Characteristics of Gases

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

2. 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 Pgas < Patm then Pgas + Ph = Patm • If Pgas > Patm then Pgas = Ph + Patm * Alternative unit for atmospheric pressure is 1 bar = 105 Pa Mullis

3. 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 MPV = MnRT MP/RT = nM/V MP/RT = mol(mass/mol)/V MP/RT = density M = dRT P Mullis

4. 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 mol mol M = dRT d= M P = 38.1 g (1 atm)______________ = 1.70 g/L P RT mol (0.0821 L-atm )(273K) ( mol-K ) OR…AT STP: 38.1 g | 1 mol = 1.70 g/L mol | 22.4 L Mullis

5. Example: Molecular Weight A 0.371 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) = 0.0116 mole RT 62.4 L-torr(373 K) mole-K MW = x g_= 0.371 g = 32.0 g/mol mol 0.0116 mol Mullis

6. 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. Pt = P1 + P2 + P3 + …. + Pn Pt = (n1 + n2 + n3 +…)RT/V = ni RT/V • Let ni = number of moles of gas 1 exerting partial pressure P1: P1 = X1P1 where X1 is the mole fraction (n1/nt) Mullis

7. 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: • Ptotal = Pgas + Pwater • The vapor pressure of water varies with temperature. Use a reference table to find. Mullis

8. 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 Mullis

9. 5 assumptions of Kinetic-molecular Theory • Gases = large numbers of tiny particles that are far apart. • Collisions between gas particles and container walls are elastic collisions (no net loss in kinetic energy). • Gas particles are always moving rapidly and randomly. • There are no forces of repulsion or attraction between gas particles. • The average kinetic energy of gas particles depends on temperature. Mullis

10. 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. Mullis

11. 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: http://user.mc.net/~buckeroo/MXDF.html Source: http://www.tannerm.com/maxwell_boltzmann.htm Mullis

12. 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. Mullis

13. Graham’s Law of Effusion • To quantify effusion rate for two gases with molar masses M1 and M2: r1 = M2 r2M1 • Only those molecules that hit the small hole will escape thru it. • Higher speed, more likely to hit hole, so r1/r2 = u1/u2 Mullis

14. Sample Problem: Molecular Speed Find the root-mean square speed of hydrogen molecules in m/s at 20º C. 1 J = 1 kg-m2/s2 R = 8.314 J/mol-K R = 8.314 kg-m2/mol-K-s2 u2= 3RT = 3(8.314 kg-m2/mol-K-s2)293K M2.016 g |1 kg___ mol |1000g u2= 3.62 x 106 m2/s2 u = 1.90 x 103 m/s Mullis

15. Example: Using Graham’s Law An unknown gas composed of homonuclear diatomic molecules effuses at a rate that is only 0.355 times that of O2 at the same temperature. What is the unknown gas? rx = MO2 0.355 = 32.0 g/mol rO2Mx 1 Mx Square both sides: 0.3552 = 32.0 g/mol Mx Mx = 32.0 g/mol = 254 g/mol  Each atom is 127 g, 0.3552 so gas is I2 Mullis

16. The van der Waals equation • Add 2 terms to the ideal-gas equation to correct for • The volume of molecules (V-nb) • Molecular attractions (n2a/V2) Where a and b are empirical constants. P + n2a (V – nb) = nRT V2 • 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. Mullis