**Chapter 6. Thermodynamic Properties**

**Chapter 6. Thermodynamic Properties**

**of Fluids**

**of Fluids**

**Introduction**

**Introduction**

### **Numerical values of thermodynamic properties are ** **most important for heat and work calculations**

**Enthalpy, Internal Energy, Entropy, …..**

**Measurable Quantities : PVT data and heat capacity **

**Properties relations are important to calculate one properties **
**from the other.**

**Generalized correlations are used when experimental values **
**are not available. **

**6.1 Properties Relations for Homogeneous Phases**

**6.1 Properties Relations for Homogeneous Phases**

### **From the 1**

^{st}**law and 2**

^{nd}**law of Thermodynamics,**

rev

rev dW

dQ )

nU (

d dW_{rev} Pd(nV)
)
nS
(
Td
dQ_{rev}

) nV ( Pd )

nS ( Td )

nU (

d

**Although this eqn. has been derived for the special case of reversible **
**process, it only contain state properties** of the system.

**Can be applied to any process **

**from one equilibrium state to another**

**The only requirements are that the system be** **closed** **and that **
**the change occur between equilibrium states.**

**6.1 Properties Relations for Homogeneous Phases**

**6.1 Properties Relations for Homogeneous Phases**

PdV TdS

dU

### **When the equations are written in molar units;**

PV U

H TS U

A TS H

G

VdP PdV

PdV TdS

) PV ( d dU

dH dH TdSVdP

SdT TdS

PdV TdS

) TS ( d dU

dA dA PdVSdT

SdT TdS

VdP TdS

) TS ( d dH

dG dG VdP SdT

**6.1 Properties Relations for Homogeneous Phases**

**6.1 Properties Relations for Homogeneous Phases**

PdV TdS

) V , S (

dU

### **Fundamental Properties Relations**

VdP TdS

) P , S (

dH

SdT PdV

) T , V (

dA

SdT VdP

) P , T (

dG

**These fundamental properties relations are general equation for **
**homogeneous fluid** **of constant composition**

### **Exact Differential Equation**

**If a function F is a property (not depending on the path)**

**The total differentiation**

**6.1 Properties Relations for Homogeneous Phases**

**6.1 Properties Relations for Homogeneous Phases**

) y , x ( F F

y dy dx F

x dF F

y x

Ndy Mdx

dF

y y x

N F x ,

M F

**6.1 Properties Relations for Homogeneous Phases**

**6.1 Properties Relations for Homogeneous Phases**

### **Exact Differential Equation**

**Condition for exactness**

y dy dx F

x dF F

y x

y x

F y

M ^{2}

x

Ndy Mdx

dF

y x

F x

N ^{2}

y

x x y

N y

M

**6.1 Properties Relations for Homogeneous Phases**

**6.1 Properties Relations for Homogeneous Phases**

### **Maxwell’s Relation**

PdV TdS

) V , S (

dU

VdP TdS

) P , S (

dH

SdT PdV

) T , V (

dA

SdT VdP

) P , T (

dG

V

S S

P V

T

P

S S

V P

T

T

V V

S T

P

T

P P

S T

V

x x y

N y

M

**Maxwell Relation**

**Maxwell Relation**

**S** **H** **P**

**U** **G**

**V** **A** **T**

PdV TdS

dU

VdP TdS

dH

SdT PdV

dA SdT VdP

dG

V

S S

P V

T

P

S S

V P

T

T

V V

S T

P

T

P P

S T

V

**Enthalpy as a Function of T and P**

**Enthalpy as a Function of T and P**

### **T and P : easily measurable**

**T and P : easily measurable**

VdP TdS

) P , S (

dH

P P

T C

H

dP

? dT

? ) P , T (

dH

T V T V

P V T S

P H

P T

T

T dP T V

V dT

C dH

P

P

**Entropy as a Function of T and P**

**Entropy as a Function of T and P**

### **T and P : easily measurable**

**T and P : easily measurable**

P P

T C

H

dP

? dT

? ) P , T (

dS

P P

P T

T S T

C H

T dP V T

C dT dS

P

P

VdP TdS

) P , S (

dH

P

T T

V P

S

**Internal Energy as a Function of T and P**

**Internal Energy as a Function of T and P**

### **T and P : easily measurable**

**T and P : easily measurable**

PV H

U

dP

? dT

? ) P , T (

dU

P P

P T

P V T

H T

U

P dP P V

T T V

T dT P V

C dU

T P

P

P

P V P V

P H P

U

T T

T

**Ideal-Gas State**

**Ideal-Gas State**

T dP T V

V dT

C dH

P

P

dP

T V T

C dT dS

P

P

P R T

) P / RT ( T

V

P P

ig

dT
C
dH _{P}

P R dP T

C dT

dS _{P}

**Alternative Form for Liquid**

**Alternative Form for Liquid**

### **For liquids,**

T V V

P

T dP T V

V dT

C dH

P

P

dP

T V T

C dT dS

P

P

###

^{1}

^{T}

###

^{VdP}

dT C

dH _{P} VdP

T C dT

dS _{P}

**Internal Energy as a Function of T and V**

**Internal Energy as a Function of T and V**

### **Sometimes T and V are more useful than T and P**

**Sometimes T and V are more useful than T and P**

* Equation of States are expressed as P = F(V,T)*
PdV

TdS )

V , S (

dU

V V

T C

U

dV

? dT

? ) V , T (

dU

T P T P

V P T S

V U

V T

T

dV T P

T P dT

C dU

V

V

**Entropy as a Function of T and V**

**Entropy as a Function of T and V**

### **Sometimes T and V are more useful than T and P**

**Sometimes T and V are more useful than T and P**

* Equation of States are expressed as P = F(V,T)*
PdV

TdS )

V , S (

dU

V V

V T

T S T C

U

dV

? dT

? ) V , T (

dS

V

T T

P V

S

T dV P T

C dT dS

V

V

**Gibbs Free Energy as a Generating Function**

**Gibbs Free Energy as a Generating Function**

### **G = G(T,P)**

**G = G(T,P)**

**T, P Canonical Variables for Gibbs energy**

**Conforms to a general rule that is both simple and clear**

SdT VdP

) P , T (

dG

###

RT dT dP H

RT V

RT dT dT S

RT dT H

RT dP S

RT V

dT TS RT H

SdT 1 RT VdP

1

RT dT dG G

RT 1 RT

d G

2

2 2 2

**Gibbs Free Energy as a Generating Function**

**Gibbs Free Energy as a Generating Function**

RT dT dP H

RT V RT

d G _{2}

**Fundamental Properties Relationship**

###

P T

RT / G RT

V

###

T P

RT / T G

RT

H

**Gibbs energy, when given as a function of T** **and P, serves as a **
**generating function****for other thermodynamic properties, and **

**implicitly represents complete property information**

**Example 6.1**

**Example 6.1**

**Determine the enthalpy and entropy changes of liquid water for a change **
**of state from 1 bar and 25 **^{o}**C to 1,000 bar and 50 **^{o}**C. Data for water are **
**given in the following table.**

T (^{o}C) P (bar) C_{p} (J/molK) V (cm^{3}/mol) (K^{-1})

25 1 75.305 18.071 256×10^{-6}

25 1,000 …… 10.012 366×10^{-6}

50 1 75.314 18.234 458×10^{-6}

50 1,000 …… 18.174 568×10^{-6}

**Example 6.1**

**Example 6.1**

###

^{1}

^{T}

###

^{VdP}

dT C

dH _{P}
T VdP

C dT

dS _{P}

) P P ( V ) T 1

( ) T T ( C

H _{P} _{2} _{1} _{2} _{2} _{1}

) P P ( T V

ln T C

S _{2} _{1}

1 2

P

**Use mean averaged values from Table**

K mol / J 310 . 2 75

314 . 75 305 .

C_{P} 75

**For P = 1 bar,**

**For T = 50 **^{o}**C,**

mol / cm 204 . 2 18

174 . 18 234 .

V 18 ^{3}

1 6

6 513 10 K

2 10 568

458

mol / J 400 , 3 ) P P ( V ) T 1

( ) T T ( C

H _{P} _{2} _{1} _{2} _{2} _{1}

K mol / J 13 . 5 ) P P ( T V

ln T C

S _{2} _{1}

1 2

P

**6.2 Residual Properties**

**6.2 Residual Properties**

**No experimental method to measure** **G****as a function of T** **and P**

**Residual Property **

**Departure function from ideal gas value**

** M****is any molar property : V, U, H, S, A or G**

id

R M M

M

id

R G G

G

P

V ZRT )

1 Z P ( RT P

V RT V

V

V^{R} ^{id}

RT dT dP H

RT V RT

d G _{2}

dT

RT dP H

RT V RT

d G _{2}

R R

R

**6.2 Residual Properties**

**6.2 Residual Properties**

**Fundamental residual-property relation**
RT dT
dP H

RT V RT

d G _{2}

R R

R

###

T R

R

P RT / G RT

V

###

P R

R

T RT / T G

RT

H

P ) dP 1 Z ( J

RTdP V RT

G RT

G ^{P}

0 P

0 R

0 P R

R ^{} ^{}_{}^{} ^{}_{}^{} ^{}

###

^{}

^{}

###

^{}

P dP T

T Z RT

H

P P

0

R ^{} ^{}

###

^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

P ) dP 1 Z P (

dP T

T Z R

S ^{P}

P 0 P

0

R ^{} ^{}

###

^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

^{}

###

^{}

**(const T)**

**(const T)**

**(const T)**

**Derive yourself!**

**(Page 209 – 210)**

**Residual Properties as P ** **0**

**Residual Properties as P**

**0**

### **A gas become ideal as P 0**

**Then, all residual properties are zero???**

- Not generally true…

ig 0

P R

0

P

### U 0 , as lim U U

### lim

### 0 ) PV ( lim U

### lim H

### lim

^{R}

0 P R

0 P R

0

P

###

0 T P 0

P R

0

P P

lim Z P RT

1 lim Z

RT V

lim

###

RT J G RT

lim G T

RT / T G

RT H

0 P R R

0 P P

R

R

**See Figure 3.9**
**(Page 87)**

**Enthalpy and Entropy from Residual Properties**

**Enthalpy and Entropy from Residual Properties**

R

ig H

H

H S S^{ig} S^{R}
dT

C H

H ^{T}

T ig P ig

0 ig

###

0

0 T

T ig P ig

0 ig

P ln P T R

C dT S

S

0

###

T R T

ig P ig

0 C dT H

H H

0

###

^{R}

0 T

T ig P ig

0 S

P ln P T R

C dT S

S

0

###

**Residual Properties**

**Residual Properties**

### **Advantages of using Residual Properties**

**Generating Function Normally function of T, P**

**Ideal Gas Property**

- P = RT/V

- C_{p} = C_{p} (T) = dH/dT , C_{v} = C_{v}(T)=dU/dT, Function of T only

**If we can use other method to evaluate residual properties, we **
**can calculate all the other properties**

- Equation of State

- Generalized Correlation

**Example 6.3**

**Example 6.3**

**Calculate the enthalpy and entropy of saturated isobutane vapor at 360 K from the **
**following information.**

1. Table 6.1 gives compressibility-factor data (Z) for isobutane vapor.

2. The vapor pressure of isobutane vapor at 360 K is 15.41 bar.

3. Set for the ideal-gas reference state at 300 K and 1 bar.

4. The ideal gas heat capacity of isobutene vapor is
*K*
*mol*
*J*

*S*
*and*
*mol*
*J*

*H*_{0}* ^{ig}* 18,115 /

_{0}

*295.976 / *

^{ig}) Kelvin in

T ( T 10 037 . 33 7765 . R 1

C_{p} _{}_{3}

**Example 6.3**

**Example 6.3**

T R T

ig P ig

0 C dT H

H H

0

###

^{R}

0 T

T ig P ig

0 S

P ln P T R

C dT S

S

0

###

P dP T

T Z RT

H

P P

0

R ^{}^{}

###

^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

P )dP 1 Z P (

dP T

T Z R

S ^{P}

P 0 P

0

R ^{}^{}

###

^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

^{}

###

^{}

P )dP 1 Z ( P and

dP T

evalute Z to

Need ^{P}

P 0 P

0

###

###

^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

^{}

**From the data in Table 6.1,**

**Fit the data Z vs. T at different P…**

2596 . P 0

)dP 1 Z (

K 10 37 . P 26

dP T

Z

P 0

1 4 P

P 0

###

###

^{}

^{}

**Then, you can perform the rest of **
**calculation!**

**6.3 Residual Properties by Equations of State**

**6.3 Residual Properties by Equations of State**

**To evaluate above expression, equation of state is to be expressed as a **
**function of P at const T.**

**Volume explicit: Z (or V) = f(P)**

**Pressure explicit: Z (or V) = f(P)**

**However, many cases are pressure explicit!**

P dP T

T Z RT

H

P P

0

R ^{} ^{}

###

^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

^{S}

_{R}

^{T}

_{T}

^{Z}

^{dP}

_{P}0

^{P}

^{(}

^{Z}

^{1}

^{)}

^{dP}

_{P}P

P 0

R ^{} ^{}

###

^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

^{}

###

^{}

^{(const T)}

1 B'P C'P^{2}
Z

_{2}

V C V

1 B Z

**6.3 Residual Properties by Equations of State**

**6.3 Residual Properties by Equations of State**

**For two-term virial equation, **

RT 1 BP Z

RT BP P

)dP 1 Z RT (

G ^{P}

0

R

###

###

^{2}

P P

0 R

T B dT

dB T

1 R

P P

dP T

T Z RT

H

dT dB R

P P

)dP 1 Z P (

dP T

T Z R

S ^{P}

P 0 P

0

R

###

**6.3 Residual Properties by Equations of State**

**6.3 Residual Properties by Equations of State**

**Pressure-explicit EOS**

**In general, EOS can be expressed as Z = PV/RT**

**By replacing V by r** **(molar density) P = ZrRT**

**Differentiation gives**

Z dZ d

P ) dP

dZ Zd

( RT

dP

r

r

r

r

Z ln 1 d Z

) 1 Z P (

) dP 1 Z RT (

G

0 P

0

R

r

r

###

^{r}

1 d Z

T T Z

RT H

0

R

r

r

r

###

rr

r r

r

###

^{r}

r

r d

) 1 Z d (

T T Z

Z R ln

S

0 0

R

**Residual Properties as a function of T and P**

**Residual Properties as a function of T and P**

**From the supplement, we can also derive the following alternative **
**expression.**

*P*
*dP*
*T*

*T* *Z*
*T* *dP*

*T* *V*
*V*
*P*

*T*
*H*
*P*

*T*
*H*
*P*

*T*
*H*

*P*
*P*

*P*

*P*
*ig*

*R*( , ) ^{} ( , )^{} ( , ) ^{}

###

0^{}

_{}

^{}

^{}

^{}

_{}

^{}

^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

^{}

^{}

###

0^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

*P*
*Z* *dP*

*P*
*dP*
*T*

*T* *Z*
*P* *dP*

*R*
*T*

*P* *V*
*T*
*S*
*P*
*T*
*S*
*P*

*T*

*S* ^{P}

*P*
*P*

*P*

*P*
*ig*

*R*( , ) ^{} ( , )^{} ( , ) ^{} ^{}

###

0^{}

_{}

^{}

^{}

_{}

^{}

^{}

_{}

^{}

_{}

^{}

^{}

^{}

_{}

^{}

^{}

^{}

###

0^{}

_{}

^{}

_{}

^{}

^{}

_{}

^{}

^{}

###

0 (^{}1)

*P*
*Z* *dP*

*V* *dP*
*V* *RT*

*P*
*T*
*G*
*P*

*T*
*G*
*P*

*T*

*G** ^{R}*( , )

^{}( , )

^{}

*( , )*

^{ig}^{}

###

0

^{P}_{}

^{ }

_{}

^{}

^{}

###

0*(*

^{P}^{}1)

**Residual Properties as a function of T,V **

**Residual Properties as a function of T,V**

**However, the application of the previous expression is not **
**straightforward, especially for complicate E.O.S.**

**From the supplement, we derived the alternative expressions, **
**which is much more useful to use for any pressure explicit E.O.S.**

) , ( )

1 (

) ,

( *P* *dV* *RT* *Z* *H* *T* *P*^{0}

*T*
*T* *P*
*V*

*T*

*H* ^{V}^{ig}

*V*

###

) , ( ln

ln )

,

( _{0} *S* *T* *P*^{0}

*P*
*R* *P*

*Z*
*R*
*V* *dV*

*R*
*T*

*V* *P*
*T*

*S* ^{V}^{ig}

*V*

###

) , ( ln

) 1 (

ln

) , ( )

, ( )

, (

0

0 *G* *T* *P*

*P*
*RT* *P*

*Z*
*RT*
*Z*

*RT*
*V* *dV*

*P* *RT*

*V*
*T*
*S*
*T*
*V*

*T*
*H*
*V*

*T*
*G*

*V* *ig*

###

**6.4 Two-Phase Systems**

**6.4 Two-Phase Systems**

### **Vapor-Liquid Transition**

**Specific Volume : large change**

**U, H, S : also large change**

**G ?**

**At constant T and P**

### P

### T

**Solid **
**Phase**

**Liquid **
**Phase**

**Vapor**
**Phase**
P_{c}

T_{c}
triple

point

**Gas**
**Vapor-Liquid Boundary**

SdT VdP

) P , T (

dG

0 ) nG (

d

G G

**6.4 Two-Phase Systems**

**6.4 Two-Phase Systems**

### **Derivation of Clapeyron Equation **

**Phase Transition **

dG dG

SdT VdP

) P , T (

dG

dT S dP

V dT

S dP

V^{} ^{sat} ^{} ^{} ^{sat} ^{}

V S V

V

S S

dT
dP^{sat}

H T S

V T

H dT

dP^{sat}

**6.4 Two-Phase Systems**

**6.4 Two-Phase Systems**

### **Derivation of Clapeyron Equation **

**Especially for vapor liquid transition**

vl vl sat

V T

H dT

dP

vl sat

vl Z

P

V RT

vl 2

vl sat

Z RT

H dT

P ln d

###

^{vl}

vl sat

Z R

H T

/ 1 d

P ln d

**Relation between**

**Vapor pressure and**
**heat of vaporization**

**Example 6.5**

**Example 6.5**

**The Clapeyron equation for vaporization may be simplified by introduction of **
**reasonable approximations, namely, that the vapor phase is an ideal gas and **
**that the molar volume of the liquid is negligible compared with the molar **
**volume of the vapor. How do these assumptions alter the Clapeyron equation?**

###

^{vl}

vl sat

Z R

H T

/ 1 d

P ln d

**From the assumption,**

1 Z P or

V RT

V^{vl} ^{v} _{sat} ^{lv}

**The Clapeyron equation becomes**

###

^{1}

^{/}

^{T}

d P ln R d H

sat vl

**the Calusius/Calpeyron equation**

T A B

P

ln ^{sat}

**6.4 Two-Phase Systems**

**6.4 Two-Phase Systems**

### **Temperature Dependence of Vapor Pressure**

Empirical equation for P vs. T

###

^{vl}

vl sat

Z R

H T

/ 1 d

P ln d

T A B

P

ln ^{sat}
**Simple Expression**

C T A B P

ln ^{sat}

**Antoine Equation**

1

D C

B P A

ln

6 3

5 . 1 sat

**Wagner Equation**

Tr

1

**Specific Temperature Range**

**6.4 Two-Phase Systems**

**6.4 Two-Phase Systems**

### **Two phase Liquid/Vapor System**

When system consist of saturated liquid and saturated vapor

Other extensive thermodynamic properties, M (M=U, H, S, A, G,..)

v v l

lV n V

n

nV

v v l

lV x V

x

V

v v l

v)V x V

x 1 (

V

v v l

v)M x M

x 1 (

M

lv v

l x M

M

M

**6.5 Thermodynamic Diagrams**

**6.5 Thermodynamic Diagrams**

**Graph showing for a particular substance a set of properties (T, P, V, H, S,…)**

**TS Diagram**

**PH Diagram (ln P vs. H)**

**HS Diagram (Mollier Diagram)**

**6.6 Tables of Thermodynamic Properties**

**6.6 Tables of Thermodynamic Properties**

**U, H, S, V,…**** Thermodynamic function**

**Tables of Thermodynamic Data**

**Tabulation of values of thermodynamic functions (U, H, V,..) at various condition (T**

**and P) **

**It is impossible to know the absolute values of U , H** **for process materials Only **

**changes are important ( *** U, *

**H,…)** Reference state

- Choose a Tand Pas a reference state and measure changes of *U*and Hfrom this reference state

tabulation

**Steam Tables **

**Steam Tables**

Compilation of physical properties of water (H, U, V)

Reference state: liquid water at triple point (0.01 ^{o}C, 0.00611 bar)

**Properties of Superheated Steam**

**Example 6.7**

**Example 6.7**

**Superheated steam originally at P**_{1}**and T**_{1}**expands through a nozzle to an**
**exhaust pressure P**_{2}**. Assuming the process is reversible and adiabatic, determine**
**the downstream state of the steam and** H for the following condition.

**P**_{1}**= 1,000 kPa, T**_{1}**= 250 **^{o}**C, and P**_{2}**= 200 kPa**

**Solution**

Since it is adiabatic process, S = 0 S_{1} = S_{2}
To calculate H, find H_{1} and H_{2}.

At the initial state, find the H_{1} and S_{1} at P_{1} = 1,000 kPa and T_{1} = 250 ^{o}C

From steam table, interpolation between 240 and 260 ^{o}C gives
**H**_{1}**= 2942 kJ/kg, S**_{1}**= 6.9252 kJ/kgK = S**_{2}

Since the entropy of saturated vapor at 200 kPa is greater than S_{2}, the
final state is the two phase liquid/vapor region!

From page 731

**Steam Table (page 731)**

**Steam Table (page 731)**

**sat.**

**liq.**

**sat.**

**vap.**

**175**
**(448.15)**

**200**
**(473.15)**

**220**
**(493.15)**

**240**
**(513.15)**

**260**
**(533.15)**

**280**
**(553.15)**

**300**
**(573.15)**

**325**
**(598.15)**

**Steam Table (page 725)**

**Steam Table (page 725)**

**sat.**

**liq.**

**sat.**

**vap.**

**300**
**(573.15)**

**350**
**(623.15)**

**400**
**(673.15)**

**450**
**(723.15)**

**500**
**(773.15)**

**550**
**(823.15)**

**600**
**(873.15)**

**650**
**(923.15)**

**Example 6.7**

**Example 6.7**

From the steam table (page 725), the final T_{2} is the saturation temperature at 200 kPa,

T_{2} = 120.23 ^{o}C
At T_{2} = 120.23 ^{o}C,

Since the entropy of saturated vapor at 200 kPa is greater than S_{2},
the final state is the two phase liquid/vapor region!

, S x S ) x 1 ( S

From ^{v} ^{l} ^{v} ^{v}

kg / kJ 3 . 706 , 2 H

, kg / kJ 7 . 504 H

K kg / kJ 1268 .

7 S

, K kg / kJ 5301 .

1 S

v 2 l

2

v 2 l

2

9640 .

0 x

1268 .

7 x 5301 .

1 ) x 1 ( 9252 .

6

v

v v

kg / kJ 9 . 315 9

. 942 , 2 0 . 627 , 2 H

H H

kg / kJ 0 . 627 , 2 3 . 706 , 2 9640 .

0 7 . 504 )

9640 .

0 1 ( H

x H

) x 1 ( H

1 2

v 2 v l

2 v 2

From page 725

**Calculation of Enthalpy and Entropy for Real Fluids**

**Calculation of Enthalpy and Entropy for Real Fluids**

**Using residual properties, H**^{R}**and S**^{R}**, and ideal gas heat capacities, we can now **
**calculate the enthalpy and entropy for real fluids at any T and P!**

**For a change from state 1 to state 2,**

R 1 T

T ig P ig

0

1 H C dT H

H

0

###

T^{T}

^{R}

^{2}

ig P ig

0

2 H C dT H

H

0

###

R 1 R

2 T

T

ig P 1

2 H C dT H H

H

H ^{2}

1

###

R 1 R

2 1

T 2 T

ig

P S S

P ln P T R

C dT
S ^{2}

1

###

**Calculation of Enthalpy and Entropy for Real Fluids**

**Calculation of Enthalpy and Entropy for Real Fluids**

**Step 1 1**^{ig}

**Step 1**^{ig}** 2**^{ig}

**Step 2**^{ig}** 2**

R 1 1

ig H H

H1 S^{ig} S_{1} S_{1}^{R}

1

dT C H

H

H ^{2}

1 1 2

T T

ig P ig

ig

ig ^{} ^{} ^{}

###

1 T 2

T ig P ig

1 ig

2 ig

P ln P T R

C dT S

S

S ^{2}

1

###

R 2 ig

2

2 H H

H S_{2} S^{ig}_{2} S^{R}_{2}

**Homework**

**Homework**

### **Problems**

**6.3, 6.11, 6.23, 6.48, 6.54, 6.74**

**For 6.54**

- use the heat capacity of n-butane from Table C.1.

- For H^{R}and S^{R}, use Pitzer correlations for the Compressibility factor (Z).

**Due:**

### **Recommend Problems**

**6.2, 6.4, 6.9, 6.12, 6.17, 6.30, 6.31, 6.49, 6.57, 6.58, 6.71, 6.72, **
**6.76, 6.80, 6.92, 6.93, 6.94**