Physics 142: Lecture 8
Today’s Agenda
z
REVIEW: The Ideal Gas Law
z
Temperature and Kinetic Energy
Î Review: Ideal Gas Law (10.3)
Î Review: Thermal Expansion (10.4)
Î Kinetic Theory of Gases (10.5)
ÎN = number of molecules
ÎT = absolute temperature (K)
Suggested problems for Ch-10: 35, 52, 56,59,79, 80
Midterm: Friday, Feb 16, 2007
12:25 – 13:25
PV = NkBT
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Alternate way to write this
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PV = nRT
În = number moles
THEATER AUDITORIUM
Assignment 1 is due this Wednesday, Jan 24
ÎR = ideal gas constant = 8.31 J/(mol-K)
P1 V1 P2 V2
=
T1
T2
if n or N remains the same
Lecture 8, Pg 1
Lecture 8, Pg 2
Expanding Rods
Thermal Expansion
z
z
When temperature rises
Îthings tend to expand
amount of expansion depends on…
Îchange in temperature
Îoriginal length
Îcoefficient of thermal expansion
» L0 + ∆L = L0 + α L0 ∆T
» ∆L = α L0 ∆T (linear expansion)
» ∆V = β V0 ∆T (volume expansion)
Temp: T
L0
Temp: T+∆T
∆L
Lecture 8, Pg 3
Lecture 8, Pg 4
Example
Amazing Water
As you heat a block of aluminum from 0 C to 100 C its density
1. Increases
2. Decreases
3. Stays the same
z
T = 100 C
T=0C
Water is very unusual in that it has a maximum density
at 4 degrees C. That is why ice floats, and we exist!
1000.00
999.95
999.90
999.85
999.80
999.75
999.70
Density
999.65
999.60
999.55
0
2
4
6
8
10
Lecture 8, Pg 5
Lecture 8, Pg 6
Example
Example
Not being a great athlete, and having lots of money to spend, Gill
Bates decides to keep the lake in his back yard at the exact
temperature which will maximize the buoyant force on him when
he swims. Which of the following would be the best choice?
1000.00
(1) 0 C
999.95
(2) 4 C
999.90
999.85
(3) 32 C
999.80
Density
(4) 100 C
999.75
999.70
(5) 212 C
An aluminum plate has a circular hole cut in it. An aluminum ball
(solid sphere) has exactly the same diameter as the hole when
both are at room temperature, and hence can just barely be
pushed through it. If both the plate and the ball are now heated
up to a few hundred degrees Celsius, how will the ball and the
hole fit ?
1. The ball wont fit through the hole any more
2. The ball will fit more easily through the hole
3. Same as at room temperature
999.65
999.60
999.55
0
2
4
6
8
Lecture 8, Pg 7
10
Lecture 8, Pg 8
Why does the hole get bigger when the plate expands ???
Imagine a plate made from 9 smaller pieces.
Each piece expands.
If you remove one piece, it will leave a “square” hole
Expansion Act
An aluminum plate has a circular hole cut in it. An iron ball
(solid sphere) has exactly the same diameter as the hole
when both are at room temperature, and hence can just
barely be pushed through it. If both the plate is cooled in
liquid nitrogen,
1. The ball wont fit the hole any longer
2. The ball will fit more easily
3. Same as at room temperature
Object at temp T
Lecture 8, Pg 9
Kinetic Theory of Gases:
The relationship between kinetic energy and
temperature for monatomic ideal gas
2
vrms
1 N
= ∑ vi2
n i =1
1
2
pV = Nmvrms
3
pV = NkbT
“rms” stands for “root-mean-square”
1
2
Nmvrms
= NkbT
3
Lecture 8, Pg 10
Example
Suppose you want the rms (root-mean-square) speed of molecules in a
sample of gas to double. By what factor should you increase the
temperature of the gas?
1. 2
2. 2
3. 4
1 2
3
mvrms = kbT = K , where K is average kinetic energy per molecule
2
2
m is the mass of one molecule
Lecture 8, Pg 11
Lecture 8, Pg 12
Problem
Internal Energy of
Ideal Monatomic Gas
(a) What is the average kinetic energy per molecule of an ideal gas
at a temperature of 27°C ?
(b) What is the average (rms) speed of the molecules if the gas is
helium? (A helium molecule consists of a single atom of mass
6.65x10-27kg)
1
3
2
2
mvrms
=
2
kbT = K
The total internal energy of an ideal gas is equal to the
total kinetic energy of all the molecules
3
3
3
⎛1 2 ⎞
U = N ⎜ mvrms
⎟ = N kbT = ( nN A ) kbT = n ( N A kb ) T
2
2
2
⎝2
⎠
3
= nRT
2
U=
3
nRT
2
total internal energy
Lecture 8, Pg 13
Lecture 8, Pg 14
Summary
Kinetic Theory
Ideal Diatomic Gas
z
Average Kinetic Energy (per molecule) of any Ideal Gas
Î
z
2
K = 1 2 mvrms
= 3 2 kbT
Total Internal Energy of an Ideal Diatomic Gas
Î
z
U=
5
nRT
2
z
Temperature is a measure of the average Kinetic
Energy of molecules
Average Kinetic Energy (per molecule) of any Ideal Gas
Î
z
Total Internal Energy of an Ideal Monatomic Gas
Î
z
U=
3
nRT
2
Total Internal Energy of an Ideal Diatomic Gas
Î
Lecture 8, Pg 15
2
K = 1 2 mvrms
= 3 2 kbT
U=
5
nRT
2
Lecture 8, Pg 16