Lecture 6
Multicomponent systems analysis
- Partial molar properties
- Chemical potential
- Fugacity
- Ideal solution
Study materials: Moran Chapter 11: 11.9
Partial Molar Property (PMP)
single phase multicomponent system
differentiate using
chain rule
setting α = 1
PMP definition
=>
Partial molar properties – apply for V, U, H and S
X – extensive property
Xi – partial molar property
Partial molar properties – apply for V, U, H and S
X – extensive property
Xi – partial molar property
Evaluate change in volume - example
Total volume of pure components before mixing:
Volume of the mixture:
Volume change on mixing is:
Change in extensive properties upon mixing
Property changes on mixing
Examples
(Sandler Fig 8.1-2)
Example – exothermic mixing
(Sandler 8.1-1)
3 moles of water and one mole of
sulfuric acid are mixed isothermally
at 0 °C. How much heat must be
absorbed or released to keep the
mixture at 0 °C?
Example – exothermic mixing
(Sandler 8.1-1)
Chemical Potential μ
-molar μ is also known as the partial molar free energy
-when T and P are constant the μ is the partial molar Gibbs free
energy
definition
Applying:
then
G – G of mixture
μi – partial molar G or μ
Chemical Potential – further relations
Taking
and
derive
Chemical Potential μ
-μ of a species is the energy that can be absorbed or released
due to a change of the particle number of the given species, e.g.
in a chemical reaction or phase change
- particle number N = nNA
n
number of moles
NA Avogrado constant
- μ of a species in a mixture is the rate of change of free energy of
the system with respect to the change in the number of atoms or
molecules of the species added to the system
=> partial derivative of free energy to amount of species
Equilibrium and Chemical Potential
- Equilibrium: A condition reflecting the absence of any net changes in
the state of a system as well as the absence of any driving force for
change
Mechanical: pressures on both sides of a piston are equal, no
motion of the piston (forces in balance, no pressure driving force)
Thermal: temperatures of two objects in contact are equal, no heat
flow (no temperature driving force for heat flow)
Phase: chemical potentials of species in multiple phases are equal,
no net flow of material between phases (no chemical driving force)
Chemical: chemical potentials of reactants and products balance,
so the net formation of products ceases (no chemical driving force)
Gibbs-Duhem Equation for G
Note: derivation on next slides
At constant P and T :
Application of Gibbs-Duhem
- a thermodynamic consistency relation – testing for consistency of data
- a restriction on the mixture equation of state
- important in minimizing amount of experimental data necessary
in evaluating thermos properties of mixtures, simplifying the description
of multi-component systems
Gibbs-Duhem Equation: derivation for G
The differential of
is
Recall
Substitution gives:
Gibbs-Duhem Equation Derivation
Recall for single component system:
Now forming the differential of:
Gibbs-Duhem Equation
Fundamental thermodynamic functions for
multicomponent systems
Similar derivations as for dG above
Single component system
counterparts
Fundamental thermodynamic functions for
multicomponent systems
The differential of
Fundamental thermodynamic functions for
multicomponent systems
And also, same for H and Ψ
Summary – fundamental relations
Multicomponent
Single component
H
Multicomponent
Single component
U
29
Summary – fundamental relations
Multicomponent
Single component
Ψ
Multicomponent
Single component
G
30
Maxwell counterparts
- mixed second derivatives
• From
• Recall single
component
• =>
And numerous relations involving μ can be obtained e.g.
(11.117)
Various expressions for chemical potential μ
*
*Note: only this one is a partial molar property
Fugacity
-A parameter to differentiate between real and ideal gas.
For an ideal gas the pressure term is used and for real gases
fugacity is used to define the behaviour of real gases, it is an
“effective pressure“ for real gases.
- Determined experimentally or estimated from various
models such as a Van der Waals gas that are closer to reality than
an ideal gas.
Fugacity
-An effective partial pressure which replaces the mechanical partial
pressure in an accurate computation of the chemical equilibrium
constant. It is equal to the pressure of an ideal gas which has the
same temperature and molar Gibbs free energy as the real gas.
-The real gas pressure and fugacity are related through
the dimensionless fugacity coefficient φ
𝑓
φ=
𝑝
- For an ideal gas, fugacity and pressure are equal and so φ = 1
- Fugacity will be used to model phase equilibrium
Fugacity – development for single component
Single component
On a mole basis
Integrate at constant T
Fugacity
real gas
ideal gas
=>
and
into
=>
𝑓
𝜑=
𝑝
Complete
fugacity
function
Fugacity – in terms of residual/excess μ
remember:
and
𝑓
𝜇𝑅 = 𝜇 − 𝜇 = 𝑅𝑇 (𝑙𝑛𝑓 − ln 𝑝) = 𝑅𝑇 ln
= 𝑅𝑇𝑙𝑛𝜑
𝑝
∗
𝑓
ln =
𝑝
𝜇−𝜇∗
𝑅𝑇
= 𝜇𝑅/𝑅𝑇
Evaluating Fugacity – using generalised EOS or chart
=>
=>
=>
In terms of Z
In terms of reduced properties
From chart:
Example - Fugacity tables to chart comparison
(Moran in-text example page 543)
Consider water at 400 °C undergoing Isothermal compression from
200 bar to 240 bar.
From chart:
But:
=>
=>
using steam table data
Fugacity for Multicomponent systems
Fugacity for Multicomponent systems
- discussion
Evaluating fugacity in a mixture
Relation between fugacity and measurable quantities
Evaluating fugacity in a mixture - derivation
In the limit as p’ tends to zero
then
Subtracting
Integrating
as p’ tends to zero