ch18 young freedman1

8/12/2019 Ch18 Young Freedman1

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Chapter 18: Thermal Properties ofMatter

Equations of State

Ideal Gas Equation

PV Diagrams

Kinetic-Molecular Model of

an Ideal Gas

Heat Capacities

Distribution of MolecularSpeeds

Phases of Matter

Topics for Discussion


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Equations of State

State Variablesphysical variables describing the

macroscopic state of the system:

P, V, T, n (or m)

Equation of State

a mathematical relationship linking thesevariables


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The Ideal Gas Equation

Properties of a gas is studied byvarying the macroscopic variables:

P, V, T, n and observing the result.

Observations:1. e.g. an air pump

2. e.g. hot air balloon

3. e.g. hot closed spray can

4. e.g. birthday ballon

1P V

V T

P T

V n


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Ideal Gas Law (summary) By putting all these observations together, we have

R Universal Gas Constant (R = 8.314 J/mol K)

(This is an important example of an Equation of State for a

gas at thermal equilibrium.)

(Thas to be in Kelvin)

An Ideal Gas (diluted): No molecular interactions besides elastic collisions

Molecular volume


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The Ideal Gas Law

Important Notes:

The relationship V vs. T (at

cont P) &P vs. T(at cont

V) are linearfor all diluted

gases. diluted gas ~ Ideal

They both extrapolate to a

singlezeropoint (absolute

zero).

Thas to be in K!

-300 -200 -100 0 100 200

-300 -200 -100 0 100 200

T = -273.15oC

Pressure

Volum

e

Temperature (oC)

P TnR

V

V TnRP


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The Ideal Gas Law (alternative form)

Instead of the number of moles (n), one can specify theamount of gas by the actual number of molecules (N).

N = n NA

whereNA is the # of molecules in a mole of materials(Avogadros number).

where kis the Boltzmann constant,

23( 6.02214 10 / )A

N molecules mole

231.381 10 /A

Rk J molecule K

N

A

NPV nRT PV RT NkT

N


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Example 18.1 (V at STP)

What is the volume of a gas (one mole) at StandardTemperature and Pressure (STP)?

STP: T = 0oC = 273.15K

P = 1 atm =

3(1 )(8.314 / )(273.15 ) 0.0224 22.4

51.013 10

nRT mole J mol K K V m L

P Pa

5

1.013 10 Pa


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Typical Usage for the Ideal Gas Law

For a fixed amount of gas (nR=const)

So, if we have a gas at two different states 1(before) and 2(after), their state variables are related simply by:

We can use this relation to solve for any unknown variableswith the others being given.

PVnR const

T

1 1 2 2

1 2

PV PV

T T

(side note: absolute pressure = gauge + atmospheric)


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Example 18.2

In an automobile engine, a mixture of air/gasoline is

being compressed before ignition.

Typical compression ration 1 to 9 Initial P = 1 atm and T = 27 oC

Find the temperature of the compressed gas if we aregiven the pressure after compression to be 21.7atm.

note

http://complex.gmu.edu/www-phys/phys262/soln/ex18.2.pdf


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The van der Waals EquationA more realistic Equation of States for gases which includes

corrections for the facts that molecules are not point particles,that they have volume, and for the attraction/repulsion thatnaturally exists between the adjacent atoms/molecules.

2

2

anP V n nRT

Vb


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Notes on Van Der Waals The volume parameter b:

It makes sense that real gas as finite size hard spheres will reduce thetotal volume of the gas by a term which is proportional to the numberof mole n.

The intra-molecular force parameter a: Intra-molecular force tends to reduce the pressure of the gas onto the

wall by pulling the molecules toward the interior of the container

This intra-molecular force acts in pairs (to the lowest order ofapproximation)

For a unit volume in front of the wall, this intra-molecular force willdepends on the number ofpairs of molecules within this unit volume

The count of molecular pairs within this unit volume ~ (n/V)2

(Note: for N molecules, # pairs =N(N-1)/2; forNlarge, ~N2.)

( )eff

nRT nRT P

V V nb

2 2

( )( )

nRT nP P a V nb nRT

V n V V

n

ba


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The Ideal Gas Law (graphical view)

P,V,Trelationship in the Ideal Gas Law can be visualize

graphically as a surface in 3D.

nRTP

V


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PV Diagrams

2D projections of the

previous 3D surface.

Evolution of a gas at

constant Twill move

along these curvescalled isotherms.

Gives P vs.Vat a various T:

1

( )P nRT V


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Phases of Matter (reading phase diagrams)


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Kinetic-Molecular Model of an Ideal Gas

An example of a successful theoretical linkage between the

micro and macro descriptions for an ideal gas.

Explicit expressions of P &Tin terms of microscopicquantities!

Macroscopic description

of gases

P, V, TIdeal Gas Law

Microscopic description

of gas molecules

v, p, F, KENewtons EqsKinetic Theory


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Kinetic Theory (assumptions) A very large #Nof identical molecules each with mass m in a container

with volume V Molecules behaves aspoint particles:

Molecule sizes


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Kinetic Theory (model)

L

A

Idea Gas in a box with V=AL

vi

-vx

-vyvf

+vx

-vy

Left Wall

before collision

after collision


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Pressure Exerted by an Ideal Gas

Pressure on left wall due to molecular

collisions

1. Momentum change in x-dir by a

molecule moving to the left at vi:

2. Duration, t, that this molecule

takes (on average) to collide withthe left wall again (diluted gas),

( ) ( ) 2f i x x x

m P P mv mv mv v

2

x

Lt

v

L

vi

-vx

-vy vf

+vx

-vy

Left Wall

m

m


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Pressure Exerted by an Ideal Gas

3.

Force exerted by this molecule on the left wall:

4. WithNmolecules, total force on wall in t:

22( )

2

x x

x

mv mvmF

t L v L

v

2 2 2

1 2

1

2

1

21

N

tot i x x Nx

i

N

ix x ai

v

mF F v v v

L

m mNv

LN v

L N

22:

x x avavNote v v

Invariantdistribution implies will be the same when

experiment is repeated.

av


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Pressure of an Ideal Gas

5. Random direction (isotropic) assumption:

(vx2)av= (vy

2)av= (vz2)av (x,y,z are the same)

Since v2

= vx2

+ vy2

+ vz2

, we have (v2

)av= 3(vx2

)av

This gives,

6. Finally, the pressure on the wall is:

2( )

3

avtot

vmNF

L

2 2( ) ( )1

3 3

tot av avF v Nm vmN

PA AL V


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Pressure of an Ideal Gas

Rewriting, we have

This tells us that P inside a container with

fixed V:

is proportional to the # of moleculesN is proportional to the avg. KE of molecules

(These are microscopic properties of the gas.)

22 1 2

3 2 3 avavPV N m v N KE

(avg KE per molecule)

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