Formulas

1. Vapor Pressure and Henry’s Law

• Henry’s Law Equation: $S<em>g = k P</em>g$

  • $S_g$: Solubility of the gas

  • $k$: Henry’s Law constant

  • $P_g$: Partial pressure of the gas above the liquid

• Concept: Gas solubility in a liquid is directly proportional to the partial pressure of that gas above the liquid. Higher gas pressure forces more gas to dissolve into the liquid phase.

2. Raoult’s Law

• Raoult’s Law Equation: PA = XA PA

  • $P_A$: Partial vapor pressure of component $A$ in solution

  • $X<em>A$: Mole fraction of component $A$, defined as: $X</em>A = \frac{\text{moles of } A}{\text{total moles}}$

  • $P_A^\circ$: Vapor pressure of pure component $A$

• Concept: The presence of a non-volatile solute lowers the vapor pressure of the solvent proportional to its mole fraction in the solution.

3. Microscopic Entropy and the Boltzmann Equation

• Boltzmann Equation: $S = k_B \ln(W)$

  • $S$: Entropy of the system

  • $k_B$: Boltzmann’s constant ($1.38 \times 10^{-23}\,J\,K^{-1}$)

  • $W$: Number of possible microstates (microscopic arrangements)

• Concept: Entropy measures the dispersion of energy across available microstates. As the number of possible microscopic arrangements increases, entropy increases ($W \uparrow \; \Rightarrow \; S \uparrow$).

4. Electrochemistry Constants and Wave Equations

• Faraday’s Constant: $F = 96{,}485\,C\,mol^{-1}$

  • Serves as the conversion factor between moles of electrons and electric charge in Coulombs ($C$).

• Speed of Light Equation: $c = \lambda \nu$

  • $c$: Speed of light in a vacuum ($3.00 \times 10^8\,m\,s^{-1}$)

  • $\lambda$: Wavelength

  • $\nu$: Frequency

5. Gibbs Free Energy and Thermodynamics

• Gibbs Free Energy Equation:
  $\Delta G = \Delta H - T \Delta S$

  • $\Delta G$: Change in Gibbs free energy (determines reaction spontaneity)

  • $\Delta H$: Change in enthalpy (heat absorbed or released)

  • $T$: Absolute temperature in Kelvin ($K$)

  • $\Delta S$: Change in entropy

• Criteria for Spontaneity:

  • $\Delta G < 0$: Spontaneous / thermodynamically favorable process

  • $\Delta G > 0$: Non-spontaneous / thermodynamically unfavorable process

  • $\Delta G = 0$: System at dynamic equilibrium

• Free Energy under Non-Standard Conditions:
  $\Delta G = \Delta G^\circ + R T \ln(Q)$

  • $\Delta G^\circ$: Standard Gibbs free energy change

  • $R$: Ideal gas constant ($8.314\,J\,mol^{-1}\,K^{-1}$)

  • $Q$: Reaction quotient evaluated at current concentrations or partial pressures

• Electrochemical Free Energy Relationship:
  $\Delta G = -nFE$

  • $n$: Moles of electrons transferred in the balanced redox reaction

  • $E$: Cell potential in Volts ($V$)

  • Key Relationship: A positive cell potential ($E > 0$) yields a negative free energy change ($\Delta G < 0$), signifying a spontaneous electrochemical reaction.

6. Strong Acids and Bases

• Strong Acids (undergo complete ionization in aqueous solution):

  • Hydrochloric acid: $HCl$

  • Hydrobromic acid: $HBr$

  • Hydroiodic acid: $HI$

  • Nitric acid: $HNO_3$

  • Sulfuric acid: $H<em>2SO</em>4$

  • Perchloric acid: $HClO_4$

• Strong Bases (undergo complete dissociation in aqueous solution):

  • Lithium hydroxide: $LiOH$

  • Sodium hydroxide: $NaOH$

  • Potassium hydroxide: $KOH$

  • Calcium hydroxide: $Ca(OH)_2$

  • Strontium hydroxide: $Sr(OH)_2$

  • Barium hydroxide: $Ba(OH)_2$

7. Spectrochemical Series

• Ligand Strength Ranking:
  $I^- < Br^- < S^{2-} < Cl^- < NO<em>3^- < F^- < OH^- < C</em>2O<em>4^{2-} < O^{2-} < H</em>2O < NH_3 < en < CN^- < CO$

• Concept: Ranks ligands by their ability to split d-orbital energy levels (crystal field splitting energy, $\Delta$) in transition metal coordination complexes.