Phase diagrams indicate which phases of a material are thermodynamically favorable given a
certain set of conditions. The Sn-Bi alloy system can produce a low melting point alloy which
could be an alternative to lead based solders. In this study, 13 Sn-Bi
Name:
Homework for Lecture 3
Consider an atom diffuses in a 3D simple cubic lattice by a random walk mechanism. The atom
jumps 10-5 times per second at 300K and 104 times per second at 600K.
1). How many times the atom will jump per second at 900K?
2). Ho
Lecture 13
Reading: Chapter V
Homework: None
Thermodynamic Relations:
Since
U
U
dU TdS PdV
dS
dV
S V
V S
H
H
dH TdS VdP
dS
dP
S P
P S
A
A
dA SdT PdV
dT
dV
T V
V T
A
G
dG SdT VdP
dT
dP
T P
P T
We have the following t
Name:
Homework for Lecture 6, 5
Consider a binary solution of A and B with the concentration of XA=0.4. The free energy of this
binary solution is given by =
G G 0 + RT ( X A X B ) . Suppose the trace diffusion coefficient of A is
three times of that of B
Name:
Homework for Lecture 5
Consider a binary solution of A and B with the concentration of XA=0.20. Assuming the Henry
activity coefficient (A) of A in this binary solution is given by
ln A = 0.50 X A 2 X B .
Determine the ratio of the chemical diffusio
Lecture 24
Reading: Chapter VIII
Homework: 8.1, 8.3, 8.5, 8.6, 8.7
Mixing of Ideal Gas under various conditions:
(1) At fixed P and T
nA, T, P
nC, T, P
nB, T, P
nD, T, P
(a) Mixing Gibbs free energy:
G'initial n AG A0 (T ) n A RT ln P nB GB0 (T ) nB RT ln
Lecture 33
Reading: Chapter X
Homework: None
Liquid solution- ideal; solid solution- Regular with > 0
The activity coefficients for a regular solution are given by
x B2
x A2
, and ln B
ln A
RT
RT
Thus, partial molar free energies (chemical potentials) i
Lecture 30
Reading: Chapter X
Homework: None
Criterion on Regular solution:
The schematic (previous note) shows that at low temperatures, the GM vs. xB curve can
be convex up. Such a trace is characterized by one maximum, two minima and two
inflexion poin
Lecture 2
Reading: Chapter One.
Homework: None.
State of a system: The most important concept in thermodynamics is the state
of a system, which has to be uniquely defined by independent state variables. Any
system contains matter made up of particles (ato
Lecture 8
Reading: Chapter Four
Homework: None
Statistical Interpretation of Entropy: Disorder at atomic scale
Consider a solid crystalline material:
At zero K: All atoms frozen at fixed positions on a periodic lattice.
Add heat to a finite temperature, S
Lecture 21
Reading: Chapter VIII
Homework: None
Real Gases: Van der Waals Model:
The pressure exerted on the container wall is reduced (compared to the ideal gas)
in the real gas due to the interaction between gas molecules. Suppose the number density
of
Lecture 17
Reading: Chapter VI
Homework: 6.3, 6.5, 6.7, 6.10
Enthalpy change involving phase changes in Reactants and/or Products:
Suppose phase changes occur in A and AB at TA and TAB, respectively.
Let, T>TAB>TA>T0
A()+BAB()
T
AB()AB()
TAB
A()A()
TA
T0
Lecture 5
Reading: Chapter Three
Homework: None
Reversible vs. Irreversible process:
Reversible - a process during which the whole system is kept at
equilibrium. The direction of the process can be reversed by reversing the external
driving force into the
Lecture 7
Reading: Chapter Three
Homework: 3.2,3.4,3.5,3.6
Heat Engines: A device converting heat into work.
First steam engine was built in 1769.
First thermodynamics analysis was done by Sadi Carnot in 1824.
q2
Heat reservoir at high
temperature T2
q1
H
Lecture 15
Reading: Chapter VI
Homework: None
Heat Capacity of Crystalline Solid:
U
At constant volume: Cv
(extensive property)
T V
H
At constant pressure: C p
T P
Introducing molar heat capacity, c v and c p (intensive property):
CV ncv ; C P nc
Lecture 31
Reading: Chapter X
Homework: None
Free energy composition diagrams phase diagrams: ideal solutions in both solid
and liquid solutions
GA
GB
s
l
s
l
TAf
T
TBf
T
Let us assume that TBf (melting point, f-fusion)>TAf
Let us also assume that heat ca
Lecture 3
Reading: Chapter Two
Homework: 2.1,2.3,2.5,2.7
First law of thermodynamics: A statement of energy conservation.
q
U
w
Change in internal energy = heat input work done
U = q w
dU = q - w
Sign conventions:
q > 0: heat input into the system from th
Lecture 19
Reading: Chapter VII
Homework: none
The variation of G with P and T (Criterion for equilibrium in a one-component
system):
Consider solid-liquid equilibrium:
l
s
We consider the entire chamber filled with liquid and solid with no void space. If
Lecture 22
Reading: Chapter VIII
Homework: None
Relation between Cp and Cv:
We have previously derived a relation between Cp and Cv. Here, we will again derive it
and then apply to Van der Waals gas.
H
U
V
Cp
P
T P T P
T P
U
Cv
T V
U
U
Lecture 28
Reading: Chapter IX
Homework: 9.1, 9.5, 9.7, 9.9, 9.10.
Regular Solutions:
Most solutions are non-ideal. The interactions between AA, BB, and AB are in general
different. In such cases, the activity coefficients are composition dependent. There
Lecture 6
Reading: Chapter Three
Homework: None
Proof of S being a state function:
A state function has exact differential. For any given function Z=Z(x,y), unique,
single-valued, and differentiable, dZ=Mdx+Ndy is an exact differential if
M
N
.
y
Lecture 14
Reading: Chapter V
Homework: 5.2, 5.8, 5.10, 5.12.
Gibbs-Helmholtz equation:
(1) Consider a process at constant pressure and composition (using G and/or H !):
dG
S
dT
dG
G
G H TS H T
S
dT
T P
dG
H G T
So,
dT
Since,
d G
G 1 dG
2
dT T
Lecture 25
Reading: Chapter IX
Homework: none
Theory of Solutions:
PA0
A(s or l)
Pure A
Equilibrium between a condensed phase (an element or compound or a multi-component
solutions) and the gas phase implies that the chemical potential of every component
Lecture 20
Reading: Chapter VII
Homework: 7.1, 7.3, 7.5, 7.8
More on vapor pressure over the solid phase:
We have shown that
dP H g (T ) H s (T )
dT
P
RT 2
What is H g (T ) H s (T ) ?
This is simply the enthalpy of sublimation (evaporation). Suppose we kn
Lecture 4
Reading: Chapter Two
Homework: 2.1,2.3,2.5,2.7
Reversible Adiabatic Process for an Ideal Gas:
q = 0, so dU = -w
In an adiabatic process,
Consider one mole of idea gas,
dU = CV dT
For a reversible process,
w = PdV.
Thus
CV dT PdV
RT
dV .
V
Integ
Lecture 29
Reading: Chapter X
Homework: None
Free energy composition diagram for a homogeneous solid solution:
We will discuss a binary solution of A and B. Although we will discuss a solid solution,
the analysis is equally applicable to a liquid solution
Lecture 10
Reading: Chapter Four
Homework: None
The effect of temperature (:
As we are considering a closed system, i.e., N ni is fixed, with a constant
i
volume, i.e. energy levels, i, are fixed. For the most probable distribution,
N
ni exp( i ) .
Z
Now,
Lecture 16
Reading: Chapter VI
Homework: None
The empirical function of constant-pressure heat capacity:
In general, the experimentally measured CP is given as an empirical function of
temperature:
C
C P a bT 2
T
The dependence of Enthalpy and Entropy on
Lecture 18
Reading: Chapter VII
Homework: None
Gibbs Phase Rule:
F= C-P+2
C= Number of components
P= Number of phases
F= Degree of Freedom
Application to a one-component system:
C=1, so, F=3-P
Thus, the maximum number of phases is 3 when F=0. When the deg
Lecture 11
Reading: Chapter Four
Homework: 4.1,4.3,4.4
Boltzmann Equation:
Consider adding a small amount of heat to a system of fixed V and T, So, w = 0.
First law:
So,
dU q w q
q dU
dS
T
T
also, ln N ln Z U / k B T
dU
dS
d (ln )
k BT k B
dS k B d (ln