. This example illustrates the dilemma that Ouellette and Williams125 presented at the beginning of their paper. Activation volumes, ΔV‡, of reactions are obtained from the dependence of their rate on external pressure and are positive if the rate diminishes with increasing pressure. Neuman126 compared ΔV‡ of the decomposition reactions of various free radical initiators in a given solvent at a given temperature with the expectations from the variation of Pint in different solvents at the same temperature with poor agreement. He concluded that there was no obvious correlation between the effects of external pressure and the internal pressure. However, other

(36)

where Vm is the molar volume of the medium. However, the resulting values (labeled P in the tables119) are excessively large, e.g., 2867 MPa for 1 M LiCl at 25 °C, and diminish with increasing concentrations (except for LiCl and NaCl), contrary to other experimental values, so that the reported Pint values must be incorrect. Chimankar et al. reported internal pressures for aqueous solutions of mixtures of amino acids and NaOH121 and of glycylglycine and NaCl.122 They did not specify how they obtained the internal pressure from the measured densities and ultrasound speeds, and its values shown in plots against the amino acid or peptide concentrations are wavy without any reason. The value for 1.5 M NaCl at 30 °C in the absence of the organic solute, 115 MPa,122 is lower than that of water at this temperature and hence must be incorrect, as are the other reported values in both studies. Kanhekar and Bichile123 obtained Pint values from eq 36 applied to NaCl and MgCl2 at varying molalities in aqueous 1 M alanine solutions at 25, 30, and 35 °C. However, the reported value for 1 M alanine in the absence of salt, 39.85 MPa at 25 °C, is lower than that of water, so the resulting Pint data in this study must be incorrect. L

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researchers came to different conclusions. For nonpolar reactions, unimolecular decompositions have ΔV‡ > 0 and ought to be retarded in solvents with larger Pint, whereas in a bimolecular reaction, such as the Diels−Alders one, ΔV‡ < 0 and their rates should be accelerated in this direction, as Dack18,127 pointed out. For polar reactions, in which charge redistribution far outweighs the effects of internal pressure, such correlations of Pint with the sign of ΔV‡ do not exist. Some authors (e.g., refs 128−130) used the concepts of internal pressure and cohesive energy density interchangeably for explanation of reaction rates, but it was pointed out that alternative explanations cannot be excluded.129 Faita and Righetti131 concluded that a relatively small (≤1 order of magnitude) rate enhancement of reactions with ΔV‡ < 0 is a useful criterion for attributing the enhancement to internal pressure effects, whereas a large rate enhancement (≥100-fold) should be due to other effects. Kumar in a series of papers109,124,132−134 examined various Diels−Alder reactions in aqueous, nonaqueous, and mixed solvents in relation to the internal pressure of the solvents. He first suggested Pint effects as an alternative explanation for the observed rate enhancement in aqueous salt solutions with different anions.109 He then considered also perchlorate solutions in nonaqueous solvents and argued that ΔPint in a salt solution is equivalent to application of a high external pressure.124 It appears that the activation volume values, ΔV‡, used in demonstrating the utility of ΔPint for explaining rate enhancement of Diels−Alder reactions were not obtained independently but were derived from the internal pressure data. Kumar’s view of equating the results of ΔPint to application of an external pressure was refuted, however, by Faita and Righetti,131 who showed that for reactions of nitrosoarenes with dimethylbutadiene much smaller external pressures lead to a given rate enhancement than the ΔPint required for perchlorate salts in acetone. Still, Kumar argued that cases where no rate enhancement of Diels−Alder reactions even by high concentrations of salts, having appreciable ΔPint values, were observed due to negligibly small (negative) ΔV‡ values. Even non-Diels− Alder reactions, such as isomerization of 1-phenylallyl chloride to cinnamyl chloride, are accelerated considerably by nonaqueous solutions of lithium perchlorate if the reactions have large negative ΔV‡ values (in this case −80 cm3 mol−1).124 In a subsequent paper Kumar132 dealt with solvent effects in the absence of salts on the rates and stereoselectivity of Diels− Alder reactions. He introduced a modification of the internal pressure of the solvent in terms of its refractive index nD divided by that of n-hexane using the variable Q = 2[Pint/(nDsolvent/ nDhexane)]1/2. For an obscure reason he termed the ratio nDsolvent/nDhexane as indicating the polarity of the solvent, whereas it relates only to its polarizability, not polarity. The relative rates of formation of the endo and exo isomers were found to be linear with Q, but the relative rates of the reactions as a whole were quadratic in Q for 75 Diels−Alder reactions in neat and mixed solvents. Melting of the DNA duplex forming two strands is affected in aqueous salt solutions, the reaction having a positive volume change, ΔV. Salts such as NaCl and MgCl2 with ΔPint > 0 in aqueous solutions raise the melting point, whereas salts such as KSCN, NaClO4, and NaCF3CO2 with ΔPint < 0 lower it, but the ΔPint values employed were not given, and their signs do not agree with reported ones (Table 11).133 In a subsequent paper Kumar and Deshpande134 conceded that although the internal pressure increment in aqueous salt solutions, ΔPint, is

an explanation for the rate enhancement or endo/exo ratios of reactions with ΔV‡ < 0, the electrostriction caused by the ions of the salt is an alternative explanation, the two concepts being interrelated, see the discussion around eqs 25−27. LeNoble135 questioned the validity of interpretations of reaction rates in terms of the internal pressure of the medium in a paper with the provocative title “internal pressure: its uses and abuses”. However, he did not add any new insights above those already available at the time. A further criticism of the conclusion of Kumar and Deshpande134 concerning the importance of the Pint of the medium with regard to Diels− Alder reactions was expressed by Graziano,136 who demonstrated that water, with the lowest Pint, has the highest reaction rate in a series of pure solvents, some dipolar and others nonpolar. On looking at all the evidence it is clear that even with the criterion proposed by Faita and Righetti131 (only moderate rate enhancement and equilibrium constant effects should be ascribed to internal pressure effects) other solvent effects ought to be excluded before the solvent internal pressure is made responsible for the observed phenomena.

6. DISCUSSION AND CONCLUSIONS Liquids have a property, the internal pressure, Pint, that can be measured by their isochoric thermal pressure coefficient T(∂P/ ∂T)V − P (eq 2) or the ratios of their isobaric expansibility to their isothermal compressibility, TαP/κT − P (eq 3). The −P term is generally negligible for liquids (but see arguments to the contrary in ref 2). This quantity has been tabulated for a large number of liquids ranging from some liquefied permanent gases (Table 8), through liquids at ambient temperatures and pressures (Table 2), room-temperature ionic liquids (Table 3), some liquid polymers (Table 4), to molten salts (Table 5), only examples of the liquids and values having been presented here. The values depend on the temperature (Tables 5 and 8 for some liquids and Table 9 for water) and pressure. The values of Pint listed in these tables are estimated to be accurate to within ±5 MPa, better than appears from the older comparison18 in Table 1, but as results from a comparison with the more accurate values for liquids at ambient conditions obtained from application of the reference equations of state.138 The internal pressure differs from the cohesive energy density, ced = (ΔVH − RT)/V, except for liquids which are held together by van der Waals forces only. Other liquids may be ‘loose’ if the repulsive forces between the molecules dominate and have n = Pint/ced > 1, whereas highly polar and hydrogen bonded liquids are ‘tight’ and have n < 1, an extreme example being water (but with increasing temperatures n increases for water and eventually, close to the critical point, exceeds unity). The internal pressure of liquids is related to other properties of them to some extent, foremost being the surface tension, σ (Figure 1 and eqs 11 and 12). Another property that should be mentioned is the free volume, Vf, that can be obtained from the speeds of ultrasound in the gas and liquid phases of the substance, eq 15. Other relationships, such as eq 19 with regard to the density, viscosity, and speed of ultrasound, that have been used by some authors in lieu of the thermodynamically sound definitions, eqs 2 and 3, probably lead to erroneous values. The internal pressure of liquid mixtures generally deviated from ideality, but care must be exercised in determination of the ideal internal pressure. This may not be the value prorated by the mole fractions of the components of the mixture nor M

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electrolytes. He has published in this and neighboring fields 8 books and over 300 papers in refereed journals as well as some 30 chapters in multiauthor books. Eight years ago he took up painting in parallel with his research, and the photo is a recent self-portrait as a lecturer.

even that prorated by their volume fractions. Correct ideal internal pressure values are those for which the isobaric expansibilities and isothermal compressibilities of each component are prorated individually with respect to the volume fractions. Excess internal pressures then agree with other excess functions for the mixtures that indicate, when positive, preferred self rather than mutual interactions of the components. The internal pressure increments, ΔPint = Pint(cE) − Pint(W), of dilute aqueous solutions of organic solutes are linear with the concentration, cE. However, the values inter- or extrapolated for 1 M (Table 10) obtained by several workers are in poor mutual agreement. Due to this situation, interpretation of the results for some solutes (having ΔPint below an arbitrary line with respect to the molar volume) as water structure enhancers and of others (above the line) as water structure breakers appears to be premature. The situation may be better with respect to the ΔPint of electrolytes (Table 11), but even here there is poor agreement of results for some salts. Nevertheless, a clear relationship has been established between ΔPint and the electrostriction caused by the electrolyte, eqs 25−27 by the intermediacy of the concept of the effective pressure of the solution. Eventually these relationships are applicable to estimation of the salting out (or in, in rare cases) of solutes by electrolyte solutions. Enhancement of the rates of reactions where no charge redistribution occurs but where the volume of activation is negative, ΔV‡ < 0, in certain series of solvents has been ascribed by some authors to the internal pressure effects. However, this view was refuted by other authors who could suggest other interpretations of the data. At present there is no firm basis for making Pint responsible for the observed facts.

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AUTHOR INFORMATION Notes

The authors declare no competing financial interest. Biography

Yizhak Marcus was born in Germany but received all his education in Jerusalem, where he obtained his Ph.D. degree from the Hebrew University in 1956. He was a researcher at the Soreq Nuclear Research Center, dealing mainly with actinide chemistry, ion exchange, and solvent extraction. From there he was called in 1965 and appointed Professor of Inorganic and Analytical Chemistry at the Hebrew University. There he has taught and done research till his retirement in 1999, but he continues with research as Professor Emeritus. His main interest is the chemistry of liquids and solutions: aqueous, nonaqueous, and mixed solvents, solutions of electrolytes and nonN

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