Chapter 3

Introduction to
Chemical Engineering
Thermodynamics
Chapter 3
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U and H are calculated from measurements of
molar volume V(T,P)
PVT relations
Equations of State
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F=2
PT Diagram
F=0
F=1
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Fluid region: can neither have vaporization
nor condensation
Transition from A to B is gradual and do not
include a vaporization step
Fluid at T>Tc is said to be supercritical
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PV Diagram
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Single phases and two-phases are regions
Triple point = horizontal line
Horizontal segment of isotherms = all possible L-V mixtures
Ranging from 100% liquid (left end) to 100% vapor (right end)
Locus of these point = Dome = BCD
BC = saturated liquid
CD = saturated vapor
For given T, horizontal segment is at the saturation pressure
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Left of BC = subcooled-liquid
Right of CD = superheated-vapor
Isotherms in the subcooled-liquid region are steep
because V is almost independent of P
Top of the Dome = critical point = L and V
indistinguishable
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Constant-volume paths in the single-phase region
EF = subcooled (tube filled with liquid is heated)
GH = superheated (tube filled with vapor is heated)
Tube partially filled with liquid
(vapor in equilibrium fills the rest of the volume)
JQ = meniscus initially near top of tube (J) and liquid expands upon
heating to fill tube (Q)
KN = meniscus initially near bottom of tube (K) and liquid vaporizes
upon heating so that meniscus recedes to bottom of tube (N)
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Equation of State
Volume expansivity
Isothermal compressibility
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β is almost always positive and κ is positive
If can assume β and κ are independent of T and P
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Virial Equations of State
A, B’, C’, etc., are constant for given T and chemical species
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Universal Gas Constant
same for all gases
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assigned arbitrarily as temperature of triple point of water
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Two Forms of the Virial Equation
compressibility factor
Virial
expansions
coefficients B’, C’ and D’ are related to B, C and D by eqs. (3.13 a-c)
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Virial coefficients are function of T only
statistical mechanics interpretation :
Two-body and three-body interactions
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Ideal Gas
P-dependency results from forces between molecules
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regardless of the kind of process causing the change
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For the three processes:
Only for process a-b and if mechanically reversible:
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Ideal gas
Mechanically reversible
Closed-system process
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Isothermal Process
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Isobaric Process
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Isochoric Process
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Adiabatic Process
monatomic gases = 1.67
diatomic gases ≈ 1.4
simple polyatomic gases=1.3
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Polytropic Process
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Isobaric process:
δ=0
Isothermal process: δ=1
Adiabatic process: δ=γ
Isochoric process: δ=∞
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Irreversible Processes
Determine W for a mechanically reversible process
Multiply or divide by the efficiency:
Process require work
Divide Wreversible by efficiency
Process produces work Multiply Wreversible by efficiency
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Application of the Virial Equation
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All isotherms originate at Z=1 for P=0
All isotherms are straight line for small P
B is substance dependent and a function of temperature.
we will see how B can be estimated
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Virial equation truncated to three terms:
cubic in volume
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Extended Virial equations
Benedict/Webb/Rubin equation
for petroleum and natural-gas industries
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Cubic Equations of State (EOS)
In order to represent both liquid and vapor
behavior, an EOS must be valid for a wide range
of Temperatures and Pressures
Cubic EOS = Polynomial equations that are
cubic in molar volume
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The van der Waals EOS
isotherm
one or three roots
saturation pressure
saturated liquid
saturated vapor
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Generic Cubic EOS
b, θ, κ, λ and η depend on T and composition (for mixtures)
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critical isotherm exhibits a horizontal inflection at the critical point:
for example one can get a and b for the van der Waals EOS
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where
are pure numbers
introduce α(Tr)
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Redlich-Kwong EOS (1949)
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ω
acentric factor
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Vapor root
iteration starts with V=RT/P
iteration starts with Z=1
where
and
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Liquid root
iteration starts with V=b
iteration starts with Z=β
where
and
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Generalized Correlations for Gases
Pitzer Correlations for Z
From data for Argon, Krypton, Xenon
two-parameter corresponding-states correlation for Z
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Lee and Kesler (1975)
Table (Appendix E) for Z0 and Z1 as a function of Tr and Pr
Valid for nonpolar or slightly polar gases
Large error for polar gases and gases that associates
Quantum gases, such as H2, require special treatment
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Pitzer Correlations for Second Virial Coefficients
for nonpolar
species
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Pitzer Correlations for Third Virial Coefficients
rapid convergence
Start with Z=1
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Validity of Ideal Gas Law
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Generalized Correlations for Liquids
Rackett equation for molar volume of saturated liquids
An estimate of liquid volume:
known volume
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from figure 3.16
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