Transcript Physics 207: Lecture 2 Notes

Lecture 24
Goals:
• Chapter 17
•
 Apply heat and energy transfer processes
 Recognize adiabatic processes
Chapter 18
 Follow the connection between temperature, thermal
energy, and the average translational kinetic energy
molecules
 Understand the molecular basis for pressure and the idealgas law.
 To predict the molar specific heats of gases and solids.
• Assignment
 HW10, Due Wednesday 9:00 AM
 For Thursday, Read through all of Chapter 18
Physics 207: Lecture 24, Pg 1
Isothermal processes
 Work done when PV = nRT = constant  P = nRT / V
W 
final
 p dV  (area under curve )
initial
Vf
Vf
Vi
Vi
W    nRT dV / V   nRT  dV / V
W  nRT n(Vf /Vi )
3
p
T1
T2
T4 T3
V
Physics 207: Lecture 24, Pg 3
Adiabatic Processes

An adiabatic process is process in which there is no thermal
energy transfer to or from a system (Q = 0)
A reversible adiabatic
process involves a
“worked” expansion in
which we can return all of
the energy transferred.
p
 In this case
PVg = const.
 All real processes are not.

4
2
T1
3
1
T2
T4 T3
V
Physics 207: Lecture 24, Pg 4
Work and Ideal Gas Processes (on system)
 Isothermal
W  nRT n(Vf /Vi )
 Isobaric
W  p (Vf - Vi )
 Isochoric
W 0
 FYI: Adiabatic (and reversible)
W   PdV  
V2
V1
V2 const
V1 g
dV
V
g
const

(V
g
1
V )
2
Physicsg207: Lecture 24, Pg 5
Combinations of Isothermal & Adiabatic Processes
All engines employ a thermodynamic cycle
W = ± (area under each pV curve)
Wcycle = area shaded in turquoise
Watch sign of the work!
Physics 207: Lecture 24, Pg 6
Relationship between energy transfer and T
Physics 207: Lecture 24, 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
If a phase transition involved then the heat transferred is
Q = ±ML+M c ΔT
Physics 207: Lecture 24, Pg 8
Q : Latent heat and specific heat
 The molar specific heat of gasses depends on the
process path
 CV= molar specific heat at constant volume
 Cp= molar specific heat at constant pressure
 Cp= CV+R (R is the universal gas constant)
g
Cp

CV
Physics 207: Lecture 24, Pg 9
Mechanical equivalent of heat
 Heating
liquid water:
 Q = amount of heat that must be supplied to
raise the temperature by an amount  T .
 [Q] = Joules or calories. 1 Cal = 4.186 J
1 kcal = 1 Cal = 4186 J
 calorie: energy to raise 1 g of water from
14.5 to 15.5 °C
(James Prescott Joule found the mechanical
equivalent of heat.)
Sign convention:
+Q : heat gained
- Q : heat lost
Physics 207: Lecture 24, Pg 10
Exercise
 The specific heat (Q = M c ΔT) of aluminum is about twice that
of iron. Consider two blocks of equal mass, one made of
aluminum and the other one made of iron, initially in thermal
equilibrium.
 Heat is added to each block at the same constant rate until it
reaches a temperature of 500 K. Which of the following
statements is true?
(a) The iron takes less time than the aluminum to reach 500
K
(b) The aluminum takes less time than the iron to reach 500 K
(c) The two blocks take the same amount of time to reach 500 K
Physics 207: Lecture 24, Pg 11
Heat and Ideal Gas Processes (on system)
 Isothermal Expansion/Contraction
ETh  0  W  Q
Q  W
 Isobaric
Q  nC p T  n(CV  R)T
 Isochoric
Q  nCV T
 Adiabatic
Q0
Physics 207: Lecture 24, Pg 12
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 = 0
D. WA = WB but neither is zero
A13
B12
B23
B 1 3
ON
W12 = 0 (isochoric)
W12 = -½ (p1+p2)(V2-V1) < 0
W23 = -½ (p2+p3)(V1-V2) > 0
= ½ (p3 - p1)(V2-V1) > 0
BY
-W12 > 0
-W23 < 0
< 0
Physics 207: Lecture 24, Pg 13
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?
(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.
Physics 207: Lecture 24, Pg 14
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
Q / t = k A T / x
where k is the thermal conductivity, which characterizes
whether the material is a good conductor of heat or a poor
conductor.
Physics 207: Lecture 24, Pg 15
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 24, Pg 16
Thermal Conductivities
J/s m °C
J/s m °C
J/s m °C
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
Physics 207: Lecture 24, Pg 17
Exercise Thermal Conduction
 Two thermal conductors (possibly
inhomogeneous) are butted together and
in contact with two thermal reservoirs
100 C
held at the temperatures shown.
 Which of the temperature vs. position
plots below is most physical?
(C)
Temperature
Temperature
Temperature
(B)
(A)
Position
300 C
Position
Position
Physics 207: Lecture 24, Pg 20
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
P = s A e T4 (power radiated)
 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
Physics 207: Lecture 24, Pg 21
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.
Vacuum
Silvered
surfaces
Hot or
cold
liquid
Physics 207: Lecture 24, Pg 22
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 = s A T4= s (4p r2) T4 = I p r2  T = [I / (4 x s )]¼
 s = 5.710-8 W/m2 K4
 T = 277 K (A little on the chilly side.)
Physics 207: Lecture 24, Pg 23
Ch. 18, Macro-micro connection
Molecular 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 24, Pg 24
Molecular Speeds and Collisions
A view of a
Fermi chopper
Physics 207: Lecture 24, Pg 25
Molecular Speeds and Collisions
Physics 207: Lecture 24, Pg 26
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 24, Pg 27
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)
2
T
 avg
3k B
2N
p
 avg
3V
vrm s  (v ) avg
2
3k BT

m
Physics 207: Lecture 24, Pg 28
Lecture 24
• Assignment
 HW10, Due Wednesday (9:00 AM)
 Tuesday review
 Reading assignment through all of Chapter 18
Physics 207: Lecture 24, Pg 29