Transcript The First Law of Thermodynamics

Physics I
The First Law of
Thermodynamics
Prof. WAN, Xin
[email protected]
http://zimp.zju.edu.cn/~xinwan/
Heat Conduction
Fourier heat conduction law
Q
dT
  t A
t
dx
Remind you of Ohm’s law?
Energy Transfer Through Two Slabs
Kinetic Theory
T
Energy exchange across plane A
1
1
Q   U R  U L    CV T
2
2
f
2
N

 V  Al  k B
dT
l
dx
1 f P
l T
t 
lvth   T
22T
UL, NL
UR, NR
Q
l
l
Q
dT
  t A
t
dx
f 1
~ 2
dm m
t for Air at Room Temperature
From last lecture
l  2.25 107 m
v  500 m/s
5
1 f P
1 5 10 N/m
t 
lvth 
 2.25 107 m  500m/s
22T
2 2 300K
 0.047 W/(m  K)
A factor less than 2 larger than the measured value of 0.026.
Not bad after so many crude approximations.
Transport in Comparison
Phenomena
Imbalance
Things being
transported
Thermal
conduction
temperature
energy
Viscosity
velocity
momentum
Diffusion
density
particle
Charge
conduction
voltage
charge
Experimental
observation
Q
dT
  t A
t
dx
dv
F  A
dy
dn
(n)
I x   DA
dx
dV
(e)
I x  A
dx
Unit of
Coefficient
W/m·K
N·s/m2
m2/s
W-1m-1
Internal 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.
Mechanical Equivalence of Heat
The amount of energy
transfer necessary to raise
the temperature of 1 g of
water from 14.5oC to 15.5oC.
Specific Heat
Note: Last time we defined
molar specific heat. In
physics, we also use
specific heat per particle.
Help Young Leonardo DiCaprio


A cowboy fires a silver bullet
with a mass of 2 g and with a
muzzle speed of 200 m/s into
the pine wall of a saloon.
Assume that all the internal
energy generated by the impact
remains with the bullet. What
is the temperature change of the bullet?
Kinetic energy 1 2 1
2
mv  2 10 3 kg 200 m/s   40 J
2
2
convert to heat
Q
40 J

T 


85
.
5
C
3

mc 2 10 kg 234 J/kg  C
Latent Heat in Phase Changes
Latent Heat
The latent heat of vaporization for a given substance is usually
somewhat higher than the latent heat of fusion. Why?
Work in Thermodynamic Processes
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
dW  Fdy  PAdy  PdV
Vf
W   PdV
Vi
Work in Thermodynamic Processes
Work done by the gas
dW  Fdy  PAdy  PdV
Vf
W   PdV
Vi
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.
Warning: 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!
Work Depends on the Path
W ( a )  Pf (V f  Vi )
W ( a )  W ( c )  W (b )
W (b)  Pi (V f  Vi )
The work done by a system depends on the initial, final, and
intermediate states of the system.
Isothermal Expansion
• The gas does work on
the piston
W 0
• Energy is transferred
slowly to the gas
Q0
An energy reservoiris 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.
Free Expansion
• No heat or energy is
transferred
Q0
• The value of the work
done is zero
W 0
Energy transfer by heat, like work done, depends
on the initial, final, and intermediate states of the
system.
The 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:
U  Q  W

The infinitesimal change:
dU  dQ  PdV
reminding you that it is path dependent
Discussion 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,
U  0, Q  W
– The net work done by the system
per cycle equals the area enclosed
by the path representing the
process on a PV diagram.
Discussion 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,
U  U (T , V )
Digression on Multivariate Calculus

If we take energy and volume as parameters,
how comes heat is path dependent?
dQ  dU  PdV

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.
Inexact Differential
x
Assume dg  dx  dy
y
 ( 2,1)  ( 2, 2)  dx  x dy   1  2 ln 2
(2,1)  y 
 (1,1)
 (1, 2)  ( 2, 2)  dx  x dy   ln 2  1
(1,2)  y 
 (1,1)
dg dx dy
Note: df 


x
x
y
Integrating factor
is an exact differential.
f ( x, y)  ln x  ln y  f 0
Ideal Gas
Experiments found
pV  NkBT
Kinetic theory found
2N 1 2
pV 
mv
3 2
1 2 3
mv  k BT
2
2
Generalized equipartition theorem (can be proved
based on statistical principles
f
U  N k BT
2
f
 U 
CV  
 NkB

 T fixed V 2
Isobaric & Isovolumetric Processes
isobaric
W  P(V f  Vi )
U  CV T
Q  U  W  CV T  PV
PV  Nk BT
  CV T  Nk B T
 Q 
CP  
 CV  NkB

 T fixed P
CV   f / 2NkB
 
CP
f 2

CV
f
Isobaric & Isovolumetric Processes
isobaric
W  P(V f  Vi )
U  CV T
Q  U  W  CV T  PV
PV  Nk BT
  CV T  Nk B T
 Q 
CP  
 CV  NkB

 T fixed P
Molar specific heat: CP  CV  R
Isobaric & Isovolumetric Processes
isobaric
W  P(V f  Vi )
U  CV T
Q  U  W  CP T
isovolumetric
W  0 U  CV T
Q  U  CV T
Isothermal Expansion
Vf
Vf
Vi
Vi
W   PdV  
Vf
Nk BT
dV  Nk BT ln
V
Vi
U  0
Q  W  Nk BT ln
Vf
Vi
Adiabatic Expansion
adiabatic
dQ  0
CV dT  dU   PdV
P
Nk BT
V
CV dT
Nk B dV
 


T
V
TV
  CP C  1  NkB C
V
V
 1
 const
or PV   const
(Adiabatic) Free Expansion
U  0
Q W  0
Questions:
• Is it possible to show the
process on the PV diagram?
• Is it reversible?
Degrees of Freedom, Again
f
CV  Nk B
2
f 2
CP 
Nk B
2

f

CP
f 2

CV
f
3
5
7
1.67 1.4 1.28
Homework
CHAP. 22 Exercises 11, 19,
21 (P540) 24, 25 (P541)