Chemical Equilibrium

Lecture: Chemical Equilibrium

Pieces to Cover:

1st Law and 2nd Law of Thermodynamics

E in terms of $ \mu $

G in terms of $ \mu $ (constant T, constant P)

• What is G? (c.f. Wikipedia)

What is $ \mu $?

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:

$ dE = \delta Q + \delta W $

(Can someone remind me of the difference between E and W/Q?)

(Why $ dE $ and not $ \delta E $? Why $ \delta Q, \delta W $ and not $ dQ, dW $?)

E = system property, state property

Q, W = path-dependent

For equilibrium chemical systems, how can we simplify $ \delta W $?

Are we considering shaft work? Electrical work?

$ \delta W = p dV $

So now the 1st Law becomes:

$ dE = \delta Q - p dV $

We can also simplify $ \delta Q $, by using the 2nd Law of Thermodynamics:

$ T dS \geq \delta Q $

and plugging this into the 1st Law gives:

$ dE \geq T dS - p dV $

or, for reversible processes,

$ dE = T dS - p dV $

Now, E is a state function

Meaning, it is completely characterized by S and V

$ E = E(S,V) $

But what about multicomponent systems? Does the energy change if the mixture changes?

Now E needs to be characterized with composition, too:

$ E = E(S, V, N_{i}) $

Recall the Gibbs Phase Rule

So if we differentiate this expression, we get:

$ dE = \left( \frac{\partial E}{\partial S} \right)_{V,N_{i}} dS + \left( \frac{\partial E}{\partial V} \right)_{S,N_{i}} dV + \displaystyle{ \sum_{i=1}^{N_{species}} \left( \frac{\partial E}{\partial N_{i}} \right)_{S,V,N_{j \neq i}} } $

So now let's use the other identity:

$ dE = T dS - p dV $

So what can we say about the relationship between $ T $ and $ \left( \frac{\partial E}{\partial S} \right)_{V,N_{i}} $?

$ T = \left( \frac{\partial E}{\partial S} \right)_{V,N_{i}} $

Same with $ P $:

$ P = \left( \frac{\partial E}{\partial V} \right)_{S,N_{i}} $

Now I'm going to define an arbitrary variable, that I'll call $ \mu_{i} $, to be equal to the last partial derivative:

$ \mu_{i} = \left( \frac{\partial E}{\partial N_{i}} \right)_{S,V,N_{j \neq i}} $

and I'm going to call $ \mu_{i} $ the chemical potential of species i.