PHY 113 C Fall 2013 Lecture 22 1 PHY 113 C General Physics I 11 AM 1215 P M MWF Olin 101 Plan for Lecture 22 Chapter 21 Ideal gas equations Molecular view of ideal gas Internal energy of ideal gas ID: 224146
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Slide1
11/14/2013
PHY 113 C Fall 2013 -- Lecture 22
1
PHY 113 C General Physics I
11 AM – 12:15
P
M MWF Olin 101
Plan for Lecture 22:
Chapter 21: Ideal gas equations
Molecular view of ideal gas
Internal energy of ideal gas
Distribution of molecular speeds in ideal
gas
Adiabatic processesSlide2
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PHY 113 C Fall 2013 -- Lecture 22
2Slide3
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PHY 113 C Fall 2013 -- Lecture 22
3
From
Webassign
(Assignment #19)
A combination of 0.250 kg of water at 20.0°C, 0.400 kg of aluminum at 26.0°C, and 0.100 kg of copper at 100°C is mixed in an insulated container and allowed to come to thermal equilibrium. Ignore any energy transfer to or from the container and determine the final temperature of the mixture.
387 J/(kg*
o
C
)
4186 J/(kg*
o
C
)
900 J/(kg*
o
C
)
(From Table 20.1)Slide4
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PHY 113 C Fall 2013 -- Lecture 22
4
From
Webassign
(Assignment #19)
A thermodynamic system undergoes a process in which its internal energy decreases by 465 J. Over the same time interval, 236 J of work is done on the system. Find the energy transferred from it by heat.
Note: Sign convention for Q
:
Q>0 system gains heat from environment
iclicker
question:
Assuming the system does not change phase, what can you say about T
F
versus T
I
for the system?
T
F
>T
ITF<TISlide5
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PHY 113 C Fall 2013 -- Lecture 22
5
From
Webassign
(Assignment #19)
A 2.20-mol sample of helium gas initially at 300 K, and 0.400
atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas.
(
a) Find the final volume of the gas
.
(b) Find the work done on the gas.
(c) Find the energy transferred by heat
.Slide6
11/14/2013
PHY 113 C Fall 2013 -- Lecture 22
6
From
Webassign
(Assignment #19)
A 2.20-mol sample of helium gas initially at 300 K, and 0.400
atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas.
(
a) Find the final volume of the gas
.Slide7
11/14/2013
PHY 113 C Fall 2013 -- Lecture 22
7
From
Webassign
(Assignment #19)
A 2.20-mol sample of helium gas initially at 300 K, and 0.400
atm is compressed isothermally to 1.80 atm. Note that the helium behaves as an ideal gas.
(
b) Find the work done on the gas.
(
c) Find the energy transferred by heat
.Slide8
11/14/2013
PHY 113 C Fall 2013 -- Lecture 22
8
From
Webassign
(Assignment #19)
One
mole of an ideal gas does 2 900 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 28.0 L. (a) Determine the initial volume of the gas.
(b) Determine the temperature of the gas.Slide9
11/14/2013
PHY 113 C Fall 2013 -- Lecture 22
9
From
Webassign
(Assignment #19)
One
mole of an ideal gas does 2 900 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 28.0 L. Determine the initial volume of the gas.
Determine
the temperature of the gas.Slide10
11/14/2013
PHY 113 C Fall 2013 -- Lecture 22
10
From
Webassign
(Assignment #19)
In
the figure, the change in internal energy of a gas
that
is taken from
A
to
C
along the blue path is +795 J. The work done on the gas along the red path ABC is -530 J. (a) How much energy must be added to the system by heat as it goes from
A through B to C?(b) If the pressure at point A is five times that of point C
, what is the work done on the system in going from C to
D?(c) What is the energy exchanged with the surroundings by heat as the gas goes from C to A along the green path?(d) If the change in internal energy in going from point D to point
A is +495 J, how much energy must be added to the system by heat as it goes from point C
to point D?Slide11
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PHY 113 C Fall 2013 -- Lecture 22
11
Review:
Consider the process described by A
BCA
iclicker
exercise:
What is the net work done on the system in this cycle?
-12000 J
12000 J
0Slide12
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PHY 113 C Fall 2013 -- Lecture 22
12
Equation of “state” for ideal gas
(from experiment)
pressure in
Pascals
volume in m
3
# of moles
temperature in K
8.314 J/(
mol
K)Slide13
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PHY 113 C Fall 2013 -- Lecture 22
13
Ideal gas -- continued
Note that at this point, the above equation for
E
int
is completely unjustified…Slide14
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PHY 113 C Fall 2013 -- Lecture 22
14
From The New Yorker Magazine, November 2003Slide15
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PHY 113 C Fall 2013 -- Lecture 22
15
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.Slide16
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PHY 113 C Fall 2013 -- Lecture 22
16
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:
d
x
v
ix
-v
ix
number of atoms
volumeSlide17
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PHY 113 C Fall 2013 -- Lecture 22
17Slide18
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PHY 113 C Fall 2013 -- Lecture 22
18
iclicker
question:
What should we call ?
Average kinetic energy of atom.
We cannot use our macroscopic equations at the atomic scale -- so this quantity will go unnamed.
We made too many approximations, so it is not worth naming/discussion.
Very boring.Slide19
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PHY 113 C Fall 2013 -- Lecture 22
19
for mono atomic ideal gasSlide20
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PHY 113 C Fall 2013 -- Lecture 22
20
Average atomic velocities:
(note <v
i
>=0)
Relationship between average atomic velocities with TSlide21
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PHY 113 C Fall 2013 -- Lecture 22
21
Periodic
table:
http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpgSlide22
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PHY 113 C Fall 2013 -- Lecture 22
22
Periodic
table:
http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg
Molecular massSlide23
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PHY 113 C Fall 2013 -- Lecture 22
23
Periodic
table:
http://www.nist.gov/pml/data/images/PT-2013-Large_2.jpg
Molecular massSlide24
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PHY 113 C Fall 2013 -- Lecture 22
24
For monoatomic ideal gas:
General form for ideal gas (including mono-, di-, poly-atomic ideal gases):Slide25
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PHY 113 C Fall 2013 -- Lecture 22
25
Macroscopic
Microscopic
8.314 J/mole
o
K
1.38 x 10
-23
J/molecule
o
KSlide26
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PHY 113 C Fall 2013 -- Lecture 22
26
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
N
2
7/5
1.41
H
2
O
4/3
1.30
Big leap!Slide27
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PHY 113 C Fall 2013 -- Lecture 22
27
Comment on “big leap” – case of diatomic molecule
v
CM
w
Note: We are assuming that molecular vibrations are not taking much energySlide28
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PHY 113 C Fall 2013 -- Lecture 22
28
Comment on “big leap” – continued
Internal energy of an ideal gas:
derived for monoatomic ideal gas
more general relation for polyatomic ideal gas
Big leap!
can be measured for each gaseous system
Note:
g
= C
P
/C
VSlide29
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PHY 113 C Fall 2013 -- Lecture 22
29
Determination of Q for various processes in an ideal gas:
Example: Isovolumetric process – (V=constant
W=0)
In terms of “heat capacity”: Slide30
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PHY 113 C Fall 2013 -- Lecture 22
30
Example: Isobaric process (P=constant):
In terms of “heat capacity”:
Note:
g
= C
P
/C
VSlide31
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PHY 113 C Fall 2013 -- Lecture 22
31
SummarySlide32
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PHY 113 C Fall 2013 -- Lecture 22
32
iclicker
question:
The previous discussion
Made me appreciate the
g
factor in thermo analyses
Made me want to scream
Put me to sleep
No problem – as long as this is not on the testSlide33
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PHY 113 C Fall 2013 -- Lecture 22
33
More examples:
Isothermal process (T=0)
D
T=0
D
E
int
= 0 Q
=-WSlide34
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34
Even more examples:
Adiabatic process (Q=0)Slide35
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PHY 113 C Fall 2013 -- Lecture 22
35Slide36
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PHY 113 C Fall 2013 -- Lecture 22
36
iclicker
question:
Suppose that an ideal gas expands adiabatically. Does the temperature
(A) Increase (B) Decrease (C) Remain the sameSlide37
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PHY 113 C Fall 2013 -- Lecture 22
37
Review of results from ideal gas analysis in terms of the specific heat ratio
g
º
C
P
/C
V
:
For an isothermal process,
D
Eint = 0 Q=-W
For an adiabatic process, Q = 0Slide38
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PHY 113 C Fall 2013 -- Lecture 22
38
Note:
It can be shown
that the work done by an ideal gas which has an initial pressure P
i and initial volume Vi when it expands adiabatically to a volume V
f
is given by
:Slide39
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PHY 113 C Fall 2013 -- Lecture 22
39
P (1.013 x 10
5
) Pa
V
i
V
f
P
i
P
f
A
B
C
D
Examples process by an ideal gas:
A
®
B
B
®
C
C
®
D
D
®
A
Q
W
0
-P
f
(
V
f
-V
i
)
0
P
i
(
V
f
-V
i
)
D
E
int
Efficiency as an engine:
e =
|
W
net
/
|/
Q
inputSlide40
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PHY 113 C Fall 2013 -- Lecture 22
40
From
Webassign
(#19)
An ideal gas initially at
P
i
,
V
i
, and
T
i is taken through a cycle as shown below. (Let the factor n = 2.6.)
(a) Find the net work done on the gas per cycle for 2.60
mol
of gas initially at 0°C.
(b) What is the net energy added by heat to the system per cycle?