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PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 31: Chapter 21: Ideal gas equations Molecular view of ideal gas Internal energy of ideal gas Distribution of molecular speeds in ideal gas. Review: Consider the process described by A BCA. iclicker exercise: - PowerPoint PPT Presentation
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PHY 113 A Fall 2012 -- Lecture 31 111/16/2012
PHY 113 A General Physics I9-9:50 AM MWF Olin 101
Plan for Lecture 31:
Chapter 21: Ideal gas equations
1. Molecular view of ideal gas
2. Internal energy of ideal gas
3. Distribution of molecular speeds in ideal gas
PHY 113 A Fall 2012 -- Lecture 31 211/16/2012
PHY 113 A Fall 2012 -- Lecture 31 311/16/2012
Review:Consider the process described by ABCA
iclicker exercise:What is the net work done on the system in this cycle?
A. -12000 JB. 12000 JC. 0
PHY 113 A Fall 2012 -- Lecture 31 411/16/2012
Equation of “state” for ideal gas(from experiment)
nRTPV
pressure in Pascals
volume in m3 # of moles
temperature in K
8.314 J/(mol K)
PHY 113 A Fall 2012 -- Lecture 31 511/16/2012
Ideal gas -- continued
..............................
diatomicfor
monoatomicfor
gas ideal of on type dependingparameter
1
1
1
1 :energy Internal
:state ofEquation
57
35
int
PVnRTE
nRTPV
Note that at this point, the above equation for Eint is completely unjustified…
PHY 113 A Fall 2012 -- Lecture 31 611/16/2012
From The New Yorker Magazine, November 2003
PHY 113 A Fall 2012 -- Lecture 31 711/16/2012
Microscopic model of ideal gas:
Each atom is represented as a tiny hard sphere of mass m with velocity v. Collisions and forces between atoms are neglected. Collisions with the walls of the container are assumed to be elastic.
PHY 113 A Fall 2012 -- Lecture 31 811/16/2012
Proof: Force exerted on wall perpendicular to x-axis by an atom which collides with it:
average over atoms
What we can show is the pressure exerted by the atoms by their collisions with the walls of the container is given by:
avgavgK
V
Nvm
V
NP
3
2
3
2 22
1
t
vm
t
pF ixiixix
2
d
x
ixvdt /2
22
2
/22
xii
ixi
i
ix
ixi
ix
ixiix
vmVN
dAvm
AF
P
dvm
vdvm
F
vix
-vix
number of atoms
volume
PHY 113 A Fall 2012 -- Lecture 31 911/16/2012
22122
22222
2222
222
22
2
3
2
3
3
that note Also
:,, along move likely toequally are molecules Since
/2
2
iiiiixi
ixiziyixi
iziyixi
iziiyiixi
ixii
ixi
i
ix
ixi
ix
ixiix
vmV
Nvm
V
Nvm
V
NP
vvvvv
vvvv
vmvmvm
zyx
vmV
N
dA
vm
A
FP
d
vm
vd
vmF
PHY 113 A Fall 2012 -- Lecture 31 1011/16/2012
iclicker question:What should we call ?
A. Average kinetic energy of atom.B. We cannot use our macroscopic equations
at the atomic scale -- so this quantity will go unnamed.
C. We made too many approximations, so it is not worth naming/discussion.
D. Very boring.
221
iivm
PHY 113 A Fall 2012 -- Lecture 31 1111/16/2012
atoms of moles ofnumber
atom) Hefor kg (0.004 massmolar thedenotes M where
)atom Hefor kg106.6( atom of mass
atoms ofnumber :Note3
2
27-
221
n
nMNm
m
N
vmV
NP
i
i
ii
atoms gas ideal of mole ofenergy kinetic average
2
3or
3
2
3
2
:law gas ideal toConnection
221
2212
21
221
i
ii
i
Mv
RTMvRTMv
nRTMvnPV
nRTE2
3int
PHY 113 A Fall 2012 -- Lecture 31 1211/16/2012
Average atomic velocities: (note <vi>=0)
M
RTv
RTMv
i
i
3
2
3
2
221
Relationship between average atomic velocities with T
PHY 113 A Fall 2012 -- Lecture 31 1311/16/2012
nRTE2
3int
For monoatomic ideal gas:
General form for ideal gas (including mono-, di-, poly-atomic ideal gases):
..............................
diatomicfor
monoatomicfor
1
1
57
35
int
nRTE
PHY 113 A Fall 2012 -- Lecture 31 1411/16/2012
RT-n
Tk-N
TkNvmNE BB 1γ1γ23
21
2int
Internal energy of an ideal gas:
derived for monoatomic ideal gas more general relation for
polyatomic ideal gas
Gas g (theory) (g exp)
He 5/3 1.67
N2 7/5 1.41
H2O 4/3 1.30
Big leap!
PHY 113 A Fall 2012 -- Lecture 31 1511/16/2012
Determination of Q for various processes in an ideal gas:
Example: Isovolumetric process – (V=constant W=0)
In terms of “heat capacity”:
WQTR-
nE
RT-
nE
1γ
1γ
int
int
fififi QTR-n
E 1γ
int
1γ
1γ
-R
C
TnCTR-n
Q
V
fiVfifi
PHY 113 A Fall 2012 -- Lecture 31 1611/16/2012
Example: Isobaric process (P=constant):
In terms of “heat capacity”:
Note: = CP/CV
fifififi WQTR-
nE
1γ int
1γγ
1γ
1γ1γ
-R
R-R
C
TnCTnRTR-n
VVPTR-n
Q
P
fiPfifiifififi
PHY 113 A Fall 2012 -- Lecture 31 1711/16/2012
More examples:
Isothermal process (T=0)
DT=0 DEint = 0 Q=-W
WQTR-
nE
RT-
nE
1γ
1γ
int
int
i
fV
V
V
V V
VnRT
V
dVnRTPdVW
f
i
f
i
ln
PHY 113 A Fall 2012 -- Lecture 31 1811/16/2012
Even more examples:
Adiabatic process (Q=0)
TnRVPPV
nRTPV
VPTR-
n
WE
1γ
int
γγγ
γ
lnln
γ
1γ
ffiii
f
i
f VPVPP
P
V
V
PP
VV
VPPVVP-TnR
PHY 113 A Fall 2012 -- Lecture 31 1911/16/2012
VVP
P
VVP
P
ii
ii
:Isotherm
:Adiabat
PHY 113 A Fall 2012 -- Lecture 31 2011/16/2012
iclicker question:
Suppose that an ideal gas expands adiabatically. Does the temperature
(A) Increase (B) Decrease (C) Remain the same
1-γ
1-γ1-γ
γγ
f
iif
ffii
i
iiiii
ffii
V
VTT
VTVT
V
TnRPnRTVP
VPVP
PHY 113 A Fall 2012 -- Lecture 31 2111/16/2012
Review of results from ideal gas analysis in terms of the specific heat ratio g º CP/CV:
For an isothermal process, DEint = 0 Q=-W
For an adiabatic process, Q = 0
1γ ;
1γ int -
RCTnCTR
-n
E VV
1γγ-R
CP
i
fii
i
fV
V V
VVP
V
VnRTPdVW
f
i
lnln
1-γ1-γ
γγ
ffii
ffii
VTVT
VPVP
PHY 113 A Fall 2012 -- Lecture 31 2211/16/2012
Note:
It can be shown that the work done by an ideal gas which has an initial pressure Pi and initial volume Vi when it expands adiabatically to a volume Vf is given by:
1γ
11γ f
i
V
V
ii
V
VVPPdVW
f
i
PHY 113 A Fall 2012 -- Lecture 31 2311/16/2012
P (
1.01
3 x
105 )
Pa
Vi Vf
Pi
Pf
A
B C
D
Examples process by an ideal gas:
A®B B®C C®D D®A
Q
W 0 -Pf(Vf-Vi) 0 Pi(Vf-Vi)
DEint
1-γ
)( ifi PPV 1-γ
)(γ iff VVP 1-γ
)( iff PPV 1-γ
)(γ ifi VVP-
1-γ
)( ifi PPV 1-γ
)( iff PPV 1-γ
)( iff VVP 1-γ
)( ifi VV-P
Efficiency as an engine:
e = Wnet/ Qinput