Sample problemp pA (non-rigid) diving bell has an air space of 3.0 m3 when on the deck ofa small boat. What is the volume of the air space when the bell has beenl d t d th f 50 ? Th d it f t i 1 025 3lowered to a depth of 50 m? The density of sea water is 1.025 g cm-3.

What do we need to assume? Ideal gas behavior for airWater temperature is unchanged at 50 m

Constant T so use Boyle: PV=cons’t Wh i “ i id” i i f i ?Constant T, so use Boyle: PV=cons t Why is “rigid” important information?

  1.0 atm f ip p gh gh

Real Gases

Compressibility Z PV PVZ Compressibility, Z PV PVZnRT RT

Z = 1 Ideal Gas behaviorZ < 1 PV less than expected Attractive forcesZ > 1 PV greater than expected Repulsive forces

Z 1

Ar

Ideal Gas Ideal Gas

P

Real Gases – data!Compressibility PVPVZ

nRT RTm

Z = 1 at all P, T Ideal Gas Behavior

Now look at real gases

Ideal gas

gat some temperature T

Look at a broader 0 – 800 atm regionWe need new eq of state for each gasThese data are many gases at one T.So next look at one gas at many T

Boyle Temperature, TBy p , B

TB is the temperature corresponding Z Z P T TB is the temperature correspondingto the greatest extentof near-ideal behavior.

,Z Z P T

,Z T Pf

We can determine TB analytically.

lim 0

,

0 t th B l t t

p

f TP

df T T0 at , the Boyle temperature Bf T T

dT

van der Waals equation of statevan der Waals equation of statePhysically-motivated corrections to Ideal Gas EoS.y yFor a real gas, both attractive and repulsive intermolecularforces are present. Empirical terms were developed to help accountfor both.

Excluded volume concept (nb)nRTP

V b

1. Repulsive forces: make pressure higher than ideal gas

Excluded volume concept (nb) V nb

Volume of one molecule of radius r is Vmol = (4/3) r3

Closest approach of two molecules with radius r is 2rClosest approach of two molecules with radius r is 2r.

ConcepTest #5ConcepTest #5

Excluded volume nRTP Excluded volume PV nb

The volume of one molecule of radius r is Vmol = 4/3 r3

Th l h f l l i h di i 2The closest approach of two molecules with radius r is 2r.What is the excluded volume for the two molecules?

A. 2Vmol

B 4VB. 4Vmol

C. 8Vmol

D 16 VD. 16 Vmol

van der Waals equation of statePhysically-motivated corrections to Ideal Gas EoS.For a real gas, both attractive and repulsive intermolecularforces are present Empirical terms were developed to help accountforces are present. Empirical terms were developed to help accountfor both.

1. Repulsive forces: make pressure higher than ideal gas(or equivalently make the volume smaller)

Do the latter: Excluded volume nRTP

(or, equivalently, make the volume smaller)

Do the latter Excluded volumeV nb

Volume of one molecule of radius r is Vmol = 4/3 r3

Closest approach of two molecules with radius r is 2rClosest approach of two molecules with radius r is 2r.The excluded volume Vexc is 23 Vmol = 8Vmol for two molecules.

So we might estimate that b 4V NSo we might estimate that Amolb 4V N

This assumes binary collisions only. Always true? NO!

van der Waals equation of statevan der Waals equation of statePhysically-motivated corrections to Ideal Gas EoS.y yFor a real gas, both attractive and repulsive intermolecularforces are present. Empirical terms were developed to help accountfor both.

Pressure depends wall collisions, both on frequencyand their force

2. Attractive forces: make pressure lower than ideal gas

and their force. Not easy to show, but we expect a pressure correction of theform –a(n/V)2, giving the van der Waals Equation of State

2

2 2

nRT an RT aPV b V V b V

2 2V nb V V b V

3D van der Waals eqn of state3D van der Waals eqn of state

T= T/Tc

Real Gases – CO2

CO2 Look at 20 C isotherm

Look at 50 C isotherm.Behavior is near ideal gas

2

ABC CompressionAt C, liquid condensation begins

Look at 20 C isotherm.

D – liquid- vapor mixture atPvap(20 C)

E l dE – last vapor condensesF – Steep rise in pressureA liquid or solid is much lessA liquid or solid is much lesscompressible than a gas

For T >T there is a single phaseFor T >Tc, there is a single phase,with no liquid formed.

van der Waals Isotherms near Tcvan der Waals Isotherms near Tc

v d W “loops” arev d W loops arenot physical. Why?

Patch up with Maxwellconstruction

van der Waals Isotherms, T/Tc

van der Waals Isotherms near Tcc

Look at one of the van der Waals isotherms at a temperat re of 0 9 T

1.5G

isotherms at a temperature of 0.9 Tc

essu

re, P

r

1.0Tr = 0.9

B

•A → D compress the gas at constant T, •F → G compress the liquid phase

Red

uced

Pre

0.5

D

B

F

C A

g, l

(steep and not very compressible)•D → F vapor condensing (gas and liquid coexist) These are stable states

R d d V l V

0.3 0.5 0.7 1 2 3 5 7 100.0

A These are stable states

F → C supercooled liquidD → B superheated gasTh t t bl t t Reduced Volume, VrThese are metastable states

C → B a non-physical artifact of vdW(patched up with Maxwell construction)

Metastable example:Use a very clean glass. Add water and heatfor a while with a microwave oven (superheat)(p p ) o a w e w t a c owave ove (supe eat)Add a drop of sand or perhaps touch with a spoon.