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Chapter 4 The Thermodynamic Properties of
Real Substance
•   Balance Equation
 Mass balance equation
n•
dN/dt =  Nk
k=1
 Energy balance equation
•        •       •
dU/dt =  Nk Hk + Q + Ws - PdV/dt
 Entropy balance equation •
n •       •
dS/dt =  Nk Sk + Q/T + Sgen
k=1
•   Equation of State
 volumetric equation of state
PV = RT for ideal gas
(P + ?)(V - ?) = RT for real gas
 thermal equation of state
dU = Cv dT for ideal gas
dH = Cp dT for ideal gas
dU = CvdT + ? for real gas
dH = Cp dT + ? for real gas

thermodyanmic problem  balance equation
 equation of state  energy change, heat
and work
Chapter 4                1
•    Phase Rule
f=2-+n
f : independent variable
 : number of phase
n : number of component

For a closed system with one homogenous pure component
 = 1 phase n = 1 component
 f = 2  T, P or T, V or P, V

S U V
H A
P G T

Two independent variables and Six dependent variables
e.x., dU = (U/T)V dT + ( U/V)T dV
i.e., dX = (X/Y)Z dY + ( X/Z)Y dZ if X = f(X, Y)
X, Y, Z  S, U, V, H, A, P, G, T (intensive variables)

•   Closed system vs. Open system
dU = (U/T)V dT + ( U/V)T dV for closed system

dU = (U/T)V ,N dT + ( U/V)T ,N dV + (U/N)T,V dN
for open system
Chapter 4                         2
•   Mathematics Operation
• (X/Y)Z,N = ( NX/Y)Z,N = N ( X/Y)Z
• (X/Y)Z,N = ( X/Y)Z
• / ZY (X/Y)Z = / YZ (X/Z)Y
• (X/Y)Z (Z/X)Y (Y/Z)X = -1
 dX = (X/Y)Z dY + ( X/Z)Y dZ
 (X/Y)X = 0 = (X/Y)Z (Y/Y)X + ( X/Z)Y
(Z/Y)X
 (Y/Y)X = 1
 (X/Y)Z (Z/X)Y (Y/Z)X = -1
• (X/K)L = (X/Y)Z (Y/K)L + (X/Z)Y (Z/K)L
When L = Z     (X/K)Z = (X/Y)Z (Y/K)Z

•   The Evaluation of Thermodynamic Partial Derivatives

Chapter 4                    3
Open system

(H = U + PV)

(A = U - TS)

(G = H - TS)

Closed system

dU = (U/S)V dS + ( U/V)S dV

dS = (S/U)V dU + ( S/V)S dV

dH = (H/S)P dS + (H/P)S dP

dA = (A/V)T dV + ( A/T)V dT

dG = (G/P)T dP + ( G/T)P dT

Chapter 4                     4
 Thermodynamic identities

S U V                      S U V
H A                        H A
P G T                      P G T

S U V                      S U V
H A                        H A
P G T                      P G T

Chapter 4            5
•   Maxwell relations

S U V
H A
P G T

S U V
H A
P G T

S U V
H A
P G T

S U V
H A
P G T
Chapter 4                  6
•   Thermodynamic functions

Chapter 4           7
Chapter 4   8
variables
Cp, Cv, T, P, V,
S, a, kT
partial derivatives
containing only T,
P and V

Chapter 4                   9
= f (T, V)

1

Chapter 4   10
Solution:
(a) ideal gas

Joule-Thomson coefficient

Chapter 4       11
Solution:
(b) Van der Waals Fluid

Chapter 4   12
Show all the state variables and thermodynamic properties
through Gibbs free energy pressure and Temperature

Chapter 4                     13
The evaluation of changes in the thermodynamic properties

What is the minimum amount of information needed to
calculate the thermodynamic properties?
Are these information available?

T, P, V  volumeric equation of state for fluid including gas
and liquid
Cp, Cv  function of T, P or V

Volumeric equation of state

van der Waals equation

a and b are material constants
 available for both liquid and vapor
 not very accurate
Redlich-Kwong equation

Peng-Robinson equation
 available for both liquid and vapor
 more accurate

general equation

 parameters b, , , , and  can depend on
temperature
Chapter 4                       14
Cubic equation of state
 Z = PV/RT       Z: Compressibility factor

Virial equation of state

 B(T), C(T) …. are the second, third …. Viral
coefficients which are temperature dependent

Heat capacity data
The change of Cp and Cv from one state to another state
can be calculated.
Cp as a function of temperature at constant pressure
is available
See next page!

Cv as a function of temperature at constant volume
is available
See next page!

Cp* is the heat capacity at very low (close to zero) pressure
Cv* is the heat capacity at very large V (approach to infinite)
Cp*=f(T) and Cv*=f(T) are available due to ideal fluid
condition
Chapter 4                          15
Chapter 4   16
Calculation of H

T            T2,P2

T1,P1

0
P
Chapter 4        17
Calculation of S and U

T                 T2,P2

T1,P1

0                             P

V=

V

V1,T1

V2,T2

0                             T
Chapter 4            18
The Principle of Corresponding States
Purpose: Predict thermodynamic properties of fluids from
generalized property correlations based on
available experimental data for similar fluids

Facts:
 Cp* and Cv* are available
 P, V and T relationships can be used to calculate all
thermodynamic properties

Critical point

x

x:

Chapter 4                         19
a and b = f(Tc, Vc)

a and b = f(Tc, Pc)

a and b = f(Vc, Pc)

reduced temp., pressure and volume

corresponding states

Chapter 4   20
Van der Waals fluid critical compressibility vs.
real fluid critical compressibility

Zc ranging from 0.23 to 0.31 is not equal to 0.375
 use Z = f (Pc, Tc, Zc) to fit experimental results

Chapter 4                      21

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