Models “Models are attempts to describe reality, that doesn’t 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 it’s 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= 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 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 How fast Fe 2+ oxidizes to Fe 3+ is a process that is affected by temperature, pH, how much O 2 is around, and how much Fe 2+ 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)

‘Pictures’ of variable changes 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) Miscibility Gap microcline orthoclase sanidine anorthoclase monalbite high albite low albite intermediate albite Orthoclase KAlSi 3 O 8 Albite NaAlSi 3 O 8 % NaAlSi 3 O 8 Temperature (ºC)

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 = x10 23 ‘units’ make up 1 mole, 1 mole of H+= x10 23 H + ions, 10 mol FeOOH = x10 24 moles Fe, x10 24 moles O, x10 24 moles OH. A mole of something is related to it’s mass by the gram formula weight  Molecular weight of S = g, so grams S has x10 23 S atoms. Molarity = moles / liter solution Molality = moles / kg solvent ppm = 1 part in 1,000,00 (10 6 ) parts by mass or volume Conversion of these units is a critical skill!!

Let’s practice! 10 mg/l K+ = ____  M K 16  g/l Fe = ____  M Fe 10  g/l PO 4 3- = _____  M P 50  m H 2 S = _____  g/l H 2 S 270 mg/l CaCO 3 = _____ M Ca 2+ FeS 2 + 2H +  Fe 2+ + H 2 S 75  M H 2 S = ____ mg/l FeS 2 GFW of Na 2 S*9H 2 O = _____ g/mol how do I make a 100ml solution of 5 mM Na 2 S??

Scientific Notation 4.517E-06 = 4.517x10 -6 = Another way to represent this: take the log = Mkdcm  np 1E E-31E-61E-91E-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 = g, volume= ml = g/l

Logarithm review 10 3 = 1000 ln = log x pH = -log [H + ]  M H+ is what pH? Antilogarithms: 10 x or e x (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 x y = y log x Log x 1/y = (1/y) log x ln transforms are the same

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 R 2 value – 1 is perfect, 0 is no correlation