Download Honors Physics Notes Nov 16, 20 Heat Persans

Honors Physics
Notes Nov 16, 20
Heat
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1
Properties of solids
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Vibrations of atoms in crystalline solids
Assuming only nearest neighbor interactions (+Hooke's law)
Fs = C (us +1 ! us ) + C (us !1 ! us )
us is the displacement of the sth plane.
The equation of motion of the sth plane is thus:
d 2 us
M
= C (us +1 + us !1 ! 2us )
2
dt
We look for solutions where all atoms move as a travelling wave e ! i" t eikx .
! M " 2 us = C (us +1 + us !1 ! 2us )
! M " 2 eiksa = C (eika + e ! ika ! 2 )eiksa
# 2C $
# 4C $ 2 ka
" =%
& (1 ! cos ka ) = %
& sin
2
'M (
'M (
2
1/ 2
# 4C $
" =%
&
'M (
sin
ka
2
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Normal Mode Frequencies (Continued)
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Topics
Heat and temperature
Kinetic model
Properties of materials
Heat capacity
Thermal expansion
Transfer 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 is
proportional to the internal energy of the
system.
What is heat?
• Heat is the energy that flows between a
system and its environment by virtue of a
temperature 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 of
the human body (~1000F).
• The Centigrade or Celsius scale assigns the
freezing point of pure water to 00C and the
boiling point to 1000C. Celsius is the metric
unit and is used worldwide, except in the US.
• The Kelvin scale is based on
thermodynamics and kinetic theory.
9
TF = TC + 32;
5
TC =
5
(TF ! 32 );
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TK = TC + 273
8
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 heat
energy needed to raise the temperature of 1
kg of water by 10C. 1kcal=4185 J
• Some US engineers insist on using the BTU
(British Thermal Unit) 1 BTU is the amount of
heat needed to raise the temperature of 1 lb
of 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:
pV = Nk BT
p = pressure; V = volume;
N = number of gas molecules in that volume;
k B = Boltzman's constant=1.38 "10!23 J/K=8.65 "10!5 eV/K
pV = nRT
n = number of moles of gas
R = universal gas constant=N A k B
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Kinetic theory: a model
1. A gas consists of molecules
2. 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 kinetic
energy
• Find the average change in momentum when a
molecule bounces elastically between two massive
walls separated by distance L.
• Assume that all molecules have the same average
speed.
• Compare the pressure we deduce with the ideal gas
law.
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!momentum = "2mvx1 ;
2mvx1
Fx1 =
(2 L / vx1 )
Fx1 mvx21
p1 =
= 3 ;
A
L
m
m 2
2
For many atoms: p = 3 # vxi = N 3 vx
L
L
v 2 = vx2 + v y2 + vz2 = 3vx2
1 m 2 2 N $1 2%
p= N v =
& mv '
3 V
3V (2
)
2 $1 2%
pV = N & mv ' and comparing to pV = Nk BT
3 (2
)
1 2 3
we conclude that : mv = k BT
2
2
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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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• Anharmonic Forces
Interatomic forces include higher order terms than linear
term in atomic displacement.
Figure 6-2 (a) Harmonic and (b) Anharmonic potential energy
functions for the interaction of two atoms. R is the atomic
separation , while E1, and E2 represent two possible vibrational
energies. For (a) an increase in energy does not result in a
change in the average atomic separation, given by the
midpoints of constant energy lines. For (b) an increase in
energy results in an increase in average separation.
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Thermal Expansion
• To a good approximation over moderate
temperature ranges, the change in length of a
solid increases proportionately to the change
in temperature.
"L # ! L"T
L = original length, ! = linear expansion coefficient
material
Al
Steel
Glass
Concrete
α (10-6/C)
25
12
3
~12
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Thermal expansion of area and
volume
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:
!L
!A(simple square) = (L + !L ) " L = 2 L!L + !L # 2 L!L = 2 A
L
!A
!L
#2
A
L
2
!V
V
3
L + !L )
(
=
L3
" L3
#3
2
2
!L
L
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Volume Thermal Expansion
• The change in volume with temperature can
be defined for solids, liquids and gases.
#V $ !V #T
! $ 3"
V = original volume, ! = volume expansion coefficient
β (10-6/C)
material
Al
Steel
Glass
Concrete
Water
Air
75
35
9
~35
210
3400
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Thermal Expansion: The special
case of ideal gases
• For many gases, it is found that the pressure,
volume, and temperature are related in a
simple way:
pV = nRT
p = pressure; V = volume;
n = number of moles of gas;
R= a constant
R = 8.314 J/(mol K)
mass (grams)
n=
molecular mass (g/mol)
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Exercise: 16 Nov 06
________________
• An ideal gas is cooled from 300C to 00C. By
what 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 on
the left is 0oC. The plates heat in the sun to 40oC
and expand. This causes a buckle as shown in the
figure 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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a) Find the differential expansion of Al and steel.
! ( Steel ) = 11#10-6 ; ! ( Al ) = 23 #10"6
$$L = L$!$T = 20m # (12 #10"6 K "1 )# 40 K = 9.6 x10"3 meters
b) Estimate the height of the buckle. Use L=10 m for one side.
2
$$L (
2
%
2
L
+
$
L
+
=
L
+
$
L
+
h
(
)
S
S
&
)
2
'
*
2
$$
L
2
L2 + $LS +
+ 2$LS L + L$$L + $L$$L
4
$$L2
2
= (L + $LS ) +
+ L$$L + $L$$L
4
h 2 + L$$L = 10m #10"2 m = 0.1m 2 , h + 0.3m
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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). In
many cases, near room temperature, the
temperature change is linear in energy
added.
!Q = C !T
C = heat capacity
• The specific heat takes the mass of material
being heated into account.
c = C / m ! "Q = mc"T
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Some approximate specific heats
•
•
•
•
•
•
•
Aluminum
Glass
Steel
Wood
Water (liquid)
Ice
Air (const V)
0.9 kJ/(kg C)
0.84
0.45
1.7
4.2
2.1
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 does
not change. The energy goes into breaking
or making bonds between atoms.
• The energy relationship is expressed using
the latent heat of melting (or condensation, or
vaporization, or freezing)
!Q = Lm
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Example: Energy required to heat
and boil a cup of water
• Estimate the amount of energy that is
required to heat a cup of water (0.2 kg) from
room temperature to boiling.
• Estimate the amount of energy that is
required to convert that cup of water to vapor.
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Example continued
To raise the temperature of 0.2 kg of water from
20°C to 100°C we use the approximate formula:
J
!Q = cm!T = 4190
• .2kg • 80 K = 67000 J = 67 kJ
kgK
To vaporize the same amount:
kJ
!Qv = Lv m = 2256 " .2kg = 451kJ
kg
The total energy is thus:
!QT = 518kJ
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Comparison of thermal and
gravitational potential energy
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
m
!KE = mgh " 70kg # 4
# 9 floors #10 2 = 25kJ (6 kcal).
floor
s
If all of this energy were converted to heat
the temperature of the water would increase by
J
m
360
10 2 # 36m
!Q mgh
gh
kg
!T "
=
=
= s
=
= 0.086 K .
J
J
cm
cm cwater
4200
4200
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 and
sensible relation:
For a slab of thickness d and area A
! TH # TC "
dQ
! dT "
H=
= kA $
% = kA $
%
dt
x
#
x
dx
&
'
C '
& H
H = rate of heat flow
! W "
k = specific thermal conductivity $
%
m
•
K
&
'
A = area;
TH (TC ) is the hot (cold) temperature
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dx
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Thermal conductivity of common
materials
Material
ρ (W/m.K)
BTU/(ft 0F hr)
Still Air
0.026
0.011
Aluminum
235
140
Glass
1.0
0.50
Concrete
0.7
0.35
Wood
0.1
0.05
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Solving a differential equation for
temperature of an object
• We can use what
we know about
heat flow and
internal energy to
deduce the
temperature of an
object as a function
of time.
Q = cmTH
# TH ! TC $
dQ
dTH
= !cm
= kA %
&
dt
dt
x
!
x
C (
' H
dTC
TC is a constant, so
=0
dt
d (TH ! TC )
kA
!
=
dt
(TH ! TC ) "xcm
final
! ln (TH ! TC ) initial
TH ! TC ) = e
(Persans
!
final
kA
=
t
"xcm initial
kA
t
"xcm
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Exercise _________________
• The temperature difference between two
sides of a 1 cm thick sheet of foam board is
T10C. The rate at which heat passes through
it 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 temperature
difference were doubled?
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Convection
• “Still air” is a very good insulator. The
problem is that air can flow and its density
depends 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 motion
is 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.
$Qemit
4
= !" ATobj
$t
! = Stephan-Boltzman constant=5.7 %10#8 W/m2 .
" = emissivity (~1 for black surface, ~0.05 for shiny metal)
• All objects absorb heat from their
environment as well.
#Qabsorb
4
= !" ATenv
#t
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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 net
pressure difference between inside and
outside a cylinder, there is work done.
Vf
W = ! pdV
Vi
For many cases we have to do the integral
because pressure varies with volume and temp.
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The p-V diagram
p
•
•
•
•
work
isothermal
adiabatic
constant volume
constant pressure
V
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First Law of Thermodynamics
• The change in internal energy (cmΔT) of a
material is the difference between the heat
added to the material system and the work
done by the system.
!Eint = Q + W
Q = heat added to the material
W = work done on the system
(W is negative if the system expands
against an external pressure)
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Using Thermodynamic Laws:
The Ideal Gas
Equation of state for an ideal gas:
pV = nRT = Nk BT
n = number of moles of gas
N = number of molecules
k B = 1.38 "10!23 J / K
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Ideal gas: Adiabatic process
• In an adiabatic process, no heat energy
enters or leaves the system.
#Q = 0 so dEint = dW = " pdV
But the internal energy only depends on the temperature
dEint = nCV dT so pdV = " nCV dT
We also know d ( pV ) = d (nRT )
so Vdp + pdV = nRdT from which Vdp = nC p dT .
dp
dV
= "!
;
p
V
!=
Cp
CV
p = AV "!
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Ideal gas: Isothermal process
No change in internal energy: Q + W = 0
nRT
p=
V
Constant volume
Work done by system must be zero.
dEint = dQ + dW = dQ
All the heat added to a system is stored
as internal energy.
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Work in special cases: Ideal gas
Constant pressure:
Vf
Vf
Vi
Vi
W p = " pdV = p " dV = p (V f ! Vi ).
Constant Temperature:
Nk BT
p=
V
# Vf $
dV
% WT = Nk BT "
= Nk BT ln & '
V
( Vi )
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Some special processes
Adiabatic process: No transfer of heat is made between the system
and the outside world. (!Q = 0) " !Eadiabatic = -W
Constant volume process: No work is done on/by the system.
W = 0;
!Ecv = !Q
Free expansion: An adiabatic process in which no work
is done on or by the system. !E fe = 0.
Cyclical process: A process in which the system returns to
a specific state or position on a p-V diagram. !Ecyclic = 0.
Work done by the system must equal energy added.
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A Heat/Work Conversion Engine
• By applying
pressure, allowing
a piston to slide,
and adding or
subtracting heat,
we can convert
heat energy into
work, or
mechanical,
energy.
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Heat engines move heat between two
reservoirs and produce or use work
!=
W
QH
(any engine)
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The Carnot Cycle
• The most efficient heat engine cycle is the
Carnot cycle, consisting of two isothermal
processes and two adiabatic processes. The
Carnot cycle is the most efficient heat engine
cycle allowed by physical laws.
! CARNOT
QH " QL
TH
=
= 1"
QH
TL
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