equations – thermal physics - mike peel physics.pdfequations – thermal physics 1 cycles carnot...
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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