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Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically Bernoulli et al. (1738) Main Postulates 1. Gas molecules in ceaseless chaotic motion. 2. Pressure (P = f/A) exerted on the container walls is due to the bombardment of the container by the gas molecules. 3. All molecular collisions are elastic, i.e., no energy loss due to friction. 4. No intermolecular forces. 5. Molecules are “point masses”, i.e., infinitesimally small molecular volumes. 6. Absolute T is proportional to the average kinetic energy of all the molecules. PV = nRT Boyle (1627 1691) PV = k 1 , [n,T] Charles (1746-1823) V/T = k 2 , [n,P] Avogadro (1776 1856) V/n = k 3 , [P,T] Gay-Lussac (1778 1850) 1 © Prof. Zvi C. Koren 21.07.10 Gas Problems: Ideal Gases: 1-13.

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Page 1: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

Kinetic Molecular Theory of Ideal Gases

Theoretical development of ideal gas laws that were determined empiricallyBernoulli et al. (1738)

Main Postulates1. Gas molecules in ceaseless chaotic motion.

2. Pressure (P = f/A) exerted on the container walls is due to the bombardment of the container by the gas molecules.

3. All molecular collisions are elastic, i.e., no energy loss due to friction.

4. No intermolecular forces.

5. Molecules are “point masses”, i.e., infinitesimally small molecular volumes.

6. Absolute T is proportional to the average kinetic energy of all the molecules.

PV = nRT

Boyle

(1627 – 1691)

PV = k1, [n,T]

Charles

(1746-1823)

V/T = k2, [n,P]

Avogadro

(1776 – 1856)

V/n = k3, [P,T]

Gay-Lussac

(1778 – 1850)

1 © Prof. Zvi C. Koren 21.07.10Gas Problems: Ideal Gases: 1-13.

Page 2: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

N identical gas molecules

m = mass of each molecule

ii

molecule of vector velocity v

kji iziyix

v v v

2222 v v v v iziyixi

.v v v v Velocity SquareMean 2222zyx

N

v

v v Averageor Mean

N2

22

iix

xx

v v v :motion randomFor 222zyx

v3 v Velocity SquareMean 22x

2222v v v v :Note zyx

Mean (or Average) Velocities

2 © Prof. Zvi C. Koren 21.07.10

Page 3: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

The Model

ℓ x

A

For simplicity, first consider:

1. Only one molecule (i) is present.

2. Molecule’s motion is only in x-direction:

only vix component.

& molecule i will be hitting wall A with velocity vix.

Then, we’ll consider N molecules.

3 © Prof. Zvi C. Koren 21.07.10

Page 4: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

ℓ x

A

fix = force exerted by molecule i on wall

fix

Note: force leads to a change of velocity (and momentum) upon collision with the wall

Rate of change of momentum

t

p

t

)(mv

t

vm ma ixixix

ix

(Newton)

For each collision cycle:

A

pix = m(+vix) – m(-vix)= 2mvix

t = d/v = 2ℓ/vix

2

ix

ix

ixixix

mv

/v2

2mv

t

p f

So, for one molecule i:

For N molecules (continued):4 © Prof. Zvi C. Koren 21.07.10

Page 5: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

2

ix

ix

ixixix

mv

/v2

2mv

t

p f

For N molecules:

So, for one molecule i (from before):

2

xvNm

vm

f f 2ix

N

iixxtotal,

N

v

v

N

i

2

2

ix

x

22

xv3 v

V

vNm

A

vNm

A

f P

22

31

31

totaltotal

2vNm PV3

1 (continued)

2vNm

3

1 f total

(“correction” of the assumption that

all movement is only in x-direction)

5 © Prof. Zvi C. Koren 21.07.10

Page 6: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

(continued) 2v m N PV3

1

N = # of molecules = n·NAvogadro, NA = 6.02x1023 molecules/mol

M = molar mass [g/mol] = NA·m

mtotal = N·m = n·M

2v M n3

1 PV

TB

k2

3 KE

,

AN

R

constantBoltzmann Bk

exp KMTKMT

M

3RT v

1/21/22

vrms = root mean square velocity

(Note the units)

Graham’s Lawof Diffusion and Effusion(for 2 gases at the same T):

2/1

1

2

2

1

1

2

M

M

R

R

t

t

nRT PV KE A

nN3

2 PV

R = 8.31 J/molK

totalKE

3

2

3

2

3

1 KE N v m ½)(2 N PV 2

“½kT” = basic unit of molecular energy

for each independent motion

kT2

1 3

(Energy of translation)

6 © Prof. Zvi C. Koren 21.07.10

Page 7: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

Boyle: PV = k1, [N,T]

Charles & Gay-Lussac: V/T = k2, [N,P]

Avogadro: V/N = k3, [P,T]

T 3

2 KE N PV

From KMT

All the empirical gas laws can be derived:

7 © Prof. Zvi C. Koren 21.07.10

Recall:

At what conditions of T or P, does a real gas behave as if it were ideal?

Page 8: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

Maxwell-Boltzmann Distribution of Molecular Velocities

= f (T, MW)

MW Effects:

> >

(continued)

Why do molecules, all at the same T, have such a wide span of velocities?

8 © Prof. Zvi C. Koren 21.07.10

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Temperature Effects 1:

(continued)9 © Prof. Zvi C. Koren 21.07.10

Page 10: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

Temperature Effects 2:

10 © Prof. Zvi C. Koren 21.07.10

Page 11: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

11 © Prof. Zvi C. Koren 21.07.10

Page 12: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

RT/M2 vmp vmp

3RT/M v3/2 2 vv mp

1/2

rms

mpv4/π v

Most-Probable, Mean, and Root-Mean-Square Velocities

v

#

Maxwell’s Distribution of Speeds:

RTMvevRT

Mvf 2/2

2/32

24)(

12 © Prof. Zvi C. Koren 21.07.10

Page 13: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

The mean speed with which one molecule approaches another

identical molecule (exact derivation is too cumbersome)

v2 vrel rendition:Qualitative

8kT/ vrel

For two dissimilar molecules approaching each other:

mm

mm μ

BA

BA

Relative Mean Speed:

reduced mass

:vrel

Oneextreme Typical

Anotherextreme

13 © Prof. Zvi C. Koren 21.07.10

v from before

Page 14: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

Collision Frequency (z) & Collision Diameter (d)

z = Average # of collisions per second made by one molecule in a

system of N molecules in a volume V:

AR/N k :recall gas), idealan (for kTP/relv z

At constant T, z P. Logical?

Collision Cross-Sections

/(nm)2Gas

0.88C6H6

0.52CO2

0.21He

0.43N2

Example:

For an N2 molecule at 1 atm and 25oC,

z 7 x 109 s-1

= ·d2 = collision cross-section(target area that a molecule presents to an incoming molecule)

d

For a sample held at constant volume, as T increases, z ______________

N/Vv z rel

inc. bec. vrel inc.

A “hit” occurs when the centers of two molecules come within a

distance “d” of each other, where “d” is the diameter of impenetrable

hard sphere molecules.

14 © Prof. Zvi C. Koren 21.07.10

Page 15: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

Mean Free Path, = The average distance a molecule travels between collisions

P)σ2kT/( /zv λ

For N2 at 1 atm: = 70 nm 103 molecular diameters

In a container of fixed volume, is dependent on T?

Summary

• A typical ideal gas molecule (N2 or O2) at 1 atm and 25oC travels at a

mean speed of 350 m/s;

• Each molecule collides within 1 ns,

• Between collisions it travels 102 – 103 molecular diameters.

• If d << , gas is nearly ideal. Why?

tfree = Time in free flight between collisions = 1/z, z = collision frequency

= Average distance traveled in free flight = freetv

15 © Prof. Zvi C. Koren 21.07.10

Gas Problems: KMT: 24, 26-31.

Page 16: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

λ/dλ

/(nm)

tfree

/(ns)

z

/(s–1)/(m/s)

/(nm)2d

/(nm)vrms

/(m/s)/(m/s)

vmp

/(m/s)Gas

C6H6

CO2

He

N2

16 © Prof. Zvi C. Koren 21.07.10

Lecture Problem #1 (to hand in next week to the Recitation instructor):

Fill in the table below at 1 atm and 25 oC:

(see next slide)

Page 17: Kinetic Molecular Theory of Ideal Gases - WordPress.com · 01.07.2010  · Kinetic Molecular Theory of Ideal Gases Theoretical development of ideal gas laws that were determined empirically

17 © Prof. Zvi C. Koren 21.07.10

Give the full reference for “d”

(author’s last name & first name initial, book title, publisher, city of

publication, year, page number; and exact website address, if from internet).

Explain all the values and interpret them.

Careful with all the units.

Don’t forget to convert “atm” to “Pascal” and that (nm)2 is 10–18 m2.

:בשני הספרים האלה dהנתונים לגבי קוטר מולקולרי http://books.google.com/books?id=vYyjL3MJq2YC&pg=PA202&lpg=PA202&dq=benzene+collision+diameter&sourc

e=bl&ots=_jAAirau-

o&sig=9y8rVIdx9n7F5EfgkhLpFAzjpso#v=onepage&q=benzene%20collision%20diameter&f=false

Bלפי פאראמטר )יש גם טבלה בספר הקלאסי הבא שחישב את הקוטר המולקולרי לפי שתי שיטות

שהיא σאבל תזהרו מהטבלה מכיוון שהוא מסמל את הקוטר עם אות , )של הגז )η)ולפי הצמיגות

.תראו תנסו ותהנו .משהו אחר במשוואות שלנוhttp://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_4.html

Notes About the Previous Table: