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.