Physics I - Lecture 25 - Heat and The 1st Law of...
Transcript of Physics I - Lecture 25 - Heat and The 1st Law of...
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General Physics IGeneral Physics I
Lecture 25: Heat and The 1st Lecture 25: Heat and The 1st Law of ThermodynamicsLaw of Thermodynamics
Prof. WAN, Xin
[email protected]://zimp.zju.edu.cn/~xinwan/
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Latent Heat in Phase ChangesLatent Heat in Phase Changes
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Latent HeatLatent Heat
The latent heat of vaporization for a given substance is usually somewhat higher than the latent heat of fusion. Why?
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Mechanical Equivalence of HeatMechanical Equivalence of Heat
The amount of energy transfer necessary to raise the temperature of 1 g of water from 14.5oC to 15.5oC.
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HeatHeat
Heat is defined as the transfer of energy across the boundary of a system due to a temperature difference between the system and its surroundings.
Heat transfer– Conduction, convection, radiation
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Heat Conduction: MicroscopicHeat Conduction: Microscopic
Q
l l
UL, NL UR, NR
T
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Heat Conduction: MacroscopicHeat Conduction: Macroscopic
dx
dTA
t
Qt
Fourier heat conduction law
Remind you of Ohm’s law?
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Energy Transfer Through Two SlabsEnergy Transfer Through Two Slabs
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Mean Free PathMean Free Path
dd
v
d
v
d
Average distance between two collisions
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During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.
Mean Free PathMean Free Path
pd
Tk
dnvtdn
vtl B
VV222
1
Tknp BV
Tkpn BV /
vtdnz V2
Volume of the cylinder
vtdV 2Average number of collisions
Mean free path
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During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.
Mean Free PathMean Free Path
pd
Tk
dntvdn
vtl B
VV222 22
1
)2(
vtdnz V2
Average number of collisions
Mean free path Relative motion vv 2
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Q&A: Collision FrequencyQ&A: Collision Frequency
Consider air at room temperature. – How far does a typical molecule (with a
diameter of 2 10-10 m) move before it collides with another molecule?
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Q&A: Collision FrequencyQ&A: Collision Frequency
Consider air at room temperature. – How far does a typical molecule (with a
diameter of 2 10-10 m) move before it collides with another molecule?
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Q&A: Collision FrequencyQ&A: Collision Frequency
Consider air at room temperature. – Average molecular separation:
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Q&A: Collision FrequencyQ&A: Collision Frequency
Consider air at room temperature. – On average, how frequently does one
molecule collide with another?
m
kT
m
kTv ~
8
l
vf
Expect ~ 500 m/s
Expect ~ 2109 /s
Try yourself!
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Kinetic TheoryKinetic Theory
Q
l l
UL, NL UR, NR
T
md
f
m
1~
2
TlvT
Pf Tltht
22
1
TCUUQ VLR 2
1
2
1
dx
dTA
t
Qt
ldx
dT BkAlV
Nf
2
Energy exchange across plane A
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tt for Air at Room Temperature for Air at Room Temperature
m1025.2 7l m/s500v
500m/sm1025.2300K
N/m10
2
5
2
1
22
1 75
tht lv
T
Pf
K)W/(m047.0
From earlier lecture
A factor less than 2 larger than the measured value of 0.026. Not bad after so many crude approximations.
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Transport in ComparisonTransport in Comparison
Phenomena Imbalance Things being transported
Experimental observation
Unit of Coefficient
Thermal conduction
temperature energy W/m·K
Viscosity velocity momentum N·s/m2
Diffusion density particle m2/s
Charge conduction
voltage charge -1m-1
dy
dvAF
dx
dTA
t
Qt
dx
dnDAI n
x )(
dx
dVAI e
x )(
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Internal Energy, Heat & WorkInternal Energy, Heat & Work
Heat is defined as the transfer of energy across the boundary of a system due to a temperature difference between the system and its surroundings.
Energy can also be transferred to or from the system by work.
Internal energy is all the energy of a system that is associated with its microscopic components —atoms and molecules —when viewed from a reference frame at rest with respect to the object.
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Work in Thermodynamic ProcessesWork in Thermodynamic Processes
PdVPAdyFdydW
Quasi-static assumption: the gas expands slowly enough to allow the system to remain essentially in thermal equilibrium at all times.
Work done by the gas
f
i
V
VPdVW
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Work in Thermodynamic ProcessesWork in Thermodynamic Processes
PdVPAdyFdydW
Work done by the gas
f
i
V
VPdVW
The work done by a gas in the expansion from an initial state to a final state is the area under the curve connecting the states in a PV diagram.
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Warning:Warning: Sign Convention Sign Convention
Historically, people are interested in the amount of work done by the expansion of gas, say, to drive a steam engine. The common treatment is
– Positive work: gas expands– Negative work: gas compressed
In mechanics we use the opposite sign, unfortunately.
But some books follow the same convention in thermal physics as in mechanics.
Trust your common sense!
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Work Depends on the PathWork Depends on the Path
)()(iff
a VVPW
)()(ifi
b VVPW
)()()( bca WWW
The work done by a system depends on the initial, final, and intermediate states of the system.
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Ideal GasesIdeal Gases
Tkmv B2
3
2
1 2
Tkf
NU B2 BV Nk
f
T
UC
2Vfixed
TNkpV B
Experiments found Kinetic theory found
2
2
1
3
2mv
NVp
Generalized equipartition theorem (can be proved based on statistical principles)
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Isothermal vs Free ExpansionIsothermal vs Free Expansion
An energy reservoir is a source of energy that is considered to be so great that a finite transfer of energy from the reservoir does not change its temperature.
An adiabatic process is one during which no energy enters or leaves the system by heat.
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Isothermal ExpansionIsothermal Expansion
i
fB V
VTNkWQ ln
i
fB
V
V
BV
V V
VTNkdV
V
TNkPdVW
f
i
f
i
ln
0U at fixed T
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(Adiabatic) Free Expansion(Adiabatic) Free Expansion
0WQ0U
Energy transfer by heat, like work done, depends on the initial, final, and intermediate states of the system.
- Is it possible to show the process on the PV diagram?
- Is the process reversible?
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The 1st Law of ThermodynamicsThe 1st Law of Thermodynamics
Although Q and W both depend on the path, the quantity Q-W is independent of the path change.
The change in the internal energy U of the system can be expressed as:
The infinitesimal change:
WQU
PdVdQdU
reminding you that it is path dependent
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Discussion on the 1st LawDiscussion on the 1st Law
The 1st law is a statement of energy conservation (now with the internal energy included).
The internal energy of an isolated system remains constant. In a cyclic process,
– The net work done by the systemper cycle equals the area enclosed by the path representing the process on a PV diagram.
WQU ,0
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Discussion on the 1st LawDiscussion on the 1st Law
On a microscopic scale, no distinction exists between the result of heat and that of work.
The internal energy function is therefore called a state function, whose value is determined by the state of the system.
– In general,
),( VTUU
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Digression on Multivariate CalculusDigression on Multivariate Calculus
If we take energy and volume as parameters, how comes heat is path dependent?
In mathematical language, dU + pdV is an inexact differential.
– In multivariate calculus, a differential is said to be exact (or perfect), as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q.
PdVdUdQ
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Inexact DifferentialInexact Differential
12ln)2,2(
)2,1(
)2,1(
)1,1(
dy
y
xdx
0lnln),( fyxyxf
2ln21)2,2(
)1,2(
)1,2(
)1,1(
dy
y
xdx
dyy
xdxdg Assume
Note: is an exact differential.
Integrating factor
y
dy
x
dx
x
dgdf
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Isobaric ProcessesIsobaric Processes
TNkTC BVTNkPV B
BVP NkCT
QC
Pfixed
TCU V)( if VVPW
VPTCWUQ V
isobaric
f
fC
CNkfCV
PBV
22/
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Isobaric vs Isovolumetric ProcessesIsobaric vs Isovolumetric Processes
0W
isovolumetric
TCU V
TCUQ V
)( if VVPW
TCU V
TCWUQ P
isobaric
Molar specific heat: RCC VP
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Degrees of Freedom, AgainDegrees of Freedom, Again
BV Nkf
C2
BP Nkf
C2
2
f
fC
CV
P2
f 3 5 7
1.67 1.4 1.28