Ch20 Young Freedman1

8/12/2019 Ch20 Young Freedman1

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Adiabatic Processes for an Ideal Gas

dU dW PdV

By definition, we have dQ=0

for any adiabatic process,

Then, from 1st Law, we have,

adiabatic expansion Tdrops

Isothermal1

PV

Adiabatic process1

PV


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Now, we use the trick that for an ideal gas, dUis the same for allprocesses with the same dT = Tf Ti. As stated previously, we can

calculate dU using,

vdU nC dT

Then 1st law gives,

v

nRT

nC dT PdV dV V

(In the last step, we used

Ideal Gas Law: PV=nRT)


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Rearrange terms, we have,

0v

dT R dV

T C V

Using the relations for the molar specific heats,

1 1p v p

v v v

C C CR

C C C

( 1) 0dT dV

T V

Then, we have,


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Integrating this equation, we have,

1

( 1) constant

ln ( 1) ln constant

ln constant

dT dV

T V

T V

TV

1 constantTV

Using the Ideal Gas Law again, we can replace Twith ,PV

nR

1 constantPV

VnR

constantPV

(alternate form)

Note: Thas to be in K.


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Work in an Adiabatic Process (Ideal Gas)

( 0)dU dW dQ We know that,

Using the same trick on dU, we can calculate the work done

in an adiabatic process if we know the changes in state

variables.

vdW dU nC dT

2 1( )vW nC T T

2 2 1 1v

CW PV PV

R or


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insulation

Adiabatic Processes: 2 examples

Quasi-static adiabatic expansion:

Expanding gas push piston up work isdone by gas W > 0 U< 0 (energyflows out of gas)

For an Ideal Gas, Uis a function of Tonly, So, U< 0 also implies T< 0

(temperature drops!)

Adiabatic free expansion (non-quasi-static): Gas expands into vacuum no work done W=0

Adiabatic Q = 0

1st law gives U= 0

Uremains unchanged and Tis a constant!

Dfire piston


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Examples

Fire Piston (demo) Example 19.68 (Comparison of processes)

Fire Piston History

http://en.wikipedia.org/wiki/Fire_piston

Fire piston calculationshttp://complex.gmu.edu/www-phys/phys262/soln/fire_piston.pdf

Example 19.68 calculationshttp://complex.gmu.edu/www-phys/phys262/soln/ex19.66.pdf

a) Isothermal

b) Adiabatic

c) Isobaric

2VGiven initial state 1 1,P V final

3 diff ways: isotherms


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Chapter 20: The 2nd

Law ofThermodynamics

Preferential Direction inThermodynamic Processes

Heat Engine and Efficiency

The 2ndLaw of

Thermodynamics The Carnot Cycle (the most

efficient heat engine)

Entropy Entropy and Disorder


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Preferred Direction of NaturalProcesses

Processes notobserved in nature:Example 1:

Ball absorbing heat energyfrom surrounding

Then, converts it into mechanicalenergy and starts to bounce and roll

Note: energy is conserved (1st Law is NOT violated): heat mechanical eng.

BUT, we dont observe this process in nature while the reverse does!


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Disorder and ThermodynamicProcesses

The2nd Law of Thermodynamics

is the physicalprinciple which will delineate the preferred directionof natural processes.

We will also see that

The direction ofpreferred natural

processes

The degree ofrandomness (disorder) of

the resulting state

All natural processes in isolation will tend toward

the state of disorder !


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The 2ndLaw of Thermodynamics

Historically, there are more than one but equivalentway to state the 2ndLaw:

To address the condition in example #2, here is theClausius Statementon the 2ndLaw:

It is impossible for any process to have as its sole

result the transfer of heat from a cooler to a hotterbody.


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The 2ndLaw of Thermodynamics

There is also the Kelvin-Plancks Statement:

It is impossible for any system to undergo a cyclicprocess in which it absorbs heat from a reservoir at agiven temperature and converts the heat completelyinto mechanical work.

This implies that all heat engines have limited efficiency !(efficiency of real mechanical engines ~ 15 to 40%)

To understand this form of the 2ndLaw, we need to look at a toy model:

heat engine


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Heat Engines Definition: A device that converts a

given amount of heatintomechanical energy.

All heat engines carry some working

substance thru a cyclic process:

Engine releases residual heat to cold

reservoir at TC

Mechanical work is done by engine

Engine absorbs heat from hotreservoir at TH

Dstering


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Work Done by an Heat Engine

The engine works in a cyclicprocess,0U

1st Law gives,0netU Q W

netQ W

where, net H C H C Q Q Q Q Q

explicit signs for heats


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Efficiency for a Heat Engine

Thermal Efficiency e is defined as the ratio of themechanical energy output to the heat energy input,

H

We

Q

what you get out

what you put in

Substituting W=QH+QC, we have

e H C

H H

Q QW

Q Q

1 1C C

H H

Q Q

Q Q

using explicit signs here


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A perfect (100% efficient) heat engine

|QH|

1perfect

e

A perfect heat engine means 100%efficiency (e=1). This means that

1 1 means 0C CH

Qe Q

Q

All heat absorbed from reservoir THisconverted into mechanical work W.No residual heat is released back.

The Kelvin-Plancks statement of the 2ndLaw does notallow this !

1realistice


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2ndLaw, Disorder, & Available Energy

Two Forms of Energy in any Thermal Process:

Internal Energy Macroscopic Mechanical Energy

In the Kinetic-Molecular Model,

this consists of the KE and PEassociated with all the randomlymoving microscopic molecules.

The pistons motion in an

automobile engine results from thecoordinatedmacroscopic motionof the molecules.

(One typically cannot control theindividual random motions of allthese molecules.)

(Energy associated with thiscoordinated [ordered] motion can

be used for useful work.)


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In a natural process (a block sliding to a stop),

e.g. vf

slightly warmerdue to friction

stopped

The coordinated motion of the block is converted into the KE & PE of theslightly more agitated random motions of the molecules in the block.

Macroscopic Mechanical Energy (KE of the block) is converted intoInternal Energy through heat exchanges as a result of friction.


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Now consider the reverse direction it is unlikely that one can coordinateALL the randomly moving molecules in a concerted fashion. In otherwords, one typically cannotconvert the internal energy of a systemcompletelyback to macroscopic mechanical energy.

However, this does not mean that internal energy is not accessible. AnHeatEngine is exactly the machine that can perform this conversion but onlypartially.

The 2nd Law of Thermodynamics is basicallya statement limiting the availability of internalenergy for useful mechanical work.


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The Carnot Cycle (The Most EfficientHeat Engine)

A reversible cycle described by Sadi Carnot in1824.

The Carnot Theorem gives the theoretical limit

to the thermal efficiency of any heat engine. The Carnot cycle consists of:

Two reversible isothermal processes

Two reversible adiabatic processes

An Ideal Gas as its working substance

Ch20 Young Freedman1

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