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Honors Physics

Notes Nov 16, 20Heat

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Properties of solids

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Vibrations of atoms in crystalline solids

( )

1 1

2

1 12

( ) ( )

2

Assuming only nearest neighbor interactions (+Hooke's law)

is the displacement of the sth plane.

The equation of motion of the sth plane is thus:

We look

s s s s s

s

s

s s s

F C u u C u u

u

d uM C u u udt

+ !

+ !

= ! + !

= + !

( )

( )

( )

2

1 1

2

2 2

1/ 2

.

2

2

2 41 cos sin

2

4sin

2

for solutions where all atoms move as a travelling wave i t ikx

s s s s

iksa ika ika iksa

e e

M u C u u u

M e C e e e

C C kaka

M M

C ka

M

"

"

"

"

"

!

+ !

!

! = + !

! = + !

# $ # $= ! =% & % &' ( ' (

# $= % &' (

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Normal Mode Frequencies (Continued)

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TopicsHeat and temperatureKinetic modelProperties of materials

Heat capacityThermal expansionTransfer of heat

First Law of thermodynamics: Heat and work

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What is temperature?• Temperature is a measure of the average

kinetic energy of the particles in a space. It isproportional to the internal energy of thesystem.

What is heat?• Heat is the energy that flows between a

system and its environment by virtue of atemperature difference between them .

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Temperature scales• Common scales are based on temperatures

at which everyday things happen.• The Fahrenheit scale is based on the freezing

point of brine (~00F) and the temperature ofthe human body (~1000F).

• The Centigrade or Celsius scale assigns thefreezing point of pure water to 00C and theboiling point to 1000C. Celsius is the metricunit and is used worldwide, except in the US.

• The Kelvin scale is based onthermodynamics and kinetic theory.

( )9 5

32; 32 ; 2735 9

F C C F K CT T T T T T= + = ! = +

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Units of Heat• Since heat is a form of energy, the SI unit is

the Joule.• Another common unit of heat is the

kilocalorie, which is the amount of heatenergy needed to raise the temperature of 1kg of water by 10C. 1kcal=4185 J

• Some US engineers insist on using the BTU(British Thermal Unit) 1 BTU is the amount ofheat needed to raise the temperature of 1 lbof water by 10F. 1BTU=0.252 kcal

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Kinetic Theory of Gases• A series of experiments on dry air led to the

following equation of state for an ideal gas:

23 5

; ;

1.38 10 8.65 10

pressure volume

number of gas molecules in that volume;

= Boltzman's constant= J/K= eV/K

number of moles of gas

universal gas constant=

B

B

A B

pV Nk T

p V

N

k

pV nRT

n

R N k

! !

=

= =

=

" "

=

=

=

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Kinetic theory: a model1. A gas consists of molecules2. The molecules are in random motion and

obey Newton’s laws.3. The total number of molecules is very large

(so we can use statistical averages)4. The average distance between molecules is

large compared to the size of the molecules.5. Molecules experience only collisions forces.6. Collisions are elastic and of short duration.

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Deducing pressure from kineticenergy

• Find the average change in momentum when amolecule bounces elastically between two massivewalls separated by distance L.

• Assume that all molecules have the same averagespeed.

• Compare the pressure we deduce with the ideal gaslaw.

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11 1

1

2

1 11 3

2 2

3 3

2 2 2 2 2

2 2

2

2

22 ;

(2 / )

;

3

1 2 1

3 3 2

2 1

3 2

1 3

2 2

For many atoms:

and comparing to

we conclude that :

xx x

x

x x

xi x

x y z x

B

mvmomentum mv F

L v

F mvp

A L

m mp v N v

L L

v v v v v

m Np N v mv

V V

pV N mv pV Nk T

mv k

! = " =

= =

= =#

= + + =

$ %= = & '

( )

$ %= =& '

( )

= BT

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Properties of common matter• Phases

– solid– liquid– gas– (plasma)

• Phase changes– melting– vaporization (boiling)– dissociation of valence electrons from atoms

• Thermal expansion

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Interatomic forces include higher order terms than linearterm in atomic displacement.

Figure 6-2 (a) Harmonic and (b) Anharmonic potential energyfunctions for the interaction of two atoms. R is the atomicseparation , while E1, and E2 represent two possible vibrationalenergies. For (a) an increase in energy does not result in achange in the average atomic separation, given by themidpoints of constant energy lines. For (b) an increase inenergy results in an increase in average separation.

• Anharmonic Forces

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Thermal Expansion• To a good approximation over moderate

temperature ranges, the change in length of asolid increases proportionately to the changein temperature.

L = original length, = linear expansion coefficient

L L T!

!

" # "

~12Concrete3Glass12Steel25Alα (10-6/C)material

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Thermal expansion of area andvolume

( )

( )

2 2 2

3 3

( ) 2 2 2

2

For solids, the amount by which the area of a plate

or the volume of a block changes can be easily estimated

from the linear expansion:

simple squareL

A L L L L L L L L AL

A L

A L

L L LV

V

!! = + ! " = ! + ! # ! =

! !#

+ ! "!=

33L

L L

!#

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Volume Thermal Expansion• The change in volume with temperature can

be defined for solids, liquids and gases.3

V = original volume, = volume expansion coefficient

V V T! ! "

!

# $ # $

~35Concrete210Water3400Air

9Glass35Steel75Alβ (10-6/C)material

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Thermal Expansion: The specialcase of ideal gases

• For many gases, it is found that the pressure,volume, and temperature are related in asimple way:

; ;pressure volume

number of moles of gas; = a constant

pV nRT

p V

n R

=

= =

=

8.314 J/(mol K)

mass (grams)

molecular mass (g/mol)

R

n

=

=

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Exercise: 16 Nov 06________________

• An ideal gas is cooled from 300C to 00C. Bywhat ratio does its volume change?

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Exercise: 16 Nov 06• Two aluminum plates are held in place by a steel

bridge as shown. The temperature for the figure onthe left is 0oC. The plates heat in the sun to 40oCand expand. This causes a buckle as shown in thefigure on the right.

20 m

a) Find the differential expansion of Al and steel.

b) Estimate the height of the buckle.

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( )

-6 6

6 1 3

( ) 11 10 ; ( ) 23 10

20 12 10 40 9.6 10

2

a) Find the differential expansion of Al and steel.

meters

b) Estimate the height of the buckle. Use L=10 m for one side.

S

Steel Al

L L T m K K x

LL L

! !

!

"

" " "

= # = #

$$ = $ $ = # # # =

$$%+ $ +&

'( )

( )

22 2

222

22

2 2 2

24

4

10 10 0.1 0.3

S

S S

S

L L h

LL L L L L L L L

LL L L L L L

h L L m m m h m"

(= + $ +)

*

$$+ $ + + $ + $$ + $ $$

$$= + $ + + $$ + $ $$

+ $$ = # = , +

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• The specific heat takes the mass of materialbeing heated into account.

Heat capacity and specific heat• When heat energy is added to or subtracted

from an object, its temperature changes(unless it undergoes a phase change). Inmany cases, near room temperature, thetemperature change is linear in energyadded.

heat capacity

Q C T

C

! = !

=

/c C m Q mc T= ! " = "

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Some approximate specific heats

• Aluminum 0.9 kJ/(kg C)• Glass 0.84• Steel 0.45• Wood 1.7• Water (liquid) 4.2• Ice 2.1• Air (const V) 1

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Latent heat of phase change• When the phase of material changes, it takes

up or gives off heat but the temperature doesnot change. The energy goes into breakingor making bonds between atoms.

• The energy relationship is expressed usingthe latent heat of melting (or condensation, orvaporization, or freezing)

Q Lm! =

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Example: Energy required to heatand boil a cup of water

• Estimate the amount of energy that isrequired to heat a cup of water (0.2 kg) fromroom temperature to boiling.

• Estimate the amount of energy that isrequired to convert that cup of water to vapor.

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Example continued

4190 .2 80 67000 67

2256 .2 451

To raise the temperature of 0.2 kg of water from

20 C to 100 C we use the approximate formula:

To vaporize the same amount:

The total ener

v v

JQ cm T kg K J kJ

kgK

kJQ L m kg kJ

kg

° °

! = ! = • • = =

! = = " =

518

gy is thus:

TQ kJ! =

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Comparison of thermal andgravitational potential energy

270 4 9 10

Let's estimate temperature change due to

conversion of gravitational potential energy to heat energy

from dropping a student from from the top of a 9 story building:

m mKE mgh kg floors

floor s! = " # # # =

2

25

36010 36

0.086 .

4200 4200

(6 kcal).

If all of this energy were converted to heat

the temperature of the water would increase by

water

kJ

Jmm

Q mgh gh kgsT KJ Jcm cm c

kgK kgK

#!

! " = = = = =

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The flow of heat

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The flow of heat• Heat energy flows from warmer volume to

colder volumes.• Convection: energy transfer by collective

motion of a macroscopic volume of the fluid.• Conduction: energy transfer by transfer of

vibrational kinetic energy on the atomic scale.• Radiative: energy transfer by emission of

electromagnetic radiation from a hot material.

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Heat Conduction• Heat transfer by conduction can be

represented using a simple andsensible relation:

•

( )

For a slab of thickness and area

= rate of heat flow

specific thermal conductivity

area;

is the hot (cold) temperature

H C

H C

H C

d A

T TdQ dTH kA kA

dt x x dx

H

Wk

m K

A

T T

! "# ! "= = =$ % $ %

# & '& '

! "= $ %

& '

=

dx

A

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Thermal conductivity of commonmaterials

0.1

0.7

1.0

235

0.026

ρ (W/m.K)

0.05Wood

0.35Concrete

0.50Glass

140Aluminum

0.011Still Air

BTU/(ft 0F hr)Material

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Solving a differential equation fortemperature of an object

• We can use whatwe know aboutheat flow andinternal energy todeduce thetemperature of anobject as a functionof time. ( )

( )

( )

( )

0

ln

is a constant, so

H

H CH

H C

CC

H C

H C

finalfinal

H C initialinitial

kAt

xcmH C

Q cmT

T TdQ dTcm kA

dt dt x x

dTT

dt

d T T kAdt

T T xcm

kAT T t

xcm

T T e!"

=

# $!= ! = % &

!' (

=

!! =

! "

! ! ="

! =

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Exercise _________________• The temperature difference between two

sides of a 1 cm thick sheet of foam board isT1

0C. The rate at which heat passes throughit is 400 W per square meter.– What would the heat flow be if the thickness were

increased to 2 cm?

– What would the heat flow be if the temperaturedifference were doubled?

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Convection• “Still air” is a very good insulator. The

problem is that air can flow and its densitydepends on temperature. (1/ρ~V/nR~T).– Cold air is denser than hot air and flows

downward.– Good insulators frequently just use air, but trap it

so in can’t convect.• Heat can then be carried around by the flow

of macroscopic volumes of air whose motionis caused by temperature differences.

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Radiative heat transfer• All objects radiate heat, but hot objects

radiate a lot more of it per unit area.4

8 25.7 10 . Stephan-Boltzman constant= W/m

(~1 for black surface, ~0.05 for shiny metal)

emitobj

QAT

t

emissivity

!"

!

"

#

$=

$

= %

=

4absorbenv

QAT

t!"

#=

#

• All objects absorb heat from theirenvironment as well.

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Heat energy and work

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Work by a system

• If we change the volume and there is a netpressure difference between inside andoutside a cylinder, there is work done.

For many cases we have to do the integral

because pressure varies with volume and temp.

f

i

V

V

W pdV= !

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The p-V diagram

p

V

work

• isothermal• adiabatic• constant volume• constant pressure

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First Law of Thermodynamics• The change in internal energy (cmΔT) of a

material is the difference between the heatadded to the material system and the workdone by the system.

int

(

heat added to the material

work done on the system

is negative if the system expands

against an external pressure)

E Q W

Q

W

W

! = +

=

=

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Using Thermodynamic Laws:The Ideal Gas

231.38 10 /

Equation of state for an ideal gas:

number of moles of gas

number of molecules

B

B

pV nRT Nk T

n

N

k J K!

= =

=

=

= "

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Ideal gas: Adiabatic process• In an adiabatic process, no heat energy

enters or leaves the system.

int

int

0

( ) ( )

.

;

so d

But the internal energy only depends on the temperature

so

We also know

so from which

V V

p

p

V

Q E dW pdV

dE nC dT pdV nC dT

d pV d nRT

Vdp pdV nRdT Vdp nC dT

Cdp dV

p V C

p AV !

! !

"

# = = = "

= = "

=

+ = =

= " =

=

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Ideal gas: Isothermal process0No change in internal energy: Q W

nRTp

V

+ =

=

Constant volume

int

Work done by system must be zero.

All the heat added to a system is stored

as internal energy.

dE dQ dW dQ= + =

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Work in special cases: Ideal gas

( ).

ln

Constant pressure:

Constant Temperature:

f f

i i

V V

p f iV V

fBT B B

i

W pdV p dV p V V

VNk T dVp W Nk T Nk T

V V V

= = = !" "

# $= % = =" & '

( )

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Some special processes

( 0) -

Adiabatic process: No transfer of heat is made between the system

and the outside world. adiabaticQ E W! = "! =

0;

Constant volume process: No work is done on/by the system.

cvW E Q= ! = !

0.

Free expansion: An adiabatic process in which no work

is done on or by the system. feE! =

0.

Cyclical process: A process in which the system returns to

a specific state or position on a p-V diagram.

Work done by the system must equal energy added.

cyclicE! =

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A Heat/Work Conversion Engine

• By applyingpressure, allowinga piston to slide,and adding orsubtracting heat,we can convertheat energy intowork, ormechanical,energy.

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Heat engines move heat between tworeservoirs and produce or use work

(any engine)H

W

Q! =

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The Carnot Cycle• The most efficient heat engine cycle is the

Carnot cycle, consisting of two isothermalprocesses and two adiabatic processes. TheCarnot cycle is the most efficient heat enginecycle allowed by physical laws.

1H L H

CARNOT

H L

Q Q T

Q T!

"= = "

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