Models Models are attempts to describe reality, that doesnt mean they necessarily have anything to do with reality Models describe some aspect(s) of a system governed by phenomena the model attempts to describe Variables In any model, looking at a process involves something that can change, a variable: Extensive variable: depends on the amount present (mass, volume) Intensive Variable: property is not additive, divisible (temperature) Models describing energy transfer fall under the study called thermodynamics
Variables For models, variables are key, and how some process changes a variable is the key to these models ex. As we heat a pool of water how does the amount of mineral dissolved change, as our car burns gas, how does its position change Describing these changes is done through differential calculus: Review of calculus principles Process (function) y driving changes in x: y=y(x), the derivative of this is dy/dx (or y(x)), is the slope of y with x By definition, if y changes an infinitesimally small amount, x will essentially not change: dy/dk=
y ( x x) y ( x) y ' ( x) lim x x 0 This derivative describes how the function y(x) changes in response to a variable, at any very small change in points it is analogous to the tangent to the curve at a point measures rate of change of a function Differential Is a deterministic (quantitative) relation between the rate of change (derivative) and a function that may be continually changing
dT q k dx dT q k 0 dx In a simplified version of heat transfer, think about heat (q) flowing from the coffee to the cup bigger T difference means faster transfer, when the two become equal, the reaction stops Partial differentials Most models are a little more complex, reflecting
the fact that functions (processes) are often controlled by more than 1 variable y y ( x x) y ( x) lim x 0 : x
x u,z u and z are constant How fast Fe2+ oxidizes to Fe3+ is a process that is affected by temperature, pH, how much O2 is around, and how much Fe2+ is present at any one time what does this function look like, how do we figure it out??? Total differential, dy, describing changes in y affected by changes in all variables (more than one, none held constant) y y
y dy dx du dz x u , z u x , z z x ,u Pictures of variable changes Temperature (C) 2 variables that affect a process: 2-axis x-y plot 3 variables that affect a process: 3 axis ternary plot (when only 2 variables are independent; know 2, automatically have #3) anorthoclase
Miscibility Gap 30 50 70 90 Orthoclase % NaAlSi O Albite 3 8 Properties derived from outer e -
Ionization potential energy required to remove the least tightly bound electron Electron affinity energy given up as an electron is added to an element Electronegativity quantifies the tendency of an element to attract a shared electron when bonded to another element. In general, first ionization potential, electron affinity, and electronegativities increase from left to right across the periodic table, and to a lesser degree from bottom to top. Ionic vs. Covalent Elements on the right and top of the periodic table draw electrons strongly
Bonds between atoms from opposite ends more ionic, diatomics are 100% covalent Bond strength Covalent>Ionic>metallic Affects hardness, melting T, solubility Bond type affects geometry of how ions are arranged More ionic vs. covalent = higher symmetry Atomic Radius A function partly of shielding, size is critical in thinking about substitution of ions, diffusion, and in coordination numbers Units review Mole = 6.02214x1023 units make up 1 mole, 1 mole of H+= 6.02214x1023 H+ ions, 10 mol FeOOH =
6.02214x1024 moles Fe, 6.02214x1024 moles O, 6.02214x1024 moles OH. A mole of something is related to its mass by the gram formula weight Molecular weight of S = 32.04 g, so 32.04 grams S has 6.02214x1023 S atoms. Molarity = moles / liter solution Molality = moles / kg solvent ppm = 1 part in 1,000,00 (106) parts by mass or volume Conversion of these units is a critical skill!! Lets practice! 10 mg/l K+ = ____ M K 16 g/l Fe = ____ M Fe 10 g/l PO43- = _____ M P 50 m H2S = _____ g/l H2S 270 mg/l CaCO3 = _____ M Ca2+ FeS2 + 2H+ Fe2+ + H2S
75 M H2S = ____ mg/l FeS2 GFW of Na2S*9H2O = _____ g/mol how do I make a 100ml solution of 5 mM Na2S?? Scientific Notation 4.517E-06 = 4.517x10-6 = 0.000004517 Another way to represent this: take the log = 10 -5.345 M k 1E+6 1000
1 d c m n p 0.1 0.01
1E-3 1E-6 1E-9 1E-12 Significant Figures Precision vs. Accuracy Significant figures number of digits believed to be precise LAST digit is always assumed to be an estimate Using numbers from 2 sources of differing precision must use lowest # of digits Mass = 2.05546 g, volume= 100.0 ml =
0.2055 g/l Logarithm review 103 = 1000 ln = 2.303 log x pH = -log [H+] 0.015 M H+ is what pH? Antilogarithms: 10x or ex (anti-natural log) pH = -log [H+] how much H+ for pH 2? Logarithmic transforms Log xy = log x + log y Log x/y = log x log y
Log xy = y log x Log x1/y = (1/y) log x ln s n a tr s m r fo he t
are me a s Line Fitting Line fitting is key to investigating experimental data and calibrating instruments for analysis Common assessment of how well a line fits is the R2 value 1 is perfect, 0 is no correlation Fe2+ oxidation log Fe2+ conc.
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