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Theory of Gases -- Their Properties and Behavior
The study of gas behavior (~1660’s) lead to many of
today’s modern theories of atomic behavior
The underpinnings of gas behavior provides insight into the physical properties of matter
Where in the world does He come from?
He is an inert, unreactive atom.
It floats to the outer reaches of our atmosphere
The “Three” States of Matter
Ordered Compressible Homogeneous Gases NO YES YES Solids YES NO YES/NO
Liquids NO NO YES
Critical Concepts of Gas Behavior
• About 0.1 % volume of gas is taken up by molecules. There is mostly empty space. In a liquid, there is only 70% empty space.
• Little interaction between neighboring gas molecules/atoms.
• Gases exert a pressure via collisions
Does our atmosphere, which is a ball of gas, exert a pressure?
With the assistance of gravity…YES.
A 1 meter diameter swath of air from ground level to outer space weighs in at 10,300 Kg
or 4,680 lbs.
1 atm = 760 mmHg = 101,325 Pascals (Pa)
The IDEAL GAS LAW
P is pressure;
V is volume;
n = moles of gas
R = constant, either 0.082058 (Lit·atm)/(mole·K) or 8.314472 (J/mole·K)
T is Temperature in Kelvin
We can use the IDEAL GAS equation to derive other gas
laws (this is the reverse of how it was ‘put together’)
Boyle’s law: P1V1 = P2V2 Temp & n held constant
Charles’ Law: 2
2
1
1
TV
TV
= Pressure & n held constant
Avagadro’s Law: 2
2
1
1
nV
nV
= Temp & Pressure constant
Gay-Lussac’s law anybody?
PV = nRT
Let’s consider Boyle’s Law, Temperature and n are held constant
So, we know that PV = nRT
In this case, T, R, and n are constants…
…this requires, nRT = Constant
PV = constant
P1V1 (for state 1) = P2V2 (for state 2) = PxVx =
constant
…as in previous slide,
Boyle’s law: P1V1 = P2V2 Temp & n are held constant
… Derivation of Charles’ Law, P & n are held constant
PV = nRT
Re-expressing (putting all constants on one side)
tConsP
nRTV tan==
tConsTV
TV
TV
x
x tan2
2
1
1 ==
Repeat this process for Avagadro’s Law,
hold P,T constant !
and don’t forget Gay-Lussac’s Law…
What are STP conditions of a gas …or anything for that matter ?
Temp 0ºC = 273 Kelvin Pressure = 1 atm = 760 mmHg
KNOW THIS !
Standard Ambient Temp and Pressure (SATP)
Temp 25ºC = 298 Kelvin
Pressure = 0.98 atm = 1 bar
STP = Standard Temperature and Pressure
What is the volume of an Ideal Gas (1 mole) at STP ? MOLAR VOLUME
Well, lets do the math! PV = nRT
(1 atm) (V) = (1 mole) (0.08206 lit atm / K mole) (273 K)
…solving for Volume (V)
V = atm1K) (273 mole)K / atmlit (0.08206 mole) (1
What causes “NON-IDEALITY” of gases ?
V = 22.414 liters/mole
Not an error
Let’s integrate the gas laws with Stoichiometry
PV = nRT remembering that n = moles of gas
…and knowing that
moles of gas =
mwmass
PV =
RTmw
mass
Combustion of a Hydrocarbon
C3H8 (g) + 5O2 (g) 3CO2 (g) + 4H2O (l)
• 15 liter bottle of propane • Pressure = 4.5 atm • Temperature = 25 C = 298 K
Question: What volume of CO2 would be produced if you burned all the propane? Relevant if you are in a well confined fish
house.
PV = nRT
4.5 atm × 15 lit = n (0.08206 lit atm/K mole) (298K)
n = 2.76 moles of C3H8 consumed (in tank)
CO2 produced = 3 × 2.76 moles C3H8 consumed = 8.28 moles CO2
This is only half the problem, remember, we want to know what volume of CO2 is
produced…
…in other words, what volume does 8.28 moles of CO2 gas occupy at 298 K and 1 atm?
PV = nRT
(1 atm) V = (8.28 mole) (0.082 lit atm/mole K) (298 K)
V = 202.5 liters of CO2
Propane burners in confined spaces consume O2, produce CO2, and in many cases CO (deadly) if
incomplete combustion occurs!
Partial Pressures and Dalton’s Law
PTOT = P1 + P2 + P3 + … + Pn at constant Volume and Temperature
Pair = PN2 + PO2 + Par + PCO2 + … + PHe
Integrating the Ideal Gas equation into the
frey…
P1 = n1
VRT
P2 = n2
VRT
Ptotal = (n1 + n2 + n3 + …) VRT
What is a mole fraction?
Mole Fraction (Xn) = mixtureinmolestotalcomponentofmoles
X 1 = Totalnn
nnn 1
21
1
...=
++
P1 = X1 · Ptotal (Dalton’s Law)
The partial pressure exerted by each
component in a gas mixture is equal to the (mole fraction) · (total pressure)
Partial Pressure Problem…
Question: What is the partial pressure and mole fractions of each of the species in the
given volume?
nG = 4 nY = 2 Ptotal = 600 mmHg nR = 6 nTotal = 12
PG = mmHgmmHg 200)600(124
=
PY = mmHgmmHg 100)600(122
=
PR = mmHgmmHg 300)600(126
=
Kinetic-Molecular Theory of Gases…let’s formalize our discussion and look at gases
from a molecular point of view…rather than an empirical point of view.
1. A gas consists of atoms/molecules moving about in a random fashion
2. The volume that the gas particles
actually occupies is significantly smaller than the volume of the gas
3. There is no attraction/repulsion
between neighboring particles of gas 4. Collisions are elastic
i.e. KE is a constant at constant temp
5. Kinetic Energy (KE) ~ Temperature
…from a “somewhat complex derivation” (wait until Physical Chemistry, ~ 4th year)
2
21
23 mv
NRTE
Ak ==
where NA is Avogadro’s number, m is mass,
v is velocity of particle,
…solving for velocity as a function of the other parameters…
=
mRTv 32
mRTv 3
=
What does this mean in a practical sense?
Life is not fair, not everybody has the same energy….
Mean Free Path for He = 2×10-7 meters or about 1000 He diameters
Problem: What is the average speed of N2 diatomics at two different temperatures (-25
°C and 37 °C).
Remember, mRTv 3
=
…at 37 °C:
( )( )( )( ) sm
moleKgKKmoleJv /525
/028.0310/314.83
==
… at – 25 °C
( )( )( )( ) sm
moleKgKKmoleJv /470
/028.0248/314.83
==
Movement of Gases Through Space
Diffusion Effusion
Gases can permeate (effuse) through porous
materials like membranes made of plastic…this affects bottling of such
things as sodas and beers…
Graham’s Law:
mwmovementofRate 1
∝
mw = molecular weight
Graham’s law applied
2
2
1
2
21
21
gasgas
mvmv
=
( ) ( ) 22
12
gasgas mvmv =
( )( ) 1
22
2
21
mm
vv
=
1
2
1
2
2
1
mm
mm
vv
==
Example: What are the relative rates of N2 (mw
= 28 amu) and C2H2 (26 amu) gases?
04.110.529.5
2628
===
… rate 1 = 1.04 × rate 2
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