“One does not meet oneself until one catches the
reflection from an eye other than human.”
- Loren Eiseley -
OFTEN, IN ORDER TO EXPLAIN THE PROPERTIES OF A
SYSTEM THAT WE ARE STUDYING, WE USE A MODEL.
FOR EXAMPLE, WHEN WE FIRST STARTED DISCUSSING THE
NATURE OF ATOMS, WE USED THE BOHR SOLAR SYSTEM
MODEL OF THE ATOM.
AS IS OFTEN THE CASE WITH MODELS, WE START OUT WITH
A SIMPLE MODEL, AND AS ADDITIONAL PROPERTIES ARE
DISCOVERED, WE CAN MODIFY OUR MODEL TO EXPLAIN
THE MORE COMPLEX PROPERTIES. THIS WAS THE CASE
WITH THE BOHR MODEL.
IN THE CASE OF GASES, WE USE THE KINETIC MOLECULAR
MODEL TO DESCRIBE AND EXPLAIN PROPERTIES OF GASES
UNDER IDEAL CONDITIONS.
AS AN ASIDE, WE CAN OFTEN LEARN AS MUCH FROM THE
FAILURES OF A MODEL AS ITS SUCCESSES.
THE KINETIC MOLECULAR THEORY OF GASES
1. GASES CONSIST OF LARGE NUMBERS OF SMALL
PARTICLES (MOLECULES OR ATOMS) THAT ARE IN
CONTINUOUS, RANDOM MOTION.
2. THE VOLUME OF THE MOLECULES IS SMALL
COMPARED TO THE VOLUME OF THE CONTAINER.
3. THE ATTRACTIVE FORCES BETWEEN THE MOLECULES
AND BETWEEN THE MOLECULES AND WALLS OF THE
CONTAINER ARE NEGLIGIBLE. IN OTHER WORDS,
COLLISIONS ARE PERFECTLY ELASTIC.
CONSEQUENCES ON K-M THEORY:
1.THE AVERAGE KINETIC ENERGY OF THE MOLECULES IN
A CONTAINER OF GAS DOES NOT CHANGE WITH TIME AS
LONG AS THE TEMPERATURE REMAINS CONSTANT. THE
MOLECULES BOUNCE AROUND AND COLLIDE WITH EACH
OTHER AND THE WALLS OF THE CONTAINER, BUT THE
COLLISIONS ARE ELASTIC (NOT STICKY). ENERGY CAN
BE TRANSFERRED, BUT NOT LOST.
2.THE AVERAGE KINETIC ENERGY OF THE MOLECULES IS
PROPORTIONAL TO ABSOLUTE TEMPERATURE. AT A
GIVEN TEMPERATURE, ALL GASES WILL HAVE THE SAME
AVERAGE KINETIC ENERGY REGARDLESS OF SIZE,
SHAPE OR MASS.
MOLECULAR PICTURE OF PRESSURE:
PRESSURE IS THE RESULT OF COLLISIONS OF THE
MOLECULES WITH THE WALLS OF THE CONTAINER.
PRESSURE IS DETERMINED BY HOW HARD AND HOW
OFTEN THE MOLECULES STRIKE THE WALLS OF THE
CONTAINER.
SEE SIMULATION.
BOYLE’S LAW STATES THAT AT CONSTANT TEMPERATURE
FOR A FIXED MASS OF GAS, THE VOLUME IS INVERSELY
PROPORTIONAL TO THE PRESSURE.
V = k/P
ANOTHER WAY OF STATING THIS IS THAT THE PRODUCT OF
THE VOLUME AND PRESSURE IS A CONSTANT.
PV = k
YOU CAN REASON THIS OUT USING THE K-M THEORY.
IF THE NUMBER OF MOLECULES STAYS THE SAME AND THE
TEMPERATURE STAYS THE SAME, AS THE VOLUME IS
REDUCED, THE NUMBER OF COLLISIONS PER UNIT TIME
WITH THE WALLS OF THE CONTAINER WILL INCREASE
(PRESSURE INCREASES).
VIRTUAL EXPERIMENT FOR HOMEWORK: GO TO
http://www.chem.iastate.edu/group/Greenbowe/sections/projec
tfolder/flashfiles/gaslaw/boyles_law_graph.html
COLLECT DATA AND SHOW THAT BOYLE’S LAW HOLDS.
SELECT YOUR GAS OF CHOICE.
MOLECULAR PICTURE OF TEMPERATURE
THE ABSOLUTE TEMPERATURE OF A GAS IS A MEASURE OF
ITS KINETIC ENERGY
KE = 1/2 MU2
DIFFERENT GASES AT THE SAME ABSOLUTE TEMPERATURE
HAVE THE SAME AVERAGE KINETIC ENERGY.
IF THE ABSOLUTE TEMPERATURE IS DOUBLED, THE
AVERAGE KINETIC ENERGY IS DOUBLED.
AS THE ABSOLUTE TEMPERATURE APPROACHES
ABSOLUTE ZERO, THE MOLECULES SLOW DOWN. AT
ABSOLUTE ZERO, ALL MOLECULAR MOTION STOPS.
NOTE: ABSOLUTE TEMPERATURE = oK = oC + 273o
THE ABSOLUTE TEMPERATURE DETERMINES THE AVERAGE
KINETIC ENERGY, BUT ALL MOLECULES IN A GAS DO NOT
HAVE THE SAME VELOCITY. SOME MOVE SLOW, AND
OTHERS MOVE VERY FAST. YOU GET A DISTRIBUTION OF
SPEEDS.
ACCORDING TO THE K-M THEORY, PRESSURE IS A
RESULT OF THE FORCE WITH WHICH THE MOLECULES
HIT THE WALLS OF THE CONTAINER.
IF THE VOLUME IS KEPT CONSTANT, AS THE ABSOLUTE
TEMPERATURE INCREASES, BOTH THE FREQUENCY
AND THE SPEED WITH WHICH THE MOLECULES IN A
GAS STRIKE THE WALLS OF THE CONTAINER WILL
INCREASE, AND THE PRESSURE INCREASES.
IF THE PRESSURE IS KEPT CONSTANT, AS THE
TEMPERATURE INCREASES, THE VOLUME WOULD
INCREASE.
THIS IS CHARLES LAW, WHICH STATES THAT AT
CONSTANT PRESSURE, THE VOLUME IS DIRECTLY
PROPORTIONAL TO THE ABSOLUTE TEMPERATURE.
V =kXT
AVOGADRO’S LAW
IN 1811, AMEDEO AVOGADRO STATED HIS GAS LAW THAT
AT CONSTANT TEMPERATURE AND PRESSURE EQUAL
VOLUMES OF GASES HAVE THE SAME NUMBER OF
MOLECULES (NUMBER OF MOLES).
THIS CAN BE STATED MATHEMATICALLY AS
V = kn
WHERE n = NUMBER OF MOLES
IF WE TAKE THESE THREE LAWS
BOYLE’S LAW V = k/P
CHARLES’ LAW V = kT
AVOGADRO’S LAW V = kn
WE CAN COMBINE THEM INTO A SINGLE EQUATION, THE
COMBINED GAS LAW EQUATION.
P1V1/T1 = P2V2/T2
WHERE n IS CONSTANT
OR, THE IDEAL GAS LAW EQUATION
PV = nRT
R = 8.31 (L kPa) / (K mol)
ONE CONSEQUENCE OF AVOGADRO’S LAW (AND THE
IDEAL GAS EQUATION) IS THAT AT STANDARD CONDITIONS
(STP) ONE MOLE OF A GAS OCCUPIES 22.4 LITERS.
STANDARD TEMPERATURE AND PRESSURE (STP) = 0o C OR
273o K AND 1 ATM PRESSURE (101.3 kPa).
THIS IS AN IMPORTANT CONCEPT, BECAUSE IT ALLOWS US
TO EXPERIMENTALLY DETERMINE THE MOLECULAR MASS
OF A GASEOUS SUBSTANCE.
IN OUR KINETIC MOLECULAR THEORY APPROACH TO
MODELING GAS BEHAVIOR, WE MAKE TWO BIG
ASSUMPTIONS.
1)THE VOLUME OF THE MOLECULES IS VERY SMALL
COMPARED TO THE VOLUME OF THE CONTAINER.
2)THE COLLISIONS OF THE MOLECULES ARE PERFECTLY
ELASTIC.
REAL GASES APPROACH IDEAL BEHAVIOR UNDER LOW
PRESSURE AND HIGH TEMPERATURE.
AT HIGH PRESSURE, THE VOLUME THAT THE MOLECULES
OCCUPY STARTS TO BECOME SIGNIFICANT COMPARED TO
THE CONTAINER VOLUME.
AT LOW TEMPERATURE, THE SPEEDS OF THE MOLECULES
ARE REDUCED, SO THE INTERMOLECULAR FORCES OF
ATTRACTION BECOME MORE SIGNIFICANT – STICKY
COLLISIONS.
EFFUSION IS THE ESCAPE OF A GAS THROUGH A TINY PIN
HOLE IN THE CONTAINER INTO A VACUUM.
TAKING THE EQUATION FOR KINETIC ENERGY
KE = 1/2 MU2
GRAHAM’S LAW OF
EFFUSION
WE COULD SHOW THAT
THE RATIOS OF THE RATES OF EFFUSION FOR TWO GASES
ARE INVERSELY PROPORTIONAL TO THEIR MOLECULAR
MASSES.
DIFFUSION IS SIMILAR TO EFFUSION. DIFFUSION IS THE
SPONTANEOUS MIXING OF TWO DISSIMILAR GASES
(FLUIDS) THAT WERE INITIALLY SEPARATED.
THE RELATIVE RATES OF DIFFUSION ARE RELATED TO
THE AVERAGE SPEEDS OF THE MOLECULES, WHICH ARE
INVERSELY PROPORTIONAL TO THE MOLECULAR
MASSES.
WHILE THE SPEEDS OF THE MOLECULES ARE RELATIVELY
HIGH (500 M/S), THE RATES OF DIFFUSION ARE
RELATIVELY LOW BECAUSE OF THE COLLISIONS
BETWEEN MOLECULES.
AT THE DENSITY OF AIR AT SEA LEVEL, THERE ARE ON
THE ORDER OF 1010 COLLIONS PER SECOND PER
MOLECULE.
THE DISTANCE THAT A MOLECULE TRAVELS BETWEEN
COLLISIONS IS CALLED THE MEAN FREE PATH.
AT SEA LEVEL, THIS IS ABOUT 60 nm. AT 100 KM
ALTITUDE, IT IS ABOUT 0.1 METER.
THE PATH THAT A GAS MOLECULE FOLLOWS IS
REFERRED TO AS RANDOM WALK.
ON A LARGER SCALE, YOU CAN OBSERVE THIS BY
LOOKING AT PARTICLES OF DUST IN A SUN BEAM.
A GOOD DISCUSSION OF THE KINETIC-MOLECULAR
THEORY IS GIVEN AT
http://www.chem.ufl.edu/~itl/2045/lectures/lec_d.html