Chemistry Proof Encyclopedia

1. Introduction

For a binary solution of volatile liquids A and B, partial pressures are: \(P_A = P_A^\circ x_A\) and \(P_B = P_B^\circ x_B\).

2. Mathematical Derivation

\(P_{\text{total}} = P_A + P_B = P_A^\circ x_A + P_B^\circ x_B\)

Since \(x_A = 1 - x_B\):

\(P_{\text{total}} = P_A^\circ (1 - x_B) + P_B^\circ x_B\)

Let \(P_1^\circ\) be vapor pressure of pure solvent and \(P_1\) be of solution.

\(P_1 = P_1^\circ x_1 \implies P_1 = P_1^\circ (1 - x_2)\)

\(P_1^\circ - P_1 = P_1^\circ x_2 \implies \frac{P_1^\circ - P_1}{P_1^\circ} = x_2\)

\(\Delta T_b = T_b - T_b^\circ\)

\(\Delta T_b = K_b \cdot m \implies \Delta T_b = K_b \cdot \frac{w_2 \cdot 1000}{M_2 \cdot w_1}\)

\(\Delta T_f = T_f^\circ - T_f\)

\(\Delta T_f = K_f \cdot m \implies \Delta T_f = K_f \cdot \frac{w_2 \cdot 1000}{M_2 \cdot w_1}\)

\(\pi = CRT = \frac{n_2}{V} RT = \frac{w_2}{M_2 V} RT\)

Dissociation: \(A \to nB\). If \(\alpha\) is degree of dissociation:

\(i = \frac{1 - \alpha + n\alpha}{1} = 1 + \alpha(n - 1)\)

Association: \(nA \to A_n\). If \(\alpha\) is degree of association:

\(i = 1 - \alpha + \frac{\alpha}{n} = 1 + \alpha\left(\frac{1}{n} - 1\right)\)

1. Basic Relation

Gibbs Free Energy: \(\Delta G = \Delta G^\circ + RT \ln Q\)

2. Link to E_cell

We know \(\Delta G = -nFE_{\text{cell}}\) and \(\Delta G^\circ = -nFE^\circ_{\text{cell}}\)

\(-nFE_{\text{cell}} = -nFE^\circ_{\text{cell}} + RT \ln Q\)

3. Final Expression

Dividing by \(-nF\): \(E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q\)

First Law: \(m \propto Q \implies m = ZQ = ZIt\)

Value of \(Z = \frac{\text{Equivalent Weight}}{F}\)

Second Law: \(\frac{m_1}{m_2} = \frac{E_1}{E_2}\) (for constant charge \(Q\)).

At infinite dilution, molar conductivity is sum of ionic contributions:

1. Rate Law

\(-\frac{d[R]}{dt} = k[R]\)

2. Integration

\(\int \frac{d[R]}{[R]} = -k \int dt \to \ln[R] = -kt + C\)

At \(t=0\), \(C = \ln[R]_0\)

3. Final Form

\(\ln \frac{[R]_0}{[R]} = kt \implies k = \frac{2.303}{t} \log \frac{[R]_0}{[R]}\)

For \(A(g) \to B(g) + C(g)\). Initial pressure \(P_i\). Total pressure \(P_t = (P_i - x) + x + x\).

\(P_t = P_i + x \implies x = P_t - P_i\). Partial pressure \(P_A = P_i - (P_t - P_i) = 2P_i - P_t\).

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

\(\log k = \log A - \frac{E_a}{2.303 RT}\)

Mass of unit cell = \(Z \times (\text{Atomic Mass } / N_A)\)

Volume of unit cell = \(a^3\)

Freundlich: \(\frac{x}{m} = k P^{1/n} \implies \log \frac{x}{m} = \log k + \frac{1}{n} \log P\)

Langmuir: Rate of Ads = Rate of Des.

1. Primary Valence (ionizable, oxidation state).

2. Secondary Valence (non-ionizable, coordination number).

Based on spin-only formula for \(n\) unpaired electrons:

Octahedral: \(\text{CFSE} = [-0.4n_{t_{2g}} + 0.6n_{e_g}] \Delta_o + nP\)

Stability rule for complexes based on Noble Gas configuration:

Tetrahedral splitting energy is always smaller than octahedral:

1. Identify metal oxidation state.

2. Determine number of unpaired electrons.

3. Check ligand strength (Strong field \(\to\) Inner orbital complex \(d^2sp^3\); Weak field \(\to\) Outer orbital complex \(sp^3d^2\)).

Proof of stability for \(Mn^{3+}/Mn^{2+}\) vs \(Cr^{3+}/Cr^{2+}\):

1. \(Mn^{2+}\) has \(d^5\) (half-filled stability).

2. \(Cr^{3+}\) has \(t_{2g}^3\) (half-filled \(t_{2g}\) stability in octahedral field).

1-step reaction. Nucleophile attacks from backside.

Transition State: Pentavalent carbon. \(sp^2\) state transition.

Inversion: Configuration flips (Walden Inversion).

Step 1: Carbocation formation (RDS). Rate = \(k[R-X]^1\).

Step 2: Fast attack by nucleophile. Racemic mixture formed.

Step 1: Protonation of alcohol (\(-OH \to -OH_2^+\)).

Step 2: Loss of \(H_2O\) to form Carbocation (RDS).

Step 3: Elimination of H+ from \(\beta\)-carbon to form Alkenes.

Reaction of Alkyl halide with Sodium Alkoxide via **SN2 mechanism**.

Requires primary alkyl halide for best yields (avoiding elimination).

Electronic structure of \(C=O\) is polar. Nucleophile attacks electrophilic Carbon.

Intermediat: Tetrahedral alkoxide ion.

1. Enolate Formation: Base abstracts \(\alpha\)-Hydrogen.

2. Nucleophilic Attack: Enolate attacks second aldehyde molecule.

3. Protonation: Forms \(\beta\)-hydroxy aldehyde (Aldol).

Convert Amide to Primary Amine with 1 carbon less.

Diazotization: Aniline + \(NaNO_2\) + HCl (\(0-5^\circ C\)) \(\to\) Benzene Diazonium Chloride.

Coupling: Diazonium salt + Phenol \(\to\) p-hydroxyazobenzene (Orange dye).

For aldehydes with **NO \(\alpha\)-Hydrogen** (e.g., Formaldehyde, Benzaldehyde).

1. Nucleophilic Attack: \(OH^-\) attacks carbonyl carbon \(\to\) Hydroxyl alkoxide.

2. Hydride Transfer: One molecule transfers \(H^-\) to another (RDS).

3. Proton Transfer: Forms Salt of Acid and Alcohol.

Halogenation of \(\alpha\)-carbon of carboxylic acids.

Requires \(\alpha\)-Hydrogen and red phosphorus catalyst.

1. **Draw Diagrams:** Always draw unit cells or mechanism arrows with a pencil.

2. **Write Units:** Don't forget \(mol L^{-1}, \Omega^{-1} cm^2 mol^{-1}\), etc.

3. **Define Terms:** Before deriving, write "Let \(w_2\) be mass of solute..." to impress the examiner.