Gate Forum Correspondence Course
3 Laws of Thermodynamics
O)
1)
2)
If two systems are in equilibrium with a third, they are in equilibrium with each other.
Conservation of energy Eth = W + Q
The entropy of an isolated system never decreases. The entropy either increases until it reaches
equilibrium, or if it’s in equilibrium, it stays the same.
 given two system w/1 > 2, heat will be spontaneously transferred from system 1 to 2.
 heat cannot be completely converted into work.
Thermodynamic basics
Partition function: Z 
e
s / 
s
Probability of being in a state w/energy  : P() 
 e
s

e / 
z
s / 
s
z
The fundamental assumption: a closed system is equally likely to be any of the quantum states
accessible to it.
g(N, U)  The multiplicity of a system with N particles and energy
S = kB = kBlogg(N, u)
Specific case: Hermonic oscillator: g(N, n) =
(N  n  1)!
n! (N  1)!
where N = #oscillators, n = quantum #
Specific case: N magnets with Sp in excess Zs = N – N : g(N, s) =
where
N
N
=
eSMB
eSMB  eSMB
N!
N !N
where B is the magnetic field and M is the magnetic moment
Kinds of energy:
d = du + pdV – N
Helmholtz Free Energy
(isothermal)
F = u –  = – logz
(isobaric)
Enthalpy
H =  + pV
(isobaric, isothermal) Gibbs Free Energy
G = F+pV = u+pV – 
 u 
  
Cv  
      ,


v

v

z  log z 
 = 
 

dF = du – d + dN
= –d – PdV + dN
dH = d + Vdp – dN
dG = –d + Vdp + dN
 u 
 v 
  
Cp  
  p        


P

p

p
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Distributions
Fermi – Dirac : Average occupancy of an orbital w/energy , for fermions
f() 
1
() / 
e
1
Bose – Einstein: Average occupancy of an orbital w/energy , bosons
f() 
1
() / 
e
–1
Plank distribution: Thermal average number of photons in a single mode a
s

1
e
/
–1
Ideal gas
PV = nRT = NkBT
 = KEavg =
3
KB T
2

P1V1 P2 V2 

 if the container is scaled

T1
T2 

( 12 kB T for each degree of freedom, note that f  potential energy, each of those
degrees of freedom gets
Per atom in a
monatomic gas
Heat capacity, constant volume :
1
2
kB T as well by the Equipartition Theorem)
3
 u 
Cv  
  N kB

2

v
( = kBT)
5
 u 
 u 
  
Cp  
  P          2 NkB


p

p

p
Partition function of an atom in a box. Z1 = nQ / n  nQV

na = M / z
z

3
2
Partition function of N atoms in a box : ZN 
 
Chemical potential :    log  n
nQ 
1 N
Z1
N!
5

Entropy S = kB = kBN log nq  
n
2

Average occupancy of an orbital of energy 
f()  e / 
where  = e/
 
Free energy: F  N log n
 1
nQ


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Reversible isothermal
Reversible isentropic
Irreversible extension into
vacuum
u2 – u1
2 – 1
V
Nlog 2
V1
O
2

 V1  3 
3

 N1 1  


2
V2  



O
Nlog
O
3 translation, 2 rotation
u
Diatomic Gas:
V2
V1
W
V
–Nlog 2
V1
Q
N log
2

 V1  3 
3

 N1 1  


2
V2  



O
O
O
V2
V1
3 kinetic / translation
3 vibration
5
kB T,
2
Solid*: u3kB T
2-D ideal gas
u = kBT,
Cv = NkB
Cp = 2NkB
Van DerWalls – attemps to modify the ideal gas law to take into account interactions between atoms
or molecules

N2a 
 P  2   V  Nb   N
V 

where a is a measure of the long-range attractive part (adds to internal pressure) of the interaction
and
b is a measure of the short-range repulsion (volume of molecules themselves)
Critical points : Pc = a
P
Pc
Vc = 3Nb,
c  8a
27b
27b2 ,
at this point, there is no separation between the vapor
and liquid phases (a horizontal point of inflection)
>c
=c
( K  K1 p290 Fig. 10.10)
<c
v
v2
v1
(For a given
v2
P
,  < c, V < V1  liquid V > V2  gas,
Pc
V1 < V < V2  both show that sum of volume of
liquid G gas = V)
Phase Diagram
Solid
Liquid
Critical point
Triple point : The one value of T and P for which all three phases
can happily coexist. Happily.
Critical point: below this point  a phase change between liquid &
gas. Above this point  phase change (fluid  continuously
between high & low density)
Triple point
T
Diffusion
2
Vrms
1
2
 3 
  
M
V2e
Mv2
2t
,
 R 2
 Clt;
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Main freepath

Main speed

1
   2
C
 ;
 M 
l
1
nd2
particle
density
diameter of particles
Maxwell velocity distribution
3
 M 
P(V)  4 

 2 
Fick’s Law
2
V2e
Mv2
2
Jn  Dn
D
cl
(diffusion constant)
3
K
1ˆ
CV cl
3

particle flux density
Fourier’s Law
Jn  K

thermal flux density
(Thermal conductivity)

heat capcity per unit volume
Carnot cycle and Work in general
2

Work done on a system =  pdV = –(area under pV curve)
1
Energy in: heat from resevoir RH (@ H)
Energy out: heat to resevoir RL (@ L < H)
For a reversible engine, H = L (if H  L, only work may be transferred)

 

Z
 1 – L 
efficiency:
QH 
H 
(heat engine)
1) compress isothermally (Q )
2) compress isentropically ( )
3) expand isothermally (Q )
4) expand isentropically ()
(for a heat pump, reverse order)
V
P









V
for the carnot cycle, efficiency is at a maximum

C
L
c  1  L
or  c  L 
H

H  4
(engine)
(pump/refrigerator)
For an ideal gas, isothermal process  QH =  = NH log
isentropic process   =  =
V2
V1
3
N  H  L 
2
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