Lecture: Chemical Equilibrium
Pieces to Cover:
1st Law and 2nd Law of Thermodynamics
E in terms of
G in terms of (constant T, constant P)
- What is G? (c.f. Wikipedia)
What is ?
What is equilibrium? (Criteria)
- Phase equilibrium analogy
- Chemical reaction equilibrium condition
- Minimization of Gibbs free energy (subject to element conservation)
NASA CEA (Chemical Equilibrium Analysis) Program
- T, P - methane combustion example
- Adiabatic flame temperature
- Condensed phase combustion (coal)
Laws of Thermodynamics
To begin a discussion of chemical equilibrium, we can start with the 1st Law of Thermodynamics:
(Can someone remind me of the difference between E and W/Q?)
(Why and not ? Why and not ?)
E = system property, state property
Q, W = path-dependent
For equilibrium chemical systems, how can we simplify ?
Are we considering shaft work? Electrical work?
So now the 1st Law becomes:
We can also simplify , by using the 2nd Law of Thermodynamics:
and plugging this into the 1st Law gives:
or, for reversible processes,
Now, E is a state function
Meaning, it is completely characterized by S and V
But what about multicomponent systems? Does the energy change if the mixture changes?
Now E needs to be characterized with composition, too:
Recall the Gibbs Phase Rule
So if we differentiate this expression, we get:
So now let's use the other identity:
So what can we say about the relationship between and ?
Same with :
Now I'm going to define an arbitrary variable, that I'll call , to be equal to the last partial derivative:
and I'm going to call the chemical potential of species i.
Okay, so, that was a lot of work - what was it for? Why did I want to get this expression for ? Why did I want it in terms of and and ?
Let me ask that another way... What does the following equation actually tell us?
What does the first term tell us?
How much the internal energy E will change, if we change the entropy of the system by a very small amount.
What about the second term?
How much the internal energy E will change, if we change the volume of the system by a very small amount.
What about the third term?
How much the internal energy E will change, if we change the composition of the system by a very small amount.
Let's say we want to go into the lab and actually measure values for these terms. We fill a piston with a gas mixture. How do we measure the first term?
How do you change the entropy a fixed amount? Can we go to the engineering supply store and buy an enthalpy meter?
No! - so, it would be nice if we could rewrite this state function in terms of quantities that we can actually measure in a lab, and control.
Which one do you think is the easiest to deal with?
Gibbs function - it's in terms of easy-to-keep-constant and easy-to-measure variables, T and P
So, let's write dG in the same form as dE:
So, can someone tell me what is equal to?
By analogy, it's equal to S
And what about ?
By analogy, it's equal to V
OK, and what about ?
Non-intuitive - but it's equal to the chemical potential... which means:
And can someone remind me of the physical meaning of chemical potential? In terms of internal energy? In terms of Gibbs energy?
The amount by which the internal energy/gibbs energy of the system changes when we change the composition by a differential amount, holding S and V/T and P constant
So does everyone see the significance of this quantity? Do we have to change the compositions? Or can they change on their own?
Chemical reactions can allow the system to change its composition on its own. This means the system can respond dynamically.
So now let's talk about the relationship between Gibbs energy and equilibrium.
What is equilibrium? Anyone remember from their thermodynamics/phase equilibria class?
Let me pose an analog question. What is the condition for two phases, e.g. vapor and liquid, to be in equilibrium?
What is the condition for a chemical reaction to be in equilibrium?
In general, the equilibrium condition is
For a mixture of different gas species, at constant temperature and constant pressure (i.e. the first two partial derivatives in the expression for above equal to zero),
For any mixture, we can solve this equation numerically and determine the amount of each species at equilibrium.
What restriction do we have - what do we have to conserve?
Are the number of moles conserved?
No - number of moles can change
Atoms must be conserved. We can make or break whatever chemical bonds - but we cannot create or destroy atoms.
NASA CEA Program Tutorial
Installation only works on Windows, it does not work on Mac OS X.
Gas Phase Reactions
Example: Reacting CH4 and air
Set (end) thermodynamic state by setting T, P
Example: adiabatic flame temperature calculation
Specify hp - we want the enthalpy change to be zero
We can specify a range of equivalence ratios to construct a plot of adiabatic flame temperature vs. equivalence ratio
Why is this plot commonly used? Why is it useful?
Tells us about the energy content of the fuel
It tells us the amount of enthalpy that goes toward heating other reactants (e.g. nitrogen - very important factor in air combustion!)
Affect of the C/H ratio in the fuel
Stoichiometric mixture required to reach the peak flame temperature
Condensed Phase Reactions
What information does CEA require for a condensed phase fuel like coal?
First: what information do you have about coal?
(Elemental composition - C, O, H, S, N)
(HHV/LHV from lab analysis)
Tuesday's lecture - elemental composition + HHV/LHV --> heat of formation of coal
So the process for condensed phase calculations is:
(MOLAR Composition and Heat of Formation) --> (NASA CEA) --> (Equilibrium products composition)
by using the equation
The molar composition tells us how many atoms we're starting with
The heat of formation tells us how much energy we're starting with
Canteraall pages on the wiki related to the Cantera combustion microkinetics and thermodynamics (a.k.a. "thermochemistry") software.
Understanding Cantera's Structure: Cantera Structure
Cantera from Matlab: Using_Cantera#Matlab
Cantera from Python: Using_Cantera#Python
Cantera from C++: Using_Cantera#C++
Cantera + Fipy (PDE Solver): Fipy and Cantera/Diffusion 1D
Cantera Gas Objects: Cantera/Gases
Cantera Gas Mixing: Cantera_Gas_Mixing
Topics in Combustion:
Sensitivity Analysis: Cantera/Sensitivity Analysis
Analysis of the Jacobian Matrix in Cantera: Jacobian_in_Cantera
Chemical Equilibrium: Chemical_Equilibrium
Kinetic Mechanisms: Cantera/Kinetic_Mechanisms
Reactor Equations: Cantera/Reactor_Equations
Differential vs. Integral Reactors: Cantera/Integral_and_Differential_Reactors
Effect of Dilution on Adiabatic Flame Temperature: Cantera/Adiabatic_Flame_Temperature_Dilution
Topics in Catalysis:
Cantera for Catalysis: Cantera_for_Catalysis
Steps for Modeling 0D Multiphase Reactor: Cantera_Multiphase_Zero-D
Reaction Rate Source Terms: Cantera/Reaction_Rate_Source_Terms
Surface coverage: Cantera/Surface_Coverage
Surface reactions: Cantera/Surface_Reactions
Cantera Input Files:
Chemkin file format: Chemkin
Pantera (monkey patches and convenience functions for Cantera): Pantera
Extending Cantera's C API: Cantera/Extending_C_API
Extending Cantera with Python Classes: Cantera/Adding Python Class
Debugging Cantera: Cantera/Debugging_Cantera
Debugging Cantera from Python: Cantera/Debugging_Cantera_from_Python
Gas Mixing Functions: Cantera_Gas_Mixing
Residence Time Reactor (new Cantera class): Cantera/ResidenceTimeReactor
Cantera Resources: Cantera Resources
Cantera Lecture Notes: Cantera_Lecture
Flags · Template:CanteraFlag · e
Installing Canteranotes on the wiki related to installing the Cantera thermochemistry software library.
Mac OS X 10.5 (Leopard): Installing_Cantera#Leopard
Mac OS X 10.7 (Lion): Installing_Cantera#Lion
Mac OS X 10.8 (Mountain Lion): Installing_Cantera#Mountain_Lion
Ubuntu 12.04 (Precise Pangolin): Installing_Cantera#Ubuntu
Windows XP: Installing_Cantera#Windows_XP
Windows 7: Installing_Cantera#Windows_7
In old versions of Cantera, a preconfig file was used to specify library locations and options.
Mac OS X 10.5 (Leopard) preconfig: Cantera_Preconfig/Leopard_Preconfig
Mac OS X 10.6 (Snow Leopard) preconfig: Cantera_Preconfig/Snow_Leopard_Preconfig
Mac OS X 10.8 (Mountain Lion) preconfig: Cantera_Config/MountainLion_SconsConfig
Ubuntu 12.04 (Precise Pangolin) preconfig: Cantera_Config/Ubuntu1204_SconsConfigFlags · Template:InstallingCanteraFlag · e