Introduction - Physical Chemistry I - Handout, Exercises of Physical Chemistry

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Introduction physical chemistry establishes and develops the principles of chemistry concepts used to explain and interpret observations on the physical and chemical properties of matter central theme: • systems • states • processes topics of physical chemistry: (1) the study of the macroscopic properties of sys- tems of many atoms or molecules (2) the study of processes which such systems can undergo (3) the study of the properties of individual atoms and molecules PChem I 1.1 (4) the study of the relationship between microscop- ic (atomic or molecular) properties and macroscopic properties main areas of physical chemistry thermodynamics (1) and (2) quantum mechanics (3) statistical mechanics (4) kinetics and transport (2) and (3) this semester: topics (1) and (2), i.e., equilibrium thermodynamics and kinetics focus on systems of many elementary building blocks, large (classical) systems N =O(NA) macroscopic viewpoint: “forget” the existence of atoms and molecules −→ physical chemistry more abstract (and more mathematical) than other chemistry courses example of a thermodynamic system: water, pure H2O, say 1 L at ambient pressure (1 atm) PChem I 1.2 further examples of thermodynamic systems: cell in a nutrient solution, mitochondrion in a cell (??) are these large systems? is thermodynamics applica- ble to these biological systems? [ATP] ∼ 10−3 mol L−1 = 10−3 M = 1 mM, size of a mi- tochondrion: diameter ∼ 0.5 µm, length ∼ 2 µm =⇒ NATP ∼ 200000; macroscopic? [H+] ∼ 10−7 M (neutral pH) =⇒ NH+ ∼ 20; macroscop- ic?? importance of fluctuations if the average ∼ N , then the variance ∼ pN =⇒ relative strength of fluctuations ∼ 1/pN (p N N = 1pN ) ATP: fluctuations ∼ 1/p200000 = 2×10−3 H+: fluctuations ∼ 1/p20 = 2×10−1 macroscopic system: sufficiently large number N of atoms, molecules, ions, . . . , such that 1/ p N ¿ 1 PChem I 1.5 Thermodynamics large systems two types of variables: intensive – extensive T , P , . . . V , m, . . . historically: empirical observations concerning rela- tions between such variables example: PV = nRT origins of thermodynamics: practical interest: heat generates motion evolved into a theory that describes transformation of states of matter in general thermodynamics is particular good in dealing with complex systems, since the exact nature of the con- stituents and microscopic processes is irrelevant two conceptual innovations of thermodynamics First Law: conservation of energy Second Law: entropy PChem I 1.6 1. The Properties of Gases Notation: (i) I use an overbar to denote molar quan- tities; the textbook uses a subscript m. Example: mo- lar volume, these notes V , textbook Vm. (ii) I use a capital P for pressure; the textbook uses p. simple system to learn the concepts and methods of thermodynamics gas: fills any container dilute gas; chemically pure macroscopic description state of pure gas: V volume of container, T temper- ature of the gas, P pressure of the gas, n amount of the gas = number of moles assume no electric or magnetic properties pressure P = F A PChem I 1.7 thermodynamic temperature scale: Θ◦C, melting point of ice: 0◦C at 1 atm; boiling point of water: 100◦C at 1 atm T K ≡ Θ◦C +273.15 thermodynamic equilibrium = thermal equilibrium, TA = TB and mechanical equilibrium, PA = PB thermodynamic equilibrium states P = f (n,V , T ) ideal gas PV = nRT SI units: V m3, P Pa, T K: R = 8.314 J K−1mol−1 PChem I 1.10 1 J = 1 N m alternative units: V L, (1 L = 1 dm3 = 10−3 m3) P atm: R = 8.206×10−2 L atm K−1mol−1 consider a system with n = const isotherm: T = const, PV = const, P ∼ 1/V , hyperbolas [Figure: Ideal gas isotherms; Atkins 9th ed., Fig. 1.4] PChem I 1.11 isobar: P = const, V ∼ T [Figure: Ideal gas isobars; Atkins 9th ed., Fig. 1.6] PChem I 1.12 T = 298.15 K, P = 1 bar =⇒ V = RT /P = 24.789 dm3mol−1 mixture of ideal gases Dalton’s law P = P1+P2+ . . . = ∑ j P j , P j = n j RT V mole fraction x j = n j n , n =∑ j n j P j = x j P for nonideal gases: P j ≡ x j P Real Gases deviations from ideal gas law: due to intermolecular forces PChem I 1.15 [Figure: Potential energy between two molecules; Atkins 9th ed., Fig. 1.13] attractive: dipole-dipole forces, H-bonds, dispersion forces repulsive: repulsion of electrons measure for deviations compression factor Z = PV RT PChem I 1.16 ideal gas (superscript ◦) V ◦ = RT P ⇒ Z ◦ ≡ 1 Z = V V ◦ [Figure: Compression factor; Atkins 9th ed., Fig. 1.14] Z > 1 ⇒V >V ◦ repulsive forces dominate Z < 1 ⇒V <V ◦ attractive forces dominate PChem I 1.17 Boyle temperature TB: B ′(TB) = 0 or B(TB) = 0 P → 0: if T = TB then Z → 1 and dZ /dP → 0 ⇒ extended range of ideal behavior [Figure: Boyle temperature; Atkins 9th ed., Fig. 1.16] Boyle temperatures for some gases: H2 109 K, CH4 510 K, C2H4 720 K, NH3 1030 K [Estrada-Torres, R.; Iglesias-Silva, G. A.; Ramos-Estrada, M. & Hall, K. R., Boyle temper- atures for pure substances, Fluid Phase Equilib., 258, 148–154 (2007), http://dx.doi.org/10.1016/j.fluid.2007.06.004] PChem I 1.20 real gas isotherms: example CO2 [Figure: CO2 isotherms; Atkins 9th ed., Fig. 1.15] phase transition: condensation, gas −→ liquid critical point: critical temperature = maximum tem- perature at which a gas can be liquefied PChem I 1.21 phase diagram critical temperature Tc, critical pressure Pc, critical molar volume V c: critical constants as the critical point is approached along the vapor pressure curve: ρ(l )−ρ(g ) → 0 or V (g )−V (l ) → 0 PChem I 1.22 continuity of states: the substance changes from liquid-like to gas-like in the supercritical region with- out ever changing phases virial equation of state good for quantitative work, B(T ), C (T ), . . . are tabulated for gases, but it does not provide understanding of the above phenomena of real gases find a “general” model; best known and most widely used: van der Waals gas repulsive interactions: hard spheres ⇒ excluded vol- ume, V →V −nb b is a material constant, equal to the volume of 1 mol of densely packed gas particles P (V −nb) = nRT or P = nRT V −nb PChem I 1.25 attractive forces: diminish pressure; pressure is the result of collisions of the gas particles with the walls; as a particle is about to hit the container wall, it is “held back”, and its impact is diminished, by the at- tractive forces from surrounding gas particles; this is a pair effect ∼ number of pairs of particles ∼ (n/V )2 = ρ2 P = nRT V −nb −a (n V )2 van der Waals equation a is a material constant; a and b are tabulated for real gases ( P + a V 2 )( V −b ) = RT compare van der Waals equation with virial equation B(T ) = b − a RT PChem I 1.26 How good is the van der Waals model? (V −b)V 2P = RT V 2−a(V −b) V 3 P −V 2bP = RT V 2−aV +ab V 3− ( b + RT P ) V 2+ a P V − ab P = 0 cubic equation ⇒ for fixed T and P , three real roots or one real root and two complex conjugate roots isotherms for the van der Waals equation for ammo- nia PChem I 1.27 between a minimum and a maximum is an inflection point (curvature changes): second derivative vanish- es critical point: at (Tc, Pc, V c) dP dV = 0 and d 2P dV 2 = 0 there are three critical constants, need one more equation: equation of state P = RT V −b − a V 2 (1) dP dV =− RT (V −b)2 + 2a V 3 = 0 at critical point (2) d2P dV 2 = 2RT (V −b)3 − 6a V 4 = 0 at critical point (3) equation (2)⇒ RTc = 2a(V c −b)2 V 3 c (4) PChem I 1.30 insert this result into equation (3) 4a(V c −b)2 V 3 c(V c −b)3 − 6a V 4 c = 0 (5) 2 V c −b − 3 V c = 0 2V c = 3V c −3b ⇒V c = 3b insert into equation (4) RTc = 2a(3b −b)2 (3b)3 = 2a ·4b 2 27b3 Tc = 8a 27bR insert into equation (1) Pc = 8a 27b 3b −b − a 9b2 = 8a 2 ·27b2 − a 9b2 Pc = a 27b2 PChem I 1.31 critical compression factor: Zc = PcV c RTc = a 27b2 ·3b 8a 27b Zc = 3 8 independent of the material constants a and b Z vdWc = 0.375; compare with experimental data range: 0.12 HF – 0.47 N2O4 mode: 0.27; 27% of all organic and inorganic sub- stances have this value of Zc 61% Zc in [0.26, 0.28]; 77% Zc in [0.25, 0.29]; 90% Zc in [0.23, 0.31] van der Waals gas: simple model that captures the essential aspects of real gases critical point is a very characteristic point for real gas- es ⇒ use critical constants as units reduced variables Pr = P Pc , V r = V V c , Tr = T Tc PChem I 1.32