10.14.2011

# 10.14.2011 - Lecture 17 Monday October 14 2011...

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Unformatted text preview: Lecture 17 Monday, October 14, 2011 Today in Chemistry 260 •  Ideal Gases •  Real Gases •  Equations of State This Week in Chemistry 260 •  Reading: (Wed.) Ch. 9, (Fri.) 12.1,12.2 •  Kinetic theory of gases •  First law of thermodynamics, heat capacities, thermodynamics of gases •  Problem set 6 due Monday Monday, Oct. 24 Image from wikipedia Jacques Charles (1746- 1823) Why study gases? Robert Boyle (1627- 1691) Joseph Gay- Lussac (1778- 1850) William Thomson (Lord Kelvin) 1824- 1907 Gases: The Phenomenological/Macroscopic approach Amedeo Avogadro (1776- 1856) John Dalton (1766- 1844) Thomas Andrews (1813- 1885) The gaseous state is the simplest state of matter (compared to solids, liquids and plasmas) What properties deYine the state of a gaseous system? Extensive properties (proportional to the amount of material): Number of moles (n), volume (V), internal energy (U),… Intensive properties (independent of the amount of material): Concentration (C), temperature (T) , pressure (P)… What happens to a gas when these properties are allowed to vary? Johannes van der Waals (1827- 1923) 1 Charles- Gay- Lussac’s Law Boyle’s Law At a given temperature, with n held constant, the pressure varies inversely with volume. At a given volume, with n held constant, the pressure varies directly with temperature. 1 P ∝ or PV = constant V P ∝T Ideal P Increasing volume T Charles- Gay- Lussac’s Law Avogadro’s Law At a given volume, with n held constant, the pressure varies directly with temperature. At a given pressure, with n held constant, the volume varies directly with temperature. P ∝T V ∝T V ∝n At a given pressure, with T held constant, the volume varies directly with the number of atoms or molecules. or V = constant n Molar volume : V = Vm = V / n = 24.79 L/mol (at 25 C and 1 bar) V Increasing pressure or decreasing temperature All lines meet at the absolute zero temperature at V=0 http://www.grc.nasa.gov/WWW/K-12/airplane/aglussac.html n 2 Ideal Gas Equation of State Boyle’s Law (Robert Boyle, 1661) P ∝ 1 V (constant n & T ) Gay-Lussac’s Law V ∝ TLouisonstant n1802)p ) (Joseph (c Gay-Lussac, & P ∝ T (constant n & V ) Charles’s Law (Joseph Louis Gay-Lussac, 1802 Jacques Charles, unpublished, 1787) Avogadro’s Law (Amedeo Avogadro, 1811) V ∝ T (constant n & p ) V ∝ n (constant T & P) Pressure PV = nRT Volume: 1Pa=1Nm -2 1m3 =103 L 5 1bar=10 Pa 6 =10 mL 1atm=1.01325bar P ∝ T (constantonmoles ) Number f & V Ideal gas constant: =760torr R = 8.31451 J K −1 mol−1 = 0.08206 L atm K -1 mol-1 Absolute temperature: T / K = t / 0C + 273.15 = ( 6.022 ×1023 mol-1 ) k B Dalton’s Law of Partial Pressures Because an ideal gas considers non- interacting atoms or molecules without volume (point particles) it doesn’t matter what the species are. Each contributes according to its number of particles. Total pressure j nA + nB + ... PV = nRT 1. The gas particles are small (size/volume is negligible) 2. The particles are inert and exert no forces on each other. 3. The particles are in constant motion; elastic collisions with the walls of the container produce pressure. Elastic collision = no loss of kinetic energy in the collision, no change in the internal energy of the atom or molecule or the container wall. LOW CONCENTRATIONS, LOW PRESSURE, HIGH TEMPERATURE Real Gases Now – real gases are not ideal. 1. Real atoms and molecules have a Yinite volume. 2. Real atoms and molecules have attractive forces, van der Waals if nothing else. 2.0 P = PA + PB + K = ∑ Pj Pj = Underlying assumptions: 3. Real collisions are not always elastic. Partial pressures nj Ideal Gas Equation of State P = xj × P Mole fraction 1.5 PV Z= nRT 1.0 0.5 Z=1 Ideal gas 0 Z=compressibility 3 Real Gases N2 as a function of pressure at three temperatures. At lower temperatures and low pressures, the attractive intermolecular forces cause a negative deviation from the Ideal Gas value. At the higher temperatures, these do not contribute as signiYicantly. At all temperatures, if the pressure is high enough, there will be a positive deviation because of repulsive interactions. Real Gases Molar volume V V = Vm = n 2.0 Z < 1 Real gas attractive forces dominate Low temperature, pressure not too high Z > 1 Real gas repulsive forces dominate High pressure and lower temperature 2.0 1.5 Z= N2 as a function of pressure at three temperatures. PV nRT 1.0 Z= Z=1 Ideal gas 0.5 0 PV RT 1.5 1.0 Z=1 Ideal gas 0.5 0 Z=compressibility Z=compressibility The van der Waals equation of state The van der Waals equation of state 2\$ ! P + a n V &!V − nb\$ = nRT % # " %" ( Measured pressure ) Measured volume The would- have- been pressure in the absence of attractive interactions Effective volume ! \$! − nb\$ # P + a n V &"V % = nRT " % Atoms and molecules 2 are not inYinitely small. At very close distances, there are repulsive interactions between the particles ( ) Measured volume Repulsive interactions (excluded volume) Effective volume Veff = V - nb nb V = Veff + nb The measured volume is bigger than expected because it is the effective volume + the volume of the atoms or molecules. Repulsive interactions (excluded volume) Note that at smaller volumes (same # of particles), that the – nb term is more important # of particles Constant for a particular gas 4 The van der Waals equation of state The van der Waals equation of state 2\$ ! P + a n V &!V − nb\$ = nRT # " %" % ( Measured pressure ( ) The would- have- been pressure in the absence of attractive interactions ⎛ྎ n a⎜ྎ ⎝ྎ V The measured pressure is smaller than expected because it is the effective pressure - the inYluence of the attractive forces. Attractive interactions Reduce the rate of collisions with the wall, Pairs of 2 particles ⎞ྏ interact ⎟ྏ ⎠ྏ # of particles/unit volume (~how close the particles are to each other) ) Peff > Pmeasured Veff < Vsystem Z = PVideal / PVreal = V / (V − nb) > 1 P = Peff – a(n/V)2 All particles have long range attractive potentials … Vdd(r), Vind- ind(r), etc.) Constant for a particular gas 2\$ ! P + a n V &!V − nb\$ = nRT # " %" % 2.0 Z= PV RT 1.5 1.0 0.5 Z=1 Ideal gas 0 a ⎞ྏ ⎛ྎ Z = PidealV / PrealV = P / ⎜ྎ P + 2 ⎟ྏ < 1 V ⎠ྏ ⎝ྎ Some relevant values: Compound a (L2atm/mol2) b (L/mol) He 0.03412 0.02370 Ne 0.2107 0.01709 H2 0.2444 0.02661 Ar O2 1.345 1.360 0.03219 0.03803 N2 1.390 0.03913 CO 1.485 0.03985 CH4 2.253 0.04278 CO2 3.592 0.04267 NH3 4.170 0.03707 5 ...
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