PARTIAL GIBBES FREE ENERGY AND GIBBES/ DUHEM EQUATION, RELATION BETWEEN BINARY

SOLUTION , RELATION BETWEEN PARTIAL PROPERTIES

PARTIAL PROPERTIES Define the partial molar property of species i

jnTPii n

nMM,,

)(

• A partial molar property is a thermodynamic quantity which indicates how an extensive property of a solution or mixture varies with changes in the molar composition of the mixture at constant temperature and pressure. 

• When one mole of water is added to a large volume of water at 25 °C, the volume increases by 18 cm3. The molar volume of pure water would thus be reported as 18 cm3 mol−1. However, addition of one mole of water to a large volume of pure ethanol results in an increase in volume of only 14 cm3. • The reason that the increase is different is that the volume occupied by a given number of water molecules depends upon the identity of the surrounding molecules. The value 14 cm3 is said to be the partial molar volume of water in ethanol.

THE GIBBS/DUHEM EQUATION

Define the partial molar property of species i:

the chemical potential and the particle molar Gibbs energy are identical:

for thermodynamic property M:

jnTPii n

nMM,,

)(

ii G,...),...,,,,( 21 innnTPMnM

i

iinPnT

dnMdTTMndP

PMnnMd

,,

)(

i

iinPnT

dnMdTTMndP

PMnnMd

,,

)(

i

iiinPnT

ndxdnxMdTTMndP

PMnMdnndM )(

,,

0,,

dnMxMndxMdTTMdP

PMdM

iii

iii

nPnT

0,,

i

iinPnT

dxMdTTMdP

PMdM and 0

iiiMxM

Calculation of mixture properties from partial properties

0 i

iiMnnM

0,,

iii

nPnT

MdxdTTMdP

PM

i

iii

ii dxMMdxdM

The Gibbs/Duhem equation

•Binary solution is a mixture of two liquids that are completely miscible one with another. The boiling point of binary solution depends upon the solution composition and there can be three cases:

1. the boiling points of solutions of all compositions lie between the boiling points of clean liquids2. the boiling points of solutions of any composition lie above the boiling points of clean liquids3. the boiling points of solutions of some compositions lie below the boiling points of clean liquids

WHAT IS BINARY SOLUTION

PARTIAL PROPERTIES IN BINARY SOLUTION For binary system

2211 MxMxM

22221111 dxMMdxdxMMdxdM

Const. P and T, using Gibbs/Duhem equation

2211 dxMdxMdM

121 xx

211

MMdxdM

1

21 dxdMxMM

112 dxdMxMM

2211 VxVxV molcmV /025.24)765.17)(7.0()632.38)(3.0( 3

molVVnt

246.83025.24

2000

moln 974.24)246.83)(3.0(1

moln 272.58)246.83)(7.0(2

3111 1017)727.40)(974.24( cmVnV t 3

222 1053)068.18)(272.58( cmVnV t

EXAMPLE : 1

The need arises in a laboratory for 2000 cm3 of an antifreeze solution consisting of 30 mol-% methanol in water. What volumes of pure methanol and of pure water at 25°C must be mixed to form the 2000 cm3 of antifreeze at 25°C? The partial and pure molar volumes are given.

Methanol (1): V1 = 38.632 cm3 /mol V1 = 40.727 cm3 /mol Water (2) : V2 = 17.765 cm3 /mol V2 = 18.068 cm3 /mol

The total number of moles required is :

Now, Volume of the each species:

Fig 11.2

The enthalpy of a binary liquid system of species 1 and 2 at fixed T and P is:

Determine expressions for and as functions of x1, numerical values for the pure-species enthalpies H1 and H2, and numerical values for the partial enthalpies at infinite dilution and

1H 2H

1H

2H

)2040(600400 212121 xxxxxxH

)2040(600400 212121 xxxxxxH

121 xx311 20180600 xxH

121 dxdHxHH

31

211 4060420 xxH

121 xx312 40600 xH

01 x

molJH 4201

11 x

molJH 6402

EXAMPLE : 2

RELATIONS AMONG PARTIAL PROPERTIES

Maxwell relation:

i

iidnGdTnSdPnVnGd )()()(

nTnP PS

TV

,,

jnTPinP

i

nnS

TG

,,,

)(

jnTPinT

i

nnV

PG

,,,

)(

ixP

i STG

,i

xT

i VPG

,

PVUH

dTSdPVGd iii

iii VPUH

Every equation that provides the linear relation among thermodynamics properties if a constant composition solution has as it’s counterpart an equation connecting corresponding partial properties of each species in that solution.

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