Lecture 23, Nov. 19
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Physics 207: Lecture 23, Pg 1
Lecture 23, Nov. 19Goals:Goals:
• Chapter 17Chapter 17 Apply heat and energy transfer processes Recognize adiabatic processes
• Chapter 18Chapter 18 Follow the connection between temperature, thermal energy, and the average translational kinetic energy molecules Understand the molecular basis for pressure and the ideal-gas law. To predict the molar specific heats of gases and solids.
• AssignmentAssignment HW10, Due Sunday (11:59 PM) For Wednesday, Read through all of Chapter 18
Physics 207: Lecture 23, Pg 2
Exam III Room assignments
613 Room 2223 Koki
601 Room 2241 Matt603 Room 2241 Heming608 Room 2241 Matt609 Room 2241 Heming607 Room 2241 Koki
And all others in Room 2103602 604605606610611612614
Physics 207: Lecture 23, Pg 3
Work and Ideal Gas Processes (on system)
Isothermal
)/VV( nRT ifnW
Isobaric
)V-V( p ifW Isochoric
0W
)( 12constconst
2
1
2
1
VVPdVW V
VVV V
dV
V
FYI: Adiabatic (and reversible)
Physics 207: Lecture 23, Pg 4
Heat and Ideal Gas Processes (on system)
Isothermal Expansion/Contraction
WQ Isobaric
Isochoric
TnCQ V
TRCnTnCQ Vp )(
Adiabatic
0Q
Physics 207: Lecture 23, Pg 5
Identify the nature of paths 1, 2, 3 and 4
(A) Isobaric
(B) Isothermal
(C) Isochoric
(D) Adiabatic
Exercise Identify processes
p
V
1
2
3
4
T1
T2
T3T4
Physics 207: Lecture 23, Pg 6
Two process are shown that take an ideal gas from state 1 to
state 3.
Compare the work done by process A to the work done by
process B.
A. WA > WB
B. WA < WB C. WA = WB = 0D. WA = WB but neither is zero
ON BYA 1 3 W12 = 0 (isochoric)B 1 2 W12 = -½ (p1+p2)(V2-V1) < 0 -W12 > 0B 2 3 W23 = -½ (p2+p3)(V1-V2) > 0 -W23 < 0B 1 3 = ½ (p3 - p1)(V2-V1) > 0 < 0
Physics 207: Lecture 23, Pg 7
Heat and Latent Heat
Latent heat of transformation L is the energy required for 1 kg of substance to undergo a phase change. (J / kg)
Q = ±ML Specific heat c of a substance is the energy required to raise the
temperature of 1 kg by 1 K. (Units: J / K kg )
Q = M c ΔT
Molar specific heat C of a gas at constant volume is the energy required to raise the temperature of 1 mol by 1 K.
Q = n CV ΔT
Physics 207: Lecture 23, Pg 8
Most people were at least once burned by hot water or steam. Assume that water and steam, initially at 100°C, are cooled down
to skin temperature, 37°C, when they come in contact with your skin. Assume that the steam condenses extremely fast, and that the specific heat c = 4190 J/ kg K is constant for both liquid water and steam.
Under these conditions, which of the following statements is true?
(a) Steam burns the skin worse than hot water because the thermal conductivity of steam is much higher than that of liquid water.
(b) Steam burns the skin worse than hot water because the latent heat of vaporization is released as well.
(c) Hot water burns the skin worse than steam because the thermal conductivity of hot water is much higher than that of steam.
(d) Hot water and steam both burn skin about equally badly.
Exercise Latent Heat
Physics 207: Lecture 23, Pg 9
Exercise Latent Heat Most people were at least once burned by hot water or steam. Assume
that water and steam, initially at 100°C, are cooled down to skin temperature, 37°C, when they come in contact with your skin. Assume that the steam condenses extremely fast, and that the specific heat c = 4190 J/ kg K is constant for both liquid water and steam.
Under these conditions, which of the following statements is true?
(b) Steam burns the skin worse than hot water because the latent heat of vaporization is released as well.
How much heat H1 is transferred to the skin by 25.0 g of steam?
The latent heat of vaporization for steam is L = 2256 kJ/kg.
H1 = 0.025 kg x 2256 kJ/kg = 63.1 kJ How much heat H2 is transferred to the skin by 25.0 g of water?
H2 = 0.025 kg x 63 K x 4190 J/ kg K = 6.7 kJ
Physics 207: Lecture 23, Pg 10
Energy transfer mechanisms Thermal conduction (or conduction) Convection Thermal Radiation
For a material of cross-section area A and length L, spanning a temperature difference ΔT = TH – TC, the rate
of heat transfer is
where k is the thermal conductivity, which characterizes whether the material is a good conductor of heat or a poor conductor.
Q / t = k A T / x
Physics 207: Lecture 23, Pg 11
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 ( J / s or W ) Through a slab of area A and
thickness x, with opposite faces at different temperatures, Tc and Th
Q / t = k A (Th - Tc ) / x
k :Thermal conductivity (J / s m °C)
Physics 207: Lecture 23, Pg 12
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 °C J/s m °C J/s m °C
Physics 207: Lecture 23, Pg 15
100 C
Two thermal conductors (possibly inhomogeneous) are butted together and in contact with two thermal reservoirs held at the temperatures shown.
Which of the temperature vs. position plots below is most physical?
Exercise Thermal Conduction
300 C
Position
Te
mp
era
ture
Position
Te
mp
era
ture
Position
Te
mp
era
ture
(A) (B) (C)
Physics 207: Lecture 23, Pg 16
Energy transfer mechanisms Convection:
Energy is transferred by flow of substance
1. Heating a room (air convection)
2. 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 Stefan’s Law
= 5.710-8 W/m2 K4 , T is in Kelvin, and A is the surface area e is a constant called the emissivity
P = A e T4 (power radiated)
Physics 207: Lecture 23, Pg 17
Minimizing 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.
VacuumVacuum
SilveredSilveredsurfacessurfaces
Hot orHot orcoldcoldliquidliquid
Physics 207: Lecture 23, Pg 18
Anti-global warming or the nuclear winter scenario
Assume P/A = I = 1340 W/m2 from the sun is incident on a thick dust cloud above the Earth and this energy is absorbed, equilibrated and then reradiated towards space where the Earth’s surface is in thermal equilibrium with cloud. Let e (the emissivity) be unity for all wavelengths of light.
What is the Earth’s temperature?P = A T4= (4 r2) T4 = I r2 T = [I / (4 x )]¼ = 5.710-8 W/m2 K4 T = 277 K (A little on the chilly side.)
Physics 207: Lecture 23, Pg 19
Ch. 18, Macro-micro connectionMolecular Speeds and Collisions
• A real gas consists of a vast number of molecules, each moving randomly and undergoing millions of collisions every second.
• Despite the apparent chaos, averages, such as the average number of molecules in the speed range 600 to 700 m/s, have precise, predictable values. • The “micro/macro” connection is built on the idea that the macroscopic properties of a system, such as temperature or pressure, are related to the average behavior of the atoms and molecules.
Physics 207: Lecture 23, Pg 20
Molecular Speeds and Collisions
A view of a Fermi chopper
Physics 207: Lecture 23, Pg 21
Molecular Speeds and Collisions
Physics 207: Lecture 23, Pg 22
Mean Free Path
If a molecule has Ncoll collisions as it travels distance L, the average distance between collisions, which is called the mean free path λ (lowercase Greek lambda), is
Physics 207: Lecture 23, Pg 23
Macro-micro connection Assumptions for ideal gas:
# of molecules N is large They obey Newton’s laws Short-range interactions with
elastic collisions Elastic collisions with walls
(an impulse…..pressure)
What we call temperature T is a direct measure of the average translational kinetic energy
What we call pressure p is a direct measure of the number density of molecules, and how fast they are moving (vrms)
avg32 VN
p
avg32 Bk
T
m
Tkvv B
rms
3)( avg
2
Physics 207: Lecture 23, Pg 24
Lecture 23, Nov. 19
• AssignmentAssignment HW10, Due Sunday (11:59 PM) For Wednesday, Read through all of Chapter 18
Following slides are for Wednesday
Physics 207: Lecture 23, Pg 25
Kinetic energy of a gas
The average kinetic energy of the molecules of an ideal gas at 10°C has the value K1. At what temperature T1 (in degrees Celsius) will the average kinetic energy of the same gas be twice this value, 2K1?
(A) T1 = 20°C
(B) T1 = 293°C
(C) T1 = 100°C
The molecules in an ideal gas at 10°C have a root-mean-square (rms) speed vrms.
At what temperature T2 (in degrees Celsius) will the molecules have twice the rms speed, 2vrms?
(A) T2 = 859°C
(B) T2 = 20°C
(C) T2 = 786°C
avg2
23
21 Tkvm B
Physics 207: Lecture 23, Pg 26
Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of 2.
1. 1. If T is doubled, what happens to the rate at which If T is doubled, what happens to the rate at which a single a single moleculemolecule in the gas has a wall bounce? in the gas has a wall bounce?
(B) x2(A) x1.4 (C) x4
22. If N is doubled, what happens to the rate at which . If N is doubled, what happens to the rate at which a a single moleculesingle molecule in the gas has a wall bounce? in the gas has a wall bounce?
(B) x1.4(A) x1 (C) x2
Exercise
PV NkBT
1
2mv 2
3
2kBT
Physics 207: Lecture 23, Pg 27
Degrees of freedom or “modes” Degrees of freedom or “modes of energy storage in the system” can
be: Translational for a monoatomic gas (translation along x, y, z axes, energy stored is only kinetic) NO potential energy
Rotational for a diatomic gas (rotation about x, y, z axes, energy stored is only kinetic)
Vibrational for a diatomic gas (two atoms joined by a spring-like molecular bond vibrate back and forth, both potential and kinetic energy are stored in this vibration)
In a solid, each atom has microscopic translational kinetic energy and microscopic potential energy along all three axes.
Physics 207: Lecture 23, Pg 28
Degrees of freedom or “modes”
A monoatomic gas only has 3 degrees of freedom (just K, kinetic)
A typical diatomic gas has 5 accessible degrees of freedom at room temperature, 3 translational (K) and 2 rotational (K)
At high temperatures there are two more, vibrational with K and U
A monomolecular solid has 6 degrees of freedom
3 translational (K), 3 vibrational (U)
Physics 207: Lecture 23, Pg 29
The Equipartition Theorem
The equipartition theorem tells us how collisions distribute the energy in the system. Energy is stored equally in each degree of freedom of the system.
The thermal energy of each degree of freedom is:
Eth = ½ NkBT = ½ nRT A monoatomic gas has 3 degrees of freedom
A diatomic gas has 5 degrees of freedom
A solid has 6 degrees of freedom
Molar specific heats can be predicted from the thermal energy, because
nRTEth 23
nRTEth 25
TnCEth nRTCV 2
3
gas Monoatomic
nRTCV 25
gas Diatomic
nRTCV 3
solid Elemental
nRTEth 3
Physics 207: Lecture 23, Pg 30
Exercise A gas at temperature T is mixture of hydrogen and helium gas.
Which atoms have more KE (on average)?
(A) H (B) He (C) Both have same KE
How many degrees of freedom in a 1D simple harmonic oscillator?
(A) 1 (B) 2 (C) 3 (D) 4 (E) Some other number
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