Kinetic Molecular Theory & Gases

Kinetic Molecular Theory & Gases An Honors/AP Chemistry Presentation

Kinetic Molecular Theory • Kinetic means motion • So the K.M.T. studies the motions of molecules. • Solids - vibrate a little • Liquids - vibrate, rotate, and translate (a little) • Gases - vibrate, rotate, and translate (a lot)!

Gases consist of large numbers of molecules in continuous random motion. The volume of the molecules is negligible compared to the total volume. Intermolecular interactions are negligible. When collisions occur, there is a transfer of kinetic energy, but no loss of kinetic energy. The average kinetic energy is proportional to the absolute temperature. Basic Assumptions of KMT

Gas Properties • Volume - amount of space (L or mL) • Temperature - relative amount of molecular motion (K) • Pressure - the amount of force molecules exert over a given area (atm, Torr, Pa, psi, mm Hg) • Moles - the number of molecules (mol)

Temperature Conversions • C = 5/9(F-32) • F = 9/5C + 32 • K = C + 273 • So what is the absolute temperature (K) of an object at -40 oF?

Answer to Temperature Conversion • -40 oF = -40 oC • -40 oC = 233 K or 230 K

Pressure Conversions • 1 atm = 760 mm Hg = 760 Torr = 101,325 Pa = 14.7 psi • How many atmospheres is 12.0 psi? • How many Torr is 1.25 atm? • How many Pascals is 720 mm Hg?

Answers to Pressure Conversions • 12.0 psi = .816 atm • 1.25 atm =950. Torr • 720 mm Hg = 96000 Pa

A Barometer • A mercury barometer measures air pressure by allowing atmospheric pressure to press on a bath of mercury, forcing mercury up a long tube. The more pressure, the higher the column of mercury.

More on the barometer • Although American meteorologists will sometimes measure the height in inches, typically this pressure is measured in mm Hg. • 1 mm Hg = 1 Torr

S.T.P. • When making comparisons we often use benchmarks or standards to compare against. • In chemistry Standard Temperature is 0 oC (273K) and Standard Pressure is 1 atm.

Boyle’s Law • If the amount and temperature of the gas are held constant, then the volume of a gas is inversely proportional to the pressure it exerts. • Mathematically this means that the pressure times the volume is a constant. • P*V = k • P1V1=P2V2

Boyle’s Law in Action

Sample Questions • The volume of a balloon is 852 cm3 when the air pressure is 1.00 atm. What is the volume if the pressure drops to .750 atm? • A gas is trapped in a 2.20 liter space beneath a piston exerting 25.0 psi. If the volume expands to 2.75 L, what is the new pressure?

The Answers are… • P1V1=P2V2 • (1atm)(852cm3)= (.750atm)*V2 V2 =1140cm3 • (25.0psi)(2.20L)=P2(2.75L) P2 = 20.0 psi

Charles’ Law • If the amount and the pressure of a gas are held constant, then the volume of a gas is directly proportional to its absolute temperature. • Mathematically, this means that the volume divided by the temperature is a constant. • V/T = k • V1/T1=V2/T2

Charles Law in Action

Sample Questions • The volume of a balloon is 5.00 L when the temperature is 20.0 oC. If the air is heated to 40.0 oC, what is the new volume? • 3.00 L of air are held under a piston at 0.00 oC. If the air is allowed to expand at constant pressure to 4.00 L, what is the new Celsius temperature of the gas?

The Answers Are… • V1/T1=V2/T2 • 5.00L/293K = V2/313K V2=5.34L • 273K/3.00L = T2/4.00L T2=364K=91oC

The Gay-Lussac Law • If the amount and volume of the gas are held constant, then the pressure exterted by the gas is directly proportional to its absolute temperature. • Mathematically this means that the pressure divided by the temperature is a constant. • P/T = k • P1/T1=P2/T2

The Gay-Lussac Law in Action

Sample Questions • A tank of oxygen is stored at 3.00 atm and -20 oC. If the tank is accidentally heated to 80 oC, what is the new pressure in the tank? • A piston is trapped in place at a temperature of 25 oC and apressure of 112 kPa. At what celcius temperature is the pressure 102 kPa?

The Answers are… • P1/T1=P2/T2 • (3atm)/(253 K)= P2/ (353 K) P2 =4.19 atm • (298 K)/(112 kPa)=T2/(102kPa) T2 = 271 K = -2 oC

Avogadro’s Law • If the temperature and the pressure of a gas are held constant, then the volume of a gas is directly proportional to the amount of gas. • Mathematically, this means that the volume divided by the # of moles is a constant. • V/n = k or V/m = k • V1/n1=V2/n2 or V1/m1=V2/m2

Avogadro’s Law in Action

Sample Questions • The volume of a balloon is 5.00 L when there is .250 mol of air. If 1.25 mol of air is added to the balloon, what is the new volume? • 3.00 L of air has a mass of about 4.00 grams. If more air is added so that the volume is now 24.0 L, what is the mass of the air now?

The Answers Are… • V1/n1=V2/n2 or V1/m1=V2/m2 • 5.00L/.250 mol = V2/1.50 mol V2=30.0 L • 4.00g/3.00L = m2/24.0L m2=32.0 g

The Combined Gas Law • This law combines the inverse proportion of Boyle’s Law with the direct proportions of Charles’, Gay-Lussac’s, and Avogadro’s Laws. • P1V1/(n1T1) = P2V2/(n2T2) • or • P1V1/T1 = P2V2/T2

Four Gas Laws in One • The combined gas law could be used in place of any of the previous 4 gas laws. • For example, in Boyle’s Law, we assume that the amount and temperature are constant. So if we cross them off of the combined gas law: • P1V1/(n1T1) = P2V2/(n2T2) • P1V1 = P2V2

Another Example • A sample of hydrogen has a volume of 12.8 liters at 104 oF and 2.40 atm. What is the volume at STP?

The answer is: • P1V1/(n1T1) = P2V2/(n2T2) • P1=2.40atm,V1=12.8L, T1=104oF=40oC=313K, T2=273K, P2=1atm, n1=n2 • (2.4atm)(12.8L)/(313K) = (1atm)V2/(273K) • V2 = 26.8 L

The Ideal Gas Law • If, P1V1/(n1T1) = P2V2/(n2T2) • Then PV/(nT) = constant • That constant is R, the ideal gas law constant. • R = .0821 L*atm/(mol*K) • R = 8.314 J/(mol*K) • So, PV=nRT

But what about… • Since n = m/M, we can substitute into PV = nRT and get • PVM = mRT • Since D = m/V, we can substitute in again and get • PM = DRT

So which one is it? • Like a good carpenter, it is good to have many tools so that you can choose the right tool for the right job. • If I am solving a gas problem with density, I use PM = DRT. • If I am solving a gas problem with moles, I use PV = nRT. • If I am solving a gas problem with mass, I use PVM = mRT.

Such as…. • Under what pressure would oxygen have a density of 8.00 g/L at 300 K? • PM = DRT • P(32 g/mol) = (8 g/L)(.0821 latm/molK)(300 K) • P = 6.16 atm

An Important Number • What is the volume of 1 mole of a gas at STP? • PV = nRT V = nRT/P • V = (1mol)*(.0821Latm/molK)(273K)/ (1atm) • V = 22.4 L • This is called the standard molar volume of an ideal gas.

Gas Stoichiometry • We had said that stoichiometry implied a ratio of molecules, or moles. Up until now we only used mole ratios. • However Avogadro said that the volume is directly proportional to the number of molecules. • This means that we can do stoichiometry with volume or moles.

Example 1 of Gas Stoichiometry • What volume of hydrogen is needed to synthesize 6.00 liters of ammonia? • N2 (g) + 3 H2 (g) --> 2 NH3 (g) • 6.00 L H2 x (2 NH3/3 H2) = 4.00 L NH3

Example 2 of Gas Stoichiometry • What mass of nitrogen is needed to synthesize 20.0 L of ammonia at 1.50 atm and 25 oC? • N2 (g) + 3 H2 (g) --> 2 NH3 (g) • 20.0 L NH3 x (1 N2/2 NH3) = 10.0 L N2 • PVM = mRT • (1.5 atm)(10 L)(28 g/mol) = m(.0821Latm/molK)(298K) • m = 17.2 g N2

Dalton’s Law • When we talk about air pressure, we need to understand that air is not oxygen. • Air is a solution of nitrogen (78.09%), oxygen (20.95%), argon (.93%), and CO2 (.03%). • So when we talk about air pressure, which gas are we talking about?

ALL OF THEM! • Dalton’s Law of Partial Pressures states that the total pressure of a system is equal to the sum of the partial (or individual) pressures of each component. • Ptotal = P1 + P2 + … Px • So if air pressure is 1 atm, then we can assume that the N2 is .78 atm, the O2 is .21 atm, and the Ar is about .01 atm.

A Corollary • If we extend Boyle’s Law and Avogadro’s Law, we could infer that, at constant temperature and volume, the pressure of a gas is directly proportional to its pressure. • P1/Ptotal = n1/ntotal

An important example • A sample of CaCO3 is heated, releasing CO2, which is collected over water (a typical practice). • The pressure in the collection bottle is the sum of the pressure of the CO2 plus the pressure of the water vapor (since some water always evaporates). • Ptotal = PCO2 + PH2O

Sample Water Vapor Pressures

So in our example • If a total pressure of 365 Torr is collected at 25 oC in a 100 ml collection bottle: • What is the partial pressure of CO2? • What mass of CaCO3 decomposed?

Here’s how it works • Ptotal = PCO2 + PH2O • 365 Torr = PCO2 + 23.8 Torr • PCO2 = 341.2 Torr = .449 atm • PVM = mRT • (.449 atm)(.100 L)(44.0 g/mol) = m(.0821Latm/molK)(298K) • m = .0807 g CO2

Corollary Problem • A gas collection bottle contains .25 mol of He, .50 mol Ar, and .75 mol of Ne. If the partial pressure of Helium is 200 Torr: • What is the total pressure in the system? • What are the partial pressures of Ne and Ar?

The answers are… • nHe = .25 mol, nAr = .50 mol, nNe = .75 mol, PHe = 200 Torr. • ntotal =1.50 mol • Ptotal/Phe = ntotal/nHe • Ptotal/200Torr = 1.50 mol/.25 mol • Ptotal = 1200 Torr • PAr/Ptotal = nAr/ntotal • Par/1200 = .50 mol/1.50 mol • PAr = 400 Torr • PNe = 1200 Torr - 400 Torr - 200 Torr • Pne = 600 Torr

Temperature and Kinetic Energy • Earlier, I stated that temperature is a relative measure of molecular motion. • By definition, Kinetic energy is a measure of the energy of motion. • Pretty similar right?

Yes they are • KEav = 3/2*R*T • The average kinetic energy depends only on the absolute temperature. • R, the Ideal Gas Law Constant, should be 8.314 J/molK, since we will want the energy in the proper SI unit of Joules.

Kinetic Molecular Theory & Gases