Statistical Mechanics

Physics 202Professor Lee

CarknerLecture 19

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

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

Liquid

Gas

Compressor (work =W)

Expansion Valve

Heatremovedfrom inside cold regionby evaporation

Heat added to room bycondensation

HighPressure

Low Pressure

QL QH

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 I1 microstate

Probability = (1/16)

Configuration II4 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