View
216
Download
1
Category
Preview:
Citation preview
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
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
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 -
Mystery of our Universe: QuarksMystery of our Universe: Quarks
Particle HuntersBig Bang Theory physics episodes
• 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
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
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
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.
Phase ChangesPhase Changes
Critical Temperature and Pressure
Phase ChangesPhase Changes
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
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
Brief Calculus Review – F13 - 3Brief Calculus Review – F13 - 3
Brief Calculus Review – F13 - 4Brief Calculus Review – F13 - 4
Brief Calculus Review – F14 -1Brief Calculus Review – F14 -1
Brief Calculus Review – F14 -2Brief Calculus Review – F14 -2
Brief Calculus Review – F14 -3Brief Calculus Review – F14 -3
Brief Calculus Review – F14 -4Brief Calculus Review – F14 -4
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?
A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm.
Mathcad-file
V r h( ) r2 h
rV r h( )d
d2 r h
hV r h( )d
d r
2
V
A right-circular cylinder has a base radius ( r ) of 2.00 cm and a height ( h ) of 5.00 cm.
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
Recommended