Mullis1 Characteristics of Gases ► Vapor = term for gases of substances that are often liquids/solids under ordinary conditions ► Unique gas properties 1.Highly compressible 2.Inverse pressure-volume relationship 3.Form homogeneous mixtures with other gases

Mullis2 Pressures of Enclosed Gases and Manometers ► Barometer: Used to measure atmospheric pressure* ► Manometer: Used to measure pressures of gases not open to the atmosphere ► Manometer is a bulb of gas attached to a U-tube containing Hg.  If U-tube is closed, pressure of gas is the difference in height of the liquid.  If U-tube is open, add correction term:  If P gas < P atm then P gas + P h = P atm  If P gas > P atm then P gas = P h + P atm * Alternative unit for atmospheric pressure is 1 bar = 10 5 Pa

Mullis3 Gas Densities and Molar Mass ► Need units of mass over volume for density (d) ► Let M = molar mass (g/mol, or mass/mol) PV = nRT M PV = M nRT M P/RT = n M /V M P/RT = mol(mass/mol)/V M P/RT = density M = dRT P

Mullis4 Sample Problem: Density 1.00 mole of gas occupies 27.0 L with a density of 1.41 g/L at a particular temperature and pressure. What is its molecular weight and what is its density at STP? M.W. = 1.41 g|27.0 L = 38.1 g___ L|1.0 molmol L|1.0 molmol M = dRT d= M P = 38.1 g (1 atm)______________ = 1.70 g/L PRT mol ( L-atm )(273K) PRT mol ( L-atm )(273K) ( mol-K ) ( mol-K ) OR…AT STP: 38.1 g | 1 mol = 1.70 g/L mol | 22.4 L mol | 22.4 L

Mullis5 Example: Molecular Weight A g sample of a pure gaseous compound occupies 310. mL at 100. º C and 750. torr. What is this compound’s molecular weight? n=PV = (750 torr)(.360L) = mole RT62.4 L-torr(373 K) RT62.4 L-torr(373 K) mole-K mole-K MW = x g_= g = 32.0 g/mol mol mol mol mol

Mullis6 Partial Pressures ► Gas molecules are far apart, so assume they behave independently. ► Dalton: Total pressure of a mixture of gases is sum of the pressures that each exerts if it is present alone. P t = P 1 + P 2 + P 3 + …. + P n P t = (n 1 + n 2 + n 3 +…)RT/V = n i RT/V ► Let n i = number of moles of gas 1 exerting partial pressure P 1 : P 1 = X 1 P 1 where X 1 is the mole fraction (n 1 /n t ) P 1 = X 1 P 1 where X 1 is the mole fraction (n 1 /n t )

Mullis7 Collecting Gases Over Water ► It is common to synthesize gases and collect them by displacing a volume of water. ► To calculate the amount of a gas produced, correct for the partial pressure of water: ► P total = P gas + P water ► The vapor pressure of water varies with temperature. Use a reference table to find.

Mullis8 Kinetic Molecular Theory ► Accounts for behavior of atoms and molecules ► Based on idea that particles are always moving ► Provides model for an ideal gas ► Ideal Gas = Imaginary: Fits all assumptions of the K.M. theory ► Real gas = Does not fit all these assumptions

Mullis9 5 assumptions of Kinetic-molecular Theory 1. Gases = large numbers of tiny particles that are far apart. 2. Collisions between gas particles and container walls are elastic collisions (no net loss in kinetic energy). 3. Gas particles are always moving rapidly and randomly. 4. There are no forces of repulsion or attraction between gas particles. 5. The average kinetic energy of gas particles depends on temperature.

Mullis10 Kinetic energy ► The absolute temperature of a gas is a measure of the average* kinetic energy. ► As temperature increases, the average kinetic energy of the gas molecules increases. ► As kinetic energy increases, the velocity of the gas molecules increases. ► Root-mean square (rms) speed of a gas molecule is u. ► Average kinetic energy, ε,is related to rms speed: ε = ½ mu 2 where m = mass of molecule *Average is of the energies of individual gas molecules.

Mullis11 Maxwell-Boltzmann Distribution ► Shows molecular speed vs. fraction of molecules at a given speed ► No molecules at zero energy ► Few molecules at high energy ► No maximum energy value (graph is slightly misleading: curves approach zero as velocity increases) ► At higher temperatures, many more molecules are moving at higher speeds than at lower temperatures (but you already guessed that) Just for fun: Link to mathematical details: Source:

Mullis12 Molecular Effusion and Diffusion ► Kinetic energy ε = ½ mu 2 ► u = 3RT Lower molar mass M, higher rms speed u M Lighter gases have higher speeds than heavier ones, so diffusion and effusion are faster for lighter gases.

Mullis13 Graham’s Law of Effusion ► To quantify effusion rate for two gases with molar masses M 1 and M 2 : r 1 = M 2 r 2 M 1 ► Only those molecules that hit the small hole will escape thru it. ► Higher speed, more likely to hit hole, so r 1 /r 2 = u 1 /u 2

Mullis14 Sample Problem: Molecular Speed Find the root-mean square speed of hydrogen molecules in m/s at 20º C. 1 J = 1 kg-m 2 /s 2 R = J/mol-K R = kg-m 2 /mol-K-s 2 u 2 = 3RT = 3(8.314 kg-m 2 /mol-K-s 2 )293K M g |1 kg___ M g |1 kg___ mol |1000g u 2 = 3.62 x 10 6 m 2 /s 2 u = 1.90 x 10 3 m/s

Mullis15 Example: Using Graham’s Law An unknown gas composed of homonuclear diatomic molecules effuses at a rate that is only times that of O 2 at the same temperature. What is the unknown gas? r x = M O = 32.0 g/mol r O2 M x 1 M x Square both sides: = 32.0 g/mol M x M x = 32.0 g/mol = 254 g/mol  Each atom is 127 g, so gas is I so gas is I 2

Mullis16 The van der Waals equation ► Add 2 terms to the ideal-gas equation to correct for 1.The volume of molecules (V-nb) 2.Molecular attractions (n 2 a/V 2 ) Where a and b are empirical constants. P + n 2 a (V – nb) = nRT V 2 V 2 ► The effect of these forces—If a striking gas molecule is attracted to its neighbors, its impact on the wall of its container is lessened.