Physics 121,Lecture 24,Pg 1
Physics 121,Lecture 24
Today’s Agenda
? Announcements
?About midterm 2,next Tuesday
?Covers Chap,6 - 13.
? Homework 10,due Friday Dec,2 @ 6:00 PM.
?Chap,10,# 2,22,24,28,34,42,49,55,and 58.
?Chap,11,# 2,9,13,27,30,34,and 43.
? Today’s topics
?Temperature and Zeroth Law of Thermodynamics
?Temperature scales and thermal expansion
?Ideal gas
?Heat and energy
?Laws of thermodynamics
Physics 121,Lecture 24,Pg 2
Chap,13,Temperature
? Temperature,measure of the motion of the individual atoms and
molecules in a gas,liquid,or solid.
?related to average kinetic energy of constituents
? High temperature,constituents are moving around energetically
?In a gas at high temperature the individual gas molecules are
moving about independently at high speeds.
?In a solid at high temperature the individual atoms of the solid
are vibrating energetically in place.
? The converse is true for a "cold" object.
?In a gas at low temperature the individual gas molecules are
moving about sluggishly.
?There is an absolute zero temperature at which the motions of
atoms and molecules practically stop.
? There is an absolute zero temperature at which the classical
motions of atoms and molecules practically stop
Physics 121,Lecture 24,Pg 3
Heat
? Solids,liquids or gases have internal energy
?Kinetic energy from random motion of molecules
? translation,rotation,vibration
?At equilibrium,it is related to temperature
? Heat,transfer of energy from one object to another as a
result of their different temperatures
? Thermal contact,energy can flow between objects
T1 T2
U1 U2
>
Physics 121,Lecture 24,Pg 4
Zeroth Law of Thermodynamics
? Thermal equilibrium,
when objects in
thermal contact cease
heat transfer
?same temperature
T1 T2
U1 U2
=
If objects A and B are separately in thermal equilibrium
with a third object C,then objects A and B are in
thermal equilibrium with each other.
A
C B
Physics 121,Lecture 24,Pg 5
Temperature scales
? Three main scales
212
Farenheit
100
Celcius
32 0 273.15
373.15
Kelvin
Water boils
Water freezes
0-273.15-459.67 Absolute Zero
Physics 121,Lecture 24,Pg 6
Some interesting facts
? In 1724,Gabriel Fahrenheit made thermometers
using mercury,The zero point of his scale is
attained by mixing equal parts of water,ice,and
salt,A second point was obtained when pure
water froze (originally set at 30oF),and a third
(set at 96oF),when placing the thermometer in
the mouth of a healthy man”,
?On that scale,water boiled at 212.
?Later,Fahrenheit moved the freezing point of
water to 32 (so that the scale had 180
increments).
? In 1745,Carolus Linnaeus of Upsula,Sweden,
described a scale in which the freezing point of
water was zero,and the boiling point 100,
making it a centigrade (one hundred steps)
scale,Anders Celsius (1701-1744) used the
reverse scale in which 100 represented the
freezing point and zero the boiling point of water,
still,of course,with 100 degrees between the two
defining points.
T (K)
108
107
106
105
104
103
100
10
1
0.1
Hydrogen bomb
Sun’s interior
Solar corona
Sun’s surface
Copper melts
Water freezes
Liquid nitrogen
Liquid hydrogen
Liquid helium
Lowest T
~ 10-9K
Physics 121,Lecture 24,Pg 7
Thermal expansion
? In most liquids or solids,when temperature rises
?molecules have more kinetic energy
? they are moving faster,on the average
?consequently,things tend to expand
? amount of expansion ?L depends on…
?change in temperature ?T
?original length L0
?coefficient of thermal expansion
? L0 + ?L = L0 + ? L0 ?T
??L = ? L0 ?T (linear expansion)
??V = ? L0 ?T (volume expansion)
L0 ?L
V
V + ?V
Physics 121,Lecture 24,Pg 8
Lecture 24,Act 1
Thermal expansion
? As you heat a block of aluminum from 0 oC to 100 oC,its density
(a) increases (b) decreases (c) stays the same
? Solution
?Here ? is positive
Volume increases
Density decreases
T = 0 C
M,V0
r0 = M / V0
T = 100 C
M,V100
r100 = M / V100
< r0
Answer,(b)
Physics 121,Lecture 24,Pg 9
Lecture 24,Act 2
Thermal expansion
? An aluminum plate (?=24?10-6) has a circular hole cut in it,A
copper ball (solid sphere,?=17?10-6) has exactly the same
diameter as the hole when both are at room temperature,and
hence can just barely be pushed through it,If both the plate and
the ball are now heated up to a few hundred degrees Celsius,
how will the ball and the hole fit?
(a) ball won’t fit (b) fits more easily (c) same as before
Physics 121,Lecture 24,Pg 10
Special system,Water
? Most liquids increase in
volume with increasing T
?water is special
?density increases from
0 to 4 oC !
?ice is less dense than
liquid water at 4 oC,
hence it floats
?water at the bottom of
a pond is the denser,
i.e,at 4 oC
Water has its maximum density at 4 degrees.
999,55
999,60
999,65
999,70
999,75
999,80
999,85
999,90
999,95
100 0,0 0
0 2 4 6 8 10
D ens ity
r(kg/m3)
T (oC)
? Reason,chemical bonds of H20 (see your chemistry courses !)
Physics 121,Lecture 24,Pg 11
Lecture 24,Act 3
? Not being a great athlete,and having lots of money to
spend,Gill Bates decides to keep the lake in his back yard
at the exact temperature which will maximize the buoyant
force on him when he swims,Which of the following would
be the best choice?
(a) 0 oC (b) 4 oC (c) 32 oC (d) 100 oC (e) 212 oC
Physics 121,Lecture 24,Pg 12
Ideal gas,Macroscopic description
? Consider a gas in a container of volume V,at pressure P,
and at temperature T
? Equation of state
?Links these quantities
?Generally very complicated,but not for ideal gas
PV = nRT R is called the universal gas constant
In SI units,R =8.315 J / mol·K n = m/M, number of moles
? Equation of state for an ideal gas
?Collection of atoms/molecules moving randomly
?No long-range forces
?Their size (volume) is negligible
Physics 121,Lecture 24,Pg 13
Boltzmann’s constant
? In terms of the total number of particles N
? P,V,and T are the thermodynamics variables
PV = nRT = (N/NA ) RT
kB is called the Boltzmann’s constant
kB = R/NA = 1.38 X 10-23 J/K
PV = N kB T
? Number of moles,n = m/M
?One mole contains NA=6.022 X 1023 particles, Avogadro’s number =
number of carbon atoms in 12 g of carbon
m=mass
M=mass of one mole
Physics 121,Lecture 24,Pg 14
What is the volume of 1 mol of gas at STP?
T = 0 oC = 273 K
p = 1 atm = 1.01 x 105 Pa
nRTpV ?
? ?
?4.22m0 2 2 4.0
Pa10x01.1
K2 7 3Km o l/J31.8
P
RT
n
V
3
5
??
?
?
?
The Ideal Gas Law
Physics 121,Lecture 24,Pg 15
Kinetic Theory of an Ideal Gas
? Assumptions for ideal gas:
?Number of molecules N is large
?They obey Newton’s laws (but move
randomly as a whole)
?Short-range interactions during elastic
collisions
?Elastic collisions with walls
?Pure substance,identical molecules
? Temperature is a direct measure of
average kinetic energy of a molecule
??
PV ? 23 N 12 m v 2?????? ??????
??
1
2 m v
2 ? 3
2 k B T
??
PV ? N k BT
? Microscopic model for a gas
? Goal,relate T and P to motion of the
molecules
Physics 121,Lecture 24,Pg 16
? Theorem of equipartition of energy
?Each degree of freedom contributes kBT/2 to the energy of a
system (e.g.,translation,rotation,or vibration)
Kinetic Theory of an Ideal Gas
??
v x2 ? v y2 ? v z2 ? 13 v 2
??
1
2 m v i
2 ? 1
2 k B T
?
? Total translational kinetic energy of a system of N molecules
?Internal energy of monoatomic gas,U = Kideal = Ktot trans
? Root-mean-square speed:
??
K t o t t r a n s ? N 12 m v 2?????? ?????? ? 32 Nk B T ? 32 n R T
??
v rm s ? v 2 ? 3 k B Tm
Physics 121,Lecture 24,Pg 17
? Consider a fixed volume of ideal gas,When N or T is
doubled the pressure increases by a factor of 2.
Nk TpV ?
1) If T is doubled,what happens to the rate at which a single
molecule in the gas has a wall bounce?
b) x2a) x1.4 c) x4
2) If N is doubled,what happens to the rate at which a single
molecule in the gas has a wall bounce?
b) x1.4a) x1 c) x2
Lecture 24,ACT 4
Physics 121,Lecture 24,Pg 18
Chap,14,Energy in Thermal Processes
? Solids,liquids or gases have internal energy
?Kinetic energy from random motion of molecules
? translation,rotation,vibration
?At equilibrium,it is related to temperature
? Heat,transfer of energy from one object to another as a
result of their different temperatures
? Thermal contact,energy can flow between objects
T1 T2
U1 U2
>
Physics 121,Lecture 24,Pg 19
Heat
? Heat,Q = C ? T
? Q = amount of heat that must be supplied to raise
the temperature by an amount ? T,
? [Q] = Joules or calories.
? energy to raise 1 g of water from 14.5 to 15.5 oC
? James Prescott Joule found mechanical equivalent
of heat.
? C, Heat capacity
1 Cal = 4.186 J
1 kcal = 1 Cal = 4186 J
? Q = c m ? T
? c,specific heat (heat capacity per units of mass)
? amount of heat to raise T of 1 kg by 1oC
? [c] = J/(kg oC)
? Sign convention,+Q, heat gained- Q, heat lost
Physics 121,Lecture 24,Pg 20
Specific Heat, examples
? You have equal masses of aluminum and copper at the
same initial temperature,You add 1000 J of heat to each of
them,Which one ends up at the higher final temperature?
a) aluminum
b) copper
c) the same
Substance c in J/(kg-C)
aluminum 900
copper 387
iron 452
lead 128
human body 3500
water 4186
ice 2000
Physics 121,Lecture 24,Pg 21
Latent Heat
? L = Q / m
? Heat per unit mass
[L] = J/kg
? Q = ? m L
+ if heat needed (boiling)
- if heat given (freezing)
? Lf, Latent heat of fusion
solid ? liquid
? Lv, Latent heat of vaporization
liquid ? gas
? Latent heat,amount of energy needed to add or to remove
from a substance to change the state of that substance.
?Phase change,T remains constant but internal energy changes
?heat does not result in change in T (latent =,hidden”)
? e.g., solid ? liquid or liquid ? gas
heat goes to breaking chemical bonds
Lf (J/kg) Lv (J/kg)
water 33.5 x 104 22.6 x 105
Physics 121,Lecture 24,Pg 22
Latent Heats of Fusion and Vaporization
Energy added (J)
T (oC)
120
100
80
60
40
20
0
-20
-40 Water
Water
+
Ice
Water + Steam Steam
62.7 396 815 3080
Physics 121,Lecture 24,Pg 23
Energy transfer mechanisms
? Thermal conduction (or conduction):
?Energy transferred by direct contact.
?E.g.,energy enters the water through
the bottom of the pan by thermal
conduction,
?Important,home insulation,etc.
? Rate of energy transfer
?through a slab of area A and
thickness ?x,with opposite faces at
different temperatures,Tc and Th
?k, thermal conductivity
?x
Th TcA
Energy
flow
? =Q/?t = k A (Th - Tc ) / ?x
Physics 121,Lecture 24,Pg 24
Thermal Conductivities
Aluminum 238 Air 0.0234 Asbestos 0.25
Copper 397 Helium 0.138 Concrete 1.3
Gold 314 Hydrogen 0.172 Glass 0.84
Iron 79.5 Nitrogen 0.0234 Ice 1.6
Lead 34.7 Oxygen 0.0238 Water 0.60
Silver 427 Rubber 0.2 Wood 0.10
J/s m 0C J/s m 0C J/s m 0C
Physics 121,Lecture 24,Pg 25
Energy transfer mechanisms
? Convection:
?Energy is transferred by flow of substance
?E.g., heating a room (air convection)
?E.g., warming of North Altantic by warm waters
from the equatorial regions
?Natural convection,from differences in density
?Forced convection,from pump of fan
? Radiation:
?Energy is transferred by photons
?E.g., infrared lamps
?Stephan’s law
?s?= 5.7?10-8 W/m2 K4,T is in Kelvin,and A is the surface area
?e is a constant called the emissivity
? = sAe T4, Power
Physics 121,Lecture 24,Pg 26
Resisting Energy Transfer
? The Thermos bottle,also called a
Dewar flask is designed to minimize
energy transfer by conduction,
convection,and radiation,The
standard flask is a double-walled
Pyrex glass with silvered walls and
the space between the walls is
evacuated.
Vacuum
Silvered
surfaces
Hot or
cold
liquid
Physics 121,Lecture 24,Pg 27
Chap,15,the Laws of Thermodynamics
0) If two objects are in thermal equilibrium with a third,they are in
equilibrium with each other.
1) There is a quantity known as internal energy that in an isolated
system always remains the same.
2) There is a quantity known as entropy that in a closed system
always remains the same (reversible) or increases (irreversible).
Physics 121,Lecture 24,Pg 28
Zeroth Law of Thermodynamics
? Thermal equilibrium,
when objects in
thermal contact cease
heat transfer
?same temperature
T1 T2
U1 U2
=
If objects A and B are separately in thermal equilibrium
with a third object C,then objects A and B are in
thermal equilibrium with each other.
A
C B
Physics 121,Lecture 24,Pg 29
1st Law,Work & Heat
? Two types of variables
?State variables,describe the system
(e.g,T,P,V,U).
?Transfer variables,describe the process
(e.g,Q,W).
=0 unless a process occurs
? change in state variables.
? Work done on gas
?W = F d cos? = -F ?y
= - PA ?y = - P ?V
?valid only for isobaric processes
(P constant)
?If not,use average force or calculus,
W = area under PV curve
PV diagram
Physics 121,Lecture 24,Pg 30
1st Law,Work & Heat
? Work:
?Depends on the path taken in the PV-diagram
?Same for Q (heat)
Physics 121,Lecture 24,Pg 31
b) = |W1|a) > |W1| c) < |W1|
i
f
p
V
2
1
? Consider the two paths,1 and 2,
connecting points i and f on the pV
diagram.
?The magnitude of the work,|W2|,
done by the system in going from i
to f along path 2 is
Lecture 24,ACT 5
Work
Physics 121,Lecture 24,Pg 32
First Law of Thermodynamics
? Isolated system
?No interaction with surroundings
?Q = W = 0 ??U = 0.
?Uf = Ui, internal energy remains constant.
? First Law of Thermodynamics
?U = Q + W
variation of internal energy
heat flow,in” (+) or,out” (-)
work done,on” the system
?Independent of path in PV-diagram
?Depends only on state of the system (P,V,T,…)
?Energy conservation statement ?only U changes
Physics 121,Lecture 24,Pg 33
Other Applications
? Cyclic process:
?Process that starts and ends at the state
?Must have ?U = 0 ? Q = -W,
? Adiabatic process:
?No energy transferred through heat ? Q = 0.
?So,?U = W,
?Important for
?expansion of gas in combustion engines
?Liquifaction of gases in cooling systems,etc.
? Isobaric process,(P is constant)
?Work is simply
??
W ? ? P d V ? ? P ( V 2V
1
V 2? ? V
1 )
Physics 121,Lecture 24,Pg 34
Other Applications (continued)
? Isovolumetric process:
?Constant volume ? W =0.
?So ?U = Q ? all heat is transferred into internal energy
?e.g,heating a,can” (no work done).
? Isothermal process:
?T is constant
?Using PV=nRT,we find P= nRT/V.
?Work becomes,
??
W ? ? P d V ? ?V
1
V 2? n R T dV
V ?V 1
V 2? n R T ln V 1
V 2
??
????
??
????
?PV is constant.
?PV-diagram,isotherm.
Physics 121,Lecture 24,Pg 35
p
V
? Identify the nature of paths A,B,C,and D
?Isobaric,isothermal,isovolumetric,and adiabatic
Lecture 24,ACT 6
Processes
A
B
C
D
T1
T2
T3T4
Physics 121,Lecture 24,Pg 36
Heat Engines
? We now try to do more than just raise the temperature of an
object by adding heat,We want to add heat to get some work
done!
? Heat engines:
?Purpose,Convert heat into work using a cyclic process
?Example,Cycle a piston of gas between hot and cold
reservoirs* (Stirling cycle)
1) hold volume fixed,raise temperature by adding heat
2) hold temperature fixed,do work by expansion
3) hold volume fixed,lower temperature by draining heat
4) hold temperature fixed,compress back to original V
Physics 121,Lecture 24,Pg 37
Heat Engines
? Example,the Stirling cycle
Gas
T=TH
Gas
T=TH
Gas
T=TC
Gas
T=TC
V
P
TC
TH
Va Vb
1 2
3
4
We can represent this
cycle on a P-V diagram:
1 1 2
34
*reservoir,large body whose
temperature does not change
when it absorbs or gives up heat
Physics 121,Lecture 24,Pg 38
? Identify whether
?Heat is ADDED or REMOVED from the
gas
?Work is done BY or ON the gas for each
step of the Stirling cycle:
V
P
TC
TH
Va Vb
1 2
3
4
ADDED
REMOVED
BY
ON
1
HEAT
WORK
step
ADDED
REMOVED
BY
ON
2
ADDED
REMOVED
BY
ON
3
ADDED
REMOVED
BY
ON
4
QU0W ???? QW0U ???? QU0W ???? QW0U ???
Heat Engines
Physics 121,Lecture 24,Pg 39
Recap for today:
? Todays’ topics
? Homework 10,due Friday Dec,2 @ 6:00 PM.
?Chap,10,# 2,22,24,28,34,42,49,55,and 58.
?Chap,11,# 2,9,13,27,30,34,and 43.
? Today’s topics
?Temperature and Zeroth Law of Thermodynamics
?Temperature scales and thermal expansion
?Ideal gas
?Heat and energy
?Laws of thermodynamics
Physics 121,Lecture 24
Today’s Agenda
? Announcements
?About midterm 2,next Tuesday
?Covers Chap,6 - 13.
? Homework 10,due Friday Dec,2 @ 6:00 PM.
?Chap,10,# 2,22,24,28,34,42,49,55,and 58.
?Chap,11,# 2,9,13,27,30,34,and 43.
? Today’s topics
?Temperature and Zeroth Law of Thermodynamics
?Temperature scales and thermal expansion
?Ideal gas
?Heat and energy
?Laws of thermodynamics
Physics 121,Lecture 24,Pg 2
Chap,13,Temperature
? Temperature,measure of the motion of the individual atoms and
molecules in a gas,liquid,or solid.
?related to average kinetic energy of constituents
? High temperature,constituents are moving around energetically
?In a gas at high temperature the individual gas molecules are
moving about independently at high speeds.
?In a solid at high temperature the individual atoms of the solid
are vibrating energetically in place.
? The converse is true for a "cold" object.
?In a gas at low temperature the individual gas molecules are
moving about sluggishly.
?There is an absolute zero temperature at which the motions of
atoms and molecules practically stop.
? There is an absolute zero temperature at which the classical
motions of atoms and molecules practically stop
Physics 121,Lecture 24,Pg 3
Heat
? Solids,liquids or gases have internal energy
?Kinetic energy from random motion of molecules
? translation,rotation,vibration
?At equilibrium,it is related to temperature
? Heat,transfer of energy from one object to another as a
result of their different temperatures
? Thermal contact,energy can flow between objects
T1 T2
U1 U2
>
Physics 121,Lecture 24,Pg 4
Zeroth Law of Thermodynamics
? Thermal equilibrium,
when objects in
thermal contact cease
heat transfer
?same temperature
T1 T2
U1 U2
=
If objects A and B are separately in thermal equilibrium
with a third object C,then objects A and B are in
thermal equilibrium with each other.
A
C B
Physics 121,Lecture 24,Pg 5
Temperature scales
? Three main scales
212
Farenheit
100
Celcius
32 0 273.15
373.15
Kelvin
Water boils
Water freezes
0-273.15-459.67 Absolute Zero
Physics 121,Lecture 24,Pg 6
Some interesting facts
? In 1724,Gabriel Fahrenheit made thermometers
using mercury,The zero point of his scale is
attained by mixing equal parts of water,ice,and
salt,A second point was obtained when pure
water froze (originally set at 30oF),and a third
(set at 96oF),when placing the thermometer in
the mouth of a healthy man”,
?On that scale,water boiled at 212.
?Later,Fahrenheit moved the freezing point of
water to 32 (so that the scale had 180
increments).
? In 1745,Carolus Linnaeus of Upsula,Sweden,
described a scale in which the freezing point of
water was zero,and the boiling point 100,
making it a centigrade (one hundred steps)
scale,Anders Celsius (1701-1744) used the
reverse scale in which 100 represented the
freezing point and zero the boiling point of water,
still,of course,with 100 degrees between the two
defining points.
T (K)
108
107
106
105
104
103
100
10
1
0.1
Hydrogen bomb
Sun’s interior
Solar corona
Sun’s surface
Copper melts
Water freezes
Liquid nitrogen
Liquid hydrogen
Liquid helium
Lowest T
~ 10-9K
Physics 121,Lecture 24,Pg 7
Thermal expansion
? In most liquids or solids,when temperature rises
?molecules have more kinetic energy
? they are moving faster,on the average
?consequently,things tend to expand
? amount of expansion ?L depends on…
?change in temperature ?T
?original length L0
?coefficient of thermal expansion
? L0 + ?L = L0 + ? L0 ?T
??L = ? L0 ?T (linear expansion)
??V = ? L0 ?T (volume expansion)
L0 ?L
V
V + ?V
Physics 121,Lecture 24,Pg 8
Lecture 24,Act 1
Thermal expansion
? As you heat a block of aluminum from 0 oC to 100 oC,its density
(a) increases (b) decreases (c) stays the same
? Solution
?Here ? is positive
Volume increases
Density decreases
T = 0 C
M,V0
r0 = M / V0
T = 100 C
M,V100
r100 = M / V100
< r0
Answer,(b)
Physics 121,Lecture 24,Pg 9
Lecture 24,Act 2
Thermal expansion
? An aluminum plate (?=24?10-6) has a circular hole cut in it,A
copper ball (solid sphere,?=17?10-6) has exactly the same
diameter as the hole when both are at room temperature,and
hence can just barely be pushed through it,If both the plate and
the ball are now heated up to a few hundred degrees Celsius,
how will the ball and the hole fit?
(a) ball won’t fit (b) fits more easily (c) same as before
Physics 121,Lecture 24,Pg 10
Special system,Water
? Most liquids increase in
volume with increasing T
?water is special
?density increases from
0 to 4 oC !
?ice is less dense than
liquid water at 4 oC,
hence it floats
?water at the bottom of
a pond is the denser,
i.e,at 4 oC
Water has its maximum density at 4 degrees.
999,55
999,60
999,65
999,70
999,75
999,80
999,85
999,90
999,95
100 0,0 0
0 2 4 6 8 10
D ens ity
r(kg/m3)
T (oC)
? Reason,chemical bonds of H20 (see your chemistry courses !)
Physics 121,Lecture 24,Pg 11
Lecture 24,Act 3
? Not being a great athlete,and having lots of money to
spend,Gill Bates decides to keep the lake in his back yard
at the exact temperature which will maximize the buoyant
force on him when he swims,Which of the following would
be the best choice?
(a) 0 oC (b) 4 oC (c) 32 oC (d) 100 oC (e) 212 oC
Physics 121,Lecture 24,Pg 12
Ideal gas,Macroscopic description
? Consider a gas in a container of volume V,at pressure P,
and at temperature T
? Equation of state
?Links these quantities
?Generally very complicated,but not for ideal gas
PV = nRT R is called the universal gas constant
In SI units,R =8.315 J / mol·K n = m/M, number of moles
? Equation of state for an ideal gas
?Collection of atoms/molecules moving randomly
?No long-range forces
?Their size (volume) is negligible
Physics 121,Lecture 24,Pg 13
Boltzmann’s constant
? In terms of the total number of particles N
? P,V,and T are the thermodynamics variables
PV = nRT = (N/NA ) RT
kB is called the Boltzmann’s constant
kB = R/NA = 1.38 X 10-23 J/K
PV = N kB T
? Number of moles,n = m/M
?One mole contains NA=6.022 X 1023 particles, Avogadro’s number =
number of carbon atoms in 12 g of carbon
m=mass
M=mass of one mole
Physics 121,Lecture 24,Pg 14
What is the volume of 1 mol of gas at STP?
T = 0 oC = 273 K
p = 1 atm = 1.01 x 105 Pa
nRTpV ?
? ?
?4.22m0 2 2 4.0
Pa10x01.1
K2 7 3Km o l/J31.8
P
RT
n
V
3
5
??
?
?
?
The Ideal Gas Law
Physics 121,Lecture 24,Pg 15
Kinetic Theory of an Ideal Gas
? Assumptions for ideal gas:
?Number of molecules N is large
?They obey Newton’s laws (but move
randomly as a whole)
?Short-range interactions during elastic
collisions
?Elastic collisions with walls
?Pure substance,identical molecules
? Temperature is a direct measure of
average kinetic energy of a molecule
??
PV ? 23 N 12 m v 2?????? ??????
??
1
2 m v
2 ? 3
2 k B T
??
PV ? N k BT
? Microscopic model for a gas
? Goal,relate T and P to motion of the
molecules
Physics 121,Lecture 24,Pg 16
? Theorem of equipartition of energy
?Each degree of freedom contributes kBT/2 to the energy of a
system (e.g.,translation,rotation,or vibration)
Kinetic Theory of an Ideal Gas
??
v x2 ? v y2 ? v z2 ? 13 v 2
??
1
2 m v i
2 ? 1
2 k B T
?
? Total translational kinetic energy of a system of N molecules
?Internal energy of monoatomic gas,U = Kideal = Ktot trans
? Root-mean-square speed:
??
K t o t t r a n s ? N 12 m v 2?????? ?????? ? 32 Nk B T ? 32 n R T
??
v rm s ? v 2 ? 3 k B Tm
Physics 121,Lecture 24,Pg 17
? Consider a fixed volume of ideal gas,When N or T is
doubled the pressure increases by a factor of 2.
Nk TpV ?
1) If T is doubled,what happens to the rate at which a single
molecule in the gas has a wall bounce?
b) x2a) x1.4 c) x4
2) If N is doubled,what happens to the rate at which a single
molecule in the gas has a wall bounce?
b) x1.4a) x1 c) x2
Lecture 24,ACT 4
Physics 121,Lecture 24,Pg 18
Chap,14,Energy in Thermal Processes
? Solids,liquids or gases have internal energy
?Kinetic energy from random motion of molecules
? translation,rotation,vibration
?At equilibrium,it is related to temperature
? Heat,transfer of energy from one object to another as a
result of their different temperatures
? Thermal contact,energy can flow between objects
T1 T2
U1 U2
>
Physics 121,Lecture 24,Pg 19
Heat
? Heat,Q = C ? T
? Q = amount of heat that must be supplied to raise
the temperature by an amount ? T,
? [Q] = Joules or calories.
? energy to raise 1 g of water from 14.5 to 15.5 oC
? James Prescott Joule found mechanical equivalent
of heat.
? C, Heat capacity
1 Cal = 4.186 J
1 kcal = 1 Cal = 4186 J
? Q = c m ? T
? c,specific heat (heat capacity per units of mass)
? amount of heat to raise T of 1 kg by 1oC
? [c] = J/(kg oC)
? Sign convention,+Q, heat gained- Q, heat lost
Physics 121,Lecture 24,Pg 20
Specific Heat, examples
? You have equal masses of aluminum and copper at the
same initial temperature,You add 1000 J of heat to each of
them,Which one ends up at the higher final temperature?
a) aluminum
b) copper
c) the same
Substance c in J/(kg-C)
aluminum 900
copper 387
iron 452
lead 128
human body 3500
water 4186
ice 2000
Physics 121,Lecture 24,Pg 21
Latent Heat
? L = Q / m
? Heat per unit mass
[L] = J/kg
? Q = ? m L
+ if heat needed (boiling)
- if heat given (freezing)
? Lf, Latent heat of fusion
solid ? liquid
? Lv, Latent heat of vaporization
liquid ? gas
? Latent heat,amount of energy needed to add or to remove
from a substance to change the state of that substance.
?Phase change,T remains constant but internal energy changes
?heat does not result in change in T (latent =,hidden”)
? e.g., solid ? liquid or liquid ? gas
heat goes to breaking chemical bonds
Lf (J/kg) Lv (J/kg)
water 33.5 x 104 22.6 x 105
Physics 121,Lecture 24,Pg 22
Latent Heats of Fusion and Vaporization
Energy added (J)
T (oC)
120
100
80
60
40
20
0
-20
-40 Water
Water
+
Ice
Water + Steam Steam
62.7 396 815 3080
Physics 121,Lecture 24,Pg 23
Energy transfer mechanisms
? Thermal conduction (or conduction):
?Energy transferred by direct contact.
?E.g.,energy enters the water through
the bottom of the pan by thermal
conduction,
?Important,home insulation,etc.
? Rate of energy transfer
?through a slab of area A and
thickness ?x,with opposite faces at
different temperatures,Tc and Th
?k, thermal conductivity
?x
Th TcA
Energy
flow
? =Q/?t = k A (Th - Tc ) / ?x
Physics 121,Lecture 24,Pg 24
Thermal Conductivities
Aluminum 238 Air 0.0234 Asbestos 0.25
Copper 397 Helium 0.138 Concrete 1.3
Gold 314 Hydrogen 0.172 Glass 0.84
Iron 79.5 Nitrogen 0.0234 Ice 1.6
Lead 34.7 Oxygen 0.0238 Water 0.60
Silver 427 Rubber 0.2 Wood 0.10
J/s m 0C J/s m 0C J/s m 0C
Physics 121,Lecture 24,Pg 25
Energy transfer mechanisms
? Convection:
?Energy is transferred by flow of substance
?E.g., heating a room (air convection)
?E.g., warming of North Altantic by warm waters
from the equatorial regions
?Natural convection,from differences in density
?Forced convection,from pump of fan
? Radiation:
?Energy is transferred by photons
?E.g., infrared lamps
?Stephan’s law
?s?= 5.7?10-8 W/m2 K4,T is in Kelvin,and A is the surface area
?e is a constant called the emissivity
? = sAe T4, Power
Physics 121,Lecture 24,Pg 26
Resisting Energy Transfer
? The Thermos bottle,also called a
Dewar flask is designed to minimize
energy transfer by conduction,
convection,and radiation,The
standard flask is a double-walled
Pyrex glass with silvered walls and
the space between the walls is
evacuated.
Vacuum
Silvered
surfaces
Hot or
cold
liquid
Physics 121,Lecture 24,Pg 27
Chap,15,the Laws of Thermodynamics
0) If two objects are in thermal equilibrium with a third,they are in
equilibrium with each other.
1) There is a quantity known as internal energy that in an isolated
system always remains the same.
2) There is a quantity known as entropy that in a closed system
always remains the same (reversible) or increases (irreversible).
Physics 121,Lecture 24,Pg 28
Zeroth Law of Thermodynamics
? Thermal equilibrium,
when objects in
thermal contact cease
heat transfer
?same temperature
T1 T2
U1 U2
=
If objects A and B are separately in thermal equilibrium
with a third object C,then objects A and B are in
thermal equilibrium with each other.
A
C B
Physics 121,Lecture 24,Pg 29
1st Law,Work & Heat
? Two types of variables
?State variables,describe the system
(e.g,T,P,V,U).
?Transfer variables,describe the process
(e.g,Q,W).
=0 unless a process occurs
? change in state variables.
? Work done on gas
?W = F d cos? = -F ?y
= - PA ?y = - P ?V
?valid only for isobaric processes
(P constant)
?If not,use average force or calculus,
W = area under PV curve
PV diagram
Physics 121,Lecture 24,Pg 30
1st Law,Work & Heat
? Work:
?Depends on the path taken in the PV-diagram
?Same for Q (heat)
Physics 121,Lecture 24,Pg 31
b) = |W1|a) > |W1| c) < |W1|
i
f
p
V
2
1
? Consider the two paths,1 and 2,
connecting points i and f on the pV
diagram.
?The magnitude of the work,|W2|,
done by the system in going from i
to f along path 2 is
Lecture 24,ACT 5
Work
Physics 121,Lecture 24,Pg 32
First Law of Thermodynamics
? Isolated system
?No interaction with surroundings
?Q = W = 0 ??U = 0.
?Uf = Ui, internal energy remains constant.
? First Law of Thermodynamics
?U = Q + W
variation of internal energy
heat flow,in” (+) or,out” (-)
work done,on” the system
?Independent of path in PV-diagram
?Depends only on state of the system (P,V,T,…)
?Energy conservation statement ?only U changes
Physics 121,Lecture 24,Pg 33
Other Applications
? Cyclic process:
?Process that starts and ends at the state
?Must have ?U = 0 ? Q = -W,
? Adiabatic process:
?No energy transferred through heat ? Q = 0.
?So,?U = W,
?Important for
?expansion of gas in combustion engines
?Liquifaction of gases in cooling systems,etc.
? Isobaric process,(P is constant)
?Work is simply
??
W ? ? P d V ? ? P ( V 2V
1
V 2? ? V
1 )
Physics 121,Lecture 24,Pg 34
Other Applications (continued)
? Isovolumetric process:
?Constant volume ? W =0.
?So ?U = Q ? all heat is transferred into internal energy
?e.g,heating a,can” (no work done).
? Isothermal process:
?T is constant
?Using PV=nRT,we find P= nRT/V.
?Work becomes,
??
W ? ? P d V ? ?V
1
V 2? n R T dV
V ?V 1
V 2? n R T ln V 1
V 2
??
????
??
????
?PV is constant.
?PV-diagram,isotherm.
Physics 121,Lecture 24,Pg 35
p
V
? Identify the nature of paths A,B,C,and D
?Isobaric,isothermal,isovolumetric,and adiabatic
Lecture 24,ACT 6
Processes
A
B
C
D
T1
T2
T3T4
Physics 121,Lecture 24,Pg 36
Heat Engines
? We now try to do more than just raise the temperature of an
object by adding heat,We want to add heat to get some work
done!
? Heat engines:
?Purpose,Convert heat into work using a cyclic process
?Example,Cycle a piston of gas between hot and cold
reservoirs* (Stirling cycle)
1) hold volume fixed,raise temperature by adding heat
2) hold temperature fixed,do work by expansion
3) hold volume fixed,lower temperature by draining heat
4) hold temperature fixed,compress back to original V
Physics 121,Lecture 24,Pg 37
Heat Engines
? Example,the Stirling cycle
Gas
T=TH
Gas
T=TH
Gas
T=TC
Gas
T=TC
V
P
TC
TH
Va Vb
1 2
3
4
We can represent this
cycle on a P-V diagram:
1 1 2
34
*reservoir,large body whose
temperature does not change
when it absorbs or gives up heat
Physics 121,Lecture 24,Pg 38
? Identify whether
?Heat is ADDED or REMOVED from the
gas
?Work is done BY or ON the gas for each
step of the Stirling cycle:
V
P
TC
TH
Va Vb
1 2
3
4
ADDED
REMOVED
BY
ON
1
HEAT
WORK
step
ADDED
REMOVED
BY
ON
2
ADDED
REMOVED
BY
ON
3
ADDED
REMOVED
BY
ON
4
QU0W ???? QW0U ???? QU0W ???? QW0U ???
Heat Engines
Physics 121,Lecture 24,Pg 39
Recap for today:
? Todays’ topics
? Homework 10,due Friday Dec,2 @ 6:00 PM.
?Chap,10,# 2,22,24,28,34,42,49,55,and 58.
?Chap,11,# 2,9,13,27,30,34,and 43.
? Today’s topics
?Temperature and Zeroth Law of Thermodynamics
?Temperature scales and thermal expansion
?Ideal gas
?Heat and energy
?Laws of thermodynamics
