Download Thermodynamics and Statistical Mechanics

Thermal Expansion



Solid
Liquid
Gas
Thermal Expansion: Linear
Linear Expansion
   0 T
   0 (1  T )
  coefficien t of linear expansion
5
Typical values range from 10  10 C
-6
-1
Thermal Expansion: Volume
Volume Expansion
DV » 3aV0 DT = bV0 DT
(follows from linear expansion and neglecting higher order terms)
b = 3a º coefficient of volume expansion
Typical values range
from 10-6 -10 -5 C°-1 for solids
from 10-4 -10 -3 C°-1 for liquids
around 3.4x10-3C°-1 for most gasses at atmospheric pressure
Thermal Expansion




When a material with a hole is warmed, does the
hole expand or contract?
A. Expand
B. Contract
C. Neither
Thermal Expansion: Exceptions


Water
Polymers (rubber band)
Kinetic Theory of Gasses
The average translati onal kinetic energy of
molecules in random motion in an ideal gas is directly
proportion al to the absolute temperatu re of the gas.
K  12 mv 2  32 kT 
v rms
3kT

m
translational degrees of freedom
2
kT
Maxwell Distribution of Speeds
 m 
f (v)  4N 

 2kT 
Consider limits :
lim v0  0
lim v  0
Discuss :
v P , v and v rms
3/ 2
2
2
v e
 12 mv
kT
Calorimetry and Conservation of
Energy
If a system (or set of subsystems ) is isolated,
then energy is conserved.
This implies that the heat flowing to or from
various subsystems must sum to zero.
Qi  0
Calorimetry and Conservation of
Energy
For solids or liquids, the heat flowing
into or out of a system is given by
Q  mcT where
m is the mass,
c is the specific heat and
ΔT  T f  Ti is the change in tempera ture.
Calorimetry and Conservation of
Energy
Phase transitions between liquid and gas
Q = ±mLV where
m is the mass,
LV is the latent heat of vaporization.
Phase transitions between liquid and solid
Q = ±mL f where
m is the mass,
L f is the latent heat of fusion.
If the transition is from a higher energy state of matter to a lower energy state
(e.g. gas to liquid or liquid to solid), then energy is being released and the sign is negative.
If the transition is from a lower energy state to a higher one, energy is being gained
by the system and the sign is positive.
Calorimetry and Conservation of
Energy
For a gas, the heat flowing into or out of a system
depends on the PROCESS.
There are two special processes with expression s
for heat that are analagous to those of solids/liq uids
Avogadro’s Hypothesis

Avogadro's Hypothesis: equal volumes of gas at
the same pressure and temperature contain equal
numbers of molecules.

specific heat for gas is process dependent and gas
dependent. However, AH implies that the specific
heat is the same for all (monatomic) gasses for a
given process. This is why we use molar specific
heat for ideal gasses.

R=“universal” gas constant
Ideal Gas Law

Assumptions:
•
•
•
Gas molecules are non-interacting
Collisions are elastic
Dimension of molecules<<average intermolecular distance

PV=nRT

R=“universal” gas constant
Calorimetry and Conservation of
Energy
For an isochoric process
Q  nCV T where
m is the mass,
C V is the molar specific heat at constant v olume and
ΔT  T f  Ti is the change in tempera ture
Calorimetry and Conservation of
Energy
For an isochoric process
Q  nCV T where
m is the mass,
C V is the molar specific heat at constant v olume and
ΔT  T f  Ti is the change in tempera ture.
For an isobarric process
Q  nC P T where
m is the mass,
C P is the molar specific heat at constant pressure and
ΔT  T f  Ti is the change in tempera ture.
Molar Specific Heats
For an isochoric process
Q  nCV T
degrees of freedom
where CV 
R
2
R is the Gas Constant 8.314 J/mol - K
For an isobarric process
Q  nC P T where
where C P  CV  R
First Law of Thermodynamics
U  Q  Wby




Change in internal energy of a system is equal to
the heat added to a system minus the work done
by the system.
Heat (Q) is the energy transferred to or from a
system due to a temperature gradient.
Work on a system (Won) is the energy transferred
to or from a system via a force across a distance
OR a pressure over a volume change.
Wby=-Won
W  PdV

Example of calculated work:
Isothermal Process for Ideal Gas
W   P (V ) dV
nRT

dV
V
dV
 nRT 
V
Vf
 nRT ln 
 Vi



First Law of Thermodynamics and
Simple Processes for an Ideal Gas
First Law
U=Q-Wby
Process
Meaning
Internal Energy
U=nCVT
U
Ideal Gas Law
PV=nRT=NAkBT
Q
Wby
First Law of Thermodynamics and
Simple Processes for an Ideal Gas
First Law
U=Q-Wby
Internal Energy
U=nCVT
Ideal Gas Law
PV=nRT=NAkBT
Process
Meaning
U
Q
Isothermal
T=0
0
Vf
Wby  nrT ln 
 Vo
Wby



Vf
nrT ln 
 Vo



First Law of Thermodynamics and
Simple Processes for an Ideal Gas
First Law
U=Q-Wby
Internal Energy
U=nCVT
Ideal Gas Law
PV=nRT=NAkBT
Process
Meaning
U
Q
Isothermal
T=0
0
Vf
Wby  nrT ln 
 Vo
Isochoric
V=0
nCV T
Wby



U  nCV T
Vf
nrT ln 
 Vo
0



First Law of Thermodynamics and
Simple Processes for an Ideal Gas
First Law
U=Q-Wby
Internal Energy
U=nCVT
Ideal Gas Law
PV=nRT=NAkBT
Process
Meaning
U
Q
Isothermal
T=0
0
Vf
Wby  nrT ln 
 Vo
Isochoric
V=0
nCV T
U  nCV T
Isobaric
P=0
nCV T
nCP T
Wby



Vf
nrT ln 
 Vo
0
PV or
nRT



First Law of Thermodynamics and
Simple Processes for an Ideal Gas
First Law
U=Q-Wby
Internal Energy
U=nCVT
Ideal Gas Law
PV=nRT=NAkBT
Process
Meaning
U
Q
Isothermal
T=0
0
Vf
Wby  nrT ln 
 Vo
Isochoric
V=0
nCV T
U  nCV T
Isobaric
P=0
nCV T
nCP T
Adiabatic
Q=0
nCV T
0
Wby



Vf
nrT ln 
 Vo
0
PV or
nRT
 U
  nCV T



More about Adiabatic Processes
For a quasistati c adiabatic process
with an ideal gas :

PV  constant
CP
where γ 
1
CV
More about Adiabatic Processes
For a quasistati c adiabatic process
with an ideal gas :

PV  constant
CP
where γ 
1
CV
Monatomic gas :   5/3
More about Adiabatic Processes
For a quasistati c adiabatic process
with an ideal gas :

PV  constant
CP
where γ 
1
CV
Monatomic gas :   5/3
Diatomic gas :   7/5
Equipartition Theorem
http://www.ux1.eiu.edu/~cfadd/1360/21KineticTheory/Equipart.html
Free Expansion




Gas is allowed to expand in volume adiabatically
without doing any work.
Q=0, Wby=0
Therefore, U=0 and T=0
The internal energy of an ideal gas does not change
during a free expansion.
Heat Engine/Efficiency need an
example for closed cycle.
Carnot Efficiency
In general :
QL
W
e
 1
QH
QH
Carnot effiency :
TL
e  1
TH
Entropy:
Second Law of Thermodynamics
Heat can flow spontaneou sly from a hot object to a cold object;
heat will not flow spontaneou sly from a cold object to a hot object.
No device is possible whose sole effect is to transform a given
amount of heat completely into work.
All reversible engines operating between th e same two constant
temperatur es TH and TL have the same efficiency . Any irreversib le
engine operating between th e same two fixed temperatu res will
have an efficiency less than this .
Entropy:
dQ
S  
T
dQ
S  
0
T
First Law of Thermodynamics and
Simple (quasi-static or reversible)
Processes for an Ideal Gas
First Law
U=Q-Wby
Internal Energy
U=nCVT
Ideal Gas Law
PV=nRT=NAkBT
S
Process
Meaning
U
Q
Isotherm
al
T=0
0
Vf 
Wby  nrT ln  
 Vo 
Isochoric
V=0
nCV T
U  nCV T
0
Isobaric
P=0
nCV T
nCP T
PV or
Q=0
nCV T
Adiabatic
0
Wby
Vf
nrT ln 
 Vo



nRT
 U
  nCV T
Q/T
First Law of Thermodynamics and
Simple (quasi-static or reversible)
Processes for an Ideal Gas
First Law
U=Q-Wby
Internal Energy
U=nCVT
Ideal Gas Law
PV=nRT=NAkBT
S
Process
Meaning
U
Q
Isotherm
al
T=0
0
Vf 
Wby  nrT ln  
 Vo 
Isochoric
V=0
nCV T
U  nCV T
0
Isobaric
P=0
nCV T
nCP T
PV or
Q=0
nCV T
Adiabatic
0
Wby
Vf
nrT ln 
 Vo



Q/T
nRT
 U
  nCV T
0
First Law of Thermodynamics and
Simple (quasi-static or reversible)
Processes for an Ideal Gas
First Law
U=Q-Wby
Internal Energy
U=nCVT
Ideal Gas Law
PV=nRT=NAkBT
S
Process
Meaning
U
Q
Isotherm
al
T=0
0
Vf 
Wby  nrT ln  
 Vo 
Isochoric
V=0
nCV T
U  nCV T
0
Isobaric
P=0
nCV T
nCP T
PV or
Q=0
nCV T
Adiabatic
0
Wby
Vf
nrT ln 
 Vo



Q/T
 Tf
nCV ln 
 Ti
nRT
 U
  nCV T
0



First Law of Thermodynamics and
Simple (quasi-static or reversible)
Processes for an Ideal Gas
First Law
U=Q-Wby
Internal Energy
U=nCVT
Ideal Gas Law
PV=nRT=NAkBT
S
Process
Meaning
U
Q
Isotherm
al
T=0
0
Vf 
Wby  nrT ln  
 Vo 
Isochoric
V=0
nCV T
U  nCV T
0
Isobaric
P=0
nCV T
nCP T
PV or
Q=0
nCV T
Adiabatic
0
Wby
Vf
nrT ln 
 Vo



Q/T
nRT
 Tf 
nCV ln  
 Ti 
 Tf 
nC P ln  
 Ti 
 U
0
  nCV T
Entropy:
Second Law of Thermodynamics
The entropy of an isolated system never decreases.
It either stays the constant (reversibl e processes)
or increases (irreversi ble processes) .
The total change in entropy of any system plus
that of the environmen t increases or stays the same.
S  S environment  Ssystem  0
Entropy:
Second Law of Thermodynamics
Statistical Description
S  k ln( W )
W  number of micro states
k  Boltzmann' s constant
Wf
S  k ln 
 Wi



Black-Body Radiation
Rate at which an object radiates
energy is observed to be proportion al
to temperatu re raised to the fourth power.
ΔQ
4
 AT
Δt
  emissivity with a value between 0 and 1
Perfrect emitter is a " black body".
Perfect emitter is a perfect absorber.
Black-Body Radiation
A black body at tempera ture T1
in an environmen t of temperatu re T2
ΔQ
4
4
 A(T1  T2 )
Δt
Planck’s radiation formula
Resolving the ultraviole t catastroph e,
Planck derived the following by assuming
that blackbody radiation was quantized.
2hc 
I ( , T )  hc / kT
e
1
2
5
Wein’s Law
Wien' s Law, first found empiricall y,
can be derived by finding the maximum of
Planck' s Radiation Formula.
 P T  2.90 x 10 3 m - K
where  P is the wavelengt h of peak intensity
T is the absolute temperatu re
Boltzmann Distribution
Consider t wo singly degenerate
energy levels with energies E 1and E 0
E1
When the system is in theral contact
with a reservoir, Boltzmann statistics
will describe the probabilit y
of level occupation .
P e  E / kT
E0
Boltzmann Distribution
P e  E / kT
E1
Relative population of levels can be
found by taking ratio of probabilit ies
P1
 ( E1  E0 ) / kT
 E / kT
e
e
P0
E0
Boltzmann Distribution
Consider limits
lim T 0
E1
P1
 E / kT

e
e 0
P0
E0
lim T 
P1
 e  E / kT  e 0  1
P0
Boltzmann Distribution:
Partition Function
P e  E / kT
What is the proportion ality factor?
E1
The factor must scale the Boltzmann factor
so that the sum of all probabilit ies adds to 1.
The sum of all Boltzmann factors is
called the partition factor (Z)
Z
e
 Ei / kT
states
Thus the proportion ality factor is
1
Z
E0
Average values
If all states are equally probable
X 
X 
X
i
all states
N
1
    X i   Pi X i
all states  N 
all states
g X
i
all X values
N
i
 gi 
    X i   P( X i ) X i
all X values  N 
all X values
Average values
If the system obeys Boltzmann statistics
e
X   Pi X i   
Z
all states
all states 
 Ei / kT

 X i 

e
 Ei / kT
all states
Z
Xi
Average values
If the system obeys Boltzmann statistics
X 
 Pi X i 
all states
e


Z
all states 
 Ei / kT

 X i 

e
 Ei / kT
Xi
all states
 e  Ei / kT
X   P ( X i ) X i    g i
Z
all X values
all X values 
Z

 X i 

g e
 Ei / kT
i
all X values
Z
Xi
Average values
If the system obeys Boltzmann statistics
 e  Ei / kT
X   Pi X i   
Z
all states
all states 

 X i 

 Ei / kT
e
Xi

all states
 e  Ei / kT
X   P ( X i ) X i    g i
Z
all X values
all X values 
Z

 X i 

 Ei / kT
g
e
Xi
 i
all X values
where Z is called the PARTITION FUNCTION
and is given by the sum of all the Boltzmann factors
Z
 Ei / kT
e


all states
 Ei / kT
g
e
 i
all energies
Z
Example: Average Energy
E
E e
i
all states
 Ei / kT

 Ei / kT
g
E
e
 i i
all Energies
Z
Z
where g is the multiplici ty
Example: Average Energy
E
E e
 Ei / kT
i

all states
 Ei / kT
g
E
e
 i i
all Energies
Z
Z
where g is the multiplici ty
and since Z 
 Ei / kT
e


all states
 (ln( Z ))
E  kT
T
2
 Ei / kT
g
e
 i
all energies