Thermal Physics - Physics & Astronomy
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Northeastern Illinois University c 2004-2020 G. Anderson Thermal Physics – slide 1 / 64 Thermal Physics Fundamentals: Energy Greg Anderson Department of Physics & Astronomy Northeastern Illinois University Spring 2020
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Northeastern
Illinois
University
c©2004-2020 G. Anderson Thermal Physics – slide 1 / 64
Thermal PhysicsFundamentals: Energy
Greg Anderson
Department of Physics & AstronomyNortheastern Illinois University
Spring 2020
Northeastern
Illinois
University
Overview
c©2004-2020 G. Anderson Thermal Physics – slide 2 / 64
Introduction
Units
Ideal Gasses
Equipartition Theorem
Heat and Work
Heat Capacity
Transport
Northeastern
Illinois
University
Introduction
IntroductionThermodynamics& StatisticalMechanicsTemperature &Equilibrium
Laws ofThermodynamics
Intensive vs.ExtensiveVariables
Units
Ideal Gasses
EquipartitionTheorem
Heat and Work
Heat Capacity
Transport
c©2004-2020 G. Anderson Thermal Physics – slide 3 / 64
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Illinois
University
Thermodynamics & Statistical Mechanics
c©2004-2020 G. Anderson Thermal Physics – slide 4 / 64
Thermal Physics = Thermodynamics + Statistical Mechanics.
Thermal physics deals with large (& 1023) number of particles.
Thermodynamics: Describes the relationship betweenmacroscopic, observable properties of bulk materials,e.g., T, P, V, . . . without microscopic assumptions.Applications: engineering, earth science, chemistry, . . . .
Statistical Mechanics: The microscopic physicsunderneath thermodynamics: Quantum mechanics ofparticles combined with the laws of statistics.Applications: solid state physics, astrophysics,chemistry, cosmology, . . . .
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University
Temperature & Equilibrium
c©2004-2020 G. Anderson Thermal Physics – slide 5 / 64
Temperature:a measure of the tendency of an object to spontaneously give upenergy to its surroundings. When two objects are in thermalcontact, the one that spontaneously loses energy is at a highertemperature.
Thermal equilibrium:A state in which all parts of a system are at the same temperature.In thermal equilibrium, two objects placed in thermal contact donot exchange net heat energy.
Relaxation time:Time to come to thermal equilibrium, aka, the time scale for thesystem to settle down so that macroscopic properties are constantin time. Examples:
• Soda or other beverage you place in the refrigerator.
• Thermometer placed in your mouth.
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University
Laws of Thermodynamics
c©2004-2020 G. Anderson Thermal Physics – slide 6 / 64
Zeroth Law: If two systems are each in thermal equilibrium with athird system, they are in thermal equilibrium with each other.
First Law (Conservation of Energy):
dU = d Q+ dW
Second Law: No process is possible whose sole result is thecomplete conversion of heat into work. The entropy of a thermallyisolated system never decreases:
dS ≥ 0
Third Law:limT→0
S(T ) = 0
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University
Intensive vs. Extensive Variables
c©2004-2020 G. Anderson Thermal Physics – slide 7 / 64
Intensive Variables: Local variables, e.g., pressure,density, temperature.
Extensive Variables: Proportional to the size of thesystem, e.g., volume, mass, number of particles,entropy, length.
Intensive Extensive / Extensive
energy density = energy / volumecharge density = charge / volumemass density = mass / volumespecific heat capacity = heat capacity / masspressure = force / area
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Illinois
University
Units
Introduction
UnitsSystemeInternationald’Unites (SI)
Temperature
T devices
T Scales
T Scales
MoleSample MolarVolumes (Vm)
Pressure
STP
Energy
Boltzmann’sConstant
Ideal Gasses
EquipartitionTheorem
Heat and Work
Heat Capacity
Transport
c©2004-2020 G. Anderson Thermal Physics – slide 8 / 64
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Illinois
University
Systeme International d’Unites (SI)
c©2004-2020 G. Anderson Thermal Physics – slide 9 / 64
We will mostly work in SI units. The seven base units ofthe SI system are:
Quantity Name Symbol
length meter mtime second smass kilogram kgelectric current ampere Atemperature kelvin Kamount of substance mole molluminous intensity candela cd
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University
Temperature
c©2004-2020 G. Anderson Thermal Physics – slide 10 / 64
Temperature is a measure of “hotness”. Order objectsin a sequence according to which gain/lose energywhen placed thermal contact.
T1
T2
T3
T1 > T2 > T3
The continuous parameter which labels this sequence istemperature.
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Illinois
University
Temperature Measuring Devices
c©2004-2020 G. Anderson Thermal Physics – slide 11 / 64
• Mercury thermometer (thermal expansion)
• Alcohol thermometer (thermal expansion)
• Electrical resistance of resistor
• Vapor pressure of liquid helium
• Magnetic susceptibility of paramagnet
• Spectrum of emitted radiation
Many common thermometers are based on thermal expansion
∆L
L≡ α∆T,
∆V
V≡ β∆T
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University
Temperature Scales
c©2004-2020 G. Anderson Thermal Physics – slide 12 / 64
• Kelvin: Abolute zero: T = 0.
• Centigrade (Celsius): SP ice melts: 0◦, water boils: 100◦.
• Fahrenheit: Ice melts: 32◦, water boils: 212◦.
• Rankine: Fahrenheit shifted for absolute zero.
Conversion between Kelvin, Celsius, Fahrenheit and Rankine
◦C =5
9(◦F − 32) , ◦R = ◦F + 459.67
K = ◦C + 273.15, ◦R =9
5K
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University
Temperature Scales
c©2004-2020 G. Anderson Thermal Physics – slide 13 / 64
water
boils
water
freezes
absolute
zero
373.15 K
273.15 K
0 K
100◦ C
0◦ C
−273.15◦ C
212◦ F
32◦ F
−459.67◦ F
Kelvin Celsius Fahrenheit
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Illinois
University
Mole
c©2004-2020 G. Anderson Thermal Physics – slide 14 / 64
A Mole “mol” is the SI unit for a macroscopic amount ofa substance.
• A mole of a substance contains Avogadro’s number ofparticles.
NA = 6.022× 1023 (Avogadro’s Number)
• By definition, 12 grams of 12C contains NA Carbon-12atoms.
• Useful approximation: (∼ 1%) The weight ingrams of a mole of a substance is equal to the numberof nucleons in an atom (molecule) of the substance.
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University
Sample Molar Volumes (Vm)
c©2004-2020 G. Anderson Thermal Physics – slide 15 / 64
Helium He 22.4136 L/molHydrogen H2 22.4135 L/molNitrogen N2 22.4131 L/molOxygen O2 22.4134 L/mol
Table 1: Molar volumes of selected gases at 273.13 K 1atm
Carbon (diamond) 3.42 cm3/molCarbon (graphite) 5.29 cm3/molIron 7.09 cm3/molCopper 7.11 cm3/molGold 10.21 cm3/mol
Table 2: Molar volumes of selected solids at 298 K, 1atm.
The molar volume of liquid water is 18.016 mL at 277 K.
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Illinois
University
Pressure
c©2004-2020 G. Anderson Thermal Physics – slide 16 / 64
Pressure is force per unit area. The SI unit of pressure is the pascal.
[P ] =[F ]
[A]=
N
m2=
kg
ms2= pascals = Pa
Other units of pressure include:
• bars
1 bar ≡ 105 Pa
• atmospheres (pressure at sea level)
1 atmosphere (atm) ≡ 1.01325× 105 Pa
• torr (mm of mercury)
1 torr = 1 mm Hg = 133.32 Pa
• pound force per square inch (psi)
1 psi = 6.895× 103 Pa
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STP
c©2004-2020 G. Anderson Thermal Physics – slide 17 / 64
Standard Temperature and Pressure (STP):
Standard Temperature:
T = 0◦C = 273.15K
Standard Pressure:
1 atmosphere = 760mmHg = 101.3 kPa
Volume of 1 mole of an ideal gas at STP: 22.4 liters
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University
Energy
c©2004-2020 G. Anderson Thermal Physics – slide 18 / 64
The SI unit of energy is the joule
1J = 1kg ·m2/s2
Historically the calorie was defined as the amount ofenergy required to raise the temperature of 1 g of waterby 1◦C. Currently the calorie is defined as:
1 cal = 4.186 J
Note that the “food calorie” (C) is a kilocalorie:
1 kilocalorie = 1C = 4186 J
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University
Boltzmann’s Constant
c©2004-2020 G. Anderson Thermal Physics – slide 19 / 64
Boltzmann’s Constant
k = 1.381× 10−23 J/K = 8.617× 10−5 eV/K
Thermal Energy (300 K = 80.33 F)
kT = 0.026 eV
(
T
300K
)
≈1
40eV
(
T
300K
)
One electron-volt: the energy it takes to move an electronacross a 1 volt potential difference:
1 eV = 1.602× 10−19 J
Northeastern
Illinois
University
Ideal Gasses
Introduction
Units
Ideal GassesEmpirical IdealGas “Law”
Ideal Gas Law
Real GassesHydrostaticPressureExponentialAtmosphericDecay
EquipartitionTheorem
Heat and Work
Heat Capacity
Transport
c©2004-2020 G. Anderson Thermal Physics – slide 20 / 64
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Empirical Ideal Gas “Law”
c©2004-2020 G. Anderson Thermal Physics – slide 21 / 64
Ideal Gas Law (1660-1809) Boyle, Gay-Lussac, Avogadro.
PV = nRT = NkT
P pressure pascalsV volume cubic metersT temperature kelvinsn # moles of gasN # molecules of gas N = NAn
R = NAk = 8.31J
mol ·K(universal constant)
k = 1.381× 10−23 J
K(Boltzmann’s constant)
Valid when space between molecules ≫ size of molecules.
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University
Ideal Gas Law
c©2004-2020 G. Anderson Thermal Physics – slide 22 / 64
P
V
isotherms
Ideal Gas Law
PV = NkT, N = nNA
P pressure pascals
V volume cubic meters
T temperature kelvins
N # molecules of gas
k = 1.381× 10−23 J
K(Boltzmann’s constant)
Valid when space between molecules ≫ size of molecules.
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University
Real Gasses
c©2004-2020 G. Anderson Thermal Physics – slide 23 / 64
Ideal GasPV = nRT
Virial Expansion
PV = nRT
(
1 +B(T )
(V/n)+
C(T )
(V/n)2+ · · ·
)
van der Waals equation of state (1873)(
P + an2
V 2
)
(V − bn) = nRT
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Illinois
University
Hydrostatic Pressure
c©2004-2020 G. Anderson Thermal Physics – slide 24 / 64
Consider the vertical variation in pressure over a thin slab of air:
z P (z)
z + dz P (z + dz)
Mg
Air mass:
M = ρV
Air volume:
V = Adz
Fnet = [P (z)− P (z + dz)]A−Mg = 0
For infinitesimal dz:
[P (z)− P (z + dz)] = −dP
dzdz
dP
dzAdz = −ρgV
dP
dz= −ρg
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Illinois
University
Exponential Atmospheric Decay
c©2004-2020 G. Anderson Thermal Physics – slide 25 / 64
z P (z)
z + dz P (z + dz)
Mg
Atmospheric density:
ρ =M
V=
Nm
V
“Average” molecule mass: m
dP
dz= −ρg = −
Nm
Vg
Using the ideal gas law PV = NkT
yields the barometric equation:
dP
dz= −
mg
kTP
∫
dP
P= −
∫
mg
kTdz
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Illinois
University
Exponential Atmospheric Decay
c©2004-2020 G. Anderson Thermal Physics – slide 25 / 64
z P (z)
z + dz P (z + dz)
Mg
Atmospheric density:
ρ =M
V=
Nm
V
“Average” molecule mass: m
dP
dz= −ρg = −
Nm
Vg
Using the ideal gas law PV = NkT
yields the barometric equation:
dP
dz= −
mg
kTP
For constant T :∫
dP
P= −
mg
kT
∫
dz
P (z) = P (0)e−mgz
kT
Northeastern
Illinois
University
Equipartition Theorem
Introduction
Units
Ideal Gasses
EquipartitionTheoremMicroscopicModel of IdealGasTheEquipartition ofEnergy
CountingDegrees offreedom
Heat and Work
Heat Capacity
Transport
c©2004-2020 G. Anderson Thermal Physics – slide 26 / 64
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Illinois
University
Microscopic Model of Ideal Gas
c©2004-2020 G. Anderson Thermal Physics – slide 27 / 64
v
vx
L
∆t = 2L/vx
P =F
A= −
m(∆vx/∆t)
A
Elastic collision:
∆vx = −2vx
Average pressure on piston
P = mv2x/V
Using the ideal gas law: PV = kT
1
2mv2x =
1
2kT
In 3D:
Ktrans =3
2kT
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University
The Equipartition of Energy
c©2004-2020 G. Anderson Thermal Physics – slide 28 / 64
Equipartition Theorem: In thermal equilibrium, at atemperature T , the average energy of any quadratic degree offreedom is:
Ui =1
2kT
For a system of N molecules with f degrees of freedom per molecule:
Uthermal = Nf1
2kT
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U =∑
i
Nifi1
2kT
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Counting Degrees of freedom
c©2004-2020 G. Anderson Thermal Physics – slide 29 / 64
Uthermal = Nf1
2kT
• Translation f = 3 (3D)
• Rotation: U = 12Iω2
– Monatomic Molecule ∆f = 0
– Diatomic Molecule ∆f = 2
– Polyatomic Molecule ∆f = 3
• Vibration in a Diatomic Molecule ∆f = 2× 1
U =1
2kx2 +
1
2mv2
• Vibration Solid Lattice 2× 3
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Illinois
University
Heat and Work
Introduction
Units
Ideal Gasses
EquipartitionTheorem
Heat and Work
Heat and Work
Heat Transfer
Heat Transfer
First Law
Quasistatic
Dissipative Work
Active learning:Changes in anIdeal gas
IsothermalCompression ofIdeal GasAdiabaticCompression
Active Learning
Adiabats andIsotherms (IdealGas)
Sucia
Heat Capacity
Transport
c©2004-2020 G. Anderson Thermal Physics – slide 30 / 64
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Heat and Work
c©2004-2020 G. Anderson Thermal Physics – slide 31 / 64
Two forms of energy transfer:
Heat: The spontaneous flow of energy from one objectto another caused by ∆T .
Work: Any other transfer of energy into or out of asystem.
Conservation of energy:
∆U = Q+W
dU = d Q+ dW
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Heat Transfer
c©2004-2020 G. Anderson Thermal Physics – slide 32 / 64
Conduction:
Energy transfer of heat by molecular contact.
Convection:
Energy transport by bulk motion of a gas or
liquid.
Radiation:
Energy transfer by EM radiation, typically
infrared.
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The First Law of Thermodynamics
c©2004-2020 G. Anderson Thermal Physics – slide 34 / 64
The first law of thermodynamics is conservation of energy:
∆U = Q+W
U
WQ
• U total energy of a system.
• ∆U change in total energy of a system.
• Q heat added to the system
• W work done on the system.
For infinitesimal energy transfer:
dU = d Q+ dW
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Quasistatic Compression of Ideal Gas
c©2004-2020 G. Anderson Thermal Physics – slide 35 / 64
quasistatic: the gas is compressed slow enough that it always staysin equilibrium, i.e. P is uniform.
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P, V
Wvpiston . vsound
For frictionless, quasistatic compression;
dW = F · dx = PAdx = −PdV
W =
∫
dW = −
∫ f
i
P (V )dV
For compression which is faster than quasistatic, or contains friction:
dW = −PdV + dWother > −PdV, dWother ≥ 0
Northeastern
Illinois
University
Dissipative Work
c©2004-2020 G. Anderson Thermal Physics – slide 36 / 64
The work done on a system can be in a reversible form(configuration) work −PdV or it can be non-reversible (dissipative)work.
dW = −PdV + dWother
Dissipative work is always done on the system dWother ≥ 0.Examples of dissipative work:
• Stirring work dWother = −τdθ
• Maintaining current in a resistor dW = I2Rdt
• Sound waves generated from compression which is not quasi-staticdW > −PdV .
Dissipative work eventually ends up as heat added to the system, butsince heat is not taken from a reservoir, the entropy of the universeincreases.
Northeastern
Illinois
University
Active learning: Changes in an Ideal gas
c©2004-2020 G. Anderson Thermal Physics – slide 37 / 64
Pressure
Volume
A
B
C
Pressure
Volume
D
E
F
G
Working in groups of 2–3, for each step above, determine if theanswer is +, −, or 0?
• W , work done on gas?
• ∆U , change in energy of the gas?
• Q, heat added to the gas?
Northeastern
Illinois
University
Isothermal Compression of Ideal Gas
c©2004-2020 G. Anderson Thermal Physics – slide 38 / 64
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bP, V, T
Heat Bath
W
−Q
P
V
Isotherms
T2
T1
Isothermal: dT = 0 ⇒ dU = 0
U =1
2fNkT (Equipartition)
First Law:
∆U = 0 = Q+W
Quasistatic compression:
dW = Fdx = −PdV
W = −∫ Vf
ViPdV
= −(NkT )∫ Vf
Vi
dVV
= NkT ln Vi
Vf= −Q
Northeastern
Illinois
University
Adiabatic Compression of Ideal Gas
c©2004-2020 G. Anderson Thermal Physics – slide 39 / 64
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P, V, T
W
Equipartition theorem:
U =1
2fNkT
Ideal gas law:
PV = NkT
First law:
dU = d Q+ dW
Adiabatic: d Q = 0
Adiabatic & quasistatic compression:
dU = dWf2NkdT = −PdV
Using the ideal gas law (PV = NkT ):
f
2
dT
T= −
dV
V
f
2
∫
dT
T= −
∫
dV
V
f
2ln
Tf
Ti
= lnVi
Vf
Northeastern
Illinois
University
Adiabatic Compression of Ideal Gas
c©2004-2020 G. Anderson Thermal Physics – slide 39 / 64
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P, V, T
W
Equipartition theorem:
U =1
2fNkT
Ideal gas law:
PV = NkT
First law:
dU = d Q+ dW
Adiabatic: d Q = 0
Adiabatic & quasistatic compression:
dU = dWf2NkdT = −PdV
Using the ideal gas law (PV = NkT ):
f
2ln
Tf
Ti
= lnVi
Vf
Tf/2f Vf = T
f/2i Vi = T f/2V
T f/2V = constant
V γP = const. γ = (f + 2)/f
Northeastern
Illinois
University
Active Learning
c©2004-2020 G. Anderson Thermal Physics – slide 40 / 64
We would like to recast this result in terms of P & V .Show that:
T f/2V = constant
implies
V γP = constant
where the adiabatic exponent is:
γ =f + 2
f, 1 < γ < 5/3
Hint, use:PV = NkT
Northeastern
Illinois
University
Adiabats and Isotherms (Ideal Gas)
c©2004-2020 G. Anderson Thermal Physics – slide 41 / 64
P
V
Adiabatic
Isothermal
PV = NkT
Isothermal compression (expansion)
PV = constant
Q = NkT lnVf
Vi= −W
Adiabatic compression (expansion)
Q = 0
V γP = constant
γ = (f + 2)/f
Pressurized Propane C3H8
Northeastern
Illinois
University
Heat Capacity
Introduction
Units
Ideal Gasses
EquipartitionTheorem
Heat and Work
Heat Capacity
Heat Capacity
CV
CP
Heat Capacities
CV for aDiatomic GasHeat Capacitiesof Solids
Latent Heat
Phases of MatterPhaseTransitionsSample LatentHeatsSodium AcetateHand Warmer
Enthalpy
Enthalpy II
Transport
c©2004-2020 G. Anderson Thermal Physics – slide 43 / 64
Northeastern
Illinois
University
Heat Capacity
c©2004-2020 G. Anderson Thermal Physics – slide 44 / 64
Heat capacity of a substance:
C =d Q
dTor d Q = CdT
Specific heat capacity: c = CM .
Note: Ambiguity for compressible substances:
dU = dW + d Q, dW = ???
Cases:
• W = 0 = dV = 0 (const. volume)
• dP = 0 (const. pressure)
• dB = 0 (const. magnetic field)
Northeastern
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University
Heat Capacity at Constant Volume
c©2004-2020 G. Anderson Thermal Physics – slide 45 / 64
Heat capacity at constant volume:
CV =
(
d Q
dT
)
V
=
(
∂U
∂T
)
V
Using equipartition:∗1
CV =
(
∂U
∂T
)
V
=
(
∂(NfkT/2)
∂T
)
V
=1
2Nfk
Examples:
CV =
{
32Nk monatomic gas3Nk solid (Dulong & Petit)
1* When f is independent of T
Northeastern
Illinois
University
Heat Capacity at Contant Pressure
c©2004-2020 G. Anderson Thermal Physics – slide 46 / 64
From the First Law:
d Q = dU − dW = dU + PdV
At constant pressure:
CP =(
d QdT
)
P=
(
dUdT
)
P−
(
dWdT
)
P
=(
∂U∂T
)
P+ P
(
∂V∂T
)
P
The extra term compensates for U lost as W .Mayer’s relation for an ideal gas:
CP =(
∂U∂T
)
P+ P
(
∂V∂T
)
P
= CV + Nk
Northeastern
Illinois
University
Heat Capacities
c©2004-2020 G. Anderson Thermal Physics – slide 47 / 64
At constant volume, all the heat added goes into raisingthe temperature.
d Q = dU = CV dT
For an ideal gas at constant pressure:
CP = CV +Nk
Thus, it takes more heat: d Q = CPdT , to achieve thesame change in temperature than it does at constantvolume. This extra heat does work.
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University
CV for a Diatomic Gas
c©2004-2020 G. Anderson Thermal Physics – slide 48 / 64
CV = CV,tr + CV,rot + CV,vib
Urot = ℓ(ℓ+ 1)kθrot. For T ≫ θrot:
CV = Nk
[
5
2+
(
θvibT
)2eθvib/T
(eθvib/T − 1)2
]
For H2: θrot = 85.4 K, θvib = 6140 K.
-1 0 1 2 3 4 5 6
1
2
3
4
10−1 100 101 102 103 104 105 1061
1.5
2.0
2.5
3.0
3.5
4.0
T (◦K)
CVNk
trans.
trans.+rot.
trans.+ rot.+ vib.
Northeastern
Illinois
University
Heat Capacities of Solids
c©2004-2020 G. Anderson Thermal Physics – slide 49 / 64
0 0.3 0.6 0.9 1.2 1.50
0.2
0.4
0.6
0.8
1.0
T/TD
CV
3Nk
Einstein Model
Debeye Model
Dulong-Petit
Northeastern
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University
Latent Heat
c©2004-2020 G. Anderson Thermal Physics – slide 50 / 64
• Phase: a system or part of a system which ishomogeneous and has definite boundaries.
• Phase transitions (phase change): when a substancechanges from a solid, liquid, or gas state to a differentstate e.g., melting ice, boiling water. The heatcapacity becomes infinite at a phase transition.
C =d Q
dT=
d Q
0= ∞
• Latent Heat: Heat required to accomplishtransformation.
L = Q/m
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University
Phases of Matter
c©2004-2020 G. Anderson Thermal Physics – slide 51 / 64
Four elements:Earth
Air
Water Fir
e
Four phases of matter:
Solid
Gas
Liquid
Plasm
a
Greek Philosopher Empedocles, Agrigentum, Sicily, 5th Century BCE
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University
Phase Transitions
c©2004-2020 G. Anderson Thermal Physics – slide 52 / 64
Plasma
Gas
Liquid SolidMelting
Vaporization D
eposition
Ionization
Condensation Sublim
ation
Freezing
Recombination
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University
Sample Latent Heats
c©2004-2020 G. Anderson Thermal Physics – slide 53 / 64
Substance Lat. Heat Melting Lat. Heat BoilingFusion kJ/kg Point ◦C Vap. kJ/kg Point ◦C
Ethyl Alcohol 108 -114 855 78.3
Ammonia 339 -75 1369 -33.34
Carbon Dioxide 184 -78 574 -57
Water 334 0 2260 100
Nitrogen 25.7 -210 200 -196
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Sodium Acetate Hand Warmer
c©2004-2020 G. Anderson Thermal Physics – slide 54 / 64
Common comercial handwarmers are made from asupersaturated solution of sodium acetate: C2H3NaO2
The latent heat of fusion is typically in the range:264-289 kJ/kg.
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University
Enthalpy
c©2004-2020 G. Anderson Thermal Physics – slide 55 / 64
Enthalpy, H, the total energy required to create a system U and tomake room for it PV , at constant pressure.
H = U + PV
In a system at constant pressure:
dH = d(U + PV ) = dU + d(PV ) = dU + PdV
Enthalpy can increase by adding energy to or expanding the system.Using the first law: dU = d Q+ dW :
dH = [d Q+ dW ] + PdV
= [d Q− PdV + dWother] + PdV
= d Q+ dWother2
2Eventually Wother ends up as heat added to the system, but unlike Q it is nottaken from a reservoir. It increases the entropy of the universe.
Northeastern
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Enthalpy II
c©2004-2020 G. Anderson Thermal Physics – slide 56 / 64
EnthapyH = U + PV, dH = d Q+ dWother
The change in enthalpy at constant pressure is the same as the heatcapacity at constant pressure.
CP =
(
∂H
∂T
)
P
Northeastern
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University
Transport
Introduction
Units
Ideal Gasses
EquipartitionTheorem
Heat and Work
Heat Capacity
Transport
Heat Transfer
Heat Conduction
Mean Free PathThermalConductivity(Ideal Gas)
DiffusionDiffusionCoefficients atSTPDiffusionCoefficients
c©2004-2020 G. Anderson Thermal Physics – slide 57 / 64
Northeastern
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University
Heat Transfer
c©2004-2020 G. Anderson Thermal Physics – slide 58 / 64
Conduction:
Energy transfer of heat by molecular contact.
Convection:
Energy transport by bulk motion of a gas or
liquid.
Radiation:
Energy transfer by EM radiation, typically
infrared.
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University
Heat Conduction
c©2004-2020 G. Anderson Thermal Physics – slide 59 / 64
Fourier’s Law of heat conduction:
∆Q
∆t= −κA
dT
dx
where κ is the thermal conductivity, and A isthe cross sectional area. In general:
∂Q
∂t= −κ
∮
S
∇T · dS
T1 T2
R-value:
R ≡∆x
κ
Northeastern
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University
Mean Free Path
c©2004-2020 G. Anderson Thermal Physics – slide 60 / 64
Consider a gas molecule of radius r which travels a distance ℓ througha volume filled by molecules of radius r with number density n.
• The molecule will scatter off any sphere with a center inside thevolume:
V = π(2r)2ℓ
ℓ
2rmolecule path
• The probability this volume is occupied by a sphere is P = nV .
• Scattering occurs for P ∼ 1 or n(4πr2ℓ) ∼ 1
• Mean free path
ℓ =1
4nπr2=
1
nσ
• Scattering cross section for hard spheres: σ = 4πr2
Northeastern
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Thermal Conductivity (Ideal Gas)
c©2004-2020 G. Anderson Thermal Physics – slide 61 / 64
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ℓ ℓ
Mean free path, thermal speed, andtime between collisions:
ℓ = v∆t
Heat transfer:
Q ∼12(U1 − U2) = −
12(U2 − U1)
= −12CV (T2 − T1)
= −12CV ℓ
dTdx
Thermal conductivity:
κ =d Q/dt
AdT/dx=
12CV ℓ
∆tA=
1
2
CV
Vvℓ
Northeastern
Illinois
University
Diffusion
c©2004-2020 G. Anderson Thermal Physics – slide 62 / 64
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J
The flow of molecules from an area ofhigh concentration to an area of lowconcentration is called diffusion. Un-der normal circumstances, diffusionprocesses obey Fick’s Law:
Jx = −Ddn
dx
where D is the diffusion coefficient. Ingeneral, the diffusion flux vector is:
J = −D∇n
Northeastern
Illinois
University
Diffusion Coefficients at STP
c©2004-2020 G. Anderson Thermal Physics – slide 63 / 64
Gas in Gas D(m2s−1)
O2 in O2 1.89× 10−5
N2 in N2 1.98× 10−5
CO2 in CO2 1.04× 10−5
O2 in Air 1.78× 10−5
CO2 in Air 1.38× 10−5
H2O in Air 2.36× 10−5
