1 The first scheduled quiz will be given next Tuesday during Lecture. It will last 15 minutes. Bring pencil, calculator, and your book. The coverage will be pp 364-424, i.e. Sections 10.0 through 11.4. • Theory developed to explain gas behavior. • Theory based on properties at the molecular level. • Kinetic molecular theory gives us a model for understanding pressure and temperature at the molecular level. • Pressure of a gas results from the number of collisions per unit time on the walls of container. 10.7 Kinetic Molecular Theory 10.7 Kinetic Molecular Theory • There is a spread of individual energies of gas molecules in any sample of gas. • As the temperature increases, the average kinetic energy of the gas molecules increases. Kinetic Molecular Theory Kinetic Molecular Theory • Assumptions: – Gases consist of a large number of molecules in constant random motion. – Volume of individual molecules negligible compared to volume of container. – Intermolecular forces (forces between gas molecules) negligible. – Energy can be transferred between molecules, but total kinetic energy is constant at constant temperature. – Average kinetic energy of molecules is proportional to temperature. 10.7 Kinetic Molecular Theory 10.7 Kinetic Molecular Theory Kinetic Molecular Theory Kinetic Molecular Theory • Magnitude of pressure given by how often and how hard the molecules strike. • Gas molecules have an average kinetic energy. • Each molecule may have a different energy. • As kinetic energy increases, the velocity of the gas molecules increases. • Root mean square speed, u, is the speed of a gas molecule having average kinetic energy. • Average kinetic energy, ε, is related to root mean square speed: Kinetic Molecular Theory Kinetic Molecular Theory 2 2 1 mu = ε
Transcript
1
The first scheduled quizwill be given next Tuesday
during Lecture.
It will last 15 minutes. Bring pencil, calculator,
and your book.
The coverage will be pp 364-424,
i.e. Sections 10.0 through 11.4.
• Theory developed to explain gas behavior.• Theory based on properties at the molecular level.• Kinetic molecular theory gives us a model for
understanding pressure and temperature at the molecular level.
• Pressure of a gas results from the number of collisions per unit time on the walls of container.
10.7 Kinetic Molecular Theory10.7 Kinetic Molecular Theory
• There is a spread of individual energies of gas molecules in any sample of gas.
• As the temperature increases, the average kinetic energy of the gas molecules increases.
Kinetic Molecular TheoryKinetic Molecular Theory
• Assumptions:– Gases consist of a large number of molecules in constant
random motion.– Volume of individual molecules negligible compared to volume
of container.– Intermolecular forces (forces between gas molecules)
negligible.– Energy can be transferred between molecules, but total kinetic
energy is constant at constant temperature.– Average kinetic energy of molecules is proportional to
temperature.
10.7 Kinetic Molecular Theory10.7 Kinetic Molecular Theory
Kinetic Molecular TheoryKinetic Molecular Theory
• Magnitude of pressure given by how often and how hard the molecules strike.
• Gas molecules have an average kinetic energy.
• Each molecule may have a different energy.
• As kinetic energy increases, the velocity of the gas molecules increases.
• Root mean square speed, u, is the speed of a gas molecule having average kinetic energy.
• Average kinetic energy, ε, is related to root mean square speed:
Kinetic Molecular TheoryKinetic Molecular Theory
221 mu=ε
2
Do you remember how to calculatevxy from vx and vy ?
( ) 2122
yxxy vvv +=
And how about v from all threecomponents?
[ ] 21222
zyx vvvv ++=
Remember these equations!! They’ll popup again in Chap. 11.
21
21
21
3Speedrms
8SpeedAverage
2SpeedProbaleMost
⎟⎠⎞
⎜⎝⎛==
⎟⎟⎠
⎞⎜⎜⎝
⎛=⟩⟨=
⎟⎠⎞
⎜⎝⎛==
MRTv
MRTv
MRTv
rms
mp
π
225.1:128.1:13:8:2::, 212
1
21
=⎟⎠⎞
⎜⎝⎛=⟩⟨πrmsmp vvvAnd
ump <u>
urms
1. Be careful of speed versus velocity. The former is the magnitudeof the latter.
2. The momentum of a molecule is p = mv. During a collision, thechange of momentum is ∆pwall = pfinal – pinitial = (-mvx) – (mvx) = 2mvx .
3. ∆t = 2ℓ / vx ∆px / ∆t = . . . = mvx2 / ℓ, where ℓ is length of the box
4. force = f = ma = m(∆v / ∆t) = ∆p / ∆t = mvx2 / ℓ = force along x
5. And for N molecules, F = N(m(vx2 )avg / ℓ )
6. But
7. And
( )( ) ...v vN
v v v vx avg x x x x xN2 2
12
22
32 21
= = + + + +
P FA
NmA
v and A V so that PV Nm vx x= = = =l
l2 2
u v v v v so that PV Nmux y z x2 2 2 2 2 1
323= + + = =
Now we have PV N mu and PV nRT= =13
2
But N = nN0 , so we can divide both sides by n to obtain
13 0
20
13
2N mu RT but N m M so M u RT= = =, ,
3
Application to Gas Laws• As volume increases at constant temperature, the average
kinetic of the gas remains constant. Therefore, u is constant. However, volume increases so the gas molecules have to travel further to hit the walls of the container. Therefore, pressure decreases.
• If temperature increases at constant volume, the average kinetic energy of the gas molecules increases. Therefore, there are more collisions with the container walls and the pressure increases.
Kinetic Molecular TheoryKinetic Molecular Theory
Molecular Effusion and Diffusion• As kinetic energy increases, the velocity of the gas
molecules increases.• Average kinetic energy of a gas is related to its mass:
• Consider two gases at the same temperature: the lighter gas has a higher rms than the heavier gas.
• Mathematically:
Kinetic Molecular TheoryKinetic Molecular Theory
221 mu=ε
MRTu 3
=
Molecular Effusion and Diffusion• The lower the molar mass, M, the higher the rms.
Kinetic Molecular TheoryKinetic Molecular Theory
Kinetic Molecular TheoryKinetic Molecular Theory
Graham’s Law of Effusion• As kinetic energy increases,
the velocity of the gas molecules increases.
• Effusion is the escape of a gas through a tiny hole (a balloon will deflate over time due to effusion).
• The rate of effusion can be quantified.
Graham’s Law of Effusion • Consider two gases with molar masses M1 and M2, the
relative rate of effusion is given by:
• Only those molecules that hit the small hole will escape through it.
• Therefore, the higher the rms the more likelihood of a gas molecule hitting the hole.
Kinetic Molecular TheoryKinetic Molecular Theory
12
21
MM=
rr
4
Graham’s Law of Effusion • Consider two gases with molar masses M1 and M2, the
relative rate of effusion is given by:
• Only those molecules that hit the small hole will escape through it.
• Therefore, the higher the rms the more likelihood of a gas molecule hitting the hole.
Kinetic Molecular TheoryKinetic Molecular Theory
12
2
121
21
3
3
MM
M
M === RT
RT
uu
rr
Diffusion and Mean Free Path • Diffusion of a gas is the spread of the gas through space.• Diffusion is faster for light gas molecules.• Diffusion is significantly slower than rms speed (consider
someone opening a perfume bottle: it takes while to detect the odor but rms speed at 25°C is about 1150 mi/hr).
• Diffusion is slowed by gas molecules colliding with each other.
Kinetic Molecular TheoryKinetic Molecular Theory
Diffusion and Mean Free Path • Average distance of a gas molecule between collisions is
called mean free path.• At sea level, mean free path is about 6 × 10-6 cm.
Kinetic Molecular TheoryKinetic Molecular Theory
• From the ideal gas equation, we have
• For 1 mol of gas, PV/nRT = 1 for all pressures.• In a real gas, PV/nRT varies from 1 significantly and is
called Z.
• The higher the pressure the more the deviation from ideal behavior.
Real Gases: Deviations Real Gases: Deviations from Ideal Behaviorfrom Ideal Behavior
1==nRTPVorn
RTPV
nRTPVZ =
• From the ideal gas equation, we have
• For 1 mol of gas, PV/RT = 1 for all temperatures.• As temperature increases, the gases behave more ideally.• The assumptions in kinetic molecular theory show where
ideal gas behavior breaks down:– the molecules of a gas have finite volume;– molecules of a gas do attract each other.
Real Gases: Deviations Real Gases: Deviations from Ideal Behaviorfrom Ideal Behavior
nRTPV
=
5
• As the pressure on a gas increases, the molecules are forced closer together.
• As the molecules get closer together, the volume of the container gets smaller.
• The smaller the container, the more space the gas molecules begin to occupy.
• Therefore, the higher the pressure, the less the gas resembles an ideal gas.
Real Gases: Deviations Real Gases: Deviations from Ideal Behaviorfrom Ideal Behavior
• As the gas molecules get closer together, the smaller the intermolecular distance.
Real Gases: Deviations Real Gases: Deviations from Ideal Behaviorfrom Ideal Behavior
• The smaller the distance between gas molecules, the more likely attractive forces will develop between the molecules.
• Therefore, the less the gas resembles and ideal gas.• As temperature increases, the gas molecules move faster
and further apart.• Also, higher temperatures mean more energy available to
break intermolecular forces.
Real Gases: Deviations Real Gases: Deviations from Ideal Behaviorfrom Ideal Behavior
• Therefore, the higher the temperature, the more ideal the gas.
Real Gases: Deviations Real Gases: Deviations from Ideal Behaviorfrom Ideal Behavior The first scheduled quiz
will be given next Tuesdayduring Lecture.
It will last 15 minutes. Bring pencil, calculator,
and your book.
The coverage will be pp 364-424,
i.e. Sections 10.0 through 11.4.
6
The van der Waals Equation• We add two terms to the ideal gas equation one to correct
for volume of molecules and the other to correct for intermolecular attractions
• The correction terms generate the van der Waals equation:
where a and b are empirical constants characteristic of each gas.
Real Gases: Deviations Real Gases: Deviations from Ideal Behaviorfrom Ideal Behavior
2
2
Van
nbVnRTP −−
=
The van der Waals Equation
• General form of the van der Waals equation:
Real Gases: Deviations Real Gases: Deviations from Ideal Behaviorfrom Ideal Behavior
In many ways, this chapter is simply acontinuation of our earlier discussion of‘real’ gases.
Remember this nice, regular behavior described by the ideal gas equation.
This plot for SO2 is a morerepresentativeone of real systems!!!
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And this is a plot for an ideal gas of the dependence of Volume on Temperature.Now this one includes a realistic one for Volume as a function of Temperature!
Why do the boiling points vary? Is there anything systematic?
London Dispersion Forces
Hydrogen Bonding
Dipole-Dipole Forces
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Intermolecular Forces -- forces between molecules --are now going to be considered.
Note that earlier chapters concentrated on Intramolecular Forces, those within the molecule.
Important ones:
ion-ion similar to atomic systems
ion-dipole (review definition of dipoles)
dipole-dipole
dipole-induced dipole
London Dispersion Forces:induced dipole-induced dipole polarizability
Hydrogen Bonding
How do you know the relative strengthsof each? Virtually impossible experimentally!!!
Most important though:Establish which are present.
London Dispersion Forces: AlwaysAll others depend on defining property
such as existing dipole for d-d.
It has been possible to calculate therelative strengths in a few cases.
Relative Energies of Various Interactions
d-d d-id disp
Ar 0 0 50
N2 0 0 58
C6H6 0 0 1086
C3H8 0.0008 0.09 528
HCl 22 6 106
CH2Cl2 106 33 570
SO2 114 20 205
H2O 190 11 38
HCN 1277 46 111
Ion-dipole interaction
Let’s take a closer look at these interactions:
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Let’s take a closer look at dipole-dipole interactions.This is the simple one.
But we also have to consider other shapes.Review hybridization and molecular shapes.
Recall the discussion of sp, sp2, and sp3
hybridization?
London dispersion forces (interactions)
A Polarized He atomwith an induced dipole molecule F2 Cl2 Br2 I2 CH4
polarizability 1.3 4.6 6.7 10.2 2.6
molecular wt. 37 71 160 254 16
Molecular Weight predicts the trends in the boiling points of atoms or molecules without dipole moments because polarizability tends to increase with increasing mass.
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But polarizability also depends on shape, as well as MW.
Water provides our best exampleof Hydrogen Bonding.
But hydrogen bonding is not limited to water:
These boiling points demonstrate the enormouscontribution of hydrogen bonding.
Water is alsounusual in the relative densities of the liquid and solid phases.
The crystal structure suggests a reason for the unusualhigh density of ice.
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But water isn’t the only substance to show hydrogen bonding!
Viscosity—the resistance to flow of a liquid, such asoil, water, gasoline, molasses, (glass !!!)
Surface Tension – tendency to minimize the surface areacompare water, mercury
Cohesive forces—bind similar molecules together
Adhesive forces – bind a substance to a surface
Capillary action results when these two are not equal
Soap reduces the surface tension, permitting onematerial to ‘wet’ another more easily
11.3 Some Properties of Liquids11.3 Some Properties of Liquids Examples of Viscosity
The unit of viscosity is poise, which is 1 g/cm-s, buttypical values are much smaller and are usuallylisted as cP = 0.01 P.
RationaleforSurfaceTension
Surface Tension• Surface molecules are only attracted inwards towards the
bulk molecules.– Therefore, surface molecules are packed more closely than bulk
molecules.
• Surface tension is the amount of energy required to increase the surface area of a liquid, in J/m2.
• Cohesive forces bind molecules to each other.• Adhesive forces bind molecules to a surface.
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Surface Tension• Meniscus is the shape of the liquid surface.
– If adhesive forces are greater than cohesive forces, the liquid surface is attracted to its container more than the bulk molecules. Therefore, the meniscus is U-shaped (e.g. water in glass).
– If cohesive forces are greater than adhesive forces, the meniscus is curved downwards.
• Capillary Action: When a narrow glass tube is placed in water, the meniscus pulls the water up the tube.
• Remember that surface molecules are only attracted inwards towards the bulk molecules.
also called
FUSION
• Sublimation: solid → gas.• Vaporization: liquid → gas.• Melting or fusion: solid → liquid.• Deposition: gas → solid.• Condensation: gas → liquid.• Freezing: liquid → solid.
