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11/18/2016

1

Physical Chemistry GTM/04 1

A quote of the week

(or camel of the week):

Minds are like parachutes – they only function when open

Thomas Dewar

2

Entropy and reaction

spontaneity Back to the II law ot thermodynamics

A spontaneous change is accompanied by an increase in the

total entropy of the system and its surroundings.

sur

syssursystotT

hssss

0 tots 0 tots 0 tots

spontaneous equilibrium externally driven

Physical Chemistry GTM/04

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Is a given chemical

reaction spontaneous? (1)

0.. sursyssi SSS

Given is reaction: S(s,rhomb) + O2(g) = SO2(g)

Can it occur spontaneously at standard conditions?

H0f,298 kJ/mol S0

298 J/(K·mol)

S(s,rhomb.) 0 31,80

O2(g) 0 205,14

SO2(g) -296,83 248,22

J/K 28,11)14,20580,31()22,248(0298, rsys SS

J 2968300

298, rH J/K 996298

296830

T

HS ukłot

0J/K 100799628,11.. siS YES, IT IS!!! Physical Chemistry GTM/04

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Chem. Fiz. TCH II/04 4

Gibbs free energy (1)

Josiah Willard

Gibbs

sursyssi SSS ..

T

HSS rrsi

..

rrsi HSTST ..

rrsi STHST ..

sspontaneou isreaction then the0 :if .. siST

TSHG STHG

5


T

HSStot

0G 0G 0G

spontaneous equilibrium externally driven

forced

spontaneous in reversed

direction

STHST tot STHG


6


STHG

exothermic Hr<0

Sr>0 always spontaneous

Sr<0 spontaneous, if |Hr| >|T Sr|

endothermic Hr>0

Sr>0 spontaneous, if |Hr| <|T Sr|

Sr<0 never spontaneous

Criteria of reaction spontaneity:

Always is spontaneous when GT,P < 0


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The standard reaction free energy G0 is the difference between

the free energies of formation of the products and the reactants

(all in their standard states).

The standard free energy of formation, G0

f , of a compound is

the standard reaction free energy per mole for its synthesis

from elements in their most stable forms. Standard free

energies of elements in their most stable forms are equal to

zero at 298K.

0

298,,,1

0

298,,,1

0

298, refi

n

iiprfi

n

iir GnGnG

8



9



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10



11

Free Helmholtz energy

Hermann Ludwig

Ferdinand von Helmholtz

For isochoric/isothermic conditions another

state function was defined, known as free

Helmholtz energy:

TSUF STUF

Criterion of spontaneity of processes (chemical reactions)

occurring at such conditions is:

FV,T < 0


12

Given is reaction: S(s,romb) + O2(g) = SO2(g)

Can it occurr spontaneously at standard conditions?

G0f,298 kJ/mol

S(s,rhomb.) 0

O2(g) 0

SO2(g) -300,19

kJ 19,3000

298,

0

298, 2 twSOr GG

YES, IT CAN!!!

rriur STHSTG .. We check calculations from part (1).

Left side:

Right side:

kJ 19,3001000/)35,1007298(.. siST

kJ 19,3001000/)28,11298(83,296 rr STH

Is a given chemical

reaction spontaneous? (2)


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13

Is a given chemical

reaction spontaneous? (3) Given is reaction: NaHCO3(s) = NaOH(s) + CO2(g)

Can it occurr spontaneously at standard conditions?

G0f,298 kJ/mol

NaHCO3(s) -851,9

NaOH(s) -379,07

CO2(g) -394,38

kJ 45,789,85145,7730

298,,

0298,,

0298,,

0298,

3

2

NaCOf

NaOHfCOfr

G

GGG

NO, IT CAN NOT!!!

CONCLUSION:

• Sodium hydrocarbonate is thermodynamically stable at standard conditions.


14

Gases

Several following slides contain sheer repetition or a reminder.

Perfect (ideal) gas

(an example of a model)

• Gas molecules remain in perpetual, chaotic movement.

• They do not interact, neither with the walls of the

container, nor with each other, except perfectly elastic

collisions (bouncing).

• The molecules do not occupy any space (their mass is

concentrated in points), at least their size is negligible,

when compared with their path between collisions.


15

Gases (2)

Gases occupy any volume uniformly.

Basic parameters:

Volume, V – units: m3, dm3 (l), cm3 (all SI)

Pressure, P – units: Pa (N/m2), hPa, kPa, MPa, Bar

(1 Bar=100kPa), Atm (1 Atm= 1.01325105 Pa), Torr

(1 mmHg).

Temperature, T – units: K (1 K=1oC, as T)

T = t + 273,15

where: t is temperature in Celsius scale. Physical Chemistry GTM/04

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16

Gas laws

Boyle’s law Charles’ law Gay-Lussac’s law

PV= const

at constant T

isothermal process

V1T2=V2T1

at constant P

isobaric process

P1T2=P2T1

at constant V

isochoric process

The laws were discovered experimentally. They are obeyed when gases

are rarefied. (not too high P or T, not to small V)



Clapeyron equation

Generalization of the 3 gas laws is equation of state of

the perfect gas, also known as Clapeyron equation:

nRTPV where: n is number of moles of gas,

R is universal gas constant equal to 8.314 J/(Kmol).

where: kB is Boltzman’s constant,

NAv is Avogadro’s number.

AvBNkR

Physical ChemistryGTM/04 18

Avogadro’s law

The same volumes of different gases contain the same

number of molecules (moles).

nconstV where: const is molar volume.

Vm=22.4 dm3/mol

at normal pressure (1.013105 Pa) and 273,15 K

(calculated from Clapeyron equation).

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19

Partial pressures and

Dalton’s law Partial pressure of a component of a gaseous mixture

is pressure of this component occupying alone the

same volume as does the mixture.

Total pressure of a gas mixture is the sum of partial

pressures of all components.

Where x is molar fraction

and (approximately)

volume fraction.

i

itotal PPtotal

iiP

Px

n

nx ii

i

ix 1

i

inn


20

Real gases vs perfect gas

The perfect gas can not be liquefied (no interactions).

Molar volumes of real gases differ from the

aforementioned value of 22.4 dm3/mol.

Real gases follow the perfect gas behavior (obey the

perfect gas law) when P 0 (gases are rarefied).

Detailed study of real gases revealed deviations

from the gas laws. Let’s see some isotherms of a

real gas (CO2).


21

Real gas isotherms

The isotherms are no more

hyperboles (inflection

points may bee seen). The

red isotherm is know as the

critical isotherm and

corresponding temperature

as the critical temperature.

The coordinates of the

inflection point of this

isotherm are: critical

pressure and critical molar

volume.


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22

Van der Waals equation

Per 1 mole of gas,

Vm – molar volume: 2

mm V

a

bV

RTP

where: b is correction for the molecules own volume,

a is a correction allowing for molecular inter-

actions (attractive).

Critical parameters:

bR

aTc

27

8 bVmc 3

227b

aPc

No real gas can be liquefied above its critical temperature.


23

Real gas isotherms (2)

P, Vm, T parameters are

given here as the reduced

parameters:

cT

T

mc

m

V

V

cP

P


8133

2


Virial Equation of State

...)'''1( 32 PDPCPBRTPVm

...1

32

mmm

mV

D

V

C

V

BRTPV

This equation was introduced by Kamerlingh Onnes (in two forms):

Coefficients B, C, D (B’, C’, D’) are known as virial

coefficients (they depend on temperature).

Frequently only second virial coefficient is used:

PBRTPVm "

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Real gases at the same reduced temperature and reduced volume have

also the same reduced pressure and the same reduced Z (compressibility)

The Principle of

Corresponding States

It originates from the reduced Van der Waals equation (gas specific

factors a and b disappear). Other equations of state (reduced) also

include this principle.

The principle cannot be applied when gas molecules are polar aor

non-spherical.

8133

2

26

Maxwell distribution

M is molar mass of the gas

v is velocity

)2/(2

2/32

24)( RTMvev

RT

Mvf


27

Maxwell distribution (2)

Mean velocity:

Important properties of the gases (viscosity, diffusion, efusion and

thermal conductivity coefficients, etc.) may be derived from the kinetic

model.

Some features of the gas molecules may be calculated,

on the basis of this distribution, like:

Mean free path:

where: z is collision frequency, (in Hz or s-1);

is active cross-section (m2).

2/18

M

RTc

PkT

zc

2


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Viscosity of Gases(1)

(1) ;Adx

dvF (2) ;

8

4

l

ΔPrv

Coefficient of internal friction, , or viscosity, one can discuss in

cathegories of the kinetic theory of gases, as exchange of momentum

between the molecules in neighbouring layers of flowing gas.

only 1/3 exchange the momentum along x; N=NA·A/Vm

v

v+ ·dv/dx

One molecule momentum: dx

dvmp 1

If in V=A there are N molecules: dx

dvNmpt 3

1

dx

dvApt

2

31


Viscosity of Gases(2)

Comparing the last equation with the Poiseuille’s one:

All this happen in time =1/z:

12

31

dx

dvAF

And because /=ĉ: Adx

dvcF 3

1

c 31

That we can further complicate introducing values from the Maxwell

distribution.

Conclusions: (experimentally verifiable) • viscosity does not depend on pressure, • viscosity does depend on the square root of temperature (how is it in liquids?)

Wzór Sutherlanda:

Tc

T

1

0


Heat Conductivity of Gases

Addx

dTdq

VV ccc 31

Heat flow depends on temperature gradient dT/dx. Amount of heat

crossing a perpendiculat to the gradient surface of area A in time d amounts to:

Heat conductivity of this gas, , means transfer of kinetic energy by

molecules in neighbouring layers. Reasoning the same way as in the

case of viscosity one can obtain:

Heat condutivity of gases is very important in gas analysis and

in GC detectors.

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Diffusion

Addx

dcDdm

cD 31

Diffusion depends on concentration gradient dc/dx. Mass of substance

crossing a surface normal to the gradient of area A, in time d amounts to (II Fick law):

Diffusion coefficient, D, is substance specific and depends on

temperature. Repeating the same of reasoning as in last cases, one can

get the authodiffusion coefficient according to the kinetic model:


Efusion

2241

P

RT

MPc

2

Pv

Efusion is flow of gas from a container through a hole (or holes) of size

smaller than the mean free path of gas molecules.

Stream of gasflowing through such a hole (mass per 1 cm2 per 1 second)

is equal to:

Efusion may be practically used in description of flow through porous

structures (separation of isotopes).

And volumetrically (cm3/(cm2 s)):

33

Cp and Cv of gases

Equipartition of energy

Each degree of freedom of motion of a molecule corresponds to

energy equal to ½kT. When we talk about a mole of a gas, each

degree of freedom means ½RT.

If, as it results from the kinetic (Maxwell) model, energy of translation

was the only kind of energy of gas molecules, then:

RCdT

duV

V

23

and from Meyer equation RRCC VP 2

5

This is true only for helium and other monoatomic gases.

The molecules of other gases must, therefore, possess also energy of

other kinds. Physical Chemistry GTM/04

11/18/2016

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34

Cp and Cv of gases (2)

For non-linear molecules (possessing three

dimensions)

Gas molecules can also rotate (rotational motion or rotation around

their axes of symmetry). Monoatomic molecules have no energy of

rotation and no degrees of freedom of rotation because all their

moments of inertia J (vs. the three axes) are negligible. Diatomic and

other linear molecules have two degrees of freedom of rotation (one

moment of inertia is negligible). For such molecules:

RTJErot 2

212

RTJErot 232

213


35


Hence, for linear molecules of gases:

It was observed, however, that at sufficiently high temperatures,

heating curves of polyatomic gases indicate yet higher heat

capacities. This is explained by vibrational excitation.

and for non-linear (3D ones):

RCV 25 RCP 2

7

RCV 3 RCP 4


36


Each normal vibration means two degrees of freedom of vibration (for

both potential and kinetic energy). Hence, at sufficiently high

temperatures, for diatomic gases:

Number of normal vibrations amounts to:

for non-linear molecules for linear molecules

There are following contributions to the internal energy of gases:

53 N 63 N

RCV 27 RCP 2

9

constRTRTRTEEEEU nucleloscrottr 23

..,


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Dulong-Petit rule:

Molar heat capacities of the elements, especially the metals, are

approximately equal to 3R at 298.15 K 25 J/(K·mol)

Crystals – oscillations only (Einstein, 1907):

Molar heat capacity of simple crystal substances increases with

temperature from zero to 3R (a set of harmonic oscillators vibrating in

three dimensions).

Molar Heat Capacities of

Liquids and Solids

There are no rules for liquids as there is no general theory of the

liquid state.


Compressibility of Gases

(1)

RT

PVZ m

0dP

dZ

Coefficient of compressibility of gases is given by the

equation:

For the perfect gas it is always equal to 1 and the derivative:

'lim ...'2'

0

BdP

dZCB

dP

dZ

P

For real gases:

However B does not have to be equal to 0, moreover, it depends on T.

There is certain temperature, known as Boyle’s temperature, at

which B = 0 for P 0, or real gases behave like the perfect one (at

low pressures).


The reason of the fact is a balance of repulsive interactions (short

distance) and attractive interactions (long distance).

Compressibility of Gases(2)

Perfect gas

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