Equilibrium (Thermodynamics: chap 1-6) (Quantum theory and …contents.kocw.net/KOCW/document/2015/pusan/limmanho1/2.pdf · 2016-09-09 · Molecular motion in gases (2015) Chemical

Contents of Atkins’ Physical Chemistry

ThermodynamicsMatter in bulk

Macroscopic view

(2015) Chemical Kinetics by M Lim 1

Quantum theoryIndividual atoms and molecules

‘structure of molecule’Microscopic (molecular) view

Statistical MechanicsProvides the link between the microscopic properties of matter

and its bulk properties

• Equilibrium (Thermodynamics: chap 1-6)

• Structure (Quantum theory and spectroscopy: Chap 7-14)

• Change (Chemical kinetics: chap 20-23)

Part 3 of Atkins: ChangeChemistry is the science of change


cf.• Thermodynamics: treat the properties of matter in bulk

interchanges among different forms of energy

• Quantum Theory (quantum mechanics, quantum chemistry): how these properties stem from the behavior of individual atoms or molecules

Rates of reactions by considering the motion of molecules

Chap 20. Molecules in motion


• The random motion of molecules of perfect gas accounts for– The pressure of a gas– The rates at which molecules and energy migrates through gases

• Molecular mobility in liquids: ionic motion in electric field -> generalize to handle neutral motion in the absence of E field

• The diffusion equation: shows how matter and energy spread through media of various kinds.

• The transport properties – Diffusion: the migration of matter down a concentration gradient– Thermal conduction: the migration of E down a T gradient– Electric conduction: the migration of electric charge along an φ gradient– Viscosity: the migration of linear momentum down a velocity gradient


Assigned ProblemsChapter 20 in the 9th edition •Numerical Problems: 1, 5, 9, 10, 12, 13, 15, 18, 19• Theoretical Problems: 22, 24, 26• Applications: 31, 32, 34, 35, 36

In the 10th edition • Problems in Chapter 19: 19A.1, 19A.3, 19A.4, 19B.6

19B.7, 19B.8, 19C.1, 19C.5, 19C.12, 19C.13

• Problems in Chapter 1: 1B.1, 1B.5, 1B.7, 1B.10

Molecular motion in gases


The kinetic model of a perfect gas as a starting point for the discussion of its transport properties

(KE of the molecules is the only contribution to E of the gas)

20.1 The kinetic model of gasesThree assumptions

1. The gas consists of molecules of mass m in ceaseless random motion2. The size of the molecule is negligible (if d << average distance travelled)3. The molecules interact only through brief, infrequent and elastic collision


20.1 (a) Pressure and molecular speeds-1a

213

pV nMc=

p of a perfect gas according to the kinetic model

( )

a molecule

2

2

Momentum change of a molecule after collision: 2

1Number of colliding molecule in : 2

= 2 =2

x

Ax

A x xtotal x

x

P mvnNt Av tV

nN Av t nMAv tP mvV V

P nMAvFt V

F nMvpA

∆ =

∆ ∆

∆ ∆∴∆

∆= =∆

= =22 2

2 2 2 2 2

3

where

xx

x y z

nM v nMcV V V

c v v v v

→ →

= = + +

1 22

AM mN

c v

=

=

2

2 33

nMcpV nR

RTcM

T

=

= =

∴


20.1 (a) Pressure and molecular speeds-1b

( )

( )

( )2

2

2

2

12

2

1direction 2

Probability having ,

2

x

x

x x

x x

mvEkT

mk

k

vT

x

Tx

mf v

v KE mv

v v

f v e e

ekTπ

−

− −

=

− =

−∞ ≤

=

∞

≤

∝

: Maxwell-Boltzmannvelocity distribution

( )23

2 2 242

MvRTMf v v e

RTπ

π− =

Maxwell distribution of speeds

molar mass, gas constant,

A

A

M mNR kN==

( )

2

2

2 1

2 1

2

x

ax

x x

mvkT

x

e dxa

f v dv

Ne dv

kTNm

makT

π

π

∞ −

−∞

∞

−∞

∞ −

−∞

=

= =

=

≡

∫

∫

∫


20.1 (a) Pressure and molecular speeds-1c

( )23

2 2 242

MvRTMf v v e

RTπ

π− =

2 2 2

2

~ ~

~ ,

si

x x x

y y y

z z z

x y z

x y z

v v dvv v dvv v dv

v v v v

dv dv dv v

++

+

= + +

=

( ) ( ) ( )22 2

2

2

32

2 2 2

322 22

0 0

32

2

n

2

sin2

42

x x y y z z

x y z

yx zx x y y z z

x y z

v dv v dv v dv

x x y y z zv v v

mvmv mvv dv v dv v dvkT kT kT

x y zv v v

mvv dvkT

v

mvk

dvd d

f v dv f v dv f v dv

m e e e dv dv dvkT

m e v dvd dkT

m ekT

π π

θ θ φ

π

θ θ φπ

ππ

+ + +

+ + + − − −

+ −

−

=

=

=

∫ ∫ ∫

∫ ∫ ∫

∫ ∫ ∫

( )

( )23

2 2 2

2

42

mv

v dv v dvT

v v

kTmf v v ekT

v dv f v dv

ππ

−

+ +

=

=

∴

∫ ∫molar mass, gas constant,

A

A

M mNR kN==

( )21

22

direction

2

xmv

kTx

x

mf v e

v

kTπ− =

−


11 22 2

12

12*

3

8

2

RTc vM

RTcM

RTcM

π

= =

=

=

Root mean square speed

Mean speed

Most probable speed

( ) ( )23

2 2 24 02

MvRTMf v v e v

RTπ

π− = ≤ ≤ ∞

( )

( )

2

2

2

2

2

0

0

2 3 2

0

320

4 5 2

0

0

2 2 2

0

12

12

41

23

8

ax

ax

ax

ax

ax

e dxa

xe dxa

x e dx a

x e dxa

x e dx a

c v vf v dv

c v v f v dv

π

π

π

∞ −

∞ −

∞ − −

∞ −

∞ − −

∞

∞

=

=

=

=

=

≡ =

≡ =

∫

∫

∫

∫

∫

∫∫

20.1 (a) Pressure and molecular speeds-2


12

2

8 where

rel

A B

A B

c c

kT m mm m

µπµ

=

= = +

Relative mean speed ( )23

2 2 242

MvRTMf v v e

RTπ

π− =

A velocity selector

Ex 20.1 of N2 molecules in air at 298K.c

20.1 (a) Pressure and molecular speeds-3

( )1 1

13 1

8 8 8.314 298 475 28.02 10

RT JK mol Kc msM kg molπ π

− −−

− −

× ×= = =

× ×


12 2

relrel

c pz c NkT

c kTz N p

σσ

λσ σ

= =

= = =

( )

( )

freeze the position of all the molecules except one

number density,

/

A

rel rel

pNV kT

pV nRT nN kT kT

z c t t N cV

σ σ

Ν≡ =

= = = Ν

Ν = ∆ ∆ =

20.1 (b) The collision frequency, z(c) The mean free path, λ

For 1 atm of N2 molecules at 298K.

( ) ( )18 2 1 2

23 1

9 1

1

9 1

0.43 10 2 475 101325

1.38 10 2987.1 10

475 67 (~ 200 times of )7.1 10

relc pzkT

m ms Nm

JK Ks

c ms nm dz s

σ

λ

− − −

− −

−

−

−

=

× × × ×=

× ×= ×

= = =×

2

collision crosssection, dσ π≡


Review 20-1

( )

( )

2

2

2

12

2

32 2 2

11 22 2

12

3

2

42

3

8

xmvkT

x

MvRT

nMcpV

mf v ekT

Mf v v eRT

RTc vM

RTcM

π

ππ

π

−

−

=

=

=

= =

=

12

relz c N

N

σ

λσ

=

=

• The collision frequency

• The mean free path


1 1 84 4 2w

kT p pZ cNm kT mkTπ π

= = =

( ) ( )

( )2

0

20 0

12

number of molecules collide with wall

14

2

12 4

x

x

x x xw

mvkT

x x x x x

NAv t

NAv t f v dvZ Nc

A tmv f v dv v e dvkT

kT cm

π

π

∞

∞ ∞ −

∆

∆= =

∆

=

= =

∫

∫ ∫

20.2 Collisions with walls and surfaces

For 1 bar of O2 molecules at 300K.

The collision flux, Zw(the number of collisions per a given time and area)

2

3 23 23 1

2 223 2 1 4 2 1

2 2

23 2 1

2100000

2 32 10 6.02 10 1.38 10 300

26893 10 10

3 10

wpZmkT

Nmkg JK K

kgms m m s cm skg kgm s

cm s

π

π

−

− − −

− −− − − − −

−

− −

= =

=× × ÷ × × × ×

= × = = × ≈ ×


0 0 00

12 2 2

A Aw

pA pA N pA NZ AmkT MRT RT Mπ π π

= = = =

0

0

0 0

mass loss in an interval due to effusion

2

2 2

w

w

tw Z A m t

w pZA m t mkT

kT w RT wpm A t M A t

π

π π

∆∆ = ∆

∆= =

∆

∆ ∆∴ = =

∆ ∆

20.3 The rate of effusion

Ex 20.2 Cs in container was heated to 773K. After 100 s, 385 mg is lost through d = 0.50 mm opening. What is the vapor pressure of Cs at 773K?

( )

0

6

23

2 2 1 2 22

2

2 8.314 773 385 100.1329 0.25 10 100

76010808 10808 81 101325

RT wpM A t

J kgkg m s

kgm s kgm s Nm Pa Torr Torrm s

π

π

π

−

−

− − − −

∆=

∆

× × ×=

× ×

= = = = = × =

Rate of effusion through a hole of area A0

The emergency of a gas from a container through a small hall

: Graham’s law of effusion

Knudsen method(vapor pressure determination for liquids and solids with low vapor pressure using effusion rate)


2

2 3

2

1

2

1 1

1

1

( )

,

matter

z

energy

z

xmomentum

xz

dNJdz

dN molecules moleculesJdz m s m m

dTJdz

dT J KJdz m s m

dvJdzdv kgms msJdz m

ms

JKms

kgmss m

gcm s P po

D

kise

κ

η−

− −

−

∝

= − =

∝

= − =

∝

= − =

≡

1 1 210 , 10gm s P cP P− − −= =

20.4 Transport properties of a perfect gas-1

Flux, J (the quantity of a property passing through unit area per unit time)ex, matter flux (diffusion), energy flux (thermal conduction)

J ∝ gradient of a property

commonly expressed in terms of a number of phenomenological equations

Diffusion coefficient

Coefficient of thermal conductivity

Coefficient of viscosity

Fick’s 1st law of diffusion



Newtonian flow (laminar flow): a series of layers moving past one another- layers tend towards a uniform velocity- interpret the retarding effect of the slow layers on the fast layers as the fluids viscosity


Review 20-21 1 8Collision flux: 4 4 2w

kT p pZ cNm kT mkTπ π

= = =

0 0

2 2Knudsen method: kT w RT wpm A t M A tπ π∆ ∆

= =∆ ∆

2

, 2 3

, 2

1 1

, 2

1 1 1 1 2 3 1

1

( ), 10 , 10 10

matter z

energy z

xmomentum z

dN molecules m moleculesJ Ddz m s s m m

dT J J KJdz m s Kms m

dv kgms kg msJdz m s ms m

gcm s P poise kgm s P cP P kgm

κ

η− −

− − − − − − −

= − =

= − =

= − =

≡ = = = 1s−



( )

( )

( ) ( )

( )

( ) ( )0 0

0 0

1 ( )41 ( )4

Net flow, 1 ( ) ( )4

1 0 04

1 1 224 2 3

1 8 ( , )3 2

T

z

z

z z

z z

J L R cN

J R L cN

J J L R J R L

J c N N

dN dNc N Ndz dz

dN dNc cdz dz

kT kTD c cmp

λ

λ

λ λ

λ λ

λ λ

λ λπσ

= =

= =

→ = −

→ =

= → − →

= − −

= − + ⋅ ⋅ ⋅ − + + ⋅ ⋅ ⋅

≈ − = − ×

∴ = = =

↑ Dp Dlarger molecule: Dσ

↑

↑ ↓

↑ ↓

The molecules within a mean free path can pass A0

Some path may not reach the wall



( )

( )

[ ]

0

0

,,

1 ( )41 ( )4

( )

1 224 3

1 1 =3 3 3 2

: independent of at high At low , if is greater than dimensions of the appar

z

V mV m

A

J L R cN

J R L cN

dTk Tdz

dTJ cN kdz

cCc kN cC A

Np p

p

ε λ

ε λ

ε λ ν λ

ν λ

κ λ ν λσ

κλ

→ = −

→ =

± = ± + ⋅⋅ ⋅

≈ − ×

∴ = =

atus depends on pκ

Each molecule carries on average energy ε = νkT

[ ]

,

For a perfect gas

V m A

A

A A

C R kNnNN

V VN n p pAN V RT kN T

ν ν= =

Ν= =

= = = =



( )

( )

0

0

1 ( )41 ( )4

( ) (0)

1 224 3

1 1 ( )3 3 2 2

: independent of

8 ( )

. for liquid,

x

x

xx x

xz

J L R cNmv

J R L cNmv

dvmv m vdz

dvJ cN mdz

mccNmN

p

kTT cm

cf T

λ

λ

λ λ

λ

η λ λσ σ

η

ηπ

η

→ = −

→ =

± = ± + ⋅⋅ ⋅

≈ − ×

∴ = = =

∝ =

↑ ↓

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Equilibrium (Thermodynamics: chap 1-6) (Quantum theory and …contents.kocw.net/KOCW/document/2015/pusan/limmanho1/2.pdf · 2016-09-09 · Molecular motion in gases (2015) Chemical