Statistical Mechanics
Physics 202
Professor Lee Carkner
Lecture 19
PAL # 18 Engines
 Engine #1 W = 10, QH = 45
e =W/QH = 0.22

 Engine #2 QL = 25, QH = 30
e = 1 – QL/QH = 0.17

 Engine #3 TH = 450 K, TL = 350 K
eC = 1 – TL/TH = 0.22

 Engine #4 W = 20, QH = 30, TH = 500, TL = 400
e = 0.66 > eC = 0.2

 Engine #5 W = 20, QH = 15
e = 1.33 > 1

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
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
Refrigerator Cycle
Compressor (work =W)
QL
Heat
removed
from inside
cold region
by evaporation
Gas
Low
Pressure
High
Pressure
Liquid
Expansion Valve
QH
Heat
added to
room by
condensation
Refrigerator Diagram
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
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
Perfect Refrigerator
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
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
Statistical Mechanics
Statistical mechanics uses microscopic
properties to explain macroscopic
properties

Consider a box with a right and left
half of equal area

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
Configurations and Microstates
Configuration I
1 microstate
Probability = (1/16)
Configuration II
4 microstates
Probability = (4/16)
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
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
Microstate Probabilities
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
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?


Arrows of Time
Three arrows of time:
Thermodynamic

Psychological

Cosmological

Entropy and Memory
When we remember things, order is
increased

A brain or a computer cannot store
information without the output of heat

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

