Elearning_Lecture 8 - Solution Thermodynamics - part 2

SOLUTION THERMODYNAMICS
AND PRINCIPLES OF PHASE EQUILIBRIA
CATIA ANGLI CURIE, MS
TEKNIK KIMIA UNIVERSITAS PERTAMINA 2018

SOLUTION THERMODYNAMICS: PART 2
Content
Fugacity & fugacity coefficient of a species in a mixture
• Ideal gas mixture & the Gibbs energy relation
• Real gas mixture & the Gibbs energy relation
• Defining fugacity & fugacity coefficient of a
species in a mixture
• Ideal solution model
• fugacity & fugacity coefficient of a species in a
mixture
Objective
• Understand fugacity & fugacity coefficient of a
component in an ideal mixture

FUGACITY & FUGACITY COEFFICIENT
2. SPECIES IN SOLUTION

IDEAL GAS MIXTURE
• For 1 mol of an ideal gas, the volume is v = RT
P
Ideal gas
Pure component
Ideal gas
Mixture of
components
At the same T & P,
Regardless of the composition, ideal
gas has the same volume
v ig =
ҧ v i
ig = vi
ig =
RT
P

IDEAL GAS MIXTURE
• The equality only holds for volume, whereas for other properties:
ഥm i
ig (T, P) = mi
ig (T, pi )
m = any property other than volume
p i = partial pressure of component i = y i P
Partial molar property (other than volume) of a component in an ideal gas mixture
is equal to the property of the pure component at the mixture T, but at a pressure equal to its
partial pressure in the mixture (Gibb’s theorem)

PARTIAL MOLAR GIBBS ENERGY IN AN IDEAL GAS MIXTURE
g = h − Ts
ig ig ig
gҧ
i = തh i − Tsiҧ
ig ig ig
Knowing the expression of തh i and siҧ
is necessary to find giҧ
ഥm i
ig (T, P) = mi
ig (T, pi )
• Partial molar enthalpy
• Enthalpy ideal gas only depends on T
തh i
ig T, P = hi
ig T, pi
= h i
ig T, P
തh i
ig = hi
ig

PARTIAL MOLAR GIBBS ENERGY IN AN IDEAL GAS MIXTURE
• Partial molar entropy
dh = Tds + vdP
ds = dh
T − vdP
T
Since
ഥm i
ig (T, P) = mi
ig (T, pi )
ig ig
s i
ҧ (T, P) = si T, P − R ln yi
For ideal gas at constant T
ds = Cp dT T − R dP P
ig
s i
ҧ =
si
ig
− R ln y i
ds = −R dP P
= −R d(ln P)
Pure-component value at the mixture T & P
Integration from p i to P
s i
ig T, P − si
ig T, pi
= −R ln P p i
= −R ln P
y i P = R ln y i

PARTIAL MOLAR GIBBS ENERGY IN AN IDEAL GAS MIXTURE
ig ig ig
gҧ
i = തh i − Tsiҧ
• Since
• Thus,
തh i
ig = hi
ig
and
ig ig
s i
ҧ = si − R ln yi
ig ig ig
gҧ
i = hi − Tsi + RT ln yi
ig ig
gҧ
i = gi + RT ln yi
Partial molar Gibbs energy in an Ideal gas mixture
• Last week we had:
• So
g i ig P = g i ig P o + RT ln P P o
ig
gҧ
i =gi ig P o + RT ln P + RT ln y P o i
=g i ig P o
+ RT ln P + RT ln y i − RT ln P o
ig
gҧ
i =RT ln(yi P) + g ig i P o
− RT ln P o
Partial molar Gibbs energy in an Ideal gas mixture

PARTIAL MOLAR GIBBS ENERGY & FUGACITY
IN A REAL GAS MIXTURE
• For species i in an ideal gas mixture we’ve had:
ig
gҧ
i =RT ln(yi P) + g ig i P o
− RT ln P o
• For species i in a real gas mixture (or in a liquid solution), the analogous expression is
gҧ
i =RT ln( መf i ) + g ig i P o − RT ln P o መf i = fugacity of component i in the solution
• Substracting the real gas with the ideal gas
gҧ
i −
ig
gҧ
i = RT ln
መf i
y i P
f
∅ i ≡ መ i
y i P
gҧ
R i = RT ln ∅ i
Replacing partial pressure y i P
ҧ g i = partial Gibbs energy = μ i (chemical potential)
∅ i = fugacity coefficient of component i in solution (mixture)
y i = mol fraction of component i in the vapor phase
P = total (mixture) pressure

PARTIAL MOLAR GIBBS ENERGY & FUGACITY
IN A REAL GAS MIXTURE
• For ideal gas mixture
∅ i ≡
መ f i
y i P
∅ i ≡ መ f i
ig
y i P = 1
or
መ f i
ig
= yi P

IDEAL SOLUTION MODEL

THE IDEAL SOLUTION MODEL
• Chemical potential for ideal gas
μ i
ig =
• Ideal solution is defined as the one for which
ig ig giҧ
= gi + RT ln yi μ id i = gҧ
id i = g i
+ RT ln x i
• The partial molar properties of ideal gas were
ig ig
vҧ
i = vi
തh i
ig = hi
ig
ig ig
s i
ҧ = si − R ln yi
• Total properties of the mixture
• Partial molar properties for ideal solution are
vҧ
id i = v i
തh i id = h i
s i ҧ
id = s i − R ln x i
• Total properties of the solution
ig
id
m = y i ഥm i
m = x i ഥm i
i
i
Where m can be any properties, eg. enthalpy (h), entropy (s), gibbs energy (g)

ҧ
THE IDEAL SOLUTION
• For component i in a gas mixture (or in a liquid • For ideal solution, the bottom-left equation become
solution), we’ve had:
g i =RT ln( መf i ) + g ig i P o − RT ln P o μ id i = g i + RT ln መ id
f i
f i
• For pure component we’ve also had:
• Comparing with the one from previous slide:
g i =RT ln(f i ) + g ig i P o − RT ln P o μ id i = gҧ
id i = g i + RT ln x i
• Substracting the first with the later gives:
• Gives
id
gҧ
i −g i = RT ln( መf i ) − RT ln(f i )
መf i id
= x
f
μ i = g i +RT ln
መ f i
or መf i = xi f i
i
i
f i
መf i = fugacity of component i in the mixture/solution
መf i
id
= fugacity of component i in an ideal mixture/solution
f i = fugacity of pure component
x i = mol fraction of component i in the mixture/solution

THE IDEAL SOLUTION
For ideal solution:
መf i
id
= xi f i
For ideal gas:
መf i
ig
= yi P
• Notice that the above equations are similar
• Ideal solution is a more general term (ideal gas is also an ideal solution), remember that for ideal gas f i = P
• Dividing the መf i
id
with xi P gives:
መf i
id
x i P = x if i
x i P
∅ i
id
= ∅i
In an ideal solution, the fugacity coefficient of component i in the solution is the same with
fugacity coefficient of pure component i at the same T & P (at mixture’s T & P)

EXERCISE
• Assuming that the mixture is an ideal solution, for a gas mixture of ethylene (1) and propylene (2), estimate
fugacity and fugacity coefficient of each component in the mixture ( መf 1 , መf 2 , ∅ 1 , ∅ 2 ) at T = 150 o C, P = 30 bar, and
y 1 =0.35.
1.
id
For ideal mixture, ∅ i = ∅i
2. Look for Tcr, Pcr, and ω of each component
3. Calculate ∅ i for each component according to the
appropriate equation (based on the phase)
4. Calculate መf i from ∅ i

REFERENCES
• Smith, J. M., Van Ness, H.C., Abbott, M. M.,”Introduction to Chemical Engineering Thermodynamics” 7th
ed., McGraw-Hill Co-Singapore. 2005.
• Atkins, de Paula., “Atkin’s Physical Chemistry”, 10th ed., Oxford University Press, 2014.
• Cengel, Y.A. dan Boles, M.A., “Thermodynamics: An Engineering Approach” 5th ed., McGraw-Hill, 2006.