Thermal Physics - Physics & Astronomy

66
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

Transcript of Thermal Physics - Physics & Astronomy

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

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Overview

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Introduction

Units

Ideal Gasses

Equipartition Theorem

Heat and Work

Heat Capacity

Transport

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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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Thermodynamics & Statistical Mechanics

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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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Temperature & Equilibrium

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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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Laws of Thermodynamics

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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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Intensive vs. Extensive Variables

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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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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

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Systeme International d’Unites (SI)

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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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Temperature

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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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Temperature Measuring Devices

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• 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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Temperature Scales

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• 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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Temperature Scales

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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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Mole

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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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Sample Molar Volumes (Vm)

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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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Pressure

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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

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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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Energy

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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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Boltzmann’s Constant

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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

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Ideal Gasses

Introduction

Units

Ideal GassesEmpirical IdealGas “Law”

Ideal Gas Law

Real GassesHydrostaticPressureExponentialAtmosphericDecay

EquipartitionTheorem

Heat and Work

Heat Capacity

Transport

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Empirical Ideal Gas “Law”

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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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Ideal Gas Law

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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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Real Gasses

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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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Hydrostatic Pressure

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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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Exponential Atmospheric Decay

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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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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

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Equipartition Theorem

Introduction

Units

Ideal Gasses

EquipartitionTheoremMicroscopicModel of IdealGasTheEquipartition ofEnergy

CountingDegrees offreedom

Heat and Work

Heat Capacity

Transport

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Microscopic Model of Ideal Gas

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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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The Equipartition of Energy

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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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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

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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

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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

Illinois

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.

Northeastern

Illinois

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

Illinois

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

Northeastern

Illinois

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

Northeastern

Illinois

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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Illinois

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

Northeastern

Illinois

University

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.

Northeastern

Illinois

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

Illinois

University

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

Illinois

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

Illinois

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.

Northeastern

Illinois

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

Illinois

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

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

• Scattering cross section for hard spheres: σ = 4πr2

Northeastern

Illinois

University

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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b

b

b

b

b

b

b

b

b

b

b

b

x

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