Exam2 Foundations of MSE

Multicomponent Material

Material whose composition involves more than one atomic element

We denote the number of components/elements by C

how many components does LiMn₂O₄ have?

C=3, ternary

Chemical potential (μ_i)

Energy cost of adding 1 mole of component i to a system

μ_i =

For example, for LiMn₂O₄, what is the chemical potential of μ_Mn?

μ_Mn = (𝜕𝑈/𝜕𝑛_Mn)_S,V,nLi,nO

Internal Energy for multicomponent system (U) is a sum of what 3 contributions?

1. Work

2. Heat

3. Mass Transfer

dU for multicomponent systems

dU = -PdV + TdS + ∑μ_idn_i

What is the expression of the differential of the free energy for LiMn₂O₄ ?

dU = -PdV + TdS + μ_Mndn_Mn + μ_Lidn_Li + μ_Odn_O

What are the conditions for thermodynamic equilibrium?

1. Intensive variables should equalize across a system, and so for all phases and components (where a and b are different phases and i is the number of components):

   1. T_a = T_b = …

   2. P_a = P_b = …

   3. μ_a,i = μ_b,i = …

Without thermodynamic equilibrium, what would happen to a system?

Heat, atoms, etc. would flow to equalzie

Conditions for thermodynamic equilibrium

For a 1-component (C=1) and 2 phase (a, b) system, when are a and b in equilibrium?

T_a = T_b

P_a = P_b

μ_a = μ_b

For a 2-component (C=2) and 2 phase (a, b) system, when are a and b in equilibrium?

T_a = T_b

P_a = P_b

μ_1,a = μ_1,b

Gibbs Phase Rule tells us what?

About the possible dimensionality of regions of a phase diagram depending on how many phases can coexist there?

What is the Gibbs Phase Rule?

DoF = C – φ + 1

for condensed phases

On a T-x phase diagram, the phases present at a given point on a phase diagram should __________ the GFE

minimize

What does T and x represent on T-x phase diagram

T = temperature

x = composition of each phase

g(x) =

G(x) / n

Molar GFE

What is the equilibrium mixture closer to pure Si?

Mostly solid b/c want to minimize g

What is the equilibrium mixture closer to pure Ge?

Mostly liquid

Where should you take the tangent of both curves to tell you the equilibrium phase compositions?

At the lowest point of each curve and connect a line

g(x) =

h(x) - Ts(x)

Why does the -Ts(x) component on a T-x curve up?

Because you introduce configurational entropy, and how many ways you can arrange the molecules in the crystal

h(x) component on T-x curve - when will it curve up?

If the two molecules “dislike” each other

h(x) component on T-x curve - when will it curve down?

If the two molecules “like” each other