(q and/or w) a closed system is one that does not exchange matter with its surroundings, although it...
TRANSCRIPT
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FUNDAMENTAL EQUATIONS
Chapter 1
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HOMOGENEOUS CLOSED SYSTEM
(Q and/or W)
(Q and/or W)
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• A closed system is one that does not exchange matter with its surroundings, although it may exchange energy.
dni = 0 (i = 1, 2, …..) (1.1)
• No internal energy transported across boundary.
• All energy exchange between a closed system and its surroundings appears as heat and work.
• The total energy change of the surroundings equals the net energy transferred to or from it as heat and work.
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First and second laws of Thermodynamics:
PdVTdSdU (1.2)
For reversible process:
PdVTdSdU (1.3)
with T dS = dQrev : heat absorbed by the system - P dV = dWrev : work done by the system
If the interaction occurs irreversibly:
PdVTdSdU (1.4)
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The internal energy change can be calculated by integrating eq. (1.2):
2
1
2
1
12
V
V
S
S
PdVTdSUUU (1.5)
For process occurring at constant S and V:
0V,SdU (1.6)
• At constant S and V, U tends toward a minimum in an actual or irreversible process in a closed system, and remains constant in a reversible process.
• Eq. (1.6) provides a criterion for equilibrium in a closed system.
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Definition: PVUH (1.7)
Differentiating eq. (1.7) yields:
VdPPdVdUdH
VdPPdVPdVTdSdH
VdPTdSdH
Combining the above equation with eq. (1.3) leads to
For a closed system at constant S and P :0dH P,S
(1.8)
(1.9)
ENTHALPY
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the Helmholtz free energy (A) is a thermodynamic potential that measures the “useful” work obtainable from a closed system at a constant temperature and volume.
– A = the maximum amount of work extractable from a thermodynamic process in which temperature and volume are held constant.
Under these conditions, it is minimized at equilibrium.
HELMHOLTZ FREE ENERGY
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Definition:
TSUA (1.10)
Differentiating eq. (1.10) yields:
SdTTdSdUdA
SdTTdSPdVTdSdA
PdVSdTdA
Combining the above equation with eq. (1.3) leads to
For a closed system at constant T and V :0dA V,T
(1.11)
(1.12)
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GIBBS FREE ENERGY
Gibbs free energy (G) is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure (isothermal, isobaric).
The Gibbs free energy is the maximum amount of non-expansion work that can be extracted from a closed system; this maximum can be attained only in a completely reversible process.
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Definition:
TSHG (1.13)
Differentiating eq. (1.13) yields:
dTSdSTdHdG
dTSdSTVdPdSTdG
SdTVdPdG
Combining the above equation with eq. (1.8) leads to
For a closed system at constant P and T :0dG P,T
(1.14)
(1.15)
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MAXWELL EQUATIONIf F = F(x, y), the total differential of F is:
dyyF
dxxF
dFxy
NdyMdxdF
yxF
M
xyF
N
with
(1.16)
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Further differentiation yields
yxF
yM 2
x
yxF
xN 2
y
yx x
NyM
Hence from equation: NdyMdxdF
we obtain:
yx xN
yM
(1.17)
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Resume:
PdVTdSdU (1.3)
VdPTdSdH (1.8)
PdVSdTdA (1.11)
SdTVdPdG (1.14)
NdyMdxdF
yx xN
yM
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According to eq. (1.17):
VS SP
VT
(1.18)
PS SV
PT
(1.19)
VT TP
VS
(1.20)
PT TV
PS
(1.21)
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Enthalpy and Entropy as Functions of T and P
(1.22)
(1.23)
Enthalpy
As a function of P and T, we may express:
P,THH
Total differential of the above equation is
dPPH
dTTH
dHTP
(H/T)P is obtained from the definition of CP:
PP T
HC
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(1.25)
(1.24)
(H/P)T is derived from fundamental equation:
VdPTdSdH (1.8)
Differentiation with respect of P at constant T yields:
VPS
TPH
TT
Combining eq. (1.24) with Maxwell equation (1.21):
PT TV
TVPH
Introducing eqs. (1.23) and (1.25) into eq. (1.22) results in :
dPTV
TVdTCdHP
P
(1.26)
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VdPTdSdH
VdPdS
TdPdH
Kalau T konstan,
VdPdS
TPH
TT
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(1.27)
Entropy
As a function of P and T, we may express:
P,TSS
Total differential of the above equation is
dPPS
dTTS
dSTP
(S/P)T is obtained from the Maxwell equation (1.21)
PT TV
PS
(1.21)
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(1.29)
(1.28)
(S/T)P is derived from fundamental equation:
VdPTdSdH (1.8)
Differentiation with respect of T at constant P yields:
PP TS
TTH
Combining eq. (1.23) with (1.28):
Introducing eqs. (1.21) and (1.29) into eq. (1.27) results in :
dPTV
TdT
CdSP
P
(1.30)
TC
TS P
P
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IDEAL GAS
PRT
Vig PR
TV
P
ig
(1.31)
dPT
VTVdTCdH
P
igigig
Pig
dPVVdTCdPPR
TVdTC igigigP
igigP
dTCdH igP
ig
Enthalpy of ideal gas
From eq. (1.26)
(1.32)
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Entropy of ideal gas
(1.33)
From eq. (1.30)
dPT
VT
dTCdS
P
igigP
ig
PdP
RT
dTCdS ig
Pig
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Kalau sistem mengalami proses dari keadaan (T1, P1) ke (T2, P2), maka perubahan entropynya adalah:
2
1
2
1
ig2
ig1
P
P
T
T
igP
S
S
ig
PdP
RT
dTCdS
Jika CP konstan maka
1
2
1
2igP
ig1
ig2
ig
PP
lnRTT
lnCSSS
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The Gibbs Energy as a Generating Function
The fundamental property relations for homogeneous fluids of constant composition given by Eqs. (1.3), (1.8), (1.11), and (1.14) show that each of the thermodynamic properties U, H, A, and G is functionally related to a special pair of variables. In particular
SdTVdPdG (1.14)
expresses the functional relation:
T,PGG
Thus the special, or canonical variables for the Gibbs energy are temperature and pressure.Since these variables can be directly measured and controlled, the Gibbs energy is a thermodynamic property of great potential utility.
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An alternative form of Eq. (1.14), a fundamental property relation, follows from the mathematical identity:
dTRT
GdG
RT1
RTG
d 2
Substitution for dG by Eq. (1.14) and for G by Eq. (1.13) gives:
dT
RTTSH
dTSdPVRT1
RTG
d 2
dTRT
HdP
RTV
RTG
d 2
(1.34)
SdTVdPdG
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T1
GdR1
RTG
d
T1
dGdGT1
R1
dT
TG
dGT1
R1
2
dTRT
GdG
RT1
2
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From eq. (1.34)
TP
RTGRTV
(1.35)
PT
RTGT
RTH
(1.36)
When G/RT is known as a function of T and P, V/RT and H/RT follow by simple differentiation. The remaining properties are given by defining equations. In particular,
RTG
RTH
RS
and RTPV
RTH
RTU
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dPdT
RTH
RTV
dPdT
RTH
dPdP
RTV
dPRTGd
22
dTRT
HdP
RTV
RTG
d 2
Untuk T konstan, dT = 0
RTV
dPRTGd
(T konstan)
dP
RTGdRTV
(T konstan)
TP
RTGRTV
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Thus, when we know how G/RT (or G) is related to its canonical variables, T and P, i.e., when we are given G/RT = g(T, P), we can evaluate all other thermodynamic properties by simple mathematical operations.
The Gibbs energy when given as a function of T and P therefore serves as a generating function for the
other thermodynamic properties, and implicitly represents complete property information.
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RESIDUAL PROPERTY
Unfortunately, no experimental method for the direct measurement of numerical values of G or G/RT is known, and the equations which follow directly from the Gibbs energy are of little practical use.
However, the concept of the Gibbs energy as a generating function for other thermodynamic properties carries over to a closely related property for which numerical values are readily obtained.
Thus, by definition the residual Gibbs energy is:
igR GGG
where G and Gig are the actual and the ideal-gas values of the Gibbs energy at the same temperature and pressure.
(1.37)
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Other residual properties are defined in an analogous way. The residual volume, for example, is:
PRT
PZRT
PRT
VVVV igR
1ZP
RTVR
The definition for the generic residual property is:
igR MMM
Where M is the molar value of any extensive thermodynamic property, e.g., V, U, H, S, or G. Note that M and Mig, the actual and ideal-gas properties, are at the same temperature and pressure.
(1.38)
(1.39)
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Equation (1.34), written for the special case of an ideal gas, becomes:
dTRTH
dPRTV
RTG
d 2
igigig
Subtracting this equation from Eq. (1.34) itself gives:
dTRTH
dPRTV
RTG
d 2
RRR
This fundamental property relation for residual properties applies to fluids of constant composition. Useful restricted forms are:
(1.40)
T
RR
PRTG
RTV
(1.41)
P
RR
TRTG
TRTH
(1.42)
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dTRT
HdP
RTV
RTG
d 2
dTRTH
dPRTV
RTG
d 2
igigig
( – )
dTRT
HHdP
RTVV
RTGG
d 2
igigig
dTRTH
dPRTV
RTG
d 2
RRR
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In addition, the defining equation for the Gibbs energy,G = H – TS, may also be written for the special case of an ideal gas,Gig = Hig – TSig; by difference,
RRR TSHG
The residual entropy is therefore:
RTG
RTH
RS RRR
(1.43)
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Thus the residual Gibbs energy serves as a generating function for the other residual properties, and here a direct link with experiment does exist. It is provided by Eq. (1.41), written:
dP
RTGdRTV RR
(constant T)
dPRTV
RTG
dRR
(constant T)
Integration from zero pressure to arbitrary pressure P yields:
P
0
RR
dPRTV
RTG
(constant T)
where at the lower limit GR/RT is equal to zero because the zero-pressure state is an ideal-gas state.
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T
RR
PRTG
RTV
dP
RTGdRTV RR
(T konstan)
(1.41)
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P
0
RG
G
R
dPRTV
RTG
dR
Rig
P
0
RRigR
dPRTV
RTG
RTG
P
0
RR
dPRTV
RTG
1ZP
RTVR
1ZP1
RTVR
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P
0
RR
dPRTV
RTG
1ZP
RTVR
P
0
R
dP1ZP
RTRT1
RTG
P
0
R
PdP
1ZRTG
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In view of Eq. (1.38):
P
0
R
PdP
1ZRTG
(constant T) (1.44)
Differentiation of Eq. (1.44) with respect to temperature in accord with Eq. (1.42) gives
P
0 P
R
PdP
TZ
TRTH
(constant T) (1.45)
The residual entropy is found by combination of Eqs. (1.43) through (1.45):
P
0
P
0 P
R
dP1ZP
dPTZ
TRTS (constant T) (1.46)