Sta$s$cal Thermodynamics
Isolated System (N, V, E)
N, V, E xed
S(N, V, E) = kB ln W(N, V, E)
W = # microstates for xed N, V, E.
W = W(N, V, E)
Fundamental thermo. eq. of state.
for process i ! f ,
S(N, V, E) = kB ln(Wf /Wi )
Spontaneous if Wf > Wi for
1
First Law of Thermodynamics (9/28/12)
Work, Heat, and Energy
The First Law of thermodynamics is expressed as
dE = dQ + dW
where the internal energy E is a state function (d: exact dierential). dQ is heat change of the system and dW is
work done on the s
1
Recitation (11/30/12)
Selected from Master Problem
29. Using the Gibbs-Duhem equation that in a dilute solution the solvent satises Raouls law and
the solute satises Henrys Law.
The Gibbs-Duhem equation provides a relationship between the dierential of
1
Phase Equilibrium (11/02/12)
Pure Substance
Gibbs phase rule
For a mixture of C components with P phases,
()
total variables: (xi , T, P ) C P + 2
()
(1) conservation:
xi
= 1 for each phase 1 P
i
()
(2) phase equilibrium: i
()
= i
(P )
= . = i for each
3
Phase Equilibrium (10/26/12)
Phase Equilibrium
The Clausius-Clapeyon equation
Two phases are said to be equilibrium if (T, P ) = (T, P ). The slope of phase boundary obeys
Clapeyon equation
dP
Htrs
=
dT
T Vtrs
Applying this to a liquid-vapor equilibr
Review of Calculus (9/7/12)
I.
(3x ) =
DIFFERENTIATION
A.
1
Single variable
Given a single variable function: f (x) and g(x), we
df
use either dx or f (x) to denote its dierentiation.
1. Product Rule:
(f g) = f g + f g
(1)
2. Chain Rule:
d
f (g(x) = f (g
1
Review Session (12/14/2012)
Simple Mixture
Chemical potential of liquid
We can write pure component i: = + RT ln Pi and component i in mixture: i = + RT ln Pi , so the
i
i
i
chemical potential can be written as
i = + RT ln
Pi
Pi
where
Pi = xi Pi
Raoult
1
Review session (11/02/12)
Review of Calculus
exact dierential
dF is an exact dierential if
y
F
x
x
=
y
x
F
y
x
y
i.e. F is a state function of the system and
b
a
dF = F (b) F (a) is independent of path.
Taylor expansion
We can approximate f (x) near x
1
Second Law of Thermodynamics (10/12/12)
Entropy Changes
The entropy is dened as a state function with
dS =
dQrev
.
T
The Second Law of thermodynamics can be expressed equivalently as
dS
dQ
T
0
dQ
T
where > applies to a spontaneous process (irreversible
1
Chemical Equilibrium (11/16/12)
Chemical Equilibrium
Chemical equilibrium minimize Gibbs free energy
One way to obtain the reaction Gibbs energy can be obtained by reaction enthalpy r H
and reaction entropy r S by r G = r H T r S. On the other hand, by
1
Fundamental Equations of Thermodynamics (10/19/12)
Fundamental Thermodynamic Equation of State
The internal energy:
U
S
U = U (S, V ) dU = T dS P dV =
U
V
dS +
V
dV
S
The enthalpy: H = U + P V
H = U (S, P ) dH = T dS + V dP =
H
S
dS +
P
H
P
dP
S
The Hol
Third Law and Beyond
Third Law of Themodynamics
Nernst Heat Theorem
In any thermodynamic process involving only
pure phases in their equilibrium states, the
entropy change
The entropy of any element in equil state is dened to
approach 0 as T approach
1
Solution for master problem set
1. An ideal gas of volume V1 expands adiabatically into a vacuum. Calculate the entropy when the
nal volume is V2 and show that the process is irreversible.
Solution:
Adiabatic expansion to vacuum: Q = 0, W = 0, and U = 0
1
Physical Chemistry 3079
G4230x
Midterm Examiation: Answer Key
Short Questions (5 pts each)
1. If one mole of a van der Waals gas, for which it can be shown that (U/V )T = a/V 2 ,
expands isothermally from 10L to V = 20L, calculate U of the transformatio
Name_
Reference Material and Scratch Paper
Selected Equations, Conversion Factors, and Constants
1. NA = 6.02 x 1023 mol-1
! !
19. F(r) =
!"! !
(Avogadros number)
2. a0 = 0.529
3. e (charge on electron) = 1.602 x 10-19 C
4. e (Eulers number) = 2.71828183
PRACTICE FINAL EXAM
1.
Daltons Atomic Theory is based on several assumptions that are listed below.
Which of these assumptions are currently known to be strictly correct?
(i) All atoms of the same element are identical.
(ii) Atoms are indivisible and unch
Physical Chemistry 3079x
Prof. B.J. Berne
Master Problem Set
1. An ideal gas of volume V1 expands adiabatically into a vacuum. Calculate the entropy when the nal volume is V2 and show that the process
is irreversible.
2. Calculate the change in entropy, H
Chem 3079
Dilute Gases
Gaseous State
Simplest state of matter.
Measuring volume is equivalent to counting number of molecules in that
volume(Avogadro) for very dilute gases.
Equation of state: In general the volume of any pure material is determined b
C3079 2012
First Law of Thermodynamics
Thermodynamics and Rate Processes
Thermodynamics is not concerned with rates of change.
Thermodynamics cannot be applied to systems not in
equilibrium
Thermodynamics tells if transformation is possible
but n
C3079 2012
Second Law of Thermodynamics
Enthalpy and Spontaneous Change!
Ball comes to rest in bowl.!
!
In nal position liberates largest!
Amount of pot. energy!
In 1878 Bertholet and Thomson suggested by analogy that setting free potential
energy was mo
1
Chemical Kinetics
Consecutive reaction:
The reaction described by
k
A P
occurs by the consecutive 2-steps mechanism:
k
k
A 1 I 2 P.
The rate law turns out coupled dierential equations
d[A]
[A] = [A]0 ek1 t
= k1 [A]
dt
k1 [A]0 k1 t
d[I]
(e
ek2 t )
[I