Week 11/Tu: Unit ‘27’ Theory for Gases

© DJMorrissey, 2o12

Unit 25, 26: Gases -- properties of gases -- gas laws: A B C -- Law D, gas mixtures (solutions) Unit 27: Kinetic Theory of Gases -- Postulates -- Diffusion, speed -- “real” gases -- vdW equation, sizes Unit 28: Intermolecular Forces -- types of forces between molecules Unit 29: Solid State Issues: Homework continues this week, due Saturday 8AM

Week 11/Tu: Kinetic Theory of Gases

© DJMorrissey, 2o12

We have a law describing macroscopic behavior (PV=nRT) We know about atoms & molecules We would like a theory based on microscopic behavior to predict the macroscopic law and other features such as average speed, collision rate, etc.

HCl (g) + NH3 (g) ! NH4Cl (s)

Acid/Base neutralization reaction in solution: NH4OH (aq) + HCl (aq) = NH4

+(aq) + Cl-

(aq) + H2O(l) However, open bottles leads to �smoke� … Gas Phase reaction !

Week 11/Tu: DEMO: 1 m Dash for Molecules

© DJMorrissey, 2o12

Solutions have strong smells:

HCl (aq) + H2O (l) = H3O+(aq) + Cl-

(aq)

NH4OH (aq) = NH3 (g) + H2O(l) mass NH3 = 17 g/mol

HCl (aq) = HCl(g) + H2O(l) mass HCl = 36.5 g/mol

NH4OH (aq) = NH4+

(aq) + OH-(aq)

Week 11/Tu: Kinetic Theory of Gases, rules

© DJMorrissey, 2o12

Postulates for a microscopic Kinetic Theory of Gases:

• Gas consists of particles in continuous motion • Forces only act during collisions and are elastic ( T > TBoilingPoint ) • Size of particles << distance between particles ( ρGas << ρLiquid ) • Particles follow straight-line paths (no external forces, e.g., gravity)

• Average KE of all depends only on the absolute temperature (K)

We have a law describing macroscopic behavior (PV=nRT) We know about atoms & molecules We would like a theory based on microscopic behavior to predict the macroscopic law and other features such as average speed, collision rate, etc.

Week 11/Tu: Kinetic Theory of Gases, result

© DJMorrissey, 2o12

Note that the average kinetic energy depends on temperature, in any true gas there is a distribution of kinetic energies … KE = ½ mv2 α T m is constant for one kind of gas, so v α √T

Note: 1 m/s ~ 2.237 mph

Distribution is not symmetric … Mode ≠ Average ≠ RMS Mode < Average < RMS

VRMS =3RTMM

Week 11/Tu: Diffusion vs. Effusion

© DJMorrissey, 2o12

Because the average kinetic energy depends on mass, a light gaseous particle will move around inside a container faster than a heavy gaseous particle ! diffusion

Also, because the average kinetic energy depends on mass, a light gaseous particle will exit through a hole in a container faster than a heavy gaseous particle ! effusion

Week 11/Tu: Kinetic Theory of Gases, mass

© DJMorrissey, 2o12

Note that the average kinetic energy depends on mass so at the same temperature, the velocity will change with the mass of the particles. KE = ½ mv2 α T if T is constant, then v α √(1/m)

T = 300K

Effusion from a container of two combustible gases: CH4 MM = 16 g/mol compared to H2 MM = 2 g/mol

Week 11/Tu: DEMO: Relative Effusion Rates

© DJMorrissey, 2o12

VRMSMethane

VRMSHydrogen =

3RTMM (CH4 )3RT

MM (H2 )

=

1MM (CH4 )

1MM (H2 )

=MM (H2 )MM (CH4 )

rateMethane

rateHydrogen=VRMS

Methane

VRMSHydrogen =

MM (H2 )MM (CH4 )

=216

=12.82

rate = distance/time ↔ time=distance/rate

timeMethane

timeHydrogen=1/VRMS

Methane

1/VRMSHydrogen =

MM (CH4 )MM (H2 )

=2.821

Graham’s Law: rate is inverse with mass of gas.

Week 11/Tu: Real Gases

© DJMorrissey, 2o12

Experimental work is not generally carried out in the limit of P=0… one might expect that T and P are generally large. Thus, we would like a description for real gases. Imagine what happens if we compress a cylinder of “real” gas at a high and then lower temperatures.

liquid mixed gas

0

50

100

150

200

0 0.1 0.2 0.3 0.4 P

ress

ure

(bar

)

Molar Volume (L/mol)

carbon dioxide at Constant Temperature

(Isotherms)

Ideal gas Real gas T→0, V→0 T→0, V→Vliguid Elastic collisions Condense at low T

Week 11/Tu: van der Waals Equation

© DJMorrissey, 2o12

An important, intuitive set of corrections to the ideal gas equation was proposed by Johannes van der Waals (Nobel Prize Physics, 1910)

1)  Particles interact with one another, actual pressure felt is larger

than applied pressure: Note that an interaction requires two particles to collide, the

probability that a particle moves into a given spot is proportional to the number density, thus density squared:

2)  Particles in gas take up space, volume that is available is smaller

in proportion to the size and number of particles

P→ P + a number density[ ] 2↔ P + a nV#

$%&

'(

2

nbVV −→

Week 11/Tu: van der Waals Equation, “a”

© DJMorrissey, 2o12

P + a nV!

"#

$

%&2'

())

*

+,,V − nb[ ] = nRT

Substitution in the Ideal Gas Equation gives the van der Waals Equation

Where a & b are empirical constants for each gas, for example: Helium a= 0.03414 L2-atm/mol2, b=0.02373 L/mol Benzene a=18.24 L2-atm/mol2, b=0.1154 L/mol

Q1: Is the attraction between two benzene molecules larger or smaller than that between two helium atoms? Q2: Compared to SO2 ? (use information in graph)

Week 11/Tu: van der Waals Equation, “b”

© DJMorrissey, 2o12

€

Vexcluded =43πd3 one pair

Vm,excluded =NA

2#

$ %

&

' ( 43πd3 pairs in a mole

b =Vm,excluded =NA

32πd3

Volume excluded by the impenetrability of gas particles

d is called the vdW�s diameter and is one way to estimate the size of a gas particle !

d#

He: b=0.02373 L/mol = NA 2 π d3 /3 d = [0.02373 L/mol * 10-3 m3/L * 3 / (NA 2 π )] 1/3 d = 2.7x10-10 m …. Diameter is ~ 3 Angstroms

Q: What is the vdW’s diameter of helium?