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CHEN 310
Chemical Engineering
Thermodynamics II
Department of Chemical Engineering
Ahmadu Bello University, Zaria
Phase Equilibrium
and Fugacity
Department of Chemical Engineering, ABU
• At the end of this week’s lecture, you should
be able to:
– Derive expression for fugacity and fugacity
coefficient
– Application of fugacity to phase equilibria
and their calculations.
CHEN 310 – CHEMICAL ENGINEERING THERMODYNAMICS II
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Department of Chemical Engineering, ABU
❑ From the Gibbs free energy differential,
❑
dG = VdP - SdT
6.1
❑ Applied to 1 mole of pure fluid i at constant T, this equation
becomes,
❑
dGi = VidP
(const T)
6.2
❑ For an ideal gas, Vi = RT/P
❑ Thus,
𝑑𝐺𝑖 =
𝑑𝑃
𝑅𝑇
𝑃
❑ or
dGi = RT dlnP
6.3
❑ Eqn 6.3 can be made universally valid by replacing P with a new
function known as fugacity, f
❑ Eqn 6.3 becomes,
❑
dGi = RT dlnfi
(const T)
6.4
❑ Where fi called the fugacity of pure i is a property of i with the units
of pressure
❑ For the special case of an ideal gas,
❑
RTdlnfi = RTdlnP
❑ Integration gives,
❑
lnfi = lnP + lnC
CHEN 310 – CHEMICAL ENGINEERING THERMODYNAMICS II
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Department of Chemical Engineering, ABU
❑ and
fi = CP
❑ Where C is a constant
❑ Putting C = 1 makes fugacity = pressure
❑ Since ideality is assumed as P
0 then the equation
❑
𝑓𝑖
𝑃→0 𝑃
lim
= 1 is valid
6.5
❑ The fugacity 𝑓መ𝑖 of a component in a solution is similarly defined as
❑
𝑑 𝐺ҧ𝑖 = 𝑅𝑇𝑑𝑙𝑛𝑓መ𝑖
(const T)
6.6
❑ For
𝑓መ𝑖
lim
𝑃→0 𝑥𝑖 𝑃
=1
❑ Thus for a mixture of gases,
❑
𝑓መ𝑖 = 𝑥𝑖 𝑃
(ideal gas)
❑ The product 𝑥𝑖𝑃 is known as the partial pressure Pi of component i in
the gas mixture
❑ i.e Pi = 𝑥𝑖𝑃
❑ and
Ʃ𝑃𝑖 = Ʃ𝑥𝑖𝑃 = 𝑃 Ʃ𝑥𝑖 = 𝑃
❑ i e the pressure of a gas mixture is equal to the sum of the partial
pressures of its individual components
CHEN 310 – CHEMICAL ENGINEERING THERMODYNAMICS II
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Department of Chemical Engineering, ABU
❑ The fugacity coefficient is defined as the ratio of the fugacity of a
material to its pressure. For a pure substance,
𝑓
❑
𝜑𝑖 = 𝑖
6.7
𝑃
❑ For a component in solution,
❑
𝜑ො 𝑖 =
𝑓መ𝑖
𝑥𝑖 𝑃
6.8
❑ 𝜑𝑖 is the fugacity coefficient and is dimensionless
❑ Values of 𝜑𝑖 and 𝜑ො 𝑖 are readily calculated from PVT data
❑ Derivation of fi and 𝝋𝒊
❑ From eqn. 6.4
❑
dGi = RT dlnfi (const T)
❑ At constant T and composition (x), 1 mole of solution, the equation
❑
𝒅(𝒏𝑮)= −(𝒏𝑺)𝒅𝑻 + (𝒏𝑽)𝒅𝑷 + Ʃ𝝁𝒊dni
❑ Becomes
dGi = Vi dP
(const T,x)
6.9
❑ Combination of eqns. 6.4 and 6.9 gives,
❑
RT dlnfi = Vi dP
6.10
❑ A logarithmic differentiation of 6.7 gives
CHEN 310 – CHEMICAL ENGINEERING THERMODYNAMICS II
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Department of Chemical Engineering, ABU
❑ A logarithmic differentiation of 6.7 gives
❑
dlnfi = dln𝜑𝑖 + dlnP
𝑑𝑃
❑
= 𝑑𝑙𝑛𝜑𝑖 +
𝑃
❑ Substituting into eqn 6.10 gives
𝑃𝑉 𝑑𝑃
𝑑𝑃
❑
𝑑𝑙𝑛𝜑𝑖 = 𝑖
−
❑ Since,
𝑍𝑖 =
𝑃𝑉𝑖
𝑅𝑇
𝑅𝑇
𝑃
(const T,x)
𝑃
6.11
we have
𝑑𝑃
❑
𝑑𝑙𝑛𝜑𝑖 = 𝑍𝑖 − 1
(const T,x)
6.12
𝑃
❑ Integrating between P = 0, where 𝜑𝑖 = 1 and P gives
𝑃
𝑑𝑃
❑
𝑙𝑛𝜑𝑖 = ‫׬‬0 𝑍𝑖 − 1
6.13
𝑃
❑ Since 𝑓𝑖 = 𝜑𝑖 𝑃, the corresponding expression for lnfi is
𝑃
𝑑𝑃
❑
𝑙𝑛𝑓𝑖 = 𝑙𝑛𝑃 + ‫׬‬0 𝑍𝑖 − 1
(const T,x)
6.14
𝑃
❑ Analogous equations to 6.13 and 6.14 for which V rather than P is
the variable of integration are easily derived They are
𝑉
𝑑𝑉𝑖
❑
𝑙𝑛𝜑𝑖 = 𝑍𝑖 − 1 − 𝑙𝑛𝑍𝑖 − ‫׬‬0 𝑍𝑖 − 1
6.15
❑
𝑙𝑛𝑓𝑖 =
𝑅𝑇
𝑙𝑛
𝑉𝑖
+ 𝑍𝑖 − 1 −
𝑉
‫׬‬0
𝑍𝑖 − 1
𝑉
𝑑𝑉𝑖
𝑉
CHEN 310 – CHEMICAL ENGINEERING THERMODYNAMICS II
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Department of Chemical Engineering, ABU
❑ For ideal gases, Z = 1 hence,
❑
𝜑𝑖 = 𝜑 = 1
❑ and
fi = f = P
❑ The analogous equation for a component i of an ideal gas mixture
(solution) are
❑
𝜑ො 𝑖 = 𝜑 = 1
❑ and
𝑓𝑖 = yi P
6.17
❑ Where yi is the mole fraction of component i in a gas phase
❑ Since the residual volume is related to the compressibility factor by
the expression
𝑅𝑇
❑
Δ𝑉𝑖=
1 − 𝑍𝑖
6.18
𝑃
❑ Hence eqn.6.13 becomes,
𝑃 𝑅𝑇
1
❑
𝑙𝑛𝜑𝑖 = −
− 𝑉𝑖 𝑑𝑃
‫׬‬0
𝑅𝑇
1
−
𝑅𝑇
❑
=
❑ Also the expression
𝑃
𝑑𝑃
❑
𝑙𝑛𝜑ො 𝑖 = ‫׬‬0 𝑍𝑖 − 1
𝑃
𝑃
𝑃
‫׬‬0 ∆𝑉𝑖 𝑑𝑃 (const T)
(const T,x) is valid
CHEN 310 – CHEMICAL ENGINEERING THERMODYNAMICS II
6.19
6,20
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Department of Chemical Engineering, ABU
❑ And since
𝑍መ𝑖 =
𝑃𝑉𝑖
𝑅𝑇
𝑃 𝑅𝑇
(const T,x)
6.21
‫׬‬0 𝑃 − 𝑉ത𝑖 𝑑𝑃
❑ The generalized correlation for fugacity coefficient, 𝜑 is
𝑃
❑
𝑙𝑛𝜑 = 𝑟 𝛽0 + 𝜔𝛽𝑙
(const T,x) is valid
❑
𝑙𝑛𝜑ො 𝑖 = −
1
𝑅𝑇
𝑇𝑟
❑ Applicable to non-polar or slightly polar gases, where 𝛽0 and 𝛽𝑙 are
given by
❑
❑ And
0.422
= 0.083 − 1.6
𝑇𝑟
0.172
𝛽𝑙 = 0.139 − 4.2
𝑇𝑟
𝛽0
❑ And also
𝑙𝑛𝜑 = 𝑙𝑛𝜑 0 + 𝜔𝑙𝑛𝜑𝑙 𝑜𝑟 𝜑 = 𝜑 0 𝜑𝑙𝜔
❑ Where 𝜑 0 , 𝜑 𝑙 , 𝛽0 , 𝜔 𝑎𝑛𝑑 𝛽𝑙 are functions of (Pr, Tr) found in literature
(see Lee Kesler chart)
❑ Practice Example
❑ Determine the values of 𝜑 and f for n butane gas at 460 K and 15 atm
by the generalized correlation for fugacity coefficient. (PC = 37 5 atm,
TC = 425.2 K, 𝜔 = 0.193)
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CHEN 310 – CHEMICAL ENGINEERING THERMODYNAMICS II
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