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Thermodynamic Properties of Gases Thermodynamic Properties of Gases - mixture of ideal gases- mixture of ideal gases
Mixture of Ideal Gases
Definition of Mole fraction: xi
Definition of partial pressure: pi
Partial molar quantities:
Pxp ii
1 mole of ideal gas @ constant T:
1
212 ln),(),(
P
PRTTPGTPG
PRTddPP
RTVdPdG ln
PRTTGTPG o ln)(),( PRTGG o ln
,,,,
'
kj nnPTii n
QQ
iiG
i
compT
i VP
G
,iiQnQ '
Byeong-Joo Lee http://cmse.postech.ac.kr
Thermodynamic Properties of Gases Thermodynamic Properties of Gases - mixture of ideal gases- mixture of ideal gases
Heat of Mixing of Ideal Gases
io
i HH
0' io
ii
iii
mix HnHnH
PRTxRTGG iio
i lnln
T
TG
T
TG io
i
)/()/(
Entropy of Mixing of Ideal Gases
Gibbs Free Energy of Mixing of Ideal Gases
mixmixmix STHG '''
iii
io
ii
iii
mix xRTnGnGnG ln'
iii
mix xRnS ln'
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fRTddG ln
1P
f as 0P
For Equation of state P
RTV
fRTdVdP ln dPRTP
fd
ln
RT
P
P
f
P
f
PPP
0
lnlnid
RTP
P
P
RT
PV
RT
Pe
P
f 1/
Thermodynamic Properties of Gases Thermodynamic Properties of Gases - Treatment of nonideal gases- Treatment of nonideal gases
Introduction of fugacity, f
fRTGG o ln
※ actual pressure of the gas is the geometric mean of the fugacity and the ideal P ※ The percentage error involved in assuming the fugacity to be equal to the
pressure is the same as the percentage departure from the ideal gas law
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dPPRT
VdP
RTP
fd
1
ln
RT
PVZ dP
P
Z
P
fd
1ln
dPP
Z
P
f PP
PPP
1ln
0
PdRTP
fdRTfRTddG lnlnln
JRTP
fRTG 112971137376150lnln
150
Thermodynamic Properties of Gases Thermodynamic Properties of Gases - Treatment of nonideal gases- Treatment of nonideal gases
Alternatively,
Example) Difference between the Gibbs energy at P=150 atm and P=1 atm for 1 mole of nitrogen at 0 oC
Byeong-Joo Lee http://cmse.postech.ac.kr
Solution Thermodynamics Solution Thermodynamics - Mixture of Condensed Phases- Mixture of Condensed Phases
Vaper A: oPA
Condensed Phase A
Vaper B: oPB
Condensed Phase B
+ →
Vaper A+ B: PA + PB
Condensed Phase A + B
condensedA
ovaporA
o GG condensedB
ovaporB
o GG condensedA
vaporA GG
condensedB
vaporB GG
io
ii
iii
mix GnGnG 'i
oi
ii p
pRTn ln for gas
Byeong-Joo Lee http://cmse.postech.ac.kr
oiie kpr )(
Solution Thermodynamics Solution Thermodynamics - ideal vs. non-ideal solution- ideal vs. non-ideal solution
1.Gibbs energy of formation 과 Gibbs energy of mixing 의 차이는 무엇인가 ?
2. Solution 에서 한 성분이 Henrian 또는 Raoultian 거동을 한다는 것을 무엇을 의미하는가 ? Molar Gibbs energy 가 다음과 같이 표현되는 A-B 2 원 Solution phase 에서 각 성분은 dilute 영역에서는 Henrian 거동을 , rich 영역에서는 Raoultian 거동을 보인다는 것을 증명하시오 .
LxxxxxxRTGxGxG BABBAABo
BAo
Am }lnln{
ExampleExample
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i
ii QdX 0
0lnln BBAA adXadX
BA
BA ad
X
Xad loglog
BA
BXXata
XataXXA adX
Xa
AAB
ABAA
log)(loglog
1log
Solution Thermodynamics Solution Thermodynamics - Use of Gibbs-Duhem Relation - I- Use of Gibbs-Duhem Relation - I
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0lnln BBAA dXdX
BA
BA d
X
Xd lnln
BA
BXXat
XatXXA dX
XAAB
ABAA
ln)(ln
ln
1ln
Solution Thermodynamics Solution Thermodynamics - Use of Gibbs-Duhem Relation - II- Use of Gibbs-Duhem Relation - II
Byeong-Joo Lee http://cmse.postech.ac.kr
2)1(
ln
i
ii X
AB
XX
XBABA dXXXAA
A
1ln
BA
BA d
X
Xd lnln
BABABB dXXdXX 2
Solution Thermodynamics Solution Thermodynamics - Introduction of - Introduction of αα-function-function
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Fe-Ni Fe-CuFe-Ni Fe-Cu
Solution Thermodynamics Solution Thermodynamics - Composition Dependence of - Composition Dependence of αα-function-function
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• Margules, 1895.Margules, 1895.
333
1222
11ln BBBA XXX
333
1222
11ln AAAB XXX
• Hildebrand, 1929. Hildebrand, 1929. (using van Laar Equation)(using van Laar Equation)