Chapter 4

Vapor Pressure

pº = Pressure of a substance in equilibrium with its pure condensed (liquid or solid) phase

Why do we care?

-spills

-pesticide application

-will lead us to Henry’s law constant

Air

WaterOctanol

A gas is a gas is a gasT, P

Fresh, salt, ground, poreT, salinity, cosolvents

NOM, biological lipids, other solvents T, chemical composition

Pure Phase(l) or (s)

Ideal behavior

PoL

Csatw

Csato

KH = PoL/Csat

w

KoaKH

Kow = Csato/Csat

w

Kow

Koa = Csato/Po

L

Ranges of pº (atm)– PCBs – 10-5 to 10-9

– n-alkanes – 100.2 to 10-16

• n-C10H22 ~ 10-2.5

• n-C20H42 ~ 10-9

– Benzene ~ 10-0.9

– toluene ~10-1.42

– Ethylbenzene ~ 10-1.90

– propyl benzene ~ 10-2.35

– carbon tetrachloride ~ 10-0.85

– methane 102.44

• Even though VP is “low”, gas phase may still be important.

Phase diagram

picture of three-phase diagram

Ideal Gas Law

nRTpV p = pressure

V = volume

n = moles of gas

R = gas constant

T = temperature (Kelvin)

Thermodynamic considerations(deriving the van’t Hoff equation)

21 dd

consider a gas: if T or P is changed and equilibrium is re-established:

the change in chemical potential in the two systems is equal

dpVdTSddpVdTSd

222

111

where S = molar entropy

and V = molar volume

12

12

21

21

)()(

VS

VVSS

dTdp

at equilibrium

STHGG

02112

substituting:

12

12

VTH

dTdp

for a liquid vaporizing, the volume change can be assumed to be equal to the volume of gas produced, since the volume of the solid or liquid is negligible

012 pRTVV gas Q. where did the n go?

A. this is molar volume

212

00 )(RTHp

dTdp

212

0lnRTH

dTpd

dxdu

udxud 1ln

recall (calculus!)

where H12 = Hvap (gas) or Hsub (solid)

= energy required to convert one mole of liquid (or solid) to gas without an increase in T.

Hvap is a function of T.

As T approaches the boiling point, Hvap increases rapidly

At T < boiling point, Hvap increases slowly

from 0-40ºC, Hvap can be assumed to be constant

The van’t Hoff equation

212

0lnRTH

dTpd

integrate assuming Hvap is constant:

aRTHp

120ln

BTAp 0ln

Antoine equation

if Hvap is not constant:

acT

bp

0lnanother Antoine equation

Using Hvap to predict VP at other temperatures

211

2 11lnTTR

Hpp vap

T

T

211

2 11lnTTR

HKK

T

T

As we saw in the thermodynamics lecture:

Specifically,

Note the change in slope when the substance is solid (sublimation)

Hsub = Hmelt (~25%) + Hvap (~75%)

still use liquid phase as reference:

Hypothetical subcooled liquid

= liquid cooled below melting point without crystallizingcompound pºs < pºL

1,4-dichlorobenzene 3.04 2.76phenol 3.59 3.4122’55’ PCB 7.60 6.6422’455’ PCB 8.02 7.40

Becomes important later when we talk about solubility

-log P

Molecular interactions affecting vapor pressure

Molecule:molecule interactions in condensed phase (L or s) have greatest affect on VP

strong interactions lead to large Hvap, low VP

weak interactions lead to small Hvap, high VP

Intermolecular interactions can be classified into three types:

van der Waals forces (nonpolar)

Polar forces

Hydrogen bonding

van der Waals forces• nonspecific• function of size (number of

electrons)• consist of:

– London dispersive energies • fleeting areas of charge

– Induced dipoles• areas of charge arising from

interactions with a polar molecule

“nonpolar”

Polar interactions:dipole-dipole interactions

• permanent areas of charge on two molecules attract

Hydrogen bondsSpecificdonors and acceptors

table 4.3

compound (class) (H-donor) (H-acceptor)alkanes 0 01-alkenes 0 0.07aliphatic ethers 0 0.45aliphatic aldehydes 0 0.45aliphatic alcohols 0.37 0.48carboxylic acids 0.60 0.45benzene 0 0.14phenol 0.6 0.31naphthalane 0 0.2f luorene 0 0.2pyrene 0 0.29DCM 0.1 0.05Water 0.82 0.35

Part of Table 4.3

Vapor Pressure Estimation Technique

5.14))((1.152149.4ln

2

2

23/2*

iiDi

DiiLiL n

nVp

based on regression of lots of VP data, best fit gives:

pressure in Pa, where:

index refractive

y)(MW/densit memolar volu

Di

iL

n

V

refractive index (response to light) is a function of polarizability.

see table 3.1, also might be available in the CRC

sizepolarizability

H-bonding ability

Refractive index

Difference between polarity and polarizability

Trouton’s ruleAt their boiling points, most organic compounds have a similar entropy of vaporization:

Svap (Tb) = 85 – 90 J/molK

We can be slightly more accurate with Kistiakowsky’s expression:

Svap (Tb) = KF(36.6 + 8.31ln(Tb)) J/molK Tb in K (eqn 4-20)

KF = 1 for most compounds

At the boiling point: vapbvap STHG 0

So if we know Tb, we can estimate Hvap (at the boiling point) fairly accurately

exception: strongly polar or H-bonding compounds

Table 4.2

Estimating VP at other T (need Hvap)

bTpaTH iLvap )(log)( 1*

1

Recognize that Hvap is not constant.

Especially if Tb is high (> 100ºC), the estimate of Hvap from Trouton/Kistiakowsky may not be valid at the temperature of interest.

Empirically, Hvap is a function of the VP:

211

2 11lnTTR

Hpp vap

T

T

FIG 4.7

From a data set of many compounds, Goss and Schwarzenbach (1999) get:

0.70)298(log80.8)298( * KpKH iLvap

Less empirically,

assume Hvap is linearly proportional to T (i.e. assume that the heat capacity, Cpvap is constant:

)()()()( TTTCTHTH bbvappbvapvap

substitute this expression into the Clausius-Clapeyron equation and integrate from Tb to T:

TT

R

TC

TT

R

TC

TTRTH

P

bbvappbbvapp

b

bvap

ln)(

1)(

11)(ln 0

don’t let notation confuse you. (Tb) means at the boiling point. You do not multiply Hvap by the boiling point

but we still need to know Cp(Tb)!

)()( bvapbbvap TSTTH Recall:

substitute:

TT

R

TC

TT

R

TC

RTS

p

bbvapp

bbvappbvap

ln)(

1)()(

ln 0

generally:)(8.0)( bvapbp TSTC ranges from 1.0 to 0.6

and Svap(Tb)~ 88 J/molK

finally!

TT

TTp bb ln5.8119ln 0

OK for liquids with Tb < 100ºC

High MW compounds, need correction for intermolecular forces (but we don’t have their boiling points anyway!) (For refinements see equation 4-33)

Can estimate boiling points, see p. 120

in atm

TT

TTTKp bb

bF ln8.018.1)ln4.4(ln 0

KF is the Fishtine factor, usually 1, but sometimes as high as 1.3 (see p 113)

the old edition gave (where KF =1):

in atmEqn 4-33

solids?those previous equations yielded the vapor pressure of the hypothetical subcooled liquid.

How can we correct this to give the true vapor pressure of a solid?

Prausnitz (1969):

Where Sfus(Tm) = entropy of fusion at melting point

unfortunately Sfus is much more variable than Svap

1

)(ln 0

0

TT

RTS

pp mmfus

L

s

)log2.192.95.56()( mfus TS J/molK

Where = number of torsional bonds and = rotational symmetry number (see p. 125)

the older edition of your book gave this simpler (but less accurate) equation:

Sfus(Tm) ~ 56.5 + 10.5(n-5) J/molK

Where n = number of flexing chain atoms.

if n<5, then ignore this term

Estimation of vapor pressures for polychlorinated biphenyls: a comparison of eleven predictive methodsLawrence P. Burkhard, Anders W. Andren, and David E. ArmstrongEnvironmental Science and Technology 1985, 19, 500 - 507

conclusions:

• non-correlative methods have poor predictive ability (error increases as VP decreases)

• correlative methods requiring a set of compounds with known P are much better

• best method: determine VP as function of GC retention times

Determination of vapor pressures for nonpolar and semipolar organic compounds from GC retention data (Hinckley et al, 1990)

• Chromatographed 2 reference compounds (eicosane and p,p’DDT) having known VP and Hvap versus a host of unknowns (PAHs, organochlorines, etc)

• Isothermal runs allow determination of RRT at several T• Comparison of RRT with reference compounds allows

determination of VP at given T• Comparison of changes in RRT with T and knowledge of

Hvap for reference compound allows calculation of Hvap

for all unknowns

Problem 4.2• In a dump site, you find an old 3-liter pressure bottle

containing FREON 12 with a pressure gauge that reads 2.7 bar. (First, you realize that this gauge was not manufactured in the US.) The temperature is 10ºC. What mass of FREON 12 is in the bottle?

• Also estimate the free energy, enthalpy, and entropy of condensation of FREON 12.

• You find the following info for FREON 12 in the CRC:

T deg C p/kPa-25 123

0 30825 65150 121675 2076

Problem 4.6

• estimate VP at 0C based on VP at 25ºC or based solely on Tb and Tm

log(p) @25C Tm (degC) Tb (degC)dimethyl phthalate 0.38 5.5 283.72,3,7,8-TCDD -6.7 305 446.5

(hint = 4)

Homework

• Do problems 4.3 and 4.4• Due 2/2/10