Chapter 9. Properties of Gases and the Kinetic Molecular Theory Gas Mixtures: Partial Pressures Pressure of a Gas The Ideal Gas Law Kinetic Molecular Theory Maxwell-Boltzmann Velocity Distribution Real Gases Department of Chemistry, KAIST 1 1
An Overview of the Physical States of Matter The Distinction of Gases from Liquids and Solids 1. Gas volume changes greatly with pressure. 2. Gas volume changes greatly with temperature. 3. Gases have relatively low viscosity. 4. Most gases have relatively low densities under normal conditions. 5. Gases are miscible.
The three states of matter
9.1 Pressure of a Gas: Barometric Principles 1643, Torricelli Force necessary to suspend the barometric fluids comes from the pressure of the atmosphere
r = m/V independent of A !
at the same P, rHghHg = rH2OhH2O Can you calculate how long the water tube would be ?? 1 atm (standard atmosphere): 760 mmHg (0 oC, dry air, sea level) 760 Torr (independent of temperature) 101,325 Nm-2 (Pa) - SI unit 1,013,250 dyne cm-2 14.70 lb in-2
Effect of atmospheric pressure on objects at the Earth’s surface.
9.2 The Ideal Gas Law Boyle’s Law Charles’s Law V a 1 P n and T are fixed V x P = constant V = constant / P Charles’s Law V a T P and n are fixed V T = constant V = constant x T
Boyle’s Law
Department of Chemistry, KAIST Boyle’s Law Department of Chemistry, KAIST
Charles’ Law relationship between volume and temperature of a gas (at constant moles and Pressure).
Boyle’s Law Charles’s Law V a 1 P n and T are fixed V x P = constant V = constant / P Charles’s Law V a T P and n are fixed V T = constant V = constant x T Gay-Lussac showed that Boyle’s law continues to hold for different T, and Charles’ law for different P PV T = constant
Gay-Lussac’s experiment & Avogadro’s Explanation 1 volume hydrogen + 1 volume of chlorine 2 volumes of hydrogen chloride Avogadro: suggesting that 1) equal volumes of any gas contain equal numbers of gas molecules 2) gaseous hydrogen and chlorine are present as diatomic molecules Avogadro’s Law V a n P, T are fixed V n = constant N NA n =
combine PV T = constant & V n = constant we get Ideal Gas Law (equation of state)
Standard temperature and pressure (STP): T = 273.15 K (0 oC), P = 1 atm
Standard molar volume Equation 9.14
9.3 Gas Mixtures: Dalton’s Law of Partial Pressures Department of Chemistry, KAIST
Mixtures of Gases Dalton’s Law of Partial Pressures P1 = n1RT / V Gases mix homogeneously in any proportions Each gas in a mixture behaves as if it were the only gas present Dalton’s Law of Partial Pressures P1 = n1RT / V P2 = n2RT / V, … Ptotal = P1 + P2 + P3 + ... P1= c1 x Ptotal where c1 is the mole fraction c1 = n1 n1 + n2 + n3 +... = n1 ntotal PO2 = (0.209)(760 Torr) = 159 Torr
9.4 The Kinetic Molecular Theory Maxwell’s idea; molecules are in constant motion (random and chaotic) molecule’s coordinate is fixed at a fixed position ~ equal to the vector distance Translational kinetic energy = (1/2)mv2 (Cartesian velocity components) Probability of molecules for v ~ v+dv = 4 πv2dv
In a container of edge length L Consider one molecule
Now consider all N molecules
When this is compared to Molar kinetic energy Average energy per molecule kB = R/N Boltzmann’s constant absolute Temp is the measure of molecular motion !!
Apparatus for studying molecular velocity distribution
Maxwell-Boltzmann Distribution of Molecular Velocity Average value of energy or velocity characterizes the group average value is derived from a distribution of values
Department of Chemistry, KAIST Velocity distribution function as Temp increases, peak velocity is shifted to higher values 2) distribution is broadened Department of Chemistry, KAIST
Fraction of molecules with speeds in a given interval Interval: a to b approximation for small x
most probable molecular speed The distribution of speeds of three different gases at the same temperature The distribution of speeds for nitrogen gas molecules at three different temperatures most probable molecular speed
ū = most probable molecular speed Average speed 8RT ∏M ū = root-mean-square (rms) speed
8RT ∏M The path traveled by a single gas molecule: ū =
Wall collision rate: - Zwall = ¼ (N/V) v A Molecular collision rate: - Zmol = 2 (N/V) v d2 The path traveled by a single gas molecule: Mean free path: - = v/Zmol = 1/[2 (N/V)d2]
M2 M1 NH4Cl NH3 17 g/mol HCl 36 g/mol Gas diffusion is the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic properties. M2 M1 r1 r2 r = flux of molecules along a concentration gradient = NH4Cl Effusion is the escape of gas through a pin hole. Effusion rate is inversely proportional to M. NH3 17 g/mol HCl 36 g/mol
A molecular description of Boyle’s Law
A molecular description of Dalton’s law of partial pressures.
A molecular description of Charles’s Law
A molecular description of Avogadro’s Law
9.5 The Behavior of Real Gases The behavior of several real gases with increasing external pressure
The effect of intermolecular attractions on measured gas pressure Pideal = P + a(n/V)2 (P: actual pressure) actual pressure is smaller than the ideal pressure
Effect of excluded volume The effect of molecular volume on measured gas volume Effect of excluded volume Videal = V - nb actual volume is greater than the ideal volume
van der Waals Constants for Some Common Gases van der Waals equation for n moles of a real gas adjusts P up adjusts V down Gas a atm*L2 mol2 b L mol 0.034 0.211 1.35 2.32 4.19 0.244 1.39 1.36 6.49 3.59 2.25 4.17 5.46 He Ne Ar Kr Xe H2 N2 O2 Cl2 CO2 CH4 NH3 H2O 0.0237 0.0171 0.0322 0.0398 0.0511 0.0266 0.0391 0.0318 0.0562 0.0427 0.0428 0.0371 0.0305 See Example 9.5
Virial Equation of State B= B(T) 2nd viral coeff. C=C(T), … B= b-a/RT if for van der Waals gas (prove) B=0 at the Boyle temp