Kinetic Molecular Theory of Ideal Gases Bernoulli et al. (1738) Theoretical development of ideal gas laws that were determined empirically Boyle Charles Gay-Lussac Avogadro (1627 – 1691) (1746-1823) (1778 – 1850) (1776 – 1856)

PV = k1, [n,T]

V/T = k2, [n,P]

V/n = k3, [P,T]

PV = nRT Main Postulates 1. 2. 3. 4. 5. 6. 1

Gas molecules in ceaseless chaotic motion. Pressure (P = f/A) exerted on the container walls is due to the bombardment of the container by the gas molecules. All molecular collisions are elastic, i.e., no energy loss due to friction. No intermolecular forces. Molecules are “point masses”, i.e., infinitesimally small molecular volumes. Absolute T is proportional to the average kinetic energy of all the molecules. Gas Problems: Ideal Gases: 1-13.

© Prof. Zvi C. Koren

21.07.10

Mean (or Average) Velocities N identical gas molecules m = mass of each molecule v velocity vector of molecule i i vixi viy j viz k 2 2 vi2 vix viy viz2 N

Mean or Average v 2x

v 2x

Mean Square Velocity v 2

v 2x

v 2y

2

vix i

N

2 Note : v 2 v 2 v 2 v 2 vz . x y z

For random motion : v 2x v 2y v 2z

Mean Square Velocity v 2 3 v 2x 2

© Prof. Zvi C. Koren

21.07.10

The Model

A ℓ

x

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

fix

A

ℓ

x

So, for one molecule i: fix = force exerted by molecule i on wall vix (mvix ) pix (Newton) maix m t t t

Rate of change of momentum

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

For each collision cycle: pix 2mvix mvix2 fix t 2/vix 4

For N molecules (continued):

pix = m(+vix) – m(-vix) = 2mvix t = d/v = 2ℓ/vix

A

© Prof. Zvi C. Koren

21.07.10

So, for one molecule i (from before): pix 2mvix mvix2 fix t 2/vix

For N molecules: N

f total, x f ix i

m 2 vix N

v 2x f total

Ptotal 5

v ix 2

m 2 Nvx

v 2 3 v 2x

i

1 Nm 2 v 3

f total A

(“correction” of the assumption that all movement is only in x-direction)

N

1 Nm v 2 1 Nm v 2 3 3

A

V

PV

1 Nm v 2 (continued) © Prof. Zvi C. Koren 3

21.07.10

(continued)

1 Nm 3

PV

v2

N = # of molecules = n·NAvogadro, NA = 6.02x1023 molecules/mol M = molar mass [g/mol] = NA·m mtotal = N·m = n·M 1 2 2 2 PV N (2 ½) m v N KE KE total 3

PV

KMT 1 nM 3

3

v2

1/2

(Note the units) v 2

exp

nRT

PV

1/2

3RT M

vrms = root mean square velocity

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

1/ 2

6

t2 R1 M2 t1 R2 M1

3 KMT 2 PV nN KE 3 A

3 KE k T 2 B k B Boltzmann constant R , N A R = 8.31 J/molK 1 3 kT (Energy of translation) 2 “½kT” = basic unit of molecular energy for each independent motion © Prof. Zvi C. Koren

21.07.10

From KMT

PV

2 N KE 3

T

All the empirical gas laws can be derived: Boyle: Charles & Gay-Lussac: Avogadro:

PV = k1, [N,T] V/T = k2, [N,P] V/N = k3, [P,T]

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

7

© Prof. Zvi C. Koren

21.07.10

Maxwell-Boltzmann Distribution of Molecular Velocities = f (T, MW) Why do molecules, all at the same T, have such a wide span of velocities? MW Effects: >

8

>

(continued)

© Prof. Zvi C. Koren

21.07.10

Temperature Effects 1:

9

(continued)

© Prof. Zvi C. Koren

21.07.10

Temperature Effects 2:

10

© Prof. Zvi C. Koren

21.07.10

11

© Prof. Zvi C. Koren

21.07.10

Most-Probable, Mean, and Root-Mean-Square Velocities Maxwell’s Distribution of Speeds: 3/ 2

M 2 Mv / 2RT f(v) 4 ve 2RT 2

#

vmp 2RT/M

vmp

v

v 4/π vmp 1/2

12

v rm s v 2

3/2 v m p 3RT/M

© Prof. Zvi C. Koren

21.07.10

Relative Mean Speed: The mean speed with which one molecule approaches another identical molecule (exact derivation is too cumbersome)

vrel Qualitative rendition: One

extreme

Typical

2v

v from before

Another extreme

v rel : For two dissimilar molecules approaching each other:

vrel 13

8kT/

mAmB μ mA mB

reduced mass © Prof. Zvi C. Koren

21.07.10

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: d = ·d2 = collision cross-section z v rel N/V

(target area that a molecule presents to an incoming molecule)

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. For a sample held at constant volume, as T increases, z ______________ inc. bec. vrel inc.

z v rel P/ kT (for an ideal gas), recall : k R/N A At constant T, z P. Logical? Collision Cross-Sections

14

Gas

/(nm)2

C6H6

0.88

CO2

0.52

He

0.21

N2

0.43

Example: For an N2 molecule at 1 atm and 25oC, z 7 x 109 s-1

© Prof. Zvi C. Koren

21.07.10

Mean Free Path, = The average distance a molecule travels between collisions tfree = Time in free flight between collisions = 1/z, = Average distance traveled in free flight =

vt

z = collision frequency

free

λ v/z kT/( 2 σ P) 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

PV = k1, [n,T]

V/T = k2, [n,P]

V/n = k3, [P,T]

PV = nRT Main Postulates 1. 2. 3. 4. 5. 6. 1

Gas molecules in ceaseless chaotic motion. Pressure (P = f/A) exerted on the container walls is due to the bombardment of the container by the gas molecules. All molecular collisions are elastic, i.e., no energy loss due to friction. No intermolecular forces. Molecules are “point masses”, i.e., infinitesimally small molecular volumes. Absolute T is proportional to the average kinetic energy of all the molecules. Gas Problems: Ideal Gases: 1-13.

© Prof. Zvi C. Koren

21.07.10

Mean (or Average) Velocities N identical gas molecules m = mass of each molecule v velocity vector of molecule i i vixi viy j viz k 2 2 vi2 vix viy viz2 N

Mean or Average v 2x

v 2x

Mean Square Velocity v 2

v 2x

v 2y

2

vix i

N

2 Note : v 2 v 2 v 2 v 2 vz . x y z

For random motion : v 2x v 2y v 2z

Mean Square Velocity v 2 3 v 2x 2

© Prof. Zvi C. Koren

21.07.10

The Model

A ℓ

x

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

fix

A

ℓ

x

So, for one molecule i: fix = force exerted by molecule i on wall vix (mvix ) pix (Newton) maix m t t t

Rate of change of momentum

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

For each collision cycle: pix 2mvix mvix2 fix t 2/vix 4

For N molecules (continued):

pix = m(+vix) – m(-vix) = 2mvix t = d/v = 2ℓ/vix

A

© Prof. Zvi C. Koren

21.07.10

So, for one molecule i (from before): pix 2mvix mvix2 fix t 2/vix

For N molecules: N

f total, x f ix i

m 2 vix N

v 2x f total

Ptotal 5

v ix 2

m 2 Nvx

v 2 3 v 2x

i

1 Nm 2 v 3

f total A

(“correction” of the assumption that all movement is only in x-direction)

N

1 Nm v 2 1 Nm v 2 3 3

A

V

PV

1 Nm v 2 (continued) © Prof. Zvi C. Koren 3

21.07.10

(continued)

1 Nm 3

PV

v2

N = # of molecules = n·NAvogadro, NA = 6.02x1023 molecules/mol M = molar mass [g/mol] = NA·m mtotal = N·m = n·M 1 2 2 2 PV N (2 ½) m v N KE KE total 3

PV

KMT 1 nM 3

3

v2

1/2

(Note the units) v 2

exp

nRT

PV

1/2

3RT M

vrms = root mean square velocity

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

1/ 2

6

t2 R1 M2 t1 R2 M1

3 KMT 2 PV nN KE 3 A

3 KE k T 2 B k B Boltzmann constant R , N A R = 8.31 J/molK 1 3 kT (Energy of translation) 2 “½kT” = basic unit of molecular energy for each independent motion © Prof. Zvi C. Koren

21.07.10

From KMT

PV

2 N KE 3

T

All the empirical gas laws can be derived: Boyle: Charles & Gay-Lussac: Avogadro:

PV = k1, [N,T] V/T = k2, [N,P] V/N = k3, [P,T]

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

7

© Prof. Zvi C. Koren

21.07.10

Maxwell-Boltzmann Distribution of Molecular Velocities = f (T, MW) Why do molecules, all at the same T, have such a wide span of velocities? MW Effects: >

8

>

(continued)

© Prof. Zvi C. Koren

21.07.10

Temperature Effects 1:

9

(continued)

© Prof. Zvi C. Koren

21.07.10

Temperature Effects 2:

10

© Prof. Zvi C. Koren

21.07.10

11

© Prof. Zvi C. Koren

21.07.10

Most-Probable, Mean, and Root-Mean-Square Velocities Maxwell’s Distribution of Speeds: 3/ 2

M 2 Mv / 2RT f(v) 4 ve 2RT 2

#

vmp 2RT/M

vmp

v

v 4/π vmp 1/2

12

v rm s v 2

3/2 v m p 3RT/M

© Prof. Zvi C. Koren

21.07.10

Relative Mean Speed: The mean speed with which one molecule approaches another identical molecule (exact derivation is too cumbersome)

vrel Qualitative rendition: One

extreme

Typical

2v

v from before

Another extreme

v rel : For two dissimilar molecules approaching each other:

vrel 13

8kT/

mAmB μ mA mB

reduced mass © Prof. Zvi C. Koren

21.07.10

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: d = ·d2 = collision cross-section z v rel N/V

(target area that a molecule presents to an incoming molecule)

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. For a sample held at constant volume, as T increases, z ______________ inc. bec. vrel inc.

z v rel P/ kT (for an ideal gas), recall : k R/N A At constant T, z P. Logical? Collision Cross-Sections

14

Gas

/(nm)2

C6H6

0.88

CO2

0.52

He

0.21

N2

0.43

Example: For an N2 molecule at 1 atm and 25oC, z 7 x 109 s-1

© Prof. Zvi C. Koren

21.07.10

Mean Free Path, = The average distance a molecule travels between collisions tfree = Time in free flight between collisions = 1/z, = Average distance traveled in free flight =

vt

z = collision frequency

free

λ v/z kT/( 2 σ P) 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