Transcript First Law of Thermodynamics {17}

Chapter 17: The First Law of Thermodynamics
Thermodynamic Systems
Interact with surroundings
Heat exchange
Q = heat added to the system(watch sign!)
Some other form of energy transfer
Mechanical Work, e.g.
W = done by the system (watch sign!)
protosystem: ideal gas
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Work done during volume changes
dW  Fdx  pAdx
Adx  dV
dW  pdV
W
V2
 pdV
V1
Depends up on process!
Work = Area under curve!
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Isobaric Expansion
expansion at constant pressure
reversed => compression
W
V2
 pdV
V1
V2
 p  dV
V1
W  p(V2  V1 )
can be rew ritten, with pV = nRT
W  nR(T2  T1 )
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Isothermal Expansion (example 17-1)
expansion at constant temperature
reversed => compression
W
V2
 pdV
V1
with pV = nRT
V2
V
2
nRT
1
W
dV  nRT  dV 
V
V
V1
V1
V2
W  nRT ln( )
V1
can be rew ritten, with p1V1  p2V2
p1
W  nRT ln( )
V2
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Example: 1 m3 of an ideal gas starting at 1 atm of pressure expands to twice its original
volume by one of two processes: isobaric expansion or isothermal expansion. How
much work is done in each case?
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Work Done Depends upon path!!!!
A
B
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Heat Transfer and “Heat Content”
Two constant Temperature processes, same initial state
slow expansion
rapid expansion
Final States (T, V and P) are the same
Heat added in first process, not in second
“Heat Content” not a valid concept
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Internal Energy U :
Sum of microscopic kinetic and potential energies
Changes in response to heat addition (Q) to the system
Changes in response to Work done (W) by the system
The First Law of Thermodynamics:
DU = Q W
or
Q = DU + W
= Conservation of energy
=> DU is independent of path!
U is a function of the state of the system (function of the
state variables). U = U(p,V,T) for an ideal gas.
Infinitesimal processes
dU = dQ - dW
dU = dQ - dW
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For an isolated system
W=Q=0
DU = 0 Uf=Ui
the internal energy of an isolated system is constant
For a cyclic process
system returns to its initial state
state variables return to their initial values
Uf = Ui
=> Wnet = Qnet
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p
Example: Thermodynamic
processes
not an ideal gas
a-b 150 J of heat added
b-d 600 J of heat added
Step
ab
bd
abd
Q
W
DU
Step
ac
cd
acd
Q
W
DU
b
d
8.0 x 104 Pa
3.0 x 104 Pa
a
c
2.0 x 10-3 m3 5.0 x 10-3 m3
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Thermodynamic Processes
Isothermal: constant temperature
generally dV 0 ; dW  0 ; dQ  0
Adiabatic: no heat transfer
Q = 0 ; dQ = 0
p
Isochoric: constant volume (isovolumetric)
dV = 0 => dW = 0 => Q = DU
V
Isobaric: constant pressure
dW = pdV => W = pDV
Polytropic processes: one generalization, not (necessarily) isoanything.
pVr = const
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Internal Energy of an ideal gas.
adiabatic free expansion
Q = 0; W = 0 => DU = 0
Dp  0 ; DV  0 ; DT = 0
=> U depends upon T only
Kinetic Theory
U = sum of microscopic kinetic and potential
energies
each microscopic DOF averages 1/2 kT
=> U depends upon T only
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Molar Heat Capacities of Ideal Gases
1  dQ
CV   
n  dT  V  const
1  dQ
Cp   
n  dT 
p  const
First Law
dQ  dU  dW
Constant Volume Process
V  const  dW  0
dQ  nCV dT  dU
 dU  nCV dT for all processes!!!
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Constant pressure process:
dQ = nCp dT = dU + dW
vs dQ = nCp dT = dU for constant volume => Cp > CV for
any material which expands upon heating
For an ideal gas:
dQ  nC p dT
dU  nCV dT (still!!)
p  const , dW  pdV
and pV  nRT
dW  pdV = d ( pV ) = d (nRT ) = nRdT
dQ  dU  dW
nC p dT  nCV dT  nRdT
C p  CV  R
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With the Equipartition Theorem
DOF
CV 
R
2
DOF
dU  CV dT 
RdT
2
3
5
5 7 
7 9 
 CV  R  R , R
 Cp  R  R, R
2 2 
2 2 
2
2
, another effective way of characterizing an ideal gas
 
Cp
CV
5  7 9
 
 , 
3  5 7
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Adiabatic Processes
dQ  0
dW  pdV   dU   nCV dT
nRT
dV   nCV dT
V
R dV dT

0
CV V
T
R C p  CV

  1
CV
CV
dV dT
(  1)

0
V
T
(  1) ln V  ln T  const
lnV  1 T   const  V  1 T  const
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Adiabatic Processes (cont’d)
V  1 T  const  ideal gas law
pV
V
 const  pV   const
nR
CV
W   DU  nCV ( T1  T2 ) 
( p1V1  p 2V2 )
R
 1
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Summary of basic thermodynamic processeses for an ideal gas
Isobaric
Isochoric
special
p  const
V  const
Isothermal
T  const
Adiabatic
Q0
TV
 1
, pV

const
W
p(V2  V1 )
0
V2
nRT ln
V1
p1
nRT ln
p2
DU
CV
( p1V1  p2V2 )
R
Q
nC p DT
nCV DT
W
0
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Polytropic process for an ideal gas
because not all processes are the basic types (iso*) or
can be approximated by one of the basic process types
polytropic exponent r
pVr = const
r = 0 > isobaric
lim r> infinity > constant volume
r =  > adiabatic
r =1 > isothermal
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