states of matter equations of state ideal gasdeviations van der waalsvirial series kinetic molecular...

States of MatterStates of Matter

Equations of State

Ideal Gas Deviations Mixtures

Van der Waals Virial Series Berth., R-K

Kinetic Molecular Model

Corresponding States

Fluids

Reduced Variables

Condensed Phases

Hi Chem.412 students, Due to a last minute appointment, there is a good chance that I will not be able to make the 9:00 a.m. class on time tomorrow (Wednesday). Therefore, I am substituting the Wednesday 9 am lecture on the next topic “Nature of Matter and Mystery of the Universe” with the following You-Tube videos: (Click on the hyperlinks to see them in sequence) Wednesday afternoon and evening labs go on as scheduled. Video #1 (explanation of the Big Bang, ~5.5 minutes) S. Hawking: Big Bang Video #2 (How to find particles, ~17 min) Particle Hunters Video #3 (A Rap on the LHC, ~4.5 min) Hadrons [Please be somewhat skeptical and don’t take any offense regarding comments after these (free) videos … these are “uncontrolled” public comments that can be at times insensitive and offensive!]Please watch them before Friday’s class since I will be skipping the beginning parts of the next powerpoint (States of Matter). Wednesday afternoon and evening labs go on as scheduled. Dr. Ng.

9/11/13 – Lec sub

http://www.youtube.com/watch?v=gs-yWMuBNr4

http://www.youtube.com/watch?v=nifTTSfqubM

http://www.youtube.com/watch?v=f6aU-wFSqt0

MatterMatter

Three States of MatterThree States of Matter

LiquidLiquid GasGas SolidSolid

MicroscopicMicroscopic MacroscopicMacroscopic

TemperatureTemperature

PressurePressure

ViscosityViscosity

DensityDensity

Molecular SizeMolecular Size

Molecular ShapeMolecular Shape

Velocity/MomentumVelocity/Momentum

Intermolecular ForcesIntermolecular Forces

CyberChem: Big BangS. Hawking: Big Bang

http://www.youtube.com/watch?v=gs-yWMuBNr4

Mystery of our Universe: A Matter of FamilyMystery of our Universe: A Matter of Family

?

QuarksQuarks

Fermions - ParticlesFermions - Particles

LeptonsLeptons

Hadrons neutron proton e- - - [ ]

nuclides atoms

elements

mixturescompounds

molecules complexes homogeneous heterogeneous

Bosons – Force carriersBosons – Force carriers

Strong (gluon)Weak (+W , -W , Z)

Electromag. (photon)Gravity (graviton)

Three families

1) u d e- e

2) c s -

3) t b -

http://www.youtube.com/watch?v=f6aU-wFSqt0

Mystery of our Universe: QuarksMystery of our Universe: Quarks

Particle HuntersBig Bang Theory physics episodes

http://www.youtube.com/watch?v=nifTTSfqubM

• We can combine these into a general gas law:

The Ideal Gas EquationThe Ideal Gas Equation

), (constant 1

TnP

V

), (constant PnTV

),(constant TPnV

• Boyle’s Law:

• Charles’s Law:

• Avogadro’s Law:

PnT

V

• R = gas constant, then

• The ideal gas equation is:

• R = 0.08206 L·atm/mol·K = 8.3145 J/mol·K• J = kPa·L = kPa·dm3 = Pa·m3

• Real Gases behave ideally at low P and high T.

The Ideal Gas EquationThe Ideal Gas Equation

P

nTRV

nRTPV

Calculate the number of air molecules in 1.00 cm3 of air at 757 torr and 21.2 oC.

Mathcad

Calculate the number of air molecules in 1.00 cm3 of air at 757 torr and 21.2 oC.

MathcadF12

Low P IdealLow P Ideal

High T IdealHigh T Ideal

Gas Densities and Molar Mass• The density of a gas behaving ideally can be determined as follows:

• The density of a gas was measured at 1.50 atm and 27°C and found to be 1.95 g/L. Calculate the molecular weight of the gas? If the gas is a homonuclear diatomic, what is this gas?

• Plotting data of density versus pressure (at constant T) can give molar mass.

Density of an Ideal-GasDensity of an Ideal-Gas

TR

MP

Mathcad

Density of an Ideal-GasDensity of an Ideal-Gas

TR

MP

Derivation of :

Plotting data of density versus pressure (at constant T) can give molar mass.

Molar Mass of an Non-Ideal Gas• Generally, density changes with P at constant T, use power series:

• First-order approximation:

• Plotting data of ρ/P vs. P (at constant T) can give molar mass.

Deviation of Density from IdealDeviation of Density from Ideal

nn PbPbPb

RT

M

P ...1 2

21

RT

MP

RT

Mb

P 1

Plotting data of ρ/P vs. P (at constant T) can give molar mass.

• Dalton’s Law: in a gas mixture the total pressure is given by the sum of partial pressures of each component:

• Each gas obeys the ideal gas equation:

Ideal Gas Mixtures and Partial PressuresIdeal Gas Mixtures and Partial Pressures

321t PPPPPi

i

VRT

nP ii i

iiavg MM

Density?

i

iiavg MM Density?

• Partial Pressures and Mole Fractions

• Let ni be the number of moles of gas i exerting a partial pressure Pi , then

where χi is the mole fraction.

Ideal Gas Mixtures and Partial PressuresIdeal Gas Mixtures and Partial Pressures

tPP ii

CyberChem (diving) video:

ii

i

t

ii n

n

n

n

ii

i

t

ii n

n

n

ntPP ii

The van der Waals Equation

• General form of the van der Waals equation:

Real Gases: Deviations from Ideal BehaviorReal Gases: Deviations from Ideal Behavior

2

2

V

annbV

nRTP

nRTnbVV

anP

2

2

Corrects for molecular volume

Corrects for molecular attraction

Real Gases: Deviations from Ideal BehaviorReal Gases: Deviations from Ideal Behavior

2

2

TV

an

nbV

nRTP

Berthelot

nbV

enRTP

RTV

na

Dieterici

)(2

1

2

nbVVT

an

nbV

nRTP

Redlick-Kwong

The van der Waals EquationThe van der Waals Equation

Calculate the pressure exerted by 15.0 g of H2 in a volume of 5.00 dm3 at 300. K .

2

2

V

an

nbV

nRTP


Calculate the molar volume of H2 gas at 40.0 atm and 300. K .

2

2

V

annbV

nRTP

nRTnbVV

anP

2

2


Can solve for P and T , but what about V?

Let: Vm = V/n { molar volume , i.e. n set to one mole}

023

P

abV

P

aV

P

RTbV mmm

Cubic Equation in V, not solvable analytically!

Use Newton’s Iteration Method:

nb

Vn

aP

nRTV

i

i

21

Mathcad: Text Solution

Mathcad: Matrix Solution

nRTnbVV

anP

2

2

nb

Vn

aP

nRTV

i

i

21

Calculate the molar volume of H2 gas at 40.0 atm and 300.K

(Newton's Iteration Method)

a 0.02479Pa m6 mol

2 b 26.60106 m

3mol

1 R 8.3145J mol1 K

1

P 40.0atm T 300K Define: Vm = V/n

Vm0R TP

Vm0 L mol1

Vm1R T

P a1

Vm0

2

b Vm1 L mol1

Vm2R T

P a1

Vm1

2

b Vm2 L mol1

Vm3R T

P a1

Vm2

2

b Vm3 L mol1

Vm4R T

P a1

Vm3

2

b Vm4 L mol1 Converged

Vm4 0.633L mol1

Picture

Postulates:– Gases consist of a large number of molecules in constant random

motion.

– Volume of individual molecules negligible compared to volume of container.

– Intermolecular forces (forces between gas molecules) negligible.

Kinetic Energy =>

Root-mean-square Velocity =>

Kinetic Molecular TheoryKinetic Molecular Theory

M

RTurms

3

TREk 2

3

Kinetic Molecular Model – Formal DerivationKinetic Molecular Model – Formal Derivation

molecule)per (2

1

direction;- xin thevelocity ;

2umump

umomentump

Preliminary note: Pressure of gas caused by collisions of molecules with rigid wall. No intermolecular forces, resulting in elastic collisions.

Consideration of Pressure:At

p

At

um

Atu

m

A

am

A

FP

11)(

Identify F=(∆p/∆t) ≡ change in momentum wrt time.

x

z

y

Wall of Unit Area A

Consider only x-direction: ( m=molecule ) ( w=wall )

Before After

pm=mu pm’=-mu

pw=0 pw’=?

moleculeper wall toed transferrMomentum

2'

)0'(

)'('

:collisions elasticFor

mup

pmumu

pppp

pp

w

w

wwmm

wm

Assumption: On average, half of the molecules are hitting wall and other not.

In unit time => half of molecules in volume (Au) hits A

If there are N molecules in volume V, then number of collisions with area A in unit time is:

And since each collision transfers 2mu of momentum, then

Total momentum transferred per unit time = pw’ x (# collisions)

2

uAVN

2a][eqn )(

1][eqn )(

2VN

2mu transMom Total

2

2

umV

N

Atp

P

umAV

N

t

p

Au

Mean Square Velocity: 2b][eqn :in Resulting2

2

2um

V

NP

i

uu i

i

In 3-D, can assume isotropic distribution:3][eqn

3:Therefore

: velocity3D Define2

2

2222

cu

wvuc

Substituting [eqn 3] into [eqn 2b] gives: 4][eqn 3

2cm

V

NP

5][eqn 2

1 :Comparing & Noting

2

cmNEk

TRnEk 3

2PV:gives 4][eqn into 5][eqn ngSubstituti

6][eqn 2

3E:Therefore k TRn

M

TRcrms

3

6][eqn 2

3E:Therefore k TRn

M

TRcrms

3

M

TRcrms

3

Mathcad

Molecular Effusion and Diffusion• The lower the molar mass, M, the higher the rms.

Kinetic Molecular TheoryKinetic Molecular Theory

Concept of Virial SeriesConcept of Virial SeriesDefine: Z = compressibility factor

n

VVwhere

TR

VPZ m

m

:

Virial Series: Expand Z upon molar concentration [ n/V ] or [ 1/Vm ]

...1432

V

nE

V

nD

V

nC

V

nBZ

B=f(T) => 2nd Virial Coeff., two-molecule interactions

C=f(T) => 3rd Virial Coeff., three-molecule interactions

Virial Series tend to diverge at high densities and/or low T.

...1111

1432

mmmm VE

VD

VC

VBZ

Concept of Virial Series – vdw exampleConcept of Virial Series – vdw example

21

2

2

mm

n

V

a

bV

TR

V

an

bnV

TRnP

Phase ChangesPhase Changes

Critical Temperature and Pressure• Gases liquefied by increasing pressure at some

temperature.• Critical temperature: the minimum temperature for

liquefaction of a gas using pressure.• Critical pressure: pressure required for liquefaction.


Critical Temperature and Pressure


Phase DiagramsPhase Diagrams

The Phase Diagrams of H2O and CO2

Phase DiagramsPhase Diagrams

Reduced VariablesReduced Variables

)(

)(

)(

volumereducedV

VV

etemperaturreducedT

TT

pressurereducedP

PP

cR

cR

cR

PVT Variations among Condensed PhasesPVT Variations among Condensed Phases

)(1

),(1

ExpansionThermaloftCoefficienT

V

Vα

alsoilityCompressibIsothermalP

V

V

P

TT

Brief Calculus ReviewBrief Calculus Review

PVT Variations among Condensed PhasesPVT Variations among Condensed Phases

Brief Calculus Review – F13 -1Brief Calculus Review – F13 -1

Mathcad

Brief Calculus Review – F13 - 2Brief Calculus Review – F13 - 2

Exact and Partial Differentials: TutorialExact and Partial Differentials: Tutorial

A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm.

(a) Find the “approximate change” in the volume ( V ) of the cylinder if r is increased by 0.30 cm and h is decreased by 0.40 cm. Express the answer in terms of cm3 . This is the “differential” volume change. Then compare to the “real” volume change from algebraic calculations of initial and final volumes.

(b)Repeat for r increase of 0.10 cm and h decrease of 0.10 cm.

(c)Repeat for r increase of 0.001 cm and h decrease of 0.001 cm.

What is your conclusion regarding the comparisons?

Mathcad-file

V r h( ) r2 h

rV r h( )d

d2 r h

hV r h( )d

d r

2

V

rh h

V

h

V

hr

V

r

V

r

0

lim&

0

limrh h

V

h

V

hr

V

r

V

r

0

lim&

0

lim

Differential Algebra

r / cm h / cm r / cm h / cm V / *cm3 V1 V2 V'=V2-V1 Diff Diff%

2.00 5.00 0.300000 -0.400000 4.400000 20.00000 24.33400 4.334000 6.6000E-02 1.52E+00

2.00 5.00 0.100000 -0.100000 1.600000 20.00000 21.60900 1.609000 9.0000E-03 5.59E-01

2.00 5.00 0.030000 -0.040000 0.440000 20.00000 20.43966 0.439664 3.3600E-04 7.64E-02

2.00 5.00 0.010000 -0.010000 0.160000 20.00000 20.16010 0.160099 9.9000E-05 6.18E-02

2.00 5.00 0.003000 -0.004000 0.044000 20.00000 20.04400 0.043997 3.0360E-06 6.90E-03

2.00 5.00 0.000300 -0.000400 4.40000E-03 20.00000 20.00440 4.39997E-03 3.0036E-08 6.83E-04

2.00 5.00 0.000030 -0.000040 4.40000E-04 20.00000 20.00044 4.40000E-04 3.0003E-10 6.82E-05

2.00 5.00 3.00E-06 -4.00E-06 4.40000E-05 20.00000 20.00004 4.40000E-05 2.9994E-12 6.82E-06

2.00 5.00 3.00E-07 -4.00E-07 4.40000E-06 20.00000 20.00000 4.40000E-06 3.3846E-14 7.69E-07

2.00 5.00 3.00E-08 -4.00E-08 4.40000E-07 20.00000 20.00000 4.40000E-07 2.6741E-15 6.08E-07

States of MatterStates of Matter

Equations of State

Ideal Gas Deviations Mixtures

Van der Waals Virial Series Berth., R-K

Kinetic Molecular Model

Corresponding States

Fluids

Reduced Variables

Condensed Phases

nRTPV

nn PbPbPb

RT

M

P ...1 2

21

nRTnbVV

anP

2

2

nb

Vn

aP

nRTV

i

i

21

...1432

V

nE

V

nD

V

nC

V

nBZ

states of matter equations of state ideal gasdeviations van der waalsvirial series kinetic molecular...

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