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10/17/2017
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Kinetic Theory and the Kinetic Theory and the Behavior of Ideal & Real Behavior of Ideal & Real
GasesGases
Why study gases?
• An understanding of real world phenomena.
• An understanding of how science “works.”
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A Gas
• Uniformly fills any container.
• Mixes completely with any other gas.
• Exerts pressure on its surroundings.
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2
o
kPa101 325Pa101 325
torr760
C)0at (measured Hg mm 760
2in lb 14.7
mb 1013 bar 1.013
kPa 101.325 Pa 101,325
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• A manometer is used to measure the pressure inside closed containers
Open-end manometer. (a) The pressure of the trapped gas, Pgas equals the atmospheric pressure, Patm. Trapped gas pressure (b) higher and (c) lower than atmospheric pressure.
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Compressing a gas increases its pressure.
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Boyle’s J-Tube Experiment
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PV
1
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Jacques Alexander Charles’ Law
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:LawCharles'
)(when
:Law sBoyle'
212211 TTVPVP
)(when //
:Law sLussac'-Gay
)(when //V
:Law Charles
212211
212211
VVTPTP
PPTVT
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Example: What will be the the final pressure of a sample of oxygen with a volume of 850 m3 at 655 torr and 25.0oC if it is heated to 80.0oC and given a final volume of 1066 m3?
ANALYSIS: Use the combined gas law with temperature in kelvins.
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SOLUTION:
1
2
2
112
T
T
V
VPP
torr619
273.2)K(25.0
K)2.2730.80(
m 1066
m 850 torr655
3
3
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• The combined gas law can be generalized to include changes in the number of moles of sample
• The ideal gas law is
nRTPV
K mol
L atm 0.0821
constant gas universal
R
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One mole of each gas occupies 22.4 at STP. Carbon dioxide is more dense that oxygen due to molar mass differences.
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Molar Mass of a Gas
Molar Mass = dRT/Pd = density of gas
T = temperature in Kelvin
P = pressure of gas
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The space above any liquid contains some of the liquid’s vapor
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Example: A sample of oxygen is collected over water at 20oC and a pressure of 738 torr. What is the partial pressure of oxygen?
ANALYSIS: The partial pressure of oxygen is less than the total pressure. Get the vapor pressure of water from table 11.2 (page p essu e o ate o tab e (page478).
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SOLUTION:
torr720torr)5417738(
torr 54.17
vaporwater
P
P
torr720. torr )54.17738( gasP
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Dalton’s Law of Partial Pressures
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• This is possible because the number of molesof each gas is directly proportional to its partial pressure
• Using the ideal gas equation for each gas
VPn A
• For a given mixture of gases, the volume and temperature is the same for all gases
• Using C=V/RT gives
RTn A
A
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• The partial pressure of a gas can be
total
A
ZBA
A
ZBA
AA
P
P
PPP
P
CPCPCP
CPX
• The partial pressure of a gas can be calculated using the total pressure and mole fraction
totalAA PXP
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• Gas volumes can be used in stoichiometry problems
)(OHl2)(Hl2
)(O volume1)(H volumes2
O(g)H2)(O)(H2
22
pressure) and re temperatu(same volumes2 volume1 volumes2
222
gg
gg
)(OH moles 2)(O mole 1
)(OH moles 2)(H moles 2
)(O mole 1)(H moles 2
asjust
)(OH volumes2)(O volumes1
)(OH volumes2)(H volumes2
22
22
22
22
22
gg
gg
gg
gg
gg
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• The behavior of ideals gases can be explained
(a) Diffusion (b) Effusion
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Kinetic Molecular Theory
• So far we have considered “what happens,” but not “why.”
• In science, “what” always comes before “why.”y
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Kinetic Molecular Theory
Postulates:
1. Gas particles are in rapid motion, colliding with container walls.
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Kinetic Molecular Theory
Postulates:
2. Gas particles have negligible size compared to the distances between them.
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Kinetic Molecular Theory
Postulates:
3. Gas particles have no attraction for one another.
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Kinetic Molecular Theory
Postulates:
4. Absolute temperature of the gas is a measure of the average kinetic energy of the gas particlesenergy of the gas particles.
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• Diffusion is the spontaneous intermingling of the molecules of one gas with another
• Effusion is the movement of gas molecules through a tiny hole into a vacuum
• The rates of both diffusion and effusion depend on the speed of the gas moleculesdepend on the speed of the gas molecules
• The faster the molecules, the faster diffusion and effusion occur
• Thomas Graham studied effusion
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• He found that the effusion rate of a gas was inversely proportional to the square root of the density (d)
• This is known as Graham’s law1
• Where Mi is the molar mass of species iA
B
A
B
M
M
d
d
B
A
TP
)( rateeffusion
)( rateeffusion
) and (constant d
1 rateeffusion
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Diffusion
• The movement of one gas through another by thermal random motion.
• Diffusion is a very slow process in air because the mean free path is very short (for N2 at STP it is 6 6x10-8 m) Given the nitrogenN2 at STP it is 6.6x10 m). Given the nitrogen molecule’s high velocity, the collision frequency is very high also (7.7x109
collisions/s).
• Diffusion also follows Graham's law:
M
1diffusionofRate
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Diffusion of agas particlethrough aspace filledwith otherparticlesparticles
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NH3(g) + HCl(g) = NH4Cl(s)
HCl = 36.46 g/mol NH3 = 17.03 g/mol
Rate =RateNH3 =
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The inverserelation betweendiffusion rate andmolar mass.
Due to it’s lightmass, ammonia travels 1.46 timesas fast as
NH3(g) + HCl(g) NH4Cl(s)
hydrogen chloride
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Relative Diffusion Rates of NH3 and HCl
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A Practical Example of Using Gas Density, Diffusion, Separation and Purification for Enriched Uranium
Gaseous Diffusion Separation of Uranium 235 / 238
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Gaseous Diffusion Separation of Uranium 235 / 238
Purified solid mixed U3O8 ,UO3 ,and, UO2 containing all uranium isotopes are converted to all isotopic forms of UF6(g)
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Gaseous Diffusion Separation of Uranium 235 / 238
Purified solid mixed U3O8 ,UO3 ,and, UO2
containing all uranium isotopes are converted to all isotopic forms of UF6(g)
235UF6 vs 238UF6
0.72 % 99.28 %
after approximately 2000 runs235UF6 is > 99% Purity
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Explain KMT
• Explain KMT on basis of the frequency of particle collisions with container walls.
E l i KMT b i f th l it f• Explain KMT on basis of the velocity of particle collisions with container walls.
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When the gas volume is made smaller going from (a) to (b), the frequency of collisions per unit area of the containers’ wall increases.
Thus the pressure increases Boyle’s Law).
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The kinetic theory and the pressure-temperature law (Gay-Lussac’s law). The pressure increases from (a) to (b) as measured by the amount of mercury that must be added to maintain a constant volume.
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The kinetic theory and the temperature-volume law (Charles’ law). The pressure is the same in both (a) and (b). At higher temperatures the volume increases because the gas molecules have higher velocities.
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Kinetic Molecular Theory
• Particles are point masses in
constant, random, straight line
motion.
• Particles are separated by great
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distances.
• Collisions are rapid and elastic.
• No force between particles.
• Total energy remains constant.
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Pressure – Assessing Collision Forces
• Translational kinetic energy,
• Frequency of collisions,
2k mu
2
1e
V
Nuv
I• Impulse or momentum transfer,
• Pressure proportional to impulse times frequency
muI
2muV
NP
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Pressure and Molecular Speed
• Three dimensional systems lead to:
2umV
N
3
1P
um is the modal speeduav is the simple average
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2u
urms
Pressure
umRT3
um3
1PV
2A
2A
N
NAssume one mole:
PV=RT so:
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M
3RTu
uM3RT
rms
2
NAm = M:
Rearrange:
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Distribution of Molecular Speeds
M
3RTurms
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Determining Molecular Speed
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Temperature
e3
2RT
)um2
1(
3
2um
3
1PV
k
22A
N
NN
A
A
Modify:
PV=RT so:
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(T)R
2
3e
Ak NSolve for ek:
Average kinetic energy is directly proportional to temperature!
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Gas Properties Relating to the Kinetic-Molecular Theory
• Diffusion– Net rate is proportional to
l l d
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molecular speed.
• Effusion– A related phenomenon.
• J. D. van der Waals corrected the ideal gas equation in a simple, but useful, way
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Plots of PV/nRT Versus P for Several Gases (200 K)
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valuegas ideal toup measured brings : 2
2
2
2
PV
an
nRTnbVV
anP measured
measuredmeasured
constants der WaalsVan theasknown are b and a
valuegas ideal to measured reduces : Vnb
Vmeasured
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0.02661 0.02444 H Hydrogen,
0.01709 0.2107 Ne Neon,
0.02370 0.03421 He Helium,mol L
mol atmL
Substance
2
122
ba
0.03049 5.464 OH Water,
0.03707 4.170 NH Ammonia,
y g ,
2
3
2
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