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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:
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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