📌 Stoichiometry

Moles from Mass:

\( n = \frac{m}{M} \)

Molar Mass:

\( M = \frac{m}{n} \)

Number of Particles:

\( N = n \cdot N_A \)

Avogadro’s Number:

\( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)

Mass Percent:

\( \text{Mass \%} = \frac{m_{\text{solute}}}{m_{\text{total}}} \times 100 \)

Mole Fraction:

\( X_i = \frac{n_i}{n_{\text{total}}} \)

Molarity:

\( M = \frac{n}{V} \) (in liters)

Molality:

\( m = \frac{n}{m_{\text{solvent}}} \) (in kg)

Normality:

\( N = \frac{\text{equivalents}}{V} \) (in liters)

Balancing Equation (Stoichiometric Ratio):

\( \frac{n_A}{a} = \frac{n_B}{b} \) (for \( aA + bB \to \text{products} \))

Limiting Reactant Yield:

\( n_{\text{product}} = n_{\text{limiting}} \cdot \frac{\text{coefficient}_{\text{product}}}{\text{coefficient}_{\text{limiting}}} \)

📌 Gas Laws

Ideal Gas Law:

\( PV = nRT \)

Boyle’s Law:

\( P_1 V_1 = P_2 V_2 \) (constant \( T \))

Charles’s Law:

\( \frac{V_1}{T_1} = \frac{V_2}{T_2} \) (constant \( P \))

Gay-Lussac’s Law:

\( \frac{P_1}{T_1} = \frac{P_2}{T_2} \) (constant \( V \))

Combined Gas Law:

\( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \)

Density of Gas:

\( \rho = \frac{P M}{RT} \)

Molar Volume (STP):

\( V_m = \frac{V}{n} = 22.414 \, \text{L/mol} \) (at 0°C, 1 atm)

Dalton’s Law of Partial Pressures:

\( P_{\text{total}} = P_1 + P_2 + \cdots + P_n \)

Partial Pressure:

\( P_i = X_i P_{\text{total}} \)

Effusion Rate (Graham’s Law):

\( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \)

Root Mean Square Speed:

\( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \)

Average Kinetic Energy of Gas:

\( KE_{\text{avg}} = \frac{3}{2} RT \)

📌 Thermodynamics

First Law of Thermodynamics:

\( \Delta U = Q - W \)

Enthalpy:

\( H = U + PV \)

Change in Enthalpy:

\( \Delta H = \Delta U + \Delta (PV) \)

Heat Capacity at Constant Volume:

\( C_V = \left( \frac{\partial U}{\partial T} \right)_V \)

Heat Capacity at Constant Pressure:

\( C_P = \left( \frac{\partial H}{\partial T} \right)_P \)

Relation Between \( C_P \) and \( C_V \):

\( C_P - C_V = R \) (ideal gas)

Work Done by Gas:

\( W = P \Delta V \) (constant pressure)

Entropy Change:

\( \Delta S = \frac{q_{\text{rev}}}{T} \)

Gibbs Free Energy:

\( G = H - T S \)

Change in Gibbs Free Energy:

\( \Delta G = \Delta H - T \Delta S \)

Standard Free Energy:

\( \Delta G^\circ = -RT \ln K \)

Hess’s Law:

\( \Delta H_{\text{total}} = \sum \Delta H_{\text{steps}} \)

📌 Reaction Kinetics

Rate of Reaction:

\( \text{Rate} = -\frac{\Delta [\text{A}]}{\Delta t} = \frac{\Delta [\text{P}]}{\Delta t} \)

Rate Law:

\( \text{Rate} = k [\text{A}]^m [\text{B}]^n \)

First-Order Rate Equation:

\( \ln [\text{A}]_t = \ln [\text{A}]_0 - k t \)

Half-Life (First-Order):

\( t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k} \)

Second-Order Rate Equation:

\( \frac{1}{[\text{A}]_t} = \frac{1}{[\text{A}]_0} + k t \)

Half-Life (Second-Order):

\( t_{1/2} = \frac{1}{k [\text{A}]_0} \)

Zero-Order Rate Equation:

\( [\text{A}]_t = [\text{A}]_0 - k t \)

Half-Life (Zero-Order):

\( t_{1/2} = \frac{[\text{A}]_0}{2k} \)

Arrhenius Equation:

\( k = A e^{-\frac{E_a}{RT}} \)

Arrhenius (Log Form):

\( \ln k = \ln A - \frac{E_a}{RT} \)

Two-Point Arrhenius:

\( \ln \left( \frac{k_2}{k_1} \right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \)

Collision Frequency Factor:

\( A = Z \cdot p \) (where \( Z \) is collision frequency, \( p \) is steric factor)

📌 Chemical Equilibrium

Equilibrium Constant (\( K_c \)):

\( K_c = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b} \) (for \( aA + bB \rightleftharpoons cC + dD \))

Equilibrium Constant (\( K_p \)):

\( K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} \)

Relation Between \( K_p \) and \( K_c \):

\( K_p = K_c (RT)^{\Delta n} \) (where \( \Delta n = \text{products} - \text{reactants} \))

Reaction Quotient (\( Q \)):

\( Q = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b} \) (at any time)

Equilibrium Shift (Le Chatelier):

\( Q < K \) (to products), \( Q > K \) (to reactants)

Equilibrium Concentration (ICE Table):

\( [\text{A}]_{\text{eq}} = [\text{A}]_0 - x \cdot \frac{a}{\text{coefficient}} \)

Quadratic Equation for \( x \):

\( ax^2 + bx + c = 0 \), roots: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Equilibrium Constant (Free Energy):

\( K = e^{-\frac{\Delta G^\circ}{RT}} \)

Van’t Hoff Equation:

\( \ln \left( \frac{K_2}{K_1} \right) = \frac{\Delta H^\circ}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \)

Pressure-Based Equilibrium:

\( K_p = K_c \left( \frac{P}{n_{\text{total}}} \right)^{\Delta n} \)

📌 Acids and Bases

pH:

\( \text{pH} = -\log_{10} [\text{H}^+] \)

pOH:

\( \text{pOH} = -\log_{10} [\text{OH}^-] \)

Water Ion Product:

\( K_w = [\text{H}^+] [\text{OH}^-] = 10^{-14} \) (at 25°C)

pH + pOH:

\( \text{pH} + \text{pOH} = 14 \) (at 25°C)

Acid Dissociation Constant (\( K_a \)):

\( K_a = \frac{[\text{H}^+] [\text{A}^-]}{[\text{HA}]} \)

Base Dissociation Constant (\( K_b \)):

\( K_b = \frac{[\text{BH}^+] [\text{OH}^-]}{[\text{B}]} \)

Relation Between \( K_a \) and \( K_b \):

\( K_a \cdot K_b = K_w \)

pKa:

\( \text{pKa} = -\log_{10} K_a \)

Henderson-Hasselbalch Equation:

\( \text{pH} = \text{pKa} + \log_{10} \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \)

Buffer Capacity:

\( \beta = \frac{d[\text{base}]}{d\text{pH}} \approx 2.303 \frac{K_a [\text{H}^+]}{(K_a + [\text{H}^+])^2} \)

Weak Acid Equilibrium:

\( [\text{H}^+] = \sqrt{K_a \cdot [\text{HA}]} \) (approximation)

📌 Electrochemistry

Nernst Equation:

\( E = E^\circ - \frac{RT}{nF} \ln Q \)

Nernst (Base 10):

\( E = E^\circ - \frac{0.0592}{n} \log_{10} Q \) (at 25°C)

Standard Cell Potential:

\( E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} \)

Free Energy and Cell Potential:

\( \Delta G^\circ = -n F E^\circ \)

Equilibrium Constant from \( E^\circ \):

\( \ln K = \frac{n F E^\circ}{RT} \)

Faraday’s Law (Mass):

\( m = \frac{Q M}{n F} \)

Faraday’s Constant:

\( F = 96485 \, \text{C/mol} \)

Current and Charge:

\( Q = I \cdot t \)

Conductivity:

\( \kappa = \frac{1}{\rho} = \frac{I}{V} \cdot \frac{l}{A} \)

Molar Conductivity:

\( \Lambda_m = \frac{\kappa}{c} \)

Kohlrausch’s Law:

\( \Lambda_m^\circ = \nu_+ \lambda_+ + \nu_- \lambda_- \)

📌 Quantum Chemistry

de Broglie Wavelength:

\( \lambda = \frac{h}{p} = \frac{h}{m v} \)

Planck’s Equation:

\( E = h f \)

Energy of Photon:

\( E = \frac{h c}{\lambda} \)

Heisenberg Uncertainty Principle:

\( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \)

Schrödinger Equation (Time-Independent):

\( -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V \psi = E \psi \)

Hydrogen Atom Energy Levels:

\( E_n = -\frac{13.6}{n^2} \, \text{eV} \)

Rydberg Formula:

\( \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), \( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \)

Particle in a Box Energy:

\( E_n = \frac{n^2 h^2}{8 m L^2} \)

Harmonic Oscillator Energy:

\( E_n = \left( n + \frac{1}{2} \right) h f \)

Radial Probability Density:

\( P(r) = 4 \pi r^2 |\psi(r)|^2 \)

Bohr Radius:

\( a_0 = \frac{\hbar^2}{m_e k e^2} \approx 5.29 \times 10^{-11} \, \text{m} \)

📌 Spectroscopy and Structure

Beer-Lambert Law:

\( A = \epsilon l c \)

Transmittance:

\( T = \frac{I}{I_0} \), \( A = -\log_{10} T \)

Wavenumber:

\( \tilde{\nu} = \frac{1}{\lambda} \) (in cm\(^{-1}\))

Vibrational Frequency:

\( f = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}} \) (where \( \mu = \frac{m_1 m_2}{m_1 + m_2} \))

Rotational Energy Levels:

\( E_J = \frac{\hbar^2}{2I} J (J + 1) \), \( I = \mu r^2 \)

Moment of Inertia:

\( I = \sum m_i r_i^2 \)

Rotational Constant:

\( B = \frac{h}{8 \pi^2 c I} \) (in cm\(^{-1}\))

Stokes Shift:

\( \Delta \tilde{\nu} = \tilde{\nu}_{\text{abs}} - \tilde{\nu}_{\text{emit}} \)

IR Absorption Frequency:

\( \tilde{\nu} = \frac{1}{2\pi c} \sqrt{\frac{k}{\mu}} \)

UV-Vis Energy Difference:

\( \Delta E = \frac{h c}{\lambda} \)