“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.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 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 MU 2 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 = o K = o C o

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 = k X T

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. P 1 V 1 /T 1 = P 2 V 2 /T 2 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) = 0 o C OR 273 o 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 MU 2 WE COULD SHOW THAT THE RATIOS OF THE RATES OF EFFUSION FOR TWO GASES ARE INVERSELY PROPORTIONAL TO THEIR MOLECULAR MASSES. GRAHAM’S LAW OF EFFUSION

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 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