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Statistical Mechanics
Physics 202Professor Lee
CarknerLecture 19
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PAL # 18 Engines Engine #1 W = 10, QH = 45
=W/QH = 0.22
Engine #2 QL = 25, QH = 30 = 1 – QL/QH = 0.17
Engine #3 TH = 450 K, TL = 350 K C = 1 – TL/TH = 0.22
Engine #4 W = 20, QH = 30, TH = 500, TL = 400 = 0.66 > C = 0.2
Engine #5 W = 20, QH = 15 = 1.33 > 1
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Engines and Refrigerators
Heat from the hot reservoir is transformed into work (+ heat to cold reservoir)
By an application of work, heat is moved from the cold to the hot reservoir
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A Refrigerator A refrigerator depends on 2 physical principles:
Boiling liquids absorb heat, condensing liquids give off heat
(heat of vaporization) Heat can be moved from a cold region to a hot region
by adjusting the pressure so that the circulating fluid boils in the cold region and condenses in the hot
n.b., the refrigerator is not the cold region (where we keep our groceries), it is the machine on the back that moves the heat
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Refrigerator Cycle
Liquid
Gas
Compressor (work =W)
Expansion Valve
Heatremovedfrom inside cold regionby evaporation
Heat added to room bycondensation
HighPressure
Low Pressure
QL QH
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Refrigerator Diagram
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Refrigerator as a Thermodynamic System
K = QL/W K is called the coefficient of performance
QH = QL + W
W = QH - QL
This is the work needed to move QL out of the cold area
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Refrigerators and Entropy We can rewrite K as:
From the 2nd law (for a reversible, isothermal
process):
So K becomes:KC = TL/(TH-TL)
Refrigerators are most efficient if they are not kept very cold and if the difference in temperature between the room and the refrigerator is small
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Perfect Refrigerator
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Perfect Systems A perfect engine converts QH directly into W
with QL = 0 (no waste heat)
Perfect refrigerators are impossible (heat won’t flow from cold to hot)
But why?
Violates the second law:
If TL does not equal TH then QL cannot equal QH Perfect systems are impossible
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Entropy
Entropy always increases for irreversible systems
Entropy always increases for any real, closed system (2nd law) Why?
The 2nd law is based on statistics
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Statistical Mechanics Statistical mechanics uses
microscopic properties to explain macroscopic properties
Consider a box with a right and left half of equal area
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Molecules in a Box There are 16 ways that the molecules can
be distributed in the box
Since the molecules are indistinguishable there are only 5 configurations Example:
If all microstates are equally probable than the configuration with equal distribution is the most probable
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Configurations and Microstates
Configuration I1 microstate
Probability = (1/16)
Configuration II4 microstates
Probability = (4/16)
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Probability
There are more microstates for the configurations with roughly equal distributions
Gas diffuses throughout a room because the probability of a configuration where all of the molecules bunch up is low
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Multiplicity The multiplicity of a configuration is the
number of microstates it has and is represented by:
W = N! /(nL! nR!)
n! = n(n-1)(n-2)(n-3) … (1)
For large N (N>100) the probability of the equal distribution configurations is enormous
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Microstate Probabilities
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Entropy and Multiplicity The more random configurations are most
probable
We can express the entropy with Boltzmann’s entropy equation as:
Where k is the Boltzmann constant (1.38 X 10-23
J/K)
ln N! = N (ln N) - N
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Irreversibility Irreversible processes move from a low
probability state to a high probability one
Increase of entropy based on statistics Why doesn’t the universe seem random?
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Arrows of Time
Three arrows of time: Thermodynamic
Psychological
Cosmological
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Entropy and Memory
When we remember things, order is increased
A brain or a computer cannot store information without the output of heat
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Fate of the Universe The universe is expanding, and there does not
seem to be enough mass in the universe to stop the expansion
Entropy keeps increasing
Stars burn out
Can live off of compact objects, but eventually will convert them all to heat