Transcript Document

Chemistry 444
Chemical Thermodynamics and Statistical Mechanics
Fall 2004 – MWF 10:00-10:50 – 217 Noyes Lab
Instructor:
Prof. Nancy Makri
Teaching Assistant:
Jay Chen
Office: A442 CLSL
E-mail: [email protected]
Office Hours: Fridays 2:00-3:00
(or by appointment)
Office: A430 CLSL
E-mail: [email protected]
Office Hours: Tuesdays 4:00-5:00
http://www.scs.uiuc.edu/~makri/444-web-page/chem-444.html
Why Thermodynamics?
 The macroscopic description of a system of ~1023 particles
may involve only a few variables!
“Simple systems”: Macroscopically homogeneous, isotropic,
uncharged, large enough that surface effects can be neglected, not
acted upon by electric, magnetic, or gravitational fields.
 Only those few particular combinations of atomic coordinates
that are essentially time-independent are macroscopically
observable. Such quantities are the energy, momentum,
angular momentum, etc.
 There are “thermodynamic” variables in addition to the
standard “mechanical” variables.
Thermodynamic Equilibrium
In all systems there is a tendency to evolve toward states whose
properties are determined by intrinsic factors and not by previously
applied external influences. Such simple states are, by definition,
time-independent. They are called equilibrium states.
Thermodynamics describes these simple static equilibrium states.
Postulate:
There exist particular states (called equilibrium states) of simple
systems that, macroscopically, are characterized completely by the
internal energy U, the volume V, and the mole numbers N1, …, Nr
of the chemical components.
The central problem of thermodynamics
is the determination of the equilibrium
state that is eventually attained after the
removal of internal constraints in a
closed, composite system.
 Laws of Thermodynamics
What is Statistical Mechanics?
 Link macroscopic behavior to atomic/molecular properties
 Calculate thermodynamic properties from “first principles”
(Uses results for energy levels etc. obtained from quantum
mechanical calculations.)
The Course
 Discovery of fundamental physical laws and concepts
 An exercise in logic (description of intricate phenomena
from first principles)
 An explanation of macroscopic concepts from our
everyday experience as they arise from the simple
quantum mechanics of atoms and molecules.
…not collection of facts and equations!!!
The Course
Prerequisites
•
•
Tools from elementary calculus
Basic quantum mechanical results
Resources
•
•
•
•
“Physical Chemistry: A Molecular Approach”, by D. A.
McQuarrie and J. D. Simon, University Science Books 1997
Lectures
(principles, procedures, interpretation, tricks, insight)
Homework problems and solutions
Course web site (links to notes, course planner)
Course Planner
o
o
o
o
o
Organized in units.
Material covered in lectures. What to focus on or review.
What to study from the book.
Homework assignments.
Questions for further thinking.
http://www.scs.uiuc.edu/~makri/444-web-page/chem-444.html/444-course-planner.html
Grading Policy
Homework
30%
(Generally, weekly assignment)
Hour Exam #1 15%
(September 29th)
Hour Exam #2 15%
(November 1st)
Final Exam
(December 13th)
40%
Please turn in homework on time! May discuss, but do not copy
solutions from any source!
10% penalty for late homework.
No credit after solutions have been posted, except in serious
situations.
Math Review
•
•
•
•
Partial derivatives
Ordinary integrals
Taylor series
Differential forms
Differential of a Function of One Variable
f
df 
df
dx
x
df
dx
dx
Differential of a Function of Two Variables
 f 
f ( x0  d x0 , y0 )  f ( x0 , y0 )    d x
 x  y
 f 
f ( x0  d x0 , y0  d y0 )  f ( x0  d x0 , y0 )    d y
 y  x
f
 f 
 f 
 f ( x0 , y0 )    d x    d y
 x  y
 y  x
f (x0, y0)
y
(x0, y0+dy)
(x0, y0)
df  f ( x0  d x0 , y0  d y0 )  f ( x0 , y0 )
(x0+dx, y0+dy)
(x0+dx, y0)
x
 f 
 f 
  dx  d y
 x  y
 y  x
Special Math Tool
If z  z ( x, y ), then
 x   y   z 
 y   z   x   1
 z  x   y
PROPERTIES OF GASES
The Ideal Gas Law
PV  nRT or PV  RT , V  V /n
Extensive vs. intensive properties
Units of pressure
Units of temperature
1 Pa  1 N m 2
1 atm  1.01325 105 Pa
1 bar = 105 Pa
1
1 torr =
atm
760
Triple point of water occurs
at 273.16 K (0.01oC)
Deviations from Ideal Gas Behavior
PV
z
RT
T=300K
“compressibility factor”
Ideal gas: z = 1
z < 1:
attractive intermolecular forces dominate
z > 1:
repulsive intermolecular forces dominate
Van der Waals equation
a 

 P  2  V  b   RT
V 

At fixed P and T, V is the solution of a
cubic equation. There may be one or three
real-valued solutions.
The set of parameters Pc, Vc, Tc for which the number of solutions changes
from one to three, is called the critical point. The van der Waals equation has
an inflection point at Tc.
Isotherms
(P vs. V at constant T)
 P 
T  Tc : 
 0

V

T
 P 

  0 : unstable region
 V T

Large V: ideal gas behavior.

Only one phase above Tc.

Unstable region: liquid+gas coexistence.
Critical Point of van der Waals Equation
RT
2a
 P 





(V  b) 2 V 3
 V T
2a (V  b) 2
 0 for V 
RT
3
(3 or 1 real roots)
 2P 
2 RT
6a



2 
3
4
 V T (V  b) V
3a (V  b)3
 0 for V 
RT
4
Both are satisfied at the critical (inflection) point, so...
Vc  3b
8a
27 Rb
a
Pc 
27b 2
Tc 
The law of corresponding states
Eliminate a and b from the van der Waals equation:
2
b  13 Vc , a  27b 2 Pc  3PV
c c ,


3 
1 8
P

V

 R
  TR
2  R
V
3
 3
R 

PR 
P
V
T
, VR  , TR 
"reduced" variables
Pc
Vc
Tc
All gases behave the same way under similar
conditions relative to their critical point.
(This is approximately true.)
Virial Coefficients
Z
PV
B
B
 1  2V  3V2 
RT
V
V
Virial expansion
B2V : second virial coefficient,
B3V : third virial coefficient,etc.
Using partition functions, one can show that
B2V (T )  2 N A 

0
e
 u ( r ) / k BT

 1 r 2 dr
u (r ): rotationally averaged intermolecular potential
Simple Models for Intermolecular Interactions
(a) Hard Sphere Model
u (r ) 
, r  
0, r  
2
 B2V (T )   N A 3
3
(b) Square Well Potential
, r  
u (r )   ,   r  
0, r  



2
B2V (T )   N A 3 1  ( 3  1) e / kBT  1 
3
(c) Hard Sphere Potential with r-6 Attraction
,
r 
u (r )  c6
 6 , r 
r
 B2V (T )

2
c 
 N A   3  6 3  (high T )
3
kBT 

Interpretation of van der Waals Parameters
From the van der Waals equation, …
B2V (T ) b 
a
RT
Comparing to the result of the square well model,
2
b   N A 3 ,
3
2
a   N A2 3   3  1 
3
Comparing to the result of the hard sphere model with r-6 attraction,
2
b   N A 3 ,
3

b
a
2
c
a   N A2 63
3

molecular diameter
molecular volume (repulsive interaction)
related to strength/range of attractive interaction
The Lennard-Jones Model
c6 
c
u (r )  4  12


12
r6 
r
Attractive term: dipole-dipole, or dipole-induced dipole, or induced dipoleinduced dipole (London dispersion) interactions.
Origin of Intermolecular Forces
Hˆ  Tˆnucl R i    Hˆ el ri  , R i  
Only Coulomb-type terms!
ri : electronic coordinates
R i : nuclear coordinates
The Born-Oppenheimer Approximation:
Electrons move much faster than nuclei. Fixing the nuclear positions,
Hˆ el ri  ,R i   n ri ;R i   En R i   n ri ;R i 
Adiabatic or electronic or
Born-Oppenheimer states
Electronic energies; form potential energy surface.
Responsible for intra/intermolecular forces.
INTRODUCTION TO STATISTICAL MECHANICS
The concept of statistical ensembles
An ensemble is a collection of a very large number of
systems, each of which is a replica of the thermodynamic
system of interest.
The Canonical Ensemble
A collection of a very large number A of systems (of volume V,
containing N molecules) in contact with a heat reservoir at temperature
T. Each system has an energy that is one of the eigenvalues Ej of the
Schrodinger equation.
A state of the entire ensemble is specified by specifying the “occupation
number” aj of each quantum state. The energy E of the ensemble is
E =
a E
j
j
j
The principle of equal a priori probabilities:
Every possible state of the canonical ensemble, i.e., every distribution of
occupation numbers (consistent with the constraint on the total energy) is
equally probable.
How many ways are there of assigning energy eigenvalues to the members
of the ensemble? In other words, how many ways are there to place a1
systems in a state with energy E1, a2 systems in a state with energy E2, etc.?
Recall binomial distribution:
The number of ways A distinguishable objects can be divided into 2 groups
containing a1 and a2 =A -a1 objects is
A !
W (a1 , a2 ) 
a1 !a2 !
Multinomial distribution:
The number of ways A distinguishable objects can be divided into groups
containing a1, a2,… objects is


a

k !

A !

W  a1 , a2 ,  
 k
a1 !a2 !
 ak !
k
3.5
3
25
A 3
A 6
20
2.5
15
W
W
2
1.5
10
1
5
0.5
0
0
0
1
2
3
0
1
2
3
4
5
6
a1
a1
1.4 1014
1000
A  12
1.2 10
14
A  50
800
1 1014
600
W
W
8 1013
6 10
400
13
4 1013
200
2 10
13
0
0
0 1 2 3 4
5 6 7 8 9 10 11 12
a1
4
14
24
a1
34
44
The Method of the Most Probable Distribution
The distribution peaks sharply about its maximum as A increases.
To obtain ensemble properties, we replace the weighted average by the
most probable distribution.
To find the most probable distribution we need to find the maximum of W
subject to the constraints of the ensemble.
This requires two mathematical tools, Stirling’s approximation and
Lagrange’s method of undermined multipliers.
Stirling’s Approximation
This is an approximation for the logarithm of the factorial of large numbers.
The results is easily derived by approximating the sum by an integral.
ln N !  N ln N  N
Lagrange’s Method of
Undetermined Multipliers
Extremize the function f ( x1 ,
, xn ) subject to the constraint g ( x1 ,
, xn )  0.
f
d x j  0.
j 1 x j
n
The function has an extremum if d f  
g
d x j  0.
j 1 x j
n
The constraint condition is satisfied at all points, so d g  
This relation connects the variations of the variables, so only n-1 of them are
independent. We introduce a parameter  and combine the two relations into
 f
g





x j
j 1  x j
n

d x j  0.

Let’s pick variable xm as the dependent one. We choose  such that
f
g

0
xm
xm
This allows us to rewrite the previous equation in the form
 f
g




x j
j  m  x j
n

d x j  0.

Because all the variables in this equation are independent, we can vary them
arbitrarily, so we conclude
f
g

for all j  m
x j
x j
Combined with the equation specifying , we have
f
g

for all j
x j
x j
Notice that Lagrange’s method doesn’t tell us how to determine .
The Boltzmann Factor
Maximize W (a1 , a2 , ) subject to the constraints
a
j
A ,
j
 

ln
W


a


a
E


k
k k   0,

a j 
k
k

a E
j
j
j
j  1, 2,
where  and  are Lagrange multipliers. Using the expression for W,
applying Stirling’s approximation and evaluating the derivative we find
aj
e
 Ej
E
It can be shown that
  1/ kBT
At a temperature T the probability that a system is in a state with quantum
mechanical energy Ej is
Pj 
Q  e
j
 Ej
e
 Ej
Q
(  1/ k BT ) canonical partition function
Thermodynamic Properties of the Canonical Ensemble
Postulate:
The ensemble average
1
P
E

Q
 j j
e
j
 Ej
E j  Q 1E  U ( N ,V , T )
j
is the observable “internal” energy.
From the above,
  ln Q 
2   ln Q 
U  

k
T
B






T

 N ,V

 N ,V
2

1  U 

ln Q 
 U 
2
cv  
 kB  
constant-volume heat capacity
 

2 
2 
kBT    N ,V
 T  N ,V
   N ,V
Separable Systems
The partition function for a system of two types of noninteracting particles,
described by the Hamiltonian
Hˆ  Hˆ (1)  Hˆ (2)
with energy eigenvalues
(2)
E jk   (1)
j  k
is
Q  Q(1)Q(2)
If the energy can be written as a sum of various (single-particle-like)
contributions, the partition function is a product of the corresponding
components.
Distinguishable vs. Indistinguishable Particles
The partition function for a system of N distinguishable particles is
Q  qN
where q is the partition function of one particle.
The partition function for a system of N indistinguishable particles is
Q  q N /N !
Partition Function for Polyatomic Molecules
The Hamiltonian of a molecule is often approximated by a sum of
translational, rotational, vibrational and electronic contributions:
Hˆ  Hˆ trans  Hˆ rot  Hˆ vib  Hˆ elec
Within this approximation the molecular partition function is
q  q trans q rot q vib qelec
Translational Partition function
Atom in cubic box of side a:
3
2
q
trans
 2 mkBT 
(V , T )  
 V
2
h


Translational energy of an ideal gas:
U trans (V , T ) 
3
RT
2
Translational contribution to the heat capacity of an ideal gas:
cvtrans 
3
R
2
Electronic Partition function
There is no general expression for electronic energies, thus one cannot write an
expression for the electronic partition function. However, electronic excitation
energies usually are large, so at ordinary temperatures
qelec (V ,T ) 1
Vibrational Partition Function for Diatomic Molecule
 vvib   v  12   (harmonic oscillator approximation)
q
vib
e   /2
e vib / 2T



,  vib 
"vibrational temperature"
1  e   1  e vib / T
kB
Vibrational energy of diatomic molecule:
1 
1
U vib  N    

2
e

1


Vibrational contribution to heat capacity of diatomic molecule:
evib / T
 vib 
R

 T  1  evib / T
2
cvvib
As T  , cvvib  R


2
Rotational Partition Function for Diatomic Molecule
 Jrot 
q
2
rot
J ( J  1)
(rigid rotor approximation, (2J +1)-fold degeneracy)
2I
8 2 Ik BT
T
h2


,  rot  2
"rotational temperature"
2
h
 rot
8 Ik B
Rotational energy of diatomic molecule:
U rot  NkBT
Rotational contribution to heat capacity of diatomic molecule:
cvrot  R
Symmetry factors:
If there are identical atoms in a molecule some rotational operations result in
identical states. We introduce the “symmetry factor”  to correct this
overcounting.
q
rot
T

 rot
For homonuclear diatomic molecules at high temperature  = ½.
Polyatomic Molecules
N atoms, 3N degrees of freedom.
Nonlinear molecules:
3
Translational degrees of freedom
3
Rotational degrees of freedom
3N-6 Vibrational degrees of freedom
Linear molecules:
3
Translational degrees of freedom
2
Rotational degrees of freedom
3N-5 Vibrational degrees of freedom

q
vib
  q vib
j ,
j 1
3n  5 (linear)

3n  6 (nonlinear)
Rotational partition function for linear polyatomic molecules
q
rot
T
h2

,  rot  2
  rot
8 Ik B
Symmetry factor:
The number of different ways the molecule can be rotated into an
indistinguishable configuration.
Asymmetric molecules:   1 (e.g. COS)
Symmetric molecules:   2 (e.g. CO2 , HC  CH)
Rotational partition function for nonlinear polyatomic molecules
Rotational properties of rigid bodies: three moments of inertia IA , IB , IC .
I A  I B  IC
spherical top (e.g. CH 4 )
I A  I B  IC
symmetric top (e.g. NH3 )
I A  I B  IC
asymmetric top (e.g. H 2O)
  8 Ik BT 
Spherical top: q 
  h 2 
2
3
2
rot
Symmetric top: q rot 
Asymmetric top: q rot 
  8 I A k BT   8 I C k BT 


2
 
h2
h


2
2
1
2
1
2
1
2
  8 I A k BT   8 I B k BT   8 I C k BT 

 
 
2
2
 
h2
h
h
 
 

2
2
2
1
2
The symmetry factor equals the number of pure rotational elements (including
the identity) in the point group of a nonlinear molecule.
The Normal Mode Transformation
2
ˆ
p
Hˆ   i +V (r1 ,r2 ,
i 1 2mi
3N
) (Cartesian atomic coordinates)
Expand the potential in a Taylor series about the minimum through quadratic terms:
V
1 3N 3N
1 T
x
K
x

x  K  x ( xi =ri - ri min Cartesian displacement coordinates)

i ij j
2 i 1 j 1
2
 2V
where Kij 
xi x j
force constant matrix
We will show in a simple way how one can obtain an independent mode form by doing a
coordinate transformation. In practice, the normal mode transformation proceeds after the
Hamiltonian in expressed in internal coordinates.
Transform to mass-weighted Cartesian coordinates qi  mi xi
3N
pi2 1 2
1 2 1 3N 3N
Then
= pi and H   pi   qi Kij q j
2m 2
2 i 1 j 1
i 1 2
2
 2V
 12  12  V
Kij 
 mi m j
qi q j
xi x j
Introduce normal mode coordinates Qi (with conjugate momenta Pi ): U  Q  q
q  K  q  QT  UT  K  U  Q  QT  Λ  Q
K U  UΛ
UT  K  U  Λ, or
U is the orthogonal matrix of eigenvectors, L is the diagonal matrix of eigenvalues.
3N
1 2 1 3N 2 2
Now H   Pi  i Qi
2 i 1
i 1 2
The Equipartition Principle
Every quadratic term in the Hamiltonian of a system contributes ½ kBT to the
internal energy U and ½ kB to the heat capacity cv at high temperature.
2
ˆ
1
p
1
ˆ
Translation in one dimension: H 
(one quadratic term)  k B
2 m
2
2
1
d
1
ˆ
Rotation about an axis: H 
(one quadratic term)  k B
2
2 d
2
(linear molecules: 2 such terms, nonlinear molecules: 3 such terms)
2
ˆ
1
p
1
Vibration: Hˆ 
 m 2 xˆ 2 (two quadratic terms)  k B
2 m 2
Diatomic molecule: 3½ kB
Linear triatomic molecule: 6½ kB
Nonlinear triatomic molecule: 6 kB
THE FIRST LAW OF THERMODYNAMICS
The first law is about conservation of energy
(in the form of work and heat)
Mechanical Work
Expansion of a gas
removing pins
m
h
piston held
down by pins
Pf , V f
Pi , Vi
Work performed by the gas:
w   Pext V
Infinitesimal volume change: d w   Pextd V
Mechanical work:
Vf
w    Pext (V )dV
Vi
Convention: work done on the
system is taken as positive.
Reversible Processes
A process is called reversible if Psystem= Pext at all times. The work expended to
compress a gas along a reversible path can be completely recovered upon reversing
the path.
 When the process is reversible the path can be reversed, so expansion and
compression correspond to the same amount of work.
 To be reversible, a process must be infinitely slow.
A process is called reversible if Psystem= Pext at all times.
Vf
w    P(V )dV
Vi
Reversible Isothermal Expansion/Compression of Ideal Gas
P
P
Pi
w   nRT ln
Pf
V
Vi
Vf
Reversible isothermal compression: minimum possible work
Reversible isothermal expansion: maximum possible work
Vf
Vi
Exact and Inexact Differentials
A state function is a property that depends solely on the state of the system. It does not
P
depend
on how the system was brought to that state.
When a system is brought from an initial to a final state, the change in a state function is
independent of the path followed.
An infinitesimal change of a state function is an exact differential.
Internal energy U : state function
dU : exact differential

i
f
dU  U f  U i  U , independent of the path
Work and heat are not state functions and do not correspond to exact differentials.
The First Law
The sum of the heat q transferred to a system and the work w performed on it equal
P
the change U in the system’s internal energy.
U  q  w
Postulate: The internal energy is a state function of the system.
Work and heat are not state functions and do not correspond to exact differentials.
dU  dq  PdV
Work and Heat along Reversible Isothermal Expansion
P
P
reversible
constant-pressure
A
PA
wAC
B reversible
isothermal
C
PC
V
VA
VC
VA
V
 nRTA ln C
VA
TB  TA
wAB   PA VB  VA 
U AC  0  qAC  nRTA ln
VC
TC
qBC  U BC   cV dT
TB
TB
U AB   cV dT
TA
q AB  U AB  wAB   cV dT  PA VB  VA 
TB
TA
wBC  0
VC
VA
Free Expansion
P
Suddenly remove the partition
No work, no heat!
U  0
 U 
dU  cV dT  
 dV

V

T
U  U (T )  T  0
 U 
For real, non-ideal gases these hold approximately, and 
 is small.
 V  T
Adiabatic Processes
A process is called adiabatic if no heat is transferred to or out of the system.
U  wad , dU  dwad
P
P
PA
dU  cV dT   PdV
reversible
constant-pressure
A
PC
VA

B reversible isothermal
nRT
dV (ideal gas)
V
C
If cV(T) is known, this can be used to determine
T (and thus also P) as a function of V.
D
For a monatomic ideal gas,
V
VC
reversible adiabatic
cV  23 nR independent of T 
3
2
 TD  VA
  
 TA  VD
Gases heat up when compressed adiabatically.
(This is why the pump used to inflate a tire becomes hot during pumping.)
Adiabatic cooling!
Enthalpy
H  U  PV
P
dH  dU  PdV  VdP  dq  VdP enthalpy function
Heat capacity at constant pressure:
Ideal gas:
 H 
cP = 

 T  P
cP  cV  nR
Heat transferred at constant pressure is enthalpy change.
Reversible Adiabatic Expansion of Ideal Gas Revisited
dq  cV dT  PdV  cP dT  VdP  0
cV dT   PdV , cP dT  VdP
cP
d ln V  d ln P
cV

If cP / cV   is constant (e.g., monatomic ideal gas),
For a monatomic ideal gas,
5
3
cP 
PV  const.
5
5
nR,  
2
3
 V2 
P1

 
P2
 V1 
ENTROPY AND THE SECOND LAW
Processes evolve toward states of minimum energy and maximum disorder.
These two tendencies are in competition.
The second law is about entropy and its role in determining whether a
process will proceed spontaneously.
Entropy
A statement of the second law:
P
No process is possible whose sole effect is the absorption of heat from a reservoir and
the conversion of this heat into work.
Postulate:
There exists a state function S called the entropy. This is such that, for a reversible
process,
dS 
dq
T
 dS  0
1/T is the integrating factor for dq.
S has units of R (or kB). For a reversible process,
dU  TdS  PdV
Isolated system is a system that cannot exchange any matter or energy with the
environment.
PThe second law: The entropy of an isolated system never decreases.
A spontaneous process that starts from a given initial condition always leads to the
same final state. This final state is the equilibrium state.
Entropy of an Ideal Gas
dU  cV dT  TdS  PdV  TdS 
nRT
dV 
V
dT
dV
 nR

T
V
dS  cV
T
V
T0
V0
S   cV (T ) d ln T  nR  d ln V  f (T )  f (T0 )  nR ln
 S (T ,V )
V
V0
independent of the path!
nRT
dH  cP dT  TdS  VdP  TdS 
dP 
P
dT
dP
dS  cP
 nR

T
P
T
P
T0
P0
S   cP (T ) d ln T  nR  d ln P  g (T )  g (T0 )  nR ln
 S (T , P)
P
P0
The Clausius Principle
The Clausius principle states that
No process is possible whose sole result is the transfer of heat from a cooler body to a
hotter body.
The Clausius principle is another statement of the second law.
dU  dU A  dU B  0 (isolated system)
A
B
dS  dS A  dS B 
 1
dU A dU B
1

 dU A   
TA
TB
 TA TB 
According to Clausius’ principle, if TA > TB then heat will flow from A to B, i.e.,
dS  0
A spontaneous process evolves in the direction of increasing entropy.
Reversible vs. Spontaneous (Irreversible) Processes
 0.
In an isolated system dq

dq
Reversible process: dS 
.
T

dq
Spontaneous (irreversible) process: dS 
.
T

dq
In general, dS 
.
T
A reversible adiabatic process is an isentropic process, dS = 0.
The Caratheodory Principle
This is yet another statement of the second law. It states that
In the neighborhood (however close) of any equilibrium state of a system (of any
number of thermodynamic coordinates) there exist states that cannot be reached by
reversible adiabatic processes.
Caratheodory’s statement is equivalent to the existence of the entropy function.
P
S1
S3
S2
V
Family of isentropic (constant S)
surfaces that don’t intersect.
Proof of Existence of Non-Intersecting Adiabatic Surfaces
Suppose B can be reached from A by a reversible adiabatic process. Let’s suppose C
can also be reached from A via a reversible adiabatic process.
Consider the process A  B  C  A.
P
A
U A B C  A  0  qB C  wA B ,C  A
qB C  0 (heat absorption) 
wA  B , C  A   q B  C  0
C
B
V
So in this cycle there is heat absorbed that is converted into work. This is in
contradiction with the second law. We arrived at this contradiction by assuming
there are two reversible adiabatic processes starting from point A.
The Carnot Cycle
P
AB: reversible isothermal at temperature T1
A
BC: reversible adiabatic
B
CD: reversible isothermal at temperature T2 < T1
DA: reversible adiabatic
D
U ABCDA  0  q AB  qCD   w  0
C
V
(the system does work)

Efficiency of Carnot engine:
Entropy changes:
 1
S AB  
T2
 1 (unless T2  0)
T1
B
A
w
q
 1  CD
q AB
q AB

dq
q
q
 AB , SCD  CD  S AB
T
T1
T2

qCD
T
 2
q AB
T1
One can never utilize all the thermal energy given to the
engine by converting it into mechanical work.
The Internal Combustion Engine
In the gasoline engine, the cycle involves six processes, four
of which require motion of the piston and are called strokes.
The idealized description of the engine is the Otto cycle.
1.
Intake stroke. A mixture of gasoline vapor and air is
drawn into the cylinder (EA).
2.
Compression stroke. The mixture of gasoline vapor and
air is compressed until its pressure and temperature rise
considerably (AB).
3.
P
C
Ingition. Combustion of the hot mixture is caused by an
electric spark. The resulting combustion products attain
a very high pressure and temperature, but the volume
remains unchanged (BC).
4.
Power stroke. The hot combustion products expand and
push the piston out, thus expanding adiabatically (CD).
5.
Valve exhaust. An exhaust valve allows some gas to
escape until the pressure drops to that of the atmosphere
(DA).
6.
Exhaust stroke. The piston pushes almost all the
remaining combustion products out of the cylinder (AE).
D
B
E
A
V
Thermodynamics of the Otto Cycle
P
Reversible adiabatic compression AB: TAVA 1  TBVB 1
C
BC is reversible absorption of heat qh from a series of
D
reservoirs whose temperatures range from TB to TC :
TC
qh   cV dT .
B
TB
E
If we assume cV is constant, qh  cV (TC  TB ).
A
V
Reversible adiabatic expansion CD: TCVC 1  TDVD 1 or TCVB 1  TDVA 1
DA is reversible rejection of heat qc to a series of reservoirs whose temperatures range
TA
from TD to TA : qc   cV dT  cV (TD  TA ).
TD
 1
q
V 
T T
  1 c  1 D A  1  B 
qh
TC  TB
 VA 
VB / VA : compression ratio
Other Ideal Gas Engines
See http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/ThermLaw2/Entropy/GasCycleEngines.html
copied in 444-web-page/Ideal Heat Engine Gas Cycles.htm
Entropy of Reversible Isothermal Expansion
of an Ideal Gas


Reversible isothermal expansion: dq
sys  dU sys  d wsys

But dU sys  0  dq
sys  PdV
Ssys  
2
1

dq
sys
T

V2
V1
V2 dV
PdV
V2
 nR 
 nR ln  0
V1 V
T
V1
The heat entering the system was absorbed from the environment. Then


dq
 Senv  Ssys , S univ  0.
env   dq sys
Entropy of Spontaneous Expansion of an Ideal Gas
Spontaneous expansion: q sys  U sys  wsys  0
Entropy is a state function, so the entropy change of the system has the
same value as that during a reversible (isothermal) expansion:
Ssys
V2
 nR ln  0
V1
Because no heat is absorbed from the environment,

dq
 Senv  0, S univ  0.
env  0
The entropy of the system increased, but the entropy of the environment
remained unchanged.
Statistical Mechanical Definition of Entropy
Sensemble  kB ln W
Completely ordered ensemble:
Maximum disorder:
a1  1, a2  a3 
a1  a2  a3 
Ssys 
 0  Sensemble  0
(set that maximizes W )
1
Sensemble
A
Ssys 
1
Sensemble
A
A !
Use W 
and apply Stirling's approximation.
a
!
 j
j
ln W  lnA !   ln a j !  A lnA  A    a j ln a j  a j 
j
j
Sensemble  k BA lnA  k B  a j ln a j
j
Use populations p j 
Sensemble
aj
A
 k BA lnA  k B A p j lnA p j
j
 k BA lnA  k BA
p
j
lnA  k BA
j
p
j
Ssys  k B  p j ln p j
j
j
ln p j
Pure and Mixed States
If all replicas of our system in a particular ensemble are in the same state n, i.e.,
pn  1, pi n  0 then S  0. This is called a pure ensemble.
Note: the quantum state n need not be an eigenstate of the Hamiltonian.
If the members of the ensemble are in different quantum states, i.e.,
pi  1 for all i, then S  0. This is called a mixed ensemble.
The canonical ensemble is a mixed ensemble.
Entropy of the Canonical Ensemble
pi  Q 1e   Ei
S   k B Q 1  e   Ei    Ei  ln Q   k B  U  k B ln Q
i
S
U
 k B ln Q
T
  ln Q 
From U  k BT 
it follows that


T

 N ,V
2
  ln Q 
S  k BT 
  k B ln Q
 T  N ,V
Entropy of Monatomic Ideal Gas
  ln Qtrans 
S  k BT 
  k B ln Qtrans
 T  N ,V
3N
  ln Qtrans 

. Using Stirling's approximation to ln N !,


 T  N ,V 2 T
 2 mk T  2 V 
5
B
S  nR  nR ln 


2
2
 N A 
 h
3
Molecular Interpretation of Work and Heat
U   pjEj,
j
pj 
e
 Ej
Q

dU   p j dE j   E j dp j
j
j
variation of energy levels
without changing populations;
can be done by changing
the volume.
change populations without
changing energy eigenvalues;
can be done by heating
or cooling
 E j 

dU   p j 
 dV   E j dp j   PdV  dq
j
j
 V  N
 Q 
 E
  E j  E j 
Q  e j , 



e





V

 N ,
j
j
 V  N
 E j 
  ln Q 
P   p j 

  k BT 

V

V

 N ,
j

N
   E dp
dq
j
j
j
Example: Monatomic ideal gas
3N
2
Q
1  2 m 
N
V


N !  h2  
N
T
P  k BT  nR
V
V
N
  ln Q 
 


 V  N ,  V
ideal gas law!
We see that the ideal gas law is obtained by using relations obtained for a gas of non-interacting
particles.
The Boltzmann Factor:
Determination of the Lagrange Multiplier
S  kB  p j ln p j
dS  kB   ln p j dp j  dp j   kB  ln p j dp j
j
because
j
 dp
j
j
0
j

dq
 
dS  kB     E j  ln Q  dp j   kB  E j dp j   kB dq

T
j
j
 kB 
1
T
THE THIRD LAW
The third law is about the impossibility of attaining the absolute zero of
temperature in a thermodynamic system.
Entropy as a Function of Temperature
 U 
 S 
 
  cV  T 


T

V
 T V
P
dU  TdS  PdV
T2
S (T2 )  S (T1 )   cV (T )d ln T under constant V
T1
 H 
 S 
dH  TdS  VdP  

c

T
P




T

T

P

P
T2
S (T2 )  S (T1 )   cP (T )d ln T under constant P
T1
The third law:
Absolute zero is not attainable via a finite series of processes.
P according to the Nernst-Simon statement,
or,
The entropy change associated with any isothermal reversible process of a condensed
system approaches zero as the temperature approaches zero.
dS 
cV
dT at constant V
T
Crystals: cV
T 3 as T  0  dS T 2dT as T  0
S  kB  p j ln p j
j
The entropy of a system that has a non-degenerate ground state vanishes at the absolute
zero.
First-Order Phase Transitions
P
Many thermodynamic variables are discontinuous across first-order phase transitions.
S

Phase I
T
T0

T
0
H
T0
cPI (T )dT 
T0
cPII (T )dT 
(constant P)
Phase II
T
Helmholtz and Gibbs “Free” Energies
A  U  TS Helmholtz free energy
dA  TdS  PdV  TdS  SdT   SdT  PdV
A  A(T ,V )
G  A  PV  U  TS  PV Gibbs free energy
dG   SdT  PdV  PdV  VdP   SdT  VdP
G  G (T , P )
Reversible isothermal process under constant volume: dA = 0
Reversible isothermal process under constant pressure: dG = 0
Legendre Transforms
It is often desirable to express a thermodynamic function in terms of different
independent variables. Most often this new desirable variable is the first
derivative of a fundamental function with respect to one of its undesirable
independent variables; for example,
U  U ( S ,V )
 U 
dU  TdS  PdV , P   

 V  S
We are seeking a general tool for finding a new function that contains the
same information as the original fundamental thermodynamic function, but
where the “undesirable” variable has been eliminated in favor of the
“desirable” one. In the previous example,
We seek a new function H  H ( S , P) that is equivalent
to U but which depends on the variable P rather than V .
General Theory of Legendre Transformation
Given a function f (x), we seek an
equivalent function (i.e., one containing
the same information as f ) whose
independent variable is df /dx.
f
The curve f (x) can be reconstructed from
the family of its tangent lines.
x
A tangent line can be specified by the slope f    (new independent variable)
and the intercept   f ( x  0).
f 
Notice   f  
   f  x f ( x) “Legendre tranform of f”
x
We solve f    ( x) for x and substitute in the above relation to obtain
   ( x). This is possible if f  is single-valued, i.e., f   0 at all x.
Application of Legendre Transform Theory
U  U ( S ,V )
dU  TdS  PdV
 U 

 T
 S V
 U 

  P
 V  S
H  H ( S , P)  U  ( P)V
dH  TdS  VdP
A  A(T ,V )  U  TS
dA   SdT  PdV
 A 

  P
 V T
G  G (T , P)  A  ( P)V
 H 

 T
 S  P
 H  TS  U  TS  PV
dG   SdT  VdP
Maxwell’s Relations
f ( x1 , x2 )
df  y1dx1  y2 dx2
 f 
y1  

 x1  x2
 f 
y2  

 x2  x1
 y2 
 y1 

 


x

x
 1  x2  2  x1
Conversely, if the Maxwell relation is satisfied, one can conclude that df is
an exact differential.
Examples
dU  TdS  PdV
 U 
T 

 S V
 U 
P  

 V  S
 T 
 P 
 



 
 V  S
 S V
dA   SdT  PdV
 A 
 A 
S  
,

P




 T V
 V T
 S   P 
 
 

 V T  T V
dH  TdS  VdP
dG   SdT  VdP
 T   V 

 

 P  S  S  P
 S   V 
   

 P T  T  P
Applications of Maxwell’s Relations
Entropy of a gas:
 S 
 S 
dS  
dT



 dV
 T V
 V T
 S 
 P 
At constant T , dS  
dV



 dV
 V T
 T V
V2  P 
S 2  S1   
 dV along an isothermal process
V1
 T V
nR
 P 
Ideal gas: 


 T V V
 S 2  S1  nR ln
V2
V1
Internal energy of a gas:
 A 
 U 
 S 
 P 
U  A  TS , 


P



P

T


P

T







 V T
 V T
 V T
 T V

 U 
 P  
At constant T , dU  
dV


P

T


  dV

 V T
 T V 

V2 
 P  
U 2  U1     P  T 
 dV along an isothermal process
V1
 T V 

We may choose a sufficiently large value of V1 such that U1 is given by
the ideal gas law, then calculate the internal energy at a different volume
where the gas does not exhibit ideal behavior.
Pfaffian Forms
Extensive variable
Intensive variable
Work
V (gas)
-P
-PdV
L (wire)
F
FdL
S
A (film)
M (dipole moment)
-SdA
B
…
B dM
…
dU  TdS  Yi dX i
…
Pfaffian form
i
X i : extensive variable
Yi : intensive variable
PHASE EQUILIBRIUM
What is the equilibrium state of a multi-component system?
Phase Diagrams
Phase diagrams
Chemical Potential
Intensive variable m such that

dw
"chemical work"
chem   m j dn j
j
(The sum is over all components of a system and nj are the mole numbers.)
dU  TdS  PdV   m j dn j
j
dH  TdS  VdP   m j dn j
j
dA   SdT  PdV   m j dn j
j
 U 
 H 
mi  




n

n
i
i

 S ,V ,n

 S , P ,n
j i
j i
 A 
 G 





n

n
 i T ,V ,n j i  i T , P ,n j i
dU   SdT  VdP   m j dn j
j
Pure substance:
G( P, T , n)  n g ( P, T )  m  g ( P, T )
The chemical potential of a pure substance is the molar Gibbs free energy.
General Conditions of Equilibrium
An isolated system tends to attain the state of maximum entropy with respect
to its internal (extensive) degrees of freedom, subject to the given external
constraints.
dS = 0, d2S < 0
Consequence:
The thermodynamic potentials attain minimum values with respect to their
internal extensive variables at equilibrium, subject to the given external
constraints.
I. Thermal Equilibrium
Constraints:
A
B
V A  const., V B  const., U  U A  U B  const.
"Internal variable": U A
rigid, diathermal wall
impermeable to matter
U A  U A ( S A ,V A ) and S  S A (U A ,V A )  S B (U B ,V B )
Since the volumes cannot change,
 S A 
 S A 
 S A 
 S A  
A
B
A
dS  
dU

dU


dU









A
B
A
B
 U V A  U V B 
 U V A
 U V B
1
 S 
From dU  TdS  PdV we find 

and therefore


U
T

V
1 
 1
dS   A  B  dU A
T 
T
At equilibrium dS  0 for any dU A . It follows that
TA TB
II. Thermal and Mechanical Equilibrium
Constraints:
A
B
V  V A  V B  const., U  U A  U B  const.
"Internal variables": U A , V A
movable, diathermal wall
impermeable to matter
 S A 
 S A 
 S A 
 S A 
B
B
A
A
dV

dU

dV

dU
dS  







B
B
A
A
 V U B
 U V B
 V U A
 U V A
 S A 
 S A 
 S A   A
 S A  
A
  B   dV

 
 dU  
A 
B 
A 
 V U A  V U B 
 U V A  U V B 
P
1  S 
 S 
and therefore

,

From dU  TdS  PdV we find 



T
V

T
U

U

V

 P A PB  A
1 
 1
A
dS   A  B  dU   A  B  dV .
T 
T 
T
T
At equilibrium dS  0 for any dU A , dV A . It follows that
T A  T B , P A  PB
III. Equilibrium with Respect to Matter Flow
A
Rigid, diathermal
wall permeable to substance 1
B
Internal variables: U A , n1A
 S A 
 S B 
 S A 
 S B 
A
A
B
B
dS  
dU

dn

dU

dn
1
1
 A
 B

A 
B 

U

n

U

n
A
A
B
B
A
A
B
B

V ,n1

V ,n1
 1 V ,U
 1 V ,U
dU  TdS  PdV   m j dn j
j
 S 
m1
 




n
T
 1 U ,V
 m1A m1B  A
1 
 1
A
dS   A  B  dU   A  B  dn1
T 
T 
T
T
At equilibrium
T A  T B , m1A  m1B
1st Order Phase Transitions:
The Clausius-Clapeyron Equation
P
Phase I
B
A
A
Phase II
Ana and Jiri Jonas
7709 Santa Margherita Way
Naples, FL 34109
T
d m I  m BI  m AI   S I dT  V I dP
d m II  m BII  m AII   S II dT  V II dP
V
II
 V  dP   S  S  dT
I
II
I
dP H

dT T V

dP S III

dT V III
Clapeyron equation
Liquid-to-vapor transition
S l  g  0, V l  g  0 
dP
0
dT
Increase in pressure causes conversion to the higher-density liquid phase.
Solid-to-liquid transition
S s l  0
If V s l  0 then
If V s l  0 then
dP
0
dT
dP
 0. This is the case with water.
dT
The Clapeyron equation is an expression of Le Chatelier’s principle.
Approximation for liquid-vapor phase transition:
Vg
V l, V g 
RT
(ideal gas), so
P
1 dP H l  g

P dT
RT 2
Clausius-Clapeyron approximation
Statistical Mechanical Calculation of Chemical Potential
 G 
 A 


  
 n  P ,T  n T ,V
m 


  ln Q 
  ln Q 
A  U  TS  kBT 
  T k BT 
  k B ln Q 

T

T

 N ,V

 N ,V


2
A  k BT ln Q
  ln Q 
  ln Q 


RT



 n T ,V
 N T ,V
m   k BT 
Chemical Potential of Ideal Gas
qN
Q
 ln Q  N ln q  ln N !  N ln q  ( N ln N  N )
N!
 ln Q
q
 ln
N
N
n
PV  nRT 
N A RT  Nk BT
NA
m   RT ln

V k BT

N
P
q k BT
q
  RT ln k BT  RT ln P 
V P
V
q kBT
P
m   RT ln

RT
ln
, or
0
V P
P0
P0 standard pressure (105 Pa)
P
m  m0  RT ln
P0
SOLUTIONS
We consider a two-component system with mole numbers n1 and n2.
dG  VdP  SdT  m1dn1  m2 dn2
dG  m1dn1  m2 dn2 at constant P and T
Scale increasing the mole numbers from 0 to their final values by varying a
dimensionless parameter :
dn1  n1d  , dn2  n2 d 
n1
n2
1
0
0
0
G   m1dn1   m 2 dn2   ( m1n1  m 2 n2 )d   m1n1  m 2 n2
G  G1  G2  G1n1  G2 n2
Gi : partial molar free energies
Gi  mi
The partial molar free energy is the chemical potential of the substance, i.e., an
intensive variable. However, the partial molar free energies generally depend on
the mole fraction n1 /(n1  n2 ) . This is so because the partial derivatives
 G 
mi  

 ni T , P ,n
j i
are functions of all the variables, including nj.
Using a similar procedure we can write
V  V1  V2  V1n1  V2n2
(Vi : partial molar volumes)
The partial molar volumes depend on the mole fraction of the particular substance
in the solution and are not additive when substances are mixed! This statement
applies generally to any extensive variable. Of course extensive variables still
scale linearly with the total number of moles, provided the mole fraction of each
substance remains fixed.
Euler Relations
The internal energy U is a function of extensive variables, U  U (S ,V , n1 , n2 , ).
Based on the previous remarks, U is a homogeneous first order property, i.e.,
U ( S , V ,  n1 ,  n2 , )  U ( S ,V , n1 , n2 , )
Differentiating with respect to ,

U ( S , V ,  n1 ,  n2 , )

U ( S , V ,  n1 ,  n2 , )  ( S ) U ( S , V ,  n1 ,  n2 , ) (V )


 ( S )

 (V )

U ( S , V ,  n1 ,  n2 , ) ( n1 )


 ( n1 )

U ( S ,V , n1 , n2 , ) 
This is true for any value of . For  = 1,
U ( S , V ,  n1 ,  n2 , )  U ( S ,V , n1 , n2 , )
U
U
U
S
V
n1 
S
V
n1
U

U  TS  PV   mi ni
i
This is called Euler’s relation for the internal energy.
Other Euler relations:
H  U  PV

H  TS   mi ni
i
A  U  TS

A   PV   mi ni
i
G  A  PV

G   mi ni
i
The Gibbs-Duhem Relation
Differentiating the Euler relation for dG,
dG   m j dn j   n j d m j
j
j
Using the relation
dG   SdT  VdP   m j dn j
j
we obtain the Gibbs-Duhem relation
SdT  VdP   n j d m j  0
j
This result can also be obtained from the Euler relation for dU:
dU  TdS  SdT  PdV  VdP   m j dn j   n j d m j
j
j
Using the relation
dU  TdS  PdV   m j dn j
j
we find
SdT  VdP   n j d m j  0
j
For a one-component system we recover the known result
 SdT  VdP  nd m  dG
Phase Equlibrium in Multicomponent Systems
n1g , n2g
n1l , n2l
Two-component liquid at equilibrium with its vapor.
At constant P and T,
 G 
 G 
l
dG   l 
dn1   g 
dn1g
 n1  P ,T ,n2l ,n1g ,n2g
 n1  P ,T ,n1l ,n2l ,n2g
 G 
 G 
l
 l 
dn2   g 
dn2g
 n2  P ,T ,n1l ,n1g ,n2g
 n2  P ,T ,n1l ,n2l ,n1g
Since dn1l  dn1g  0, dn2l  dn2g  0, and
 G 
l

m
, etc.
1
 l
 n1  P ,T ,n2l ,n1g ,n2g
m1l  m1g , m2l  m2g
Assuming the vapor behaves as an ideal gas, the chemical potential of substance
j in the solution is
m  m  m (T )  RT ln
l
j
g
j
0
j
Pj
P0
For the pure substance j,
m lj*  m gj *  m 0j (T )  RT ln
Pj*
P0
because m 0j (T ) doesn't depend on mole fractions. It follows that
m  m  RT ln
l
j
l*
j
Pj
Pj*
Ideal Solutions and Raoult’s Law
If the partial vapor pressure of each component in a solution obeys the relation
Pj  x j Pj*
where x j 
nj
 ni
is the mole fraction of component j in the liquid phase,
i
the solution is called ideal. Ideal solutions follow Raoult’s law,
*
m sol

m
j
j  RT ln x j
Here m sol
j is the chemical potential of (liquid) component j in the solution
and m *j is the chemical potential of the pure substance.
Vapor Pressure of Ideal Two-Component Solutions
P  P1  P2  x1P1*  x2 P2*  x1P1*  (1  x1 )P2*  x1 (P1*  P2* )  P2*
P
*
1
P
P
P1
P2
x1
The mole fraction of
component 1 in the liquid
phase is
P  P2*
x1  *
P1  P2*
P2*
0
(linear in x1 )
1
Calculate the mole fraction y1 of component 1 in the vapor phase at a given
value P of the vapor pressure using Dalton’s law of partial pressures:
P1
y1 
P
(Dalton's law)
x1 P1*
P  P2* P1*
P1*  P2* 

 *
 *
1 
*
*
*
* 
x1 P1  x2 P2 P1  P2 P P1  P2 
P
liquid-vapor coexistence
x1 depends linearly on P.
P vs. x1
y1 depends nonlinearly
(hyperbolically) on P (and on x1 )!
P1*
P vs. y1
P2*
0
x1 or y1
1
How much liquid vs. vapor is there at a pressure PC, given that the overall mole
fraction of component 1 is x1B ?
Mole fractions in liquid and vapor phases:
n1l
n1l
x  l
 l,
l
n1  n2 n
n1g
n1g
y  g
 g
g
n1  n2 n
E
1
n1l  n1g
x  l
n  ng
B
1
n x  x
l
B
1
E
1
F
1
or n1l  nl x1E , n1g  n g y1F
 x1B  nl  n g   n1l  n1g  nl x1E  n g y1F
n y
g
F
1
x
B
1

A
"lever rule"
P vs. x1
*
2
P
P1*
B
E
C
F
D
P vs. y1
0
x1 or y1
1
Non-Ideal Solutions
Pj  x j Pj*
P
P1*
P
P
P1
P2*
P1*
P1
P2*
P2
P2
0
1
x1
attractive interactions between
different molecules dominate
Pj
0
x1
repulsive interactions between
different molecules dominate
x j Pj* as x j  1 only
1
Temperature-Composition Diagrams
P
vapor
T1*
Tb vs. y1
T2*
Tb vs. x1
liquid
0
1
Substance labeled 2 is assumed to have a lower boiling point. The vapor is
richer than the solution in the more volatile substance 2, thus y1 < x1..
Fractional distillation exploits this principle.
Azeotropes
P
vapor
T1*
Tb vs. y1
T2*
Tb vs. x1
liquid
0
x1
1
Fractional distillation cannot separate the two components.
Activity
For any solution (ideal or not),
m  m  RT ln
l
j
For ideal solutions
Pj
*
j
P
For nonideal solutions
l*
j
Pj
Pj*
 xj
Pj
*
j
P
 a j "activity of component j in the solution"
m lj  m lj*  RT ln a j
Solid-Liquid Solutions
A
B
A: water
B: water + sugar
Solutions separated by membrane
permeable to water only.
m Aj  m Bj for every substance that appears on both sides and
m wA  m w*
m wB  m w*  RT ln aw
These cannot be equal with aw  1 unless m wA  m wB .
The only way for this to happen is to have P A  P B .
Osmotic Pressure
m wA (T , P)  m w* (T , P)
B
A
m wB (T , P  )  m w* (T , P  )  RT ln aw
 m w* 
Pw
*
aw  * . But 

V
w , so

Pw
 P T
m w* (T , P   )  m w* (T , P)  
Assuming Vw is independent of pressure for the liquid,

P 
P
Vw ( P)dP  Vw
so Vw  RT ln aw  0
P 
P
Vw ( P)dP
For a dilute solution
aw  xw  1  xs
ln aw  ln(1  xs )   xs
 Vw  RTxs
xs 
ns
nw

 Vw  RTns
It follows that the osmotic pressure of dilute solutions is given by the relation
  cRT
Additional Definitions
Coefficient of Thermal Expansion

1  V 


V  T  P
Isothermal Compressibility Factor
1  V 
T   

V  P T
Special Math Tool
If z  z ( x, y ), then
 x   y   z 
 y   z   x   1
 z  x   y
KINETIC THEORY OF GASES
Root-Mean-Square Velocity
u   ux , u y , uz  velocity vector of a gas molecule
p2
3
 kBT
2m
2
From the equipartition principle,
u
2
1
2
 urms 
3kBT
m

Velocity Distribution
Recall that the probability of having translational energy
E
1
m  u x2  u y2  u z2 
2
is given by the Boltzmann factor
P( E ) e


 m ux2 u 2y uz2 / 2 k BT
The probability of having a velocity component ux in the x direction is Gaussian:
m
 mu x2 / 2 k BT
px (u x ) 
e
2 k BT
From this, u x  0 and
u
2
x

  u x2 px (u x ) 

k BT
m
The Maxwell-Boltzmann Distribution
Calculate the probability distribution f (u) for a molecule to have a velocity
modulus
u  u  u  u
2
x
2
y
2
z

1
2
by converting to spherical polar coordinates and integrating over angles:

2
0
0
f (u )du  u du  sin  d 
2
3
2
 m   mu 2 / 2 kBT
d 
e
 2 k BT 
3
2
 m  2  mu 2 / 2 kBT
f (u )  4 
 ue
 2 k BT 

Average speed: u   u f (u )du 

8kBT
m

u 
8kBT
m
Most Probable Speed
The maximum of the Maxwell-Boltzmann distribution lies at
d
f (u )  0
du
which is satisfied for
ump
2 k BT

m
Velocity Distribution and Reaction Rates
The shape of the Maxwell-Boltzmann distribution has important implications
for chemical reactions. Even though the maximum of the curve depends
weakly on temperature, the fraction of molecules with velocities higher than a
critical value depends exponentially on the temperature. Thus a relatively
small increase of temperature can have a large effect on the rate of a chemical
reaction.
REACTION RATES
Chemical Reactions
 A A  BB 
  CC   D D 
1 d [A]
1 d [B] 1 d[C] 1 d[D]





 A dt
 B dt  C dt  D dt
 k[A] [B]
Exponents: reaction order
d [A]
 k[A] [A]  [A]0 e  kt
dt
d [A]
1
1
2
Second order reactions: 
 k[A]

 kt
dt
[A] [A]0
First order reactions: 
1st Order Reactions
A
k
k
d [A]
d [B]
  k [A]+k [B]  
dt
dt
Steady state (dynamic equilibrium):
[B]eq
d [A]
k
0 
K 
dt
[A]eq
k
At all times, [A]  [B]  [A]0
B
k [A]eq  k [B]eq  k [A]0  [A]eq  
 k  k [A]eq  k [A]0
At all times, [A]  [B]  [A]0
Steady state (dynamic equilibrium):
[B]eq
d [A]
k
0 
K 
dt
[A]eq
k
k [A]eq  k [B]eq  k [A]0  [A]eq  
 k  k [A]eq  k[A]0
d [A]
 k [A]  k [A]0  [A]    k  k [A]   k  k [A]eq
dt
d [A]

  k  k  [A]  [A]eq 
dt
d
 [A]  [A]eq    k  k  [A]  [A]eq 
dt
[A]  [A]eq  [A]0  [A]eq  e ( k k )t
Temperature Dependence of Rate Constants
In most cases under common conditions, k  A e  Ea / RT Arrhenius equation
Ea : activation energy
reactants
products
reaction coordinate
Eb
Transition State Theory
Assumption: All trajectories that reach the barrier top lead to products.
k
TST









Numerator =
p
d ( x  xb ) ( p)
m
dp e  H ( x , p )
dx  dp e  H ( x , p )


0
dp e
dx 


  H ( xb , p )

p
e  Eb
  p 2 /2 m   Eb p
  dp e
e

0
m
m

Denominator = Qrcl (partition function of reactants)
Assuming the potential is harmonic about the minimum,
cl
r
Q




dx  dp e
 p2 1

 
 m02 ( x  x0 )2 
 2 m 2


k TST 

2
0
0   E
e
2
b
Taking into account degrees of freedom orthogonal to the reaction coordinate,
†
k TST
k BT Q †   Eb

e
h Qr
†
Q † : partition function of stable modes at the transition state
Qr : partition function of reactants
Corrections to Classical Transition State Theory
1.
2.
Recrossing of transition state region (negative corrections)
Quantum mechanical effects (primarily tunneling)
ln k
activated crossing
tunneling regime
1/T
STABILITY CRITERIA AND PHASE TRANSITIONS
Concavity of the Entropy
Imagine a system whose entropy function
of a system has the shape shown in the
figure. Consider two identical such
systems, each with internal energy U0 and
entropy 2S(U0). Suppose we remove
energy U from the first system and put
it in the second system. The new entropy
will be
S (U  U )  S (U  U )  2S (U )
S
U0 - U
U0 U0 + U
U
Since this rearrangement of the energy results in a larger entropy, it should occur
spontaneously if the two systems are connected through a diathermal wall. This way
the system will break up into two systems of different thermodynamic properties. This
process is a phase transition.
The instability leading to phase separation is a consequence of the assumed convex
shape of S over a range of U. In stable thermodynamic systems the entropy function is
a concave function, i.e., d 2 S < 0 with respect to the extensive variables U and V.
Stability Conditions for Thermodynamic Potentials
The concavity condition for the entropy implies the convexity of the internal energy
function with respect to its extensive variables S and V, as illustrated in the figure.
Adapted from H. B. Callen, Thermodynamics and
an Introduction to Thermostatistics, 2nd Edition.
V
V
The other thermodynamic potentials are functions of extensive as well as intensive
variables. Because intensive variables are introduced through negative terms in the
Legendre transform of the internal energy, the resulting thermodynamic potentials
are concave functions of their intensive variables (but they are still convex functions
of their extensive variables). For example,
 2 A 
  2G 
 2   0,  2   0
 V T
 T  P
A consequence of the first of these relations is
2
 P    A 

   2   0  T  0
 V T  V T
V
First Order Phase Transitions
G or U
Failure of stability criteria:
If the fundamental thermodynamic
function of a system is unstable,
fluctuations may take the system over the
local maximum, and the system breaks up
into more than one phases.
S or V
Second Order Phase Transitions and Critical Phenomena
The two stable minima responsible for a first order phase transition coalesce at the
critical point, giving rise to a second order phase transition.
G or U
Critical phenomena are
accompanied by huge density
fluctuations, which give rise to
the observed “critical
opalescence”.
S or V
SUPERFLUIDITY AND BOSE-EINSTEIN CONDENSATION
The History of Superfluid 4He
1908: 4He was first liquified (5.2 K).
Unusual properties were observed: strange flow, expansion upon cooling below 2.2
K.
1928: Sharp maximum in the density with a discontinuity in slope at ~2.2 K. Two phases.
1932: The specific heat diverges at 2.17 K; the curve has a  shape (“lambda transition”).
Normal and superfluid phases identified.
1930s-1940s:
Remarkable transport properties of superfluid 4He studied extensively.
• Viscosity drops by many orders of magnitude; the system flows through capillaries.
• The superfluid forms extended thin films over large surfaces.
• The superfluid does not rotate upon rotating its container.
• It appears the superfluid flows without friction!
The Phenomenon of Superfluidity
A group of phenomena including:
• Frictionless flow
• Persistent current
• Heat transfer without a thermal gradient
Bose-Einstein Condensation (BEC)
Einstein predicted that if a gas of bosons were cooled to a sufficiently low temperature, all the
atoms would gather in the lowest energy state.
In 1995, Cornell and Wieman produced the first condensate of 2000 Rb atoms at 20 nK.
Ketterle produced a condensate of Na with more atoms and observed interference patterns.
BEC is intimately connected with superfluidity, but is not a necessary condition for this group
of phenomena.
Recent Nobel prizes:
2001: Cornell, Wieman and Ketterle for BEC
2003: Leggett for theory of superfluids
Condensate fraction:
n0 
N0
N
(N0: number of particles in the zero momentum state)
In the strongly interacting 4He superfluid the condensate fraction is small (about 7% at T = 0)
The Quantum-Classical Isomorphism
A single quantum mechanical particle is isomorphic to a “necklace” of N classical “beads”
that are connected with one another via harmonic springs and which experience a
potential equal to 1/N of the actual potential felt by the quantum particle.
Quantum statistical effects of identical bosons or fermions manifest themselves in the
exchange of beads, which causes the necklaces of different particles to cross-link.
A snapshot of 4He at 1.2 K. Each 4He atoms is represented in the simulation through
20 “pair-propagator” beads. The blue beads correspond to linked necklaces.
The End