states of matter equations of state ideal gasdeviations van der waalsvirial series kinetic molecular...
TRANSCRIPT
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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
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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
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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
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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 -
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Mystery of our Universe: QuarksMystery of our Universe: Quarks
Particle HuntersBig Bang Theory physics episodes
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• 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
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• 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
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Calculate the number of air molecules in 1.00 cm3 of air at 757 torr and 21.2 oC.
Mathcad
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Calculate the number of air molecules in 1.00 cm3 of air at 757 torr and 21.2 oC.
MathcadF12
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Low P IdealLow P Ideal
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High T IdealHigh T Ideal
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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
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Density of an Ideal-GasDensity of an Ideal-Gas
TR
MP
Derivation of :
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Plotting data of density versus pressure (at constant T) can give molar mass.
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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
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Plotting data of ρ/P vs. P (at constant T) can give molar mass.
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• 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?
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i
iiavg MM Density?
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• 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
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ii
i
t
ii n
n
n
ntPP ii
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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
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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
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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
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The van der Waals EquationThe van der Waals Equation
Calculate the molar volume of H2 gas at 40.0 atm and 300. K .
2
2
V
annbV
nRTP
nRTnbVV
anP
2
2
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The van der Waals EquationThe van der Waals Equation
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
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nRTnbVV
anP
2
2
nb
Vn
aP
nRTV
i
i
21
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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
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Picture
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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
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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.
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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
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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
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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
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6][eqn 2
3E:Therefore k TRn
M
TRcrms
3
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M
TRcrms
3
Mathcad
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Molecular Effusion and Diffusion• The lower the molar mass, M, the higher the rms.
Kinetic Molecular TheoryKinetic Molecular Theory
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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
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Concept of Virial Series – vdw exampleConcept of Virial Series – vdw example
21
2
2
mm
n
V
a
bV
TR
V
an
bnV
TRnP
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Phase ChangesPhase Changes
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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.
Phase ChangesPhase Changes
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Critical Temperature and Pressure
Phase ChangesPhase Changes
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Phase DiagramsPhase Diagrams
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The Phase Diagrams of H2O and CO2
Phase DiagramsPhase Diagrams
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Reduced VariablesReduced Variables
)(
)(
)(
volumereducedV
VV
etemperaturreducedT
TT
pressurereducedP
PP
cR
cR
cR
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PVT Variations among Condensed PhasesPVT Variations among Condensed Phases
)(1
),(1
ExpansionThermaloftCoefficienT
V
Vα
alsoilityCompressibIsothermalP
V
V
P
TT
Brief Calculus ReviewBrief Calculus Review
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PVT Variations among Condensed PhasesPVT Variations among Condensed Phases
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PVT Variations among Condensed PhasesPVT Variations among Condensed Phases
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Brief Calculus Review – F13 -1Brief Calculus Review – F13 -1
Mathcad
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Brief Calculus Review – F13 - 2Brief Calculus Review – F13 - 2
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Brief Calculus Review – F13 - 3Brief Calculus Review – F13 - 3
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Brief Calculus Review – F13 - 4Brief Calculus Review – F13 - 4
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Brief Calculus Review – F14 -1Brief Calculus Review – F14 -1
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Brief Calculus Review – F14 -2Brief Calculus Review – F14 -2
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Brief Calculus Review – F14 -3Brief Calculus Review – F14 -3
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Brief Calculus Review – F14 -4Brief Calculus Review – F14 -4
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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?
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A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm.
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Mathcad-file
V r h( ) r2 h
rV r h( )d
d2 r h
hV r h( )d
d r
2
V
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A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm.
![Page 66: States of Matter Equations of State Ideal GasDeviations Van der WaalsVirial Series Kinetic Molecular Model Corresponding States Fluids Reduced Variables](https://reader036.vdocuments.net/reader036/viewer/2022062408/56649f1c5503460f94c330aa/html5/thumbnails/66.jpg)
rh h
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V
hr
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r
V
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0
lim&
0
limrh h
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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
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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
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P ...1 2
21
nRTnbVV
anP
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nb
Vn
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i
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V
nE
V
nD
V
nC
V
nBZ