FUNDAMENTAL EQUATIONS

Chapter 1

HOMOGENEOUS CLOSED SYSTEM

(Q and/or W)

(Q and/or W)

• 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.

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)

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.

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

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

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)

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.

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)

MAXWELL EQUATIONIf F = F(x, y), the total differential of F is:

dyyF

dxxF

dFxy

NdyMdxdF

yxF

M

xyF

N

with

(1.16)

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)

Resume:

PdVTdSdU (1.3)

VdPTdSdH (1.8)

PdVSdTdA (1.11)

SdTVdPdG (1.14)

NdyMdxdF

yx xN

yM

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)

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

(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)

VdPTdSdH

VdPdS

TdPdH

Kalau T konstan,

VdPdS

TPH

TT

(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)

(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

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)

Entropy of ideal gas

(1.33)

From eq. (1.30)

dPT

VT

dTCdS

P

igigP

ig

PdP

RT

dTCdS ig

Pig

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

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.

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

T1

GdR1

RTG

d

T1

dGdGT1

R1

dT

TG

dGT1

R1

2

dTRT

GdG

RT1

2

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

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

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.

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)

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)

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)

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

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)

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.

T

RR

PRTG

RTV

dP

RTGdRTV RR

(T konstan)

(1.41)

P

0

RG

G

R

dPRTV

RTG

dR

Rig

P

0

RRigR

dPRTV

RTG

RTG

P

0

RR

dPRTV

RTG

1ZP

RTVR

1ZP1

RTVR

P

0

RR

dPRTV

RTG

1ZP

RTVR

P

0

R

dP1ZP

RTRT1

RTG

P

0

R

PdP

1ZRTG

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)