Equations – Thermal Physics

1

Cycles Carnot Engine

1) Isothermal compression at Tc= T

1= T

2

2) Adiabatic compression 3) Isothermal expansion at T

H= T

3= T

4

4) Adiabatic expansion

!carnot

= 1"TC

TH

QH

TH

=QC

TC

Otto Cycle: ! = 1"V2

V1

#

$%&

'(

) "1

= 1"T1

T2

Heat Engine Efficiency:

!rev

engine =We

QH

=QH "QC

QH

=what you get

what you pay

! is always less than 1.

Pump Efficiency: !rev

pump =QH

W=

1

nrev

engine

This is greater than 1 by definition Refrigerator Efficiency:

!rev

fridge =QC

We

=QC

QH "QC

=what you pay

what you get

This is usually greater than 1. Expansions Adiabatic:

PV!= const

TV! "1

= const T

!P1"!( )

= const Isothermal

Constant temperature:

W = !nrTdV

VV1

V2

"

PV = const

Isothermal Compressability: !T= "

1

V

#V#P

$%&

'()T

Energies

Energy: E =!!"

F

kBT

#

$%&

'(= )

! lnZ!"

#$%

&'(V ,N

Internal Energy per molecule/particle:

U =n

2kT = E

kin

Helmholtz Free Energy: F = E ! TS

dF = !SdT ! PdV + µdN F = !k

BT lnZ

Gibbs Free Energy:

Specific Gibbs Free Energy when g =G

m,

where m is the mass: G = E ! TS + PV dG = !SdT +VdP + µdN

Chemical Potential Can be seen as the Gibbs Free Energy per molecule:

µ =!E!N

"#$

%&'S ,V

=!F!N

"#$

%&'T ,V

=!G!N

"#$

%&'P,T

= g

Enthalpy: H = E + PV dH = dU + PdV +VdP

= dQ +VdP

= TdS +VdP

H T( ) = H T0( ) + Cp T( )dT

T0

T

!

Entropy If not isothermal, consider the start and end states to obtain !S

!S =dQ

T"=

ncdT

T"

dS =!S!T

"#$

%&'V

dT +!S!V

"#$

%&'T

dV

S = kBln!

S = !"F"T

#$%

&'(V ,N

!S!E

"#$

%&'V

=1

T

!S!V

"#$

%&'E

=P

T

Sackur-Tetrode Equation

S = NxBln

V

N

!"#

$%&+3

2lnT

3

2ln

MkB

2'!2!"#

$%&+5

2

(

)*

+

,-

Heat Capacities

C is the overall capacity, while c is the specific heat capacity, and is per mole or kg

c =dQ

rev

dT

cP=

!Q!T

"#$

%&'P

=!H!T

"#$

%&'P

= T!S!T

"#$

%&'P

Equations – Thermal Physics

2

cv=

!Q!T

"#$

%&'V

=!E!T

"#$

%&'V

= T!S!T

"#$

%&'V

=n

2R (

1

T2

cp! c

v= nR = VT

"2

#T

> 0

cp

cv= ! =

5

3monatomic( ) =

7

3diatomic( )

Low-temperature specific heat: c = !T + "T

32 + #T 3

Terms from: electron gas, disturbances in magnetic order, and Debye model

Availability A = E ! T

oS + P

oV( ) " 0

This is maximized in equilibrium Maxwell Relations:

!T!V

"#$

%&'S

= (!P!S

"#$

%&'V

!S!V

"#$

%&'T

= (!P!T

"#$

%&'V

!T!P

"#$

%&'S

= (!V!S

"#$

%&'P

!S!P

"#$

%&'T

= (!V!T

"#$

%&'P

Gases Clausius-Clapeyron Equation

L is the Latent heat dP

dT=

L

T!V

dp

dT=SL ! SS

VL !VS

Conduction: ! =1

3ndcvv"

Density of State: D k( ) =Vk

2

2!2

Effusion:

n1

n2

!

"#$

%&after

=n1

n2

!

"#$

%&before

m2

m1

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 =

1

D

"T

"t

Thermal diffusivity: D =k

Cp

D =1

2v!

Ideal Gas Law: PV = nRT = NkBT

P = !"F"V

#$%

&'(N ,T

Z =PV

m

RT= 1 for ideal gas

Ideal Gas Pressure: P =1

3mn

dv2=1

3!v2

Isobaric Thermal Expansivity: ! =1

V

"V"T

#$%

&'(P

Mean Free Path: ! =1

2nd"d

2

Quantum Concentration !T

is the de Broglie wavelength for a particle with thermal energy k

BT and mass

m

nQ=1

!T

3

!T=

h

2"MkBT

Speed (Mean): vmean

=8kT

!m=

8RT

!M

Speed (Most probable):

vprobable =2kT

m=

2RT

M

Speed (RMS): vrms

=3kT

m=

3RT

M

Van der Waal’s Law:

P +an

2

V2

!"#

$%&V ' nb( ) = nRT

Lenard-Jones Potential (for Van der Waal’s solids):

U r( ) = 4!a

r

"#$

%&'12

(a

r

"#$

%&'6)

*++

,

-..

Viscosity: ! =1

3ndmv"

Work: W = F !dx" = pdV" Thermal (volume) expansion coefficient:

! "dV

dT

#$%

&'(p

1

Vk)1*+ ,-

Equations – Thermal Physics

3

Compressibility:

X ! "dV

dp

#$%

&'(T

1

VPa

"1,bar

"1)* +, =1

B

( B = bulk modulus)

Tension coefficient: ! "dP

dT

#$%

&'(V

1

P

Fluids Bernoulli’s Equation (conservation of flow

along a pipe):

p1+ !gh

1+1

2!v

1

2= p

2+ !gh

2+1

2!v

2

2

Poiseuille’s equation (flow of fluid in a pipe): dV

dt=!8

R4

"#$%

&'(Pa) P

b

L

#$%

&'(

Stoke’s Law (laminar flow): = 6!"rv

Solids Typical equation of state for a solid:

V = V01+ ! T " T

0( ) "# P " P0( )( )

Solid State Physics Number of states:

g !( )d! = " k( )dk

g !( ) = " k( )dk

d!

2D: ! k( ) =Ak

2" 2

3D: ! k( ) =Vk

2

2" 2

kBT =

!2!2

2ML2"2+ m

2+ n

2( )

Single Particle Partition Function:

! = e

"# k( )

kBT

all k

states

$

Grand Single-state partition function:

!G= 1+ e

" µ#$( ) = eNs" µ#$( )

Ns =0

%

&

N particle partition function:

Z = e

!E

kBT

all

microstates

" =# N

N !

Grand Canonical Partition Function ZG = e

! µNs "Es( )

Ns ,S

# = $Gall singleparticle states

%

E = ! f !( )g !( )d!0

"

#

N = f !( )g !( )d!0

"

#

Energy:

Particles:

! =k2!2

2M

Photons: E = pc = !ck

Bosons:

f !( ) =1

e

! "µ

kBT "1

Fermions:

f !( ) =1

e

! "µ

kBT +1

Number of particles (fixed):

N =1

e! " #µ( ) ±1all

states

$

Fermi wavenumber (etc):

k f = 3! 2N

V

"#$

%&'

13

! f =1

2Mvf

2

pf = !k f

! f =2"

k f

Tf =! f

kB

E = n +1

2

!"#

$%&!'

Mean number of excited quanta:

n =1

e!!" #1

Debye frequency

!D=6N

V" 2#

$%&'(

13

v

!D=2"

kD

=2"

#D

v

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.

Equations – Thermal Physics

4

First Law !E = Q +W dE = dQ + dW

dE =!E!T

"#$

%&'V

dT +!E!V

"#$

%&'T

dV

Q is heat added to the system. W is work done on the system. Asides: Joule’s Law: dQ = cdT Work Done: dW

rev= !PdV

Stretched string: dW = !dl ! = tension in string Stretched surface: dW = ! dA where ! = surface tension dE = TdS ! PdV + µdN

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.” (Kelvin-Planck) “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

= !µoVM "dB = !VM "dH

V is Volume M is Magnetic Moment per unit volume

Radiation

Planck Distribution: E =hc

!e

hc

!kT"#$

%&' (1

)

*++

,

-..

(1

Planck Distribution Function:

I !( )d! =2"hc2

!5 e

hc

!kT#$%

&'( )1

*

+,,

-

.//

d!

Stefan’s Law: I = !T

4 dQ

dt= Ae!T

4

Wein’s Law: !mT = 2.9x10

"3k #m

Statistics Macrostates

This is the bulk motion of the system, i.e. an overall view. Calculated by averaging over all microstates, e.g.

x = piXi

i

! =1

"Xi

i

! , for an isolated

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 ! =

nCr.

Average: Q =Qf Q( )dQ

!"

"

#f Q( )dQ

!"

"

#

Binomial: P r

n; p( ) = n

Cr pr1! p( )

n! r( )

=n!

r! n ! r( )!pr1! p( )

n! r( )

Boltzmann Distribution: P E( )dE = Ae!E

kT dE Gaussian Distribution:

Generally: p x( ) =1

2!"e

#x# x( )

2

2" 2

$

%

&&

'

(

))

For gases:

f v( ) =4

!m

2kBT

"

#$%

&'

32

v2e(mv

2

2kBT

=4

!M

2RT

"#$

%&'

32

v2e(Mv

2

2RT

Gibbs Distribution

Pi=e! "

i!µN

i( )#

!

This is the same as the Boltzmann distribution, except it includes the number of particles.

Equations – Thermal Physics

5

Grand Potential: !

G= F " µN = "kT ln! = "PV

Mean:

x =1

Nxi! = xP x( )dx

"#

#

$

x2= x

2P x( )dx

!"

"

#

Normalisation: P x( )dx!"

"

# = 1

Partition Function g !( ) is the degeneracy of that energy ZN ,dist

is the total partition function for distinguishable particles, while Z

N ,indist is

for indistinguishable particles. Z = g !( )e"!#

!

$

Z1= Vn

Q

ZN ,dist

= Z1( )

N

ZN ,indist

=Z1( )

N

N !

Grand Partition Function:

! = e! "

i!µN( )#

i

$

Poisson Distribution: p r;!( ) =!re"!

r!

Scale Height: ! z + z

o( )! z( )

= e"1

Standard Deviation: ! 2= x

2" x

2 Sterling’s Approximation: lnN != N lnN ! N Temperatures Centigrade System:

Tcentigrade =lim

p! 0

PV

PV( )triple point

" 273.16k