Equations – Thermal Physics
Cycles
Carnot Engine
1) Isothermal compression at Tc  T1  T2
2) Adiabatic compression
3) Isothermal expansion at TH  T3  T4
4) Adiabatic expansion
T
carnot  1  C
TH
QH QC

TH TC
V 
Otto Cycle:   1   2 
V 
1
 1
 1
T1
T2
Heat Engine Efficiency:
W
Q  QC what you get
 rev engine  e  H

QH
QH
what you pay
 is always less than 1.
Q
1
Pump Efficiency:  rev pump  H  rev
W n engine
This is greater than 1 by definition
Refrigerator Efficiency:
Q
QC
what you pay
 rev fridge  C 

We QH  QC what you get
This is usually greater than 1.
Expansions
Adiabatic:
PV   const
TV  1  const
T  P 1    const
Isothermal
Constant temperature:
V2 dV
W  nrT 
V1 V
PV  const
Isothermal Compressability:  T  
1  V 


V  P  T
dF  SdT  PdV  dN
F  kBT ln Z
Gibbs Free Energy:
Specific Gibbs Free Energy when g 
G
,
m
where m is the mass:
G  E  TS  PV
dG  SdT  VdP  dN
Chemical Potential
Can be seen as the Gibbs Free Energy per
molecule:
 E 
 F 
 G 
      g
 N  S,V  N  T ,V  N  P,T
Enthalpy:
H  E  PV
dH  dU  PdV  VdP
 dQ  VdP
 TdS  VdP
H T   H T0    C p T dT
T
T0
Entropy
If not isothermal, consider the start and end
states to obtain S
dQ
ncdT
S  

T
T
 S 
 S 
dS    dT  
dV
 T  V
 V  T
S  kB ln
 F 
S   
 T  V , N
1
 S 
  
E V T
P
 S 
  
V E T
Sackur-Tetrode Equation
3  MkB  5 
 V 3
S  NxB  ln    lnT ln 

2  2 2  2 
  N 2
Energies
Energy: E 
  ln Z 
  F 
 


   V , N
  kBT 
Internal Energy per molecule/particle:
n
U  kT  Ekin
2
Helmholtz Free Energy:
F  E  TS
Heat Capacities
C is the overall capacity, while c is the
specific heat capacity, and is per mole or
kg
dQ rev
c
dT
 Q 
 H 
 S 
cP  

T 


 T  P  T  P
 T  P
1
Equations – Thermal Physics
n
1
 Q 
 E 
 S 
cv  
  T   R 2

 T  V  T  V
 T  V 2
T
c p  cv  nR  VT
cp

0
T
5
7
monatomic   diatomic 
cv
3
3
Low-temperature specific heat:
3
c   T   T 2  T 3
Terms from: electron gas, disturbances in
magnetic order, and Debye model
Availability
A  E  To S  PoV   0
This is maximized in equilibrium
Maxwell Relations:
 T 
 P 

    
V S
S V
 S 
 P 

    
V T
T V
 T 
 V 
    

P S
S  P
 S 
 V 
    

P T
T  P
1
v
2
Ideal Gas Law: PV  nRT  NkBT
 F 
P  
 V  N ,T
PVm
Z
 1 for ideal gas
RT
1
1
Ideal Gas Pressure: P  mnd v 2   v 2
3
3
1  V 
Isobaric Thermal Expansivity:   

V  T  P
1
Mean Free Path:  
2nd  d 2
Quantum Concentration
T is the de Broglie wavelength for a
particle with thermal energy kBT and mass
m
1
nQ  3
T
T 
2
m2
m1
before
Enrichment Factor
Gibb’s Phase Law: F  C  P  2
F  # of degrees of freedom
C  # of components
P  # of phases
Heat flow: j  kT
Diffusion equation:  2T 
h
2 MkBT
Speed (Mean): vmean 
Gases
Clausius-Clapeyron Equation
L is the Latent heat
dP
L

dT T V
dp SL  SS

dT VL  VS
1
Conduction:   nd cv v
3
Vk 2
Density of State: D k  
2 2
n 
n 
 1
Effusion:  1 
n 
n 
after
k
Cp
D
2
 
2
Thermal diffusivity: D 
1 T
D t
8kT
8RT

m
M
Speed (Most probable):
2kT
2RT
v probable 

m
M
3kT
3RT
Speed (RMS): vrms 

m
M
Van der Waal’s Law:

an 2 
 P  V 2  V  nb   nRT
Lenard-Jones Potential (for Van der Waal’s
solids):
 a  12  a  6 
U r   4       
 r  
 r 
Viscosity:  
1
nd mv
3
Work: W   F  dx   pdV
Thermal (volume) expansion coefficient:
 dV  1 1
k 
  
 dT  p V  
2
Equations – Thermal Physics
Compressibility:
 dV  1
1
 Pa 1,bar 1  
X  

 dp  T V
B
( B  bulk modulus)
 dP  1
Tension coefficient:   
 dT  V P
Fluids
Bernoulli’s Equation (conservation of flow
along a pipe):
1
1
p1  gh1  v12  p2  gh2  v2 2
2
2
Poiseuille’s equation (flow of fluid in a pipe):
dV   R 4   Pa  Pb 
  

dt
8   L 
Stoke’s Law (laminar flow):
 6rv
Solids
Typical equation of state for a solid:
V  V0 1   T  T0   P  P0 
Solid State Physics
Number of states:
g  d   k dk
dk
g     k 
d
Ak
2D:  k  
2 2
Vk 2
3D:  k  
2 2
kBT 
2

2ML
2
2
2

e
 m2  n2

 k 
kB T

 G  1  e     
e
N s     
Ns  0
N particle partition function:
E

N
Z   e kB T 
N!
all
microstates
Grand Canonical Partition Function
N s ,S
0
Energy:
k2 2
2M
Photons: E  pc  ck
Bosons:
1
f      
Particles:  
e kB T  1
Fermions:
f   
1

e kB T  1
Number of particles (fixed):
1
N       
1
all e
states
Fermi wavenumber (etc):
N

k f   3 2 

V
1
 f  Mv f 2
2
pf  kf
1
3
2
kf
f
kB
1

E  n   

2
Mean number of excited quanta:
1
n 
e 1
Debye frequency
1
Grand Single-state partition function:
 e 
N   f  g  d
Tf 
all k
states
ZG 
0

f 
Single Particle Partition Function:


E    f  g  d
N s  Es 


all single
particle states
G
 6N 2  3
D  
  v
 V

2 2
D 

v
kD  D
Laws of Thermodynamics
Zeroth Law:
If two systems are separately in
equilibrium with a third system, then they
must be in thermal equilibrium with each
other.
3
Equations – Thermal Physics
First Law
E  Q  W
dE  dQ  dW
 E 
 E 
dE    dT  
dV
 T  V
 V  T
Q is heat added to the system.
W is work done on the system.
Asides:
Joule’s Law: dQ  cdT
Work Done: dWrev  PdV
Stretched string: dW  dl   tension in
string
Stretched surface: dW   dA where
  surface tension
dE  TdS  PdV  dN
Second Law:
“It is impossible to construct an engine
which, operating in a cycle, will produce
no other effect than the extraction of heat
from a reservoir and the performance of an
equivalent amount of work.” (KelvinPlanck)
“It is impossible to construct a refrigerator
which, operating in a cycle, will produce
no other effect than the transfer of heat
from a cooler body to a hotter one.”
(Clausius)
dQ
 T 0
Third Law
Absolute Zero, T  0 , is unobtainable.
Para-Magnets
Energy: dE  TdS  VMdB
Work Done
dWrev   oV M  dB  V M  dH
V is Volume
M is Magnetic Moment per unit volume
Radiation
 hc 

hc    kT 
 1
Planck Distribution: E   e
 

Planck Distribution Function:
2 hc 2
I  d  
d
 hc 






 5  e  kT  1


1
Stefan’s Law:
I  T 4
dQ
 Ae T 4
dt
Wein’s Law: mT  2.9x103 k  m
Statistics
Macrostates
This is the bulk motion of the system, i.e.
an overall view. Calculated by averaging
over all microstates, e.g.
1
x   pi Xi   Xi , for an isolated
 i
i
system.
Equilibrium is when the macrostate has the
maximum possible number microstates.
Microstates
This is a description of the system at a
microscopic level, where the position and
momentum (or quantum state) of each
particle is specified. Total number
  n Cr .

Average: Q 


Qf Q dQ



f Q dQ
Binomial:
n  r 
P r n ; p  n Cr p r 1  p 
n!
n  r 

p r 1  p 
r!n  r !


Boltzmann Distribution: P E dE  Ae
Gaussian Distribution:
 xx 2 
 

 2 2 
1

Generally: p x  
e
2
For gases:
4  m 
f v  
  2kBT 
4  M 



  2RT 
Gibbs Distribution
e i   Ni 
Pi 
3
3
2
2

ve
2
2
ve


E
kT
dE
mv 2
2 kB T
Mv 2
2 RT
This is the same as the Boltzmann
distribution, except it includes the number
of particles.
4
Equations – Thermal Physics
Grand Potential:
G  F   N  kT ln  PV
Mean:

1
x   xi   xP x dx

N

x 2   x 2 P x dx


Normalisation:


P x dx  1
Partition Function
g   is the degeneracy of that energy
Z N ,dist is the total partition function for
distinguishable particles, while Z N ,indist is
for indistinguishable particles.
Z   g  e 

Z1  VnQ
Z N ,dist  Z1 
N
Z N ,indist
N
Z1 


N!
Grand Partition Function:
  e i   N 
i
Poisson Distribution: p r;   
Scale Height:
 z  zo  1
e
 z 
 r e 
r!
Standard Deviation:  2  x 2  x 2
Sterling’s Approximation: ln N !  N ln N  N
Temperatures
Centigrade System:
lim
PV
Tcentigrade 
 273.16k
p  0 PV triple point
5