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0. Introduction

physical chemistry establishes and develops the

principles of chemistry

concepts used to explain and interpret observations

on the physical and chemical properties of matter

central theme:

systems states

processes

topics of physical chemistry:

(1) the study of the macroscopic properties of sys-

tems of many atoms or molecules

(2) the study of processes which such systems can

undergo

(3) the study of the properties of individual atoms and

molecules

PChem I 1.1


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states: ice (solid) 32F

water (liquid)212F

vapor (gas)

processes: heat the system; sharp transition from iceto water at 32

F and from water to vapor at 212

F

macroscopic properties change, no change in the

molecules or the forces between them!

Can we understand these changes? Can we pre-

dict the transition temperature, i.e., the melting tem-perature Tm and the boiling temperature Tb, or the

macroscopic properties of the different phases, e.g.,

the molar volumes, or the dependence of the transi-

tion temperatures on pressure P, on impurities (e.g.

salt), etc.

Can two or more different phases exist for the same

external conditions?

carbon: 1) diamond: transparent, colorless, hard;

2) graphite: black, slippery, soft; 3) buckminster-

fullerene: ?

properties: topic (1)

processes (transformations, reactions): melting ice,

evaporating water, burning methane, . . .

PChem I 1.3


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Tobacco Mosaic Virus (TMV)

2130 identical subunits in protein coat, closelypacked in a helical array around a single-stranded

RNA molecule (6390 nucleotides)

TMV can be dissociated by acetic acid into coat sub-

units and RNA

spontaneously reassemble under suitable con-

ditions into virus particles

indistinguishable from original TMV in structure and

infectivity

self-assembly

process

1) Can it occur?

2) Will it occur spontaneously?

3) How fast will it occur?

thermodynamics: 1) and 2); kinetics: 3)

How can we influence a process?

applications: reactor design, catalysts, corrosion,. . .

PChem I 1.4


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toluene+methanol

mxylene 0.50oxylene 0.25pxylene 0.25

pxylene oxidized terepthalic acid

ethylene glycol

polyester (Dacron)

How can we increase the yield of p-xylene?

zeolite: crystalline aluminosilicate framework, exten-

sive 3-D network of SiO44

and AlO45

ions and Si4+

andAl3+ ions

ZSM-5: channels and chambers 1 nm

PChem I 1.5


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Thermodynamics

large systemstwo types of variables:

intensive extensive

T,P, . . . V,m, . . .

historically: empirical observations concerning rela-

tions between such variables

example: PV= nRT

origins of thermodynamics:

practical interest: heat generates motion

evolved into a theory that describes transformationof states of matter in general

thermodynamics is particular good in dealing with

complex systems, since the exact nature of the con-

stituents and microscopic processes is irrelevant

two conceptual innovations of thermodynamics

First Law: conservation of energy

Second Law: entropy

PChem I 1.6


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1. The Properties of Gases

Notation: (i) I use an overbar to denote molar quan-

tities; the textbook uses a subscript m. Example: mo-

lar volume, these notes V, textbook Vm. (ii) I use a

capitalPfor pressure; the textbook usesp.

simple system to learn the concepts and methods ofthermodynamics

gas: fills any container

dilute gas; chemically pure

macroscopic description

state of pure gas: V volume of container, T temper-

ature of the gas, Ppressure of the gas, n amount of

the gas = number of moles

assume no electric or magnetic properties

pressure P= FA

PChem I 1.7


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Fforce perpendicular to area A

units: Pa=Nm2

,1 N= 1 k g m s2

1bar 100.kPa= 105 Pa= P standard pressure1atm 101.325kPa1atm 760torr= 760mmHg

hydrostatic pressure P= g h; density of fluid; ggravitational acceleration;hheight of fluid column

temperature

gas A gas B

diathermic wall

diathermic wall: thermal contact between the twocompartments, flow of energy (heat) possible

ast: thermal equilibrium

PChem I 1.8


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mechanical properties of each compartment, e.g., P,

do no longer change with time!

fortsufficiently large: PA(t) PA,PB(t)PBfor a fixed, rigid wall, in general PA = PB

Zeroth Law of Thermodynamics

If A is in thermal equilibrium with B andif B is in ther-

mal equilibrium with C, then A is in thermal equilibri-um with C.

(empirical observations)

[Figure: Thermal equilibrium and the Zeroth Law; Atkins 9th ed., Fig. 1.3]

= existence of a property common to all systemsthat are in thermal equilibrium with each other: tem-

perature

PChem I 1.9


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thermodynamic temperature scale:

C, melting

point of ice: 0

C at 1 atm; boiling point of water:

100C at 1 atm

T

K C+273.15

thermodynamic equilibrium

= thermal equilibrium,TA = TBand

mechanical equilibrium,PA = PB

thermodynamic equilibrium states

P=f(n,V,T)

ideal gas

PV= nRTSI units:Vm3,PPa,TK: R= 8.314JK1mol1

PChem I 1.10


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1 J= 1 N malternative units:

VL, (

1 L = 1 dm3

= 103

m

3)

Patm:

R= 8.206102 LatmK1mol1

consider a system withn= constisotherm: T= const,P V= const,P1/V, hyperbolas

[Figure: Ideal gas isotherms; Atkins 9th ed., Fig. 1.4]

PChem I 1.11


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isobar: P=const,V T

[Figure: Ideal gas isobars; Atkins 9th ed., Fig. 1.6]

PChem I 1.12


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isochore:V= const,PT

[Figure: Ideal gas isochores; Atkins 9th ed., Fig. 1.7]

PChem I 1.13


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surface of equilibrium states

[Figure: Ideal gas: surface of possible states; Atkins 9th ed., Fig. 1.8]

standard temperature and pressure (STP): =0 C,P=1atm = V= RT/P=22.414dm

3

mol1

= 22.414Lmol1

standard ambient temperature and pressure (SATP):

PChem I 1.14


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T= 298.15 K, P= 1 bar = V= RT/P= 24.789dm3mol1

mixture of ideal gases

Daltons law P=P1+P2+ . . .=j Pj, Pj=njRT

V

mole fraction xj=nj

n, n=

j

nj

Pj= xjP

for nonideal gases: Pj xjP

Real Gases

deviations from ideal gas law: due to intermolecular

forces

PChem I 1.15


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[Figure: Potential energy between two molecules; Atkins 9th ed., Fig. 1.13]

attractive: dipole-dipole forces, H-bonds, dispersion

forces

repulsive: repulsion of electrons

measure for deviations

compression factor Z= PVRT

PChem I 1.16


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ideal gas (superscript

) V = RT

P Z 1

Z= VV

[Figure: Compression factor; Atkins 9th ed., Fig. 1.14]

Z> 1V> V repulsive forces dominateZ< 1V< V attractive forces dominate

PChem I 1.17


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equation of state for real gases? no general form!

Try: PV= RT f(P), f(P)to be determinedf(P) 1for ideal gasesas P0, all gases follow the ideal gas law= f(P)1asP0

try power-law ansatz for f(P), i.e., expand f(P) in

powers ofP

PV= RT[1+B(T)P+C(T)P2+ ...]

virial expansionorvirial equation of state

B(T) second virial coefficient (depends only on pair

interactions), C(T) third virial coefficient (note: the does not denote a derivative; it is simply used to

distinguish these coefficients from those in the alter-

native form)

alternative, but equivalent form of the virial equation

of state

ideal gas law is valid for dilute gases, i.e., small or

V large ( =m/V=M/V or 1/V)

PChem I 1.18


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PV= RT f(V), f(V) 1 for ideal gas

expand f(V)in powers of1/V

PV= RT

1+B(T) 1V+C(T) 1

V2+ . . .

left as an exercise to show that

B(T)= B(T)RT

, C(T)= C(T)B(T)2

(RT)2

Z= 1+B(T)P+C(T)P2+ . . .dZ

dP=B(T)

+2C(T)P

+. . .

PChem I 1.19


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Boyle temperatureTB: B(TB)= 0orB(TB)= 0

P 0: if T= TB then Z 1 and dZ/dP 0 extended range of ideal behavior

[Figure: Boyle temperature; Atkins 9th ed., Fig. 1.16]

Boyle temperatures for some gases: H2 109 K,

CH4 510 K, C2H4 720 K, NH3 1030 K [Estrada-Torres,

R.; Iglesias-Silva, G. A.; Ramos-Estrada, M. & Hall, K. R., Boyle temper-

atures for pure substances, Fluid Phase Equilib., 258, 148154 (2007),

http://dx.doi.org/10.1016/j.fluid.2007.06.004]

PChem I 1.20


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real gas isotherms: exampleCO2

[Figure: CO2 isotherms; Atkins 9th ed., Fig. 1.15]

phase transition: condensation, gas liquidcritical point: critical temperature = maximum tem-

perature at which a gas can be liquefied

PChem I 1.21


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phase diagram

critical temperature Tc, critical pressure Pc, critical

molar volumeVc: critical constants

as the critical point is approached along the vapor

pressure curve: (l)(g) 0orV(g)V(l) 0

PChem I 1.22


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[Figure: Approaching the critical point; Atkins 9th ed., Fig. 4.6]

critical opalescence: strong density fluctuations near

the critical point, (l) (g); characteristic length of

fluctuations on the order of the wavelength of visible

light = strong scattering = milky appearance

law of rectilinear diameter (Cailletet and Mathias

1886)

12

((l)+(g))= A+BT[Figure: rectilinear diameter]

PChem I 1.23


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continuity of states: the substance changes from

liquid-like to gas-like in the supercritical region with-

out ever changing phases

virial equation of state good for quantitative work,

B(T), C(T), . . . are tabulated for gases, but it does

not provide understanding of the above phenomena

of real gases

find a general model; best known and most widely

used: van der Waals gas

repulsive interactions: hard spheresexcluded vol-ume,V Vnb

bis a material constant, equal to the volume of 1 molof densely packed gas particles

P(Vnb)= nRT

or

P= nRTVnb

PChem I 1.25


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attractive forces: diminish pressure; pressure is the

result of collisions of the gas particles with the walls;

as a particle is about to hit the container wall, it is

held back, and its impact is diminished, by the at-

tractive forces from surrounding gas particles; this is

a pair effect number of pairs of particles (n/V)2 =

2

P= nRTVnba

n

V

2

van der Waals equation

ais a material constant;aand bare tabulated for real

gases

P+ a

V2

Vb

=RT

compare van der Waals equation with virial equation

B(T)= b aRT

PChem I 1.26


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How good is the van der Waals model?

(Vb)V2P=RT V2a(Vb)

V3

PV2bP=RT V2aV+ab

V3

b+ RT

P

V

2+ aP

V abP= 0

cubic equation for fixed T and P, three real rootsor one real root and two complex conjugate roots

isotherms for the van der Waals equation for ammo-

nia

PChem I 1.27


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T< Tc: between the minimum and the maximum ofthe isotherm we have

dPdV

> 0, and since dPdV

=

dVdP

1

we have

dV

dP>0

= unstable!!

replace the unphysical van der Waals loop by the

Maxwell equal area construction

PChem I 1.28


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branch BB and AA are metastable states

AA: fast compression: supersaturated gas, cloud

chamber

BB: fast expansion: superheated liquid, bubble

chamber

critical point = coalescence of minimum and maxi-mum= disappearance of van der Waals loop

minimum or maximum: first derivative vanishes

PChem I 1.29


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between a minimum and a maximum is an inflection

point (curvature changes): second derivative vanish-

es

critical point: at (Tc,Pc,Vc)

dddP

dddV= 0 and ddd

2P

dddV

2= 0

there are three critical constants, need one more

equation: equation of state

P=RT

Vba

V2 (1)

dP

dV= RT

(Vb)2+ 2a

V3= 0 at critical point (2)

d2P

dV

2= 2RT

(V

b)3 6a

V

4= 0 at critical point (3)

equation (2)RTc=2a(Vcb)2

V3

c

(4)

PChem I 1.30


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insert this result into equation (3)

4a(Vcb)2

V3

c(Vcb)3 6a

V4

c

= 0 (5)

2

Vcb 3

Vc= 0

2Vc=

3Vc

3b

Vc=3b

insert into equation (4)

RTc=2a(3bb)2

(3b)3 = 2a4b

2

27b3

Tc= 8a27bR

insert into equation (1)

Pc=

8a27b

3bba

9b2 =8a

2 27b2 a

9b2

Pc=a

27b2

PChem I 1.31


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critical compression factor:

Zc= PcVcRTc

=a

27b2 3b8a

27b

Zc=3

8

independent of the material constants aandb

ZvdWc = 0.375; compare with experimental datarange: 0.12 HF 0.47N2O4

mode: 0.27; 27% of all organic and inorganic sub-

stances have this value ofZc

61% Zc in [0.26, 0.28]; 77% Zc in [0.25, 0.29]; 90%

Zcin [0.23, 0.31]

van der Waals gas: simple model that captures the

essential aspects of real gases

critical point is a very characteristic point for real gas-

es

use critical constants as units

reduced variables Pr=P

Pc, Vr=

V

Vc, Tr=

T

Tc

PChem I 1.32


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P= RTV

b a

V

2

PrPc=RTrTc

VrVcb a

V2

rV2

c

Pra

27b2=

8a27b

Tr

3bVrb a

V2

r9b2

Pr=8

3Tr

Vr 13 3V

2

r

material independent

law of corresponding states

universality

PChem I 1.33


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[Figure: Law of corresponding states; Atkins 9th ed., Fig. 1.21]

PChem I 1.34


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