# Gateforum Correspondence Course Gateforum All India Test Series Gateforum Classroom Coaching

Μηχανική

27 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

94 εμφανίσεις

Gate Forum Correspondence Course

Gateforum Correspondence Course

Gateforum All India Test Series

Gateforum Classroom Coaching

#17, 30
th

Cross
, Tilak Nagar, Bangalore

560041. Call us: 080
-

41310203, Email
:
support@gateforum.com

www
.
gateforum
.
com

1

3 Laws of Thermodynamics

O)

If two systems are in equilibrium with a third, they are in equilibrium with each other.

1)

Conservation of energy

E
th

= W + Q

2)

The entropy of an isolated system never decreases. The entropy either increases until it reaches
equilibr
ium, or if it’s in equilibrium, it stays the same.

given two system w/

1

>

2
, heat wi
ll be spontaneously transferred from system 1 to 2.

heat cannot be completely converted into work.

Thermodynamic basics

Partition function:

Probability of being in a state w/energy

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 = k
B

= k
B
logg(N, u)

Specific case: Hermonic oscillator: g(N, n) =

w
here N = #oscillators, n = quantum #

Specific case: N magnets with S
p

in excess Zs =
N

N

:

g(N, s) =

where
=
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

dF = du

d

+

dN

=

d

PdV +

dN

(isobaric)

Enthalpy

H =

+ pV

dH =

d

+ Vdp

dN

(isobaric, isothermal)

Gibbs Free Energy

G = F
+
pV = u+pV

dG =

d

+ Vdp +

dN

,

Gate Forum Correspondence Course

Gateforum Correspondence Course

Gateforum All India Test Series

Gateforum Classroom Coaching

#17, 30
th

Cross
, Tilak Nagar, Bangalore

560041. Call us: 080
-

41310203, Email
:
support@gateforum.com

www
.
gateforum
.
com

2

Distributions

Fermi

Dirac : Average

occupancy of an orbital w/energy

, for fermions

Bose

Einstein: Average occupancy of an orbital w/energy

, bosons

Plank distr
ibution: Thermal average number of photons in a single mode a

Ideal gas

PV
=
n
RT

= Nk
B
T

= KE
avg

=

(

for each degree of freedom, note

that f

potential energy, each of those
degrees of freedom gets

as well by the Equipartition Theorem)

Heat capacity, constant volume :

(

= k
B
T)

Partition fun
ction of an atom in a box. Z
1

=

n
a

=

Partition function of N atoms in a box :

Entropy S = k
B

= k
B
N

Chemical potential :

Average occupancy of an orbital of energy

where

= e

/

Free energy:

Per atom in a
monatomic gas

Gate Forum Correspondence Course

Gateforum Correspondence Course

Gateforum All India Test Series

Gateforum Classroom Coaching

#17, 30
th

Cross
, Tilak Nagar, Bangalore

560041. Call us: 080
-

41310203, Email
:
support@gateforum.com

www
.
gateforum
.
com

3

u
2

u
1

2

1

W

Q

O

O

O

O

O

O

Diatomic Gas:

2
-
D ideal gas

u = k
B
T,

C
v

= Nk
B

C
p

= 2Nk
B

Van DerWalls

attemps to modify the
ideal gas law to take into account interactions between atoms
or molecules

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 re
pulsion (volume of molecules themselves)

Critical points : Pc =

Vc = 3Nb,

at this point, there is no separation between the vapor

and liquid phases (a horizontal point of inflection)

( K

K
1

p
290

Fig. 10.10)

(For a given
,

<

c
, V < V
1

liquid V > V
2

gas,

V
1

< V < V
2

both show that sum of volume of

liquid G gas = V)

Phase Diagram

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 densit
y)

Diffusion

,

Reversible isothermal

Reversible isentropic

Irreversible extension into
vacuum

v
1

v
2

<

c

=

c

>

c

Solid

Liquid

Critical point

Triple point

T

Gate Forum Correspondence Course

Gateforum Correspondence Course

Gateforum All India Test Series

Gateforum Classroom Coaching

#17, 30
th

Cross
, Tilak Nagar, Bangalore

560041. Call us: 080
-

41310203, Email
:
support@gateforum.com

www
.
gateforum
.
com

4

Maxwell velocity distribution

Fick’s Law

(diffusion constant)

Four
ier’s Law

Carnot cycle and Work in general

Work done on a system =
=

(area under pV curve)

Energy in: heat from resevoir R
H

(@

H
)

Energy out: heat to resevoir R
L

(@

L

<

H
)

For a reversible engine,

H

=

L
(if

H

L,
only work may be transferred)

e
fficiency:

1)

compress isothermally (Q

)

2)

compress isentropically (

)

3)

expand isothermally (Q

)

4)

expand isentropically (

)

(for
a heat pump, reverse or
der)

for the carnot cycle, efficiency is at a maximum

or

(engine)

(pump/refrigerator)

For an ideal gas, isothermal process

Q
H

=

= N

H

log

isentropic process

=

=

V

P

V