Physical Chemistry: Equations and Important Facts

Atoms, Molecules, and Quantum Mechanics

• Avogadro’s number = 6.022 × 10²³
• Cations are smaller than the original ions; anions are larger than the original ions
• Five periodic trends
  • Ionization energy, electron affinity, and electronegativity increase from left to right, bottom to top
  • Atomic radius and metallic character increase from right to left, top to bottom
• Percent yield = 100 × (actual yield/theoretical yield)
• Quantum numbers
  • Principal quantum number, n, designates the shell level
  • Azimuthal quantum number, l, designates the subshell
  • Magnetic quantum number, m_l, designates the precise orbital of a given subshell
  • Electron spin quantum number, m_s, designates the spin number (½ and -½)
• Pauli exclusion principle states that no two electrons in the same atom can have the same four quantum numbers
• Heisenberg Uncertainty Principle states that there exists an inherent uncertainty in the product of the position of a particle and its momentum; Δx Δp ○ h
• Aufbau principle states that with each new proton added to create a new element, a new electron is added as well, to the lowest energy orbital available
• Hund’s rule states that electrons will not fill any orbital in the same subshell until all orbitals in that subshell contain at least one electron, and the unpaired electrons will have parallel spins
• Planck’s quantum theory: ΔE = hf, E_photon = hf, Planck’s constant = h = 6.63 × 10^-34 J s

Gases, Kinetics, and Chemical Equilibrium

• Standard temperature and pressure, STP is 0°C or 273 K, and 1 atm
• Mean free path is the distance traveled by a gas molecule between collisions
• Kinetic molecular theory, characteristics of an ideal gas
  • Gas molecules have zero volume
  • Gas molecules exert no forces other than repulsive forces due to collisions
  • Gas molecules make completely elastic collisions
  • The average kinetic energy of gas molecules is directly proportional to the temperature of the gas
• Ideal gas law: PV = nRT, universal gas constant = R = 0.08206 L atm/K mol = 8.314 J/K mol
• At STP, one mole of gas, behaving ideally, occupies the standard molar volume of 22.4 liters
• Partial pressure: P_a = X_a P_total
• Dalton’s law: P_total = P₁ + P₂ + P₃ +…
• KE_avg = 3/2 RT
• Graham’s law: v₁/v₂ = √m₂/√m₁
• Effusion is the spreading of a gas from a high pressure to a very low pressure through a pinhole (opening much smaller than the average distance between gas molecules
  • Effusion: effusion rate₁/effusion rate₂ = √M₂/√M₁
• Diffusion is the spreading of one gas into another gas or into empty space; can be calculated much like effusion rates
• Real gases deviate from ideal behavior when their molecules are close together
  • High pressures, low temperatures cause more deviations; generally deviate at pressures above ten atmospheres and temperatures near their boiling points
  • V_real > V_ideal, where PV_ideal = nRT
  • P_real < P_ideal, where P_idealV = nRT
• Two requirements for a given collision to create new molecules in a reaction
  • The relative kinetic energies of the colliding molecules must reach a certain activation energy
  • The molecules must have a proper spatial orientation
• The rate of a reaction increases with temperature mainly because more collisions with sufficient energy occur each second
• Elementary reactions are those that occur in a single step
• Intermediates are species that are products of one reaction and reactants of a later reaction in a chain
• Balanced equation: aA + bB → cC + dD
  • Rate law: rate_forward = k_f [A]^α[B]^β, where α and β are the orders of each respective reactant and the sum of α + β is the overall reaction order
  • If the reaction is elementary, α = a and β = b
• Irreversible first order reaction: A → products, rate = k_f [A]
  • A graph of ln[A] and time t gives a straight line with a slope of -k_f
• Irreversible second order reaction 2A → products, rate = k_f [A]²
  • A graph of 1/[A] and time t gives a straight line with a slope of k_f
• Second order reaction A + B → products, rate = k_f [A][B]
  • Does not have same graph as other second order reaction
• Irreversible third order reaction 3A → products, rate = k­_f [A]³
  • A graph of 1/2[A]² and time t gives a straight line with a slope of k_f
• In any multi-step reaction, the slow step determines the rate
• A catalyst increases the rate of the reaction without being consumed or permanently altered; most work by lowering the activation energy
• Equilibrium constant = K_c = [products]^coefficients/[reactants]^coefficients
• K_p = K_c (RT)^Δn, where Δn is the sum of the product coefficients minus the sum of the reactant coefficients
• Reaction quotient = Q = [products]^coefficients/[reactants]^coefficients
  • If Q is equal to K, then the reaction is at equilibrium
  • If Q is greater than K, there is a leftward shift towards the reactants
  • If Q is less than K, there is a rightward shift towards the products
• Le Châtelier’s principle states that when a system at equilibrium is stressed, the system will shift in a direction that will reduce that stress
  • Addition or removal of a product or reactant: adding a product or removing a reactant favors the reactant side, and vice versa
  • Changing the pressure of the system: increased pressure favors side with fewer moles and vice versa
  • Heating or cooling the system: heating favors the side without heat and vice versa

Thermodynamics

• The system is the macroscopic body under study, and the surroundings are everything else
• Three types of systems
  • Open systems exchange both mass and energy
  • Closed systems exchange energy but not mass
  • Isolated systems do not exchange energy or mass
• Extensive properties are proportional to the size of the system
• Intensive properties are independent of the size of the system
• Divide one extensive property by another to get an intensive property
• State functions are pathway independent and describe the state of the system
• Two ways to transfer energy between systems are heat q and work w
• Heat has three forms
  • Conduction is thermal energy transfer via molecular collisions; thermal conductivity: Q/t = kA T_h – T_c/L; heat flow is constant across any number of slabs between two heat reservoirs in a steady state system
  • Convection is thermal energy transfer via fluid movements
  • Radiation is thermal energy transfer via electromagnetic waves; Stefan-Boltzmann law: P = σεAT⁴, Stefan-Boltzmann constant = σ = 5.67 × 10^-8 W/m²K⁴, ε is the emissivity constant between 0 and 1
• PV work is work done at constant pressure, w = PΔV
• Work is path-dependent
• The first law of thermodynamics states that the energy of the system and the surroundings are always conserved
  • Energy change = ΔE = q + w, w = work done on the system
• The second law of thermodynamics states that heat cannot be changed completely into work in a cyclical process
  • Heat engines convert heat to work, a reversed heat engine is a refrigerator, which requires work
  • Carnot engines represent the most efficient engines possible, with efficiency = e = 1 – T_C/T_H
• Seven state functions
  • Internal energy = U
  • Temperature = T
  • Pressure = P
  • Volume = V
  • Enthalpy = H
  • Entropy = S
  • Gibbs free energy = G
• Internal energy is all the possible forms of energy imaginable on the molecular scale
• In a closed system at rest with no electric or magnetic fields, the only energy change will be in internal energy, ΔU = q + w; this becomes ΔU = q if there is no change in volume and no work of any kind
• The zeroth law of thermodynamics states that two systems in thermal equilibrium with a third system are in thermal equilibrium with each other; defined temperature
• Average kinetic energy of molecule in fluid: KE_avg = 3/2 kT, Boltzmann constant = k = 1.38 × 10^-23 J/K
• Enthalpy is a man-made property that accounts for the extra capacity in a system at rest to do PV work
  • H ≡ U + PV
  • ΔH = ΔU + PΔV, at constant pressure
• For a pure liquid or solid, the standard state is a reference form of a system at any chosen temperature T and a pressure of 1 bar or about 750 torr
• An element in its standard state at 25°C or 298 K has an enthalpy value of 0 J/mol
• The standard enthalpy of formation, ΔH°_f, is the change in enthalpy for a reaction that creates one mole of that compound from its raw elements in their standard state
• For reactions involving no change in pressure, ΔH = q
• Heat of reaction: ΔH°_reaction = ΔH_f°_products – ΔH_f°_reactants
• Hess’s law states that the sum of the enthalpy changes for each step is equal to the total enthalpy change regardless of the path chosen
• If ΔH is positive, the reaction is endothermic; if ΔH is negative, the reaction is exothermic
• Entropy is nature’s tendency toward disorder and its tendency to create the most probable situation occur with a system
• The second law of thermodynamics also states that the entropy of an isolated system will never decrease
• ΔS_system + ΔS_surroundings = ΔS_universe ≥ 0
• The third law of thermodynamics assigns by convention an entropy value of 0 J/K to any pure substance at absolute zero and in internal equilibrium
• ΔS = dq_rev/T
• Gibbs free energy is an extensive property that represents the maximum non-PV work available from a reaction
  • If ΔG is negative, the reaction is spontaneous and vice versa
• ΔG = ΔH – TΔS
  • Positive ΔH and negative ΔS always results in positive ΔG, and vice versa
  • Positive ΔH and ΔS results in negative ΔG at higher temperatures and vice versa
  • Negative ΔH and ΔS results in positive ΔG at higher temperatures and vice versa

Solutions

• A solution is a homogenous mixture of two or more compounds in a single phase
• The compound of which there is more is called the solvent, and the compound of which there is less is called the solute
• The ideal solution are solutions made from compounds that have similar properties
• Ideally, dilute solutions are solutions in which the solute molecules are completely separated by solvent molecules so that they have no interaction with each other
• Nonideal solutions violate the conditions of both ideal and ideally dilute solutions
• Particles larger than small molecules can form mixtures over time; if these do not settle out due to gravity over time, this mixture is termed a colloidal system or colloid
  • The Tyndall effect causes colloidal suspensions to scatter light
  • Colloids can be separated via coagulation followed by filtration, or using a semipermeable membrane, in a process called dialysis
• Like solutes dissolve in like solutions, so that nonpolar solvents dissolve nonpolar solutes and polar solvents dissolve polar solutes
• Hydrogen bonds and dipole-dipole interactions hold together polar molecules
• London dispersion forces hold together non-polar molecules via instantaneous dipole moments
• Solvation is the process by which ionic compounds break apart into cations and anions in solution and are surrounded by oppositely charged ends of the polar solvent, called hydration when water is the solvent
• Something that is hydrated is said to be in aqueous phase
• Electrolytes are compounds which form ions in aqueous solution; strong electrolytes conduct electricity well and vice versa
• Molarity = M = moles of solute/volume of solution
• Molality = m = moles of solute/kilograms of solvent
• Mole fraction = X = moles of solute/total moles of all solutes and solvent
• Mass percentage = mass % = (mass of solute/total mass of solution) × 100
• Parts per million = ppm = (mass of solute/total mass of solution) × 10⁶
• Normality measures the number of equivalents in solution
• Formation of a solution
  • The breaking of intermolecular forces between solute molecules
  • The breaking of intermolecular forces between solvent molecules
  • The formation of intermolecular bonds between solvent and solute molecules
• ΔH_sol = ΔH₁ + ΔH₂ + ΔH₃, where the first two steps are endothermic and positive, and the third step is exothermic and negative; a negative value of ΔH_sol results in stronger intermolecular bonds, and vice versa
• Solution formation has positive entropy
• Vapor pressure is the pressure created by molecules that escape from the liquid phase into the open space within a closed container
  • Vapor pressures increases with temperature
  • Boiling occurs when the vapor pressure of a liquid equals the atmospheric pressure
  • Melting occurs when the vapor pressure of the solid phase equals the vapor pressure of the liquid phase
• Raoult’s law: P_v = X_aP_a, applicable to a nonvolatile solute, which has no vapor pressure
• Modified Raoult’s law: P_v = X_aP_a + X_bP_b, applicable to a volatile solute, which has a vapor pressure
• Negative heats of solution form stronger bonds and lower vapor pressure, and vice versa, for nonideal solutions
• Solubility is a solute’s tendency to dissolve in solution
• The reverse reaction of dissolution is called precipitation
  • When the rates of dissolution and precipitation are equal, the solution is said to be saturated, and is in dynamic equilibrium
• Solubility product = K_sp = [products]^coefficients/[reactants]^coefficients, where the reactants and products must be aqueous or gaseous
  • The solubility product changes only with temperature
• Some solubility guidelines
  • Nearly all compounds containing nitrate (NO₃^–), ammonium (NH₄⁺), and alkali metals (Li⁺, Na⁺, K⁺,…) are soluble
  • Ionic compounds containing halogens (Cl^–, Br^–, I^–) are soluble, except for silver (Ag⁺), mercury (Hg₂²⁺), and lead (Pb²⁺)
  • Sulfate compounds (SO₄^2-) are soluble, except for mercury (Hg₂²⁺), lead (Pb²⁺), and the heavier alkali earth metals (Ca²⁺, Sr²⁺, Ba²⁺)
  • Compounds containing the heavier alkali earth metals (Ca²⁺, Sr²⁺, Ba²⁺) are soluble when paired with sulfides (S^2-) and hydroxides (OH^–)
  • Carbonates (CO₃^2-), phosphates (PO₄^3-), sulfides (S^2-), and hydroxides (OH^–) are generally insoluble except in the cases mentioned above

Heat Capacity, Phase Changes, and Colligative Properties

• Heat capacity C is a measure of the energy change required to change the temperature of a substance
  • C = q/ΔT
• Constant pressure capacity C_P is greater than constant volume capacity C_V­
• q = CΔT
• Specific heat: q = mcΔT, c = specific heat of a substance
  • c_water = 1 cal/g°C
• A coffee cup calorimeter is an example of a constant pressure calorimeter because it measures energy change at atmospheric pressure, also measures heats of reaction
• A bomb calorimeter measures energy change at constant volume
• Normal melting point and normal boiling point occur where a given substance changes phase at 1 atm
• Heat of fusion is the heat associated with melting, heat of vaporization is associated with boiling
• A phase diagram indicates the phases of a substance at different temperatures and pressures
  • There is only point where all three phases exist, called the triple point
  • The critical temperature is the temperature above which a substance cannot be liquefied regardless of the pressure applied
  • The pressure required for liquefaction at the critical temperature is called critical pressure
  • The critical pressure and critical temperature define the critical point
• Colligative properties are those that depend solely on the number of particles, regardless of the type
• Boiling point elevation: ΔT = k_bmi, van’t Hoff factor = i = the number of particles into which a single solute particle will dissociate into in solution
• Freezing point depression: ΔT = k_fmi
• Osmotic pressure: Π = iMRT

Acids and Bases

• An Arrhenius acid is anything that produces hydrogen ions in an aqueous solution, and an Arrhenius base is anything that produces hydroxide ions in aqueous solutions
• A Brønsted-Lowry acid is anything that donates a proton, and a Brønsted-Lowry base is anything that accepts a proton
• A Lewis acid is anything that accepts an electron pair, and a Lewis base is anything that donates an electron pair
• pH = -log [H⁺]
• The conjugate acid and base in a reaction correspond, respectively, the original base and acid
• The stronger the acid is, the weaker its conjugate base is, and vice versa
• K_w = K_aK_b­, K_w = 10^-14
• Amphoteric substances are those that can act as either acids or bases, depending on their environments
• Strong acids include hydroiodic acid (HI), hydrobromic acid (HBr), hydrochloric acid (HCl), nitric acid (HNO₃), perchloric acid (HClO₄), chloric acid (HClO₃), and sulfuric acid (H₂SO₄)
• Strong bases include sodium hydroxide (NaOH), potassium hydroxide (KOH), amide ion (NH₂^–), hydride ion (H^–), calcium hydroxide (Ca(OH)₂), sodium oxide (Na₂O), calcium oxide (CaO)
• Polyprotic acids are those that donate more than one proton
  • In general, if the K_a­ values of the protons differ by more than 10³, the second proton can be ignored
• Acid dissociation decreases with acid concentration but acid strength increases, because although the relative percentage dissociation decreases, the absolute number of dissociated ions increases
• Factors in molecular structure that determine what hydrogen-containing compounds act like acids
  • The strength of the bond holding the hydrogen to the molecule
  • The polarity of the bond
  • The stability of the conjugate base
• In a series of oxyacids, more oxygens mean a stronger acid
• Hydrides increase in acidity on the periodic table, from left to right and top to bottom
• The autoionization of water is represented by H₂O + H₂O → H₃O⁺ + OH^–
• pH + pOH = pK_w, pH + pOH = 14 at 25°C
• The acid dissociation constant K_a is the equilibrium constant of an acid in water
  • For HA + H₂O → H₃O⁺ + A^–, K_a = [H⁺][A^–]/[HA]
• The base dissociation constant K_b is the equilibrium constant of the corresponding conjugate base in water
  • For A^– + H₂O → OH^– + HA, K_b = [OH^–][HA]/[A^–]
• Salts are ionic compounds that dissociate in water
  • All cations, except those of alkali metals and the heavier alkali earth metals, act as weak Lewis acids in aqueous solutions
• A titration is a drop-by-drop mixing of an acid and a base
• The equivalence point for a monoprotic acid is the point in the titration when there are equal equivalents of acid and base in the solution
• The half equivalence point is where exactly one half of the acid has been neutralized by the base
  • The concentration of the acid is equal to the concentration of its conjugate base
  • Shows the point in the titration where the solution is most well buffered
  • Here, the pH of the solution is equal to the pK_a of the acid
• Henderson-Hasselbalch equation: pH = pK_a + log [A^–]/[HA]
• Indicators for pH change color at the end point of a titration

Electrochemistry

• In a redox reaction, electrons are transferred from one atom to another
  • The atom that loses electrons is oxidized, and the atom that gains electrons is reduced
• Oxidation states
  • Atoms in their elemental state have an oxidation state of 0
  • Fluorine has an oxidation state of -1
  • Hydrogen has an oxidation state of +1 normally, -1 when bonded to a metal
  • Oxygen has an oxidation state of -2 normally, except in peroxides like H₂O₂
  • Group 1 elements, alkali metals, have an oxidation state of +1
  • Group 2 elements, alkali earth metals, have an oxidation state of +2
  • Group 15 elements, nitrogen family, have an oxidation state of -3
  • Group 16 elements, oxygen family, have an oxidation state of -2
  • Group 17 elements, halogens, have an oxidation state of -1
• The atom being reduced is also the oxidizing agent, and vice versa
• There is an electric potential E associated with any redox reaction
  • The oxidation and reduction components are individually referred to as half reactions
  • Each half-reaction can only have one electric potential and are usually listed as either oxidation or reduction potentials in table; the signs of these values can be reversed to find the value for the opposite half reaction
  • 2H⁺ + 2e^– → H₂, E° = 0.00 V
  • Reduction potentials are intensive properties, so half reaction potentials are not multiplied by the number of times they occur
• Galvanic, or voltaic, cells are used to turn chemical energy into electrical energy
  • The two electrodes are called the cathode and anode, where reduction and oxidation take place, respectively; also, the cathode is positive and the anode is negative
  • Current moves in the opposite direction of electrons, which move through the load from anode to cathode
  • In some types of galvanic cells, a salt bridge is used to minimize the potential difference and prevent solutions from mixing and short circuiting the cell
• ΔG = -nFE_max, Faraday’s constant = F = 96486 C/mol, n = number of moles of electrons transferred in the balanced redox reaction
• A positive cell potential indicates a negative ΔG and a spontaneous reaction
• ΔG = ΔG° + RT ln(Q)
  • Also seen as ΔG = ΔG° + 2.3RT log(Q)
• ΔG° = -RT ln(K)
  • If K = 1, then ΔG° = 0
  • If K > 1, then ΔG° < 0
  • If K < 1, then ΔG° > 0
• Nernst equation: E = E° – RT/nF ln(Q)
• In a given concentration cell, current will flow from the more concentrated side to the less concentrated side, in order to create more entropy
• In an electrolytic cell, the cathode is negative and the anode is positive
  • These cells are used for metal plating