Principles of Physical Chemistry

CHAPTER 1: Some Basic Concepts of Chemistry

SOME USEFUL CONVERSION FACTORS

• \(1\text{ \AA} = 10^{-10}\text{ m}\)
• \(1\text{ nm} = 10^{-9}\text{ m}\)
• \(1\text{ pm} = 10^{-12}\text{ m}\)
• \(1\text{ litre} = 10^{-3}\text{ m}^3 = 1\text{ dm}^3\)
• \(1\text{ atm} = 760\text{ mm or torr} = 101325\text{ Pa or Nm}^{-2}\)
• \(1\text{ bar} = 10^5\text{ Nm}^{-2} = 10^5\text{ Pa}\)
• \(1\text{ calorie} = 4.184\text{ J}\)
• \(1\text{ electron volt (eV)} = 1.6022 \times 10^{-19}\text{ J}\)
• \((1\text{ J} = 10^7\text{ ergs})\)
• \((1\text{ cal} > 1\text{ J} > 1\text{ erg} > 1\text{ eV})\)

ATOMIC MASS OR MOLECULAR MASS

Mass of one atom or molecule in a.m.u.

• \(C \rightarrow 12\text{ amu}\)
• \(NH_3 \rightarrow 17\text{ amu}\)

Actual Mass

Mass of one atom or molecule in grams

• \(C \rightarrow 12 \times 1.6 \times 10^{-24}\text{ g}\)
• \(CH_4 \rightarrow 16 \times 1.6 \times 10^{-24}\text{ g}\)

Relative Atomic Mass or Relative Molecular Mass

Mass of one atom or molecule w.r.t. \(1/12^{th}\) of \(^{12}C\) atom.

• \(C \rightarrow 12\)
• \(CH_4 \rightarrow 16\)

It is unitless.

GRAM ATOMIC MASS OR GRAM MOLECULAR MASS

Mass of one mole of atom or molecule.

• \(C \rightarrow 12\text{ g}\)
• \(CO_2 \rightarrow 44\text{ g}\)

It is also called molar mass.

DEFINITION OF MOLE

One mole is a collection of that many entities as there are number of atoms exactly in 12 gm of C-12 isotope.

The number of atoms present in exactly 12 gm of C-12 isotope is called Avogadro's number \([N_A = 6.022 \times 10^{23}]\).

\[1\text{ u} = 1\text{ amu} = (1/12)^{th} \text{ of mass of 1 atom of } C^{12} = \frac{1\text{ g}}{N_A} = 1.66 \times 10^{-24}\text{ g}\]

For Elements

• 1 g atom = 1 mole of atoms = \(N_A\) atoms
• g atomic mass (GAM) = mass of \(N_A\) atoms in g
• \(Mole of atoms = \frac{Mass (g)}{GAM or molar mass}\)

For Molecule

• 1 g molecule = 1 mole of molecule = \(N_A\) molecule
• g molecular mass (GMM) = mass of \(N_A\) molecule in g
• \(Mole of molecule = \frac{Mass (g)}{GMM or molar mass}\)

1 Mole of Substance

• Contains \(6.022 \times 10^{23}\) particles.
• Weighs as much as molecular mass/atomic mass/ionic mass in grams.
• If it is a gas, one mole occupies a volume of 22.4 L at 1 atm & 273 K or 22.7 L at STP.

For Ionic Compounds

• 1 g formula unit = 1 mole of formula unit = \(N_A\) formula unit.
• g formula mass (GFM) = mass of \(N_A\) formula unit in g.
• \(Mole of formula unit = \frac{Mass (g)}{GMM or molar mass}\)

VAPOUR DENSITY

Ratio of density of vapour to the density of hydrogen at similar pressure and temperature.

\[Vapour density = \frac{Molar mass}{2}\]

Figure: Flowchart for Mole Conversions (Mass, Moles, Volume at STP, Particles).

STOICHIOMETRY BASED CONCEPT

\[aA + bB \rightarrow cC + dD\]

• a, b, c, d represent the ratios of moles, volumes (for gaseous) or molecules in which reactants react or products form.
• a, b, c, d does not represent the ratio of masses.
• Relation: \[\frac{Moles of A reacted}{a} = \frac{Moles of B reacted}{b} = \frac{Moles of C reacted}{c} = \frac{Moles of D reacted}{d}\]

Concept of limiting reagent

If data of more than one reactant is given, first convert all the data into moles then divide the moles of reactants with their respective stoichiometric coefficient. The reactant having minimum ratio will be Limiting Reagent (L.R.). Find the moles of product formed or excess reagent left by comparing it with L.R. through stoichiometric concept.

Percentage Purity

\[\%purity = \frac{Actual mass of compound}{Total mass of sample} \times 100\]

If impurity is unknown, it is always considered as inert (unreactive) material.

EMPIRICAL AND MOLECULAR FORMULA

• Empirical formula: Formula depicting constituent atoms in their simplest ratio.
• Molecular formula: Formula depicting actual number of atoms in one molecule of the compound.
• Molecular formula = Empirical formula \(\times n\)
• \[n = \frac{molecular formula mass}{empirical formula mass}\]

Dulong's & Petit's Law

For determination of atomic mass: \(Atomic weight of metal \times specific heat capacity (cal/gm-^{\circ}C) \approx 6.4\).

This is an empirical observation and gives an approximate value, performing better for heavier solid elements at high temperatures.

CONCENTRATION TERMS

Concentration Type	Mathematical Formula	Concept
Percentage by mass (\(\frac{W}{W} \%\))	\(\frac{Mass of solute}{Mass of solution} \times 100\)	Mass of solute (in gm) present in 100 gm of solution.
Volume percentage (\(\frac{V}{V} \%\))	\(\frac{Volume of solute}{Volume of solution} \times 100\)	Volume of solute (in \(cm^3\)) present in 100 \(cm^3\) of solution.
Mass-volume percentage (\(\frac{W}{V} \%\))	\(\frac{Mass of solute}{Volume of solution} \times 100\)	Mass of solute (in gm) present in 100 \(cm^3\) of solution.
Parts per million (ppm)	\(\frac{Mass of solute}{Mass of solution} \times 10^6\)	Parts by mass of solute per million parts by mass of the solution.
Mole fraction (\(X_A\))	\(X_A = \frac{n_A}{n_A + n_B + \dots}\)	Ratio of number of moles of one component to the total number of moles.
Molarity (M)	\(M = \frac{Mole of solute}{Volume of solution (in L)}\)	Moles of solute in one liter of solution.
Molality (m)	\(m = \frac{Mole of solute}{Mass of solvent (Kg)}\)	Moles of solute in one kg of solvent.

MIXING OF SOLUTIONS

Based on law of conservation of moles.

1. Two solutions having same solute: \(Final molarity = \frac{M_1V_1 + M_2V_2}{V_1 + V_2}\)
2. Dilution Effect: Final molarity \(M_2 = \frac{M_1V_1}{V_1 + V_2}\)

Volume Strength of \(H_2O_2\) Solutions

Labelled as 'volume \(H_2O_2\)' means volume of \(O_2\) (in litre) at 1 bar & 273 K that can be obtained from 1 litre of such a sample when it decomposes as: \(2H_2O_2 \rightarrow 2H_2O + O_2\).

\[Volume Strength of H_2O_2 solution = 11.35 \times molarity\]

PERCENTAGE LABELLING OF OLEUM

Labelled as '% oleum' means maximum amount of \(H_2SO_4\) that can be obtained from 100 gm of such oleum (mixture of \(H_2SO_4\) and \(SO_3\)) by adding sufficient water. Example: 109% oleum means 100 gm oleum + 9 gm water gives 109 gm \(H_2SO_4\).

% labelling of oleum sample = \((100 + x)\%\) where \(x\) is mass of \(H_2O\) required for complete conversion of \(SO_3\) in \(H_2SO_4\).

\[\% of free SO_3 in oleum = (\frac{40}{9} \times x)\%\]

EUDIOMETRY

1. Gay-Lussac's law: Volumes of gaseous reactants and products, at same T and P, bear a simple ratio.
2. Volumes of solids or liquids are considered negligible in comparison to gas volume.
3. Air is considered a mixture of 21% \(O_2\) and 79% \(N_2\) (approximately).
4. Nitrogen gas is considered non-reactive.

Solvent	Gases absorbed
KOH	\(CO_2, SO_2, Cl_2\)
Ammonical \(Cu_2Cl_2\)	CO
Turpentine oil	\(O_3\)
Alkaline pyrogallol	\(O_2\)
Water	\(NH_3, HCl\)
\(CuSO_4 / CaCl_2\)	\(H_2O\)

CHAPTER 2: Atomic Structure

IMPORTANT DEFINITIONS

Property	Proton (\(m_p\))	Neutron (\(m_n\))	Electron (\(m_e\))
Mass (kg)	\(1.67 \times 10^{-27}\)	\(1.67 \times 10^{-27}\)	\(9.1 \times 10^{-31}\)
Mass (amu)	1.00750	1.00850	0.000549
e/m value	Depends on gas	-	Independent of gas

REPRESENTATION OF AN ELEMENT

\[^{A}_{Z}X\]

• Atomic Number (Z) = No. of protons.
• Mass number (A) = Total number of neutrons + protons.
• Isotopes: Same Z, different A (e.g., \(^{12}_{6}C, ^{14}_{6}C\)).
• Isobars: Same A, different Z (e.g., \(^{40}_{18}Ar, ^{40}_{20}Ca\)).
• Isotones: Same number of neutrons.
• Isoelectronic: Species having same no. of electrons.

ATOMIC MODELS

• Rutherford Model: Nucleus size is \(10^{-13}\) cm, atom size is \(10^{-8}\) cm. Radius of nucleus \(R_N = R_0(A)^{1/3}\) where \(R_0 = 1.33 \times 10^{-13}\) cm.

ELECTROMAGNETIC SPECTRUM

Order: Radiowaves \(\rightarrow\) Microwaves \(\rightarrow\) IR \(\rightarrow\) Visible \(\rightarrow\) UV \(\rightarrow\) X-rays \(\rightarrow\) Cosmic rays.

\[E = h\nu = \frac{hc}{\lambda}\]

\[E(eV) = \frac{12400}{\lambda(\text{\AA})}\]

BOHR'S ATOMIC MODEL

• Radius: \(r = 0.529 \times \frac{n^2}{Z}\text{ \AA}\)
• Velocity: \(v = 2.188 \times 10^6 \frac{Z}{n}\text{ ms}^{-1}\)
• Total Energy: \(E = -13.6 \times \frac{Z^2}{n^2}\text{ eV/atom}\)
• Relationship: \(PE = 2TE, KE = -TE\).
• Energy difference between levels: \[\Delta E = 13.6 Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\text{ eV/atom}\]

HYDROGEN SPECTRUM

Rydberg's Equation: \[\frac{1}{\lambda} = \bar{\nu} = R_H \left[\frac{1}{n_1^2} - \frac{1}{n_2^2}\right] \times Z^2\] where \(R_H \approx 109700\text{ cm}^{-1}\).

Number of spectral lines (\(n_2\) to \(n_1\)): \[\frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2}\]

DE-BROGLIE HYPOTHESIS

\[\lambda = \frac{h}{mv} = \frac{h}{p} = \frac{h}{\sqrt{2mKE}}\]

HEISENBERG UNCERTAINTY PRINCIPLE

\[\Delta x \cdot \Delta p \ge \frac{h}{4\pi}\]

QUANTUM NUMBERS

• Principal (n): Indicates size and energy. Values: 1, 2, 3...
• Azimuthal (l): Indicates subshell shape. Values: 0 to (n-1). \(l=0(s), l=1(p), l=2(d), l=3(f)\).
• Magnetic (m): Indicates orientation. Values: \(-l \dots 0 \dots +l\).
• Spin (s): Indicates electron spin. Values: \(\pm 1/2\).

RULES FOR FILLING ORBITALS

• Aufbau principle: Orbitals filled in increasing order of energy.
• (n+l) rule: Lower (n+l) fills first; if same, lower n fills first.
• Pauli exclusion: No two electrons in an atom have same four quantum numbers.
• Hund's rule: Maximum multiplicity; pairing starts after each orbital is singly filled.

CHAPTER 3: Gaseous State

GAS LAWS

• Boyle's law: \(P_1V_1 = P_2V_2\) (constant n, T).
• Charle's law: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) (constant n, P).
• Ideal gas equation: \(PV = nRT\).
• R values: 0.0821 L atm \(mol^{-1}K^{-1}\); 8.314 J \(K^{-1}mol^{-1}\).

GRAHAM'S DIFFUSION LAW

\[r \propto \frac{1}{\sqrt{d}} \propto \frac{1}{\sqrt{M_w}}\]

DALTON'S LAW OF PARTIAL PRESSURE

\[P_{total} = P_1 + P_2 + P_3 + \dots\]

\[P_A = X_A P_T\]

KINETIC THEORY

Average KE (\(n\) moles) = \(\frac{3}{2}nRT\).

• \(v_{rms} = \sqrt{\frac{3RT}{M_w}}\)
• \(v_{av} = \sqrt{\frac{8RT}{\pi M_w}}\)
• \(v_{mp} = \sqrt{\frac{2RT}{M_w}}\)
• Ratio: \(v_{mp} : v_{av} : v_{rms} = 1 : 1.128 : 1.224\)

VANDERWAAL'S EQUATION

\[\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT\]

• 'a' measures intermolecular attraction.
• 'b' represents effective volume of molecules.
• Compressibility factor \(Z = \frac{PV_m}{RT}\). If \(Z=1\) (Ideal), \(Z>1\) (+ve deviation), \(Z<1\) (-ve deviation).

CHAPTER 4: Thermodynamics

BASIC DEFINITIONS

• System: Part of universe under investigation.
• State Function: Depends on initial/final state (U, H, S, G).
• Path Function: Depends on path (q, w).
• Extensive: Depends on mass (Volume, Enthalpy).
• Intensive: Independent of mass (T, P, Density).

FIRST LAW OF THERMODYNAMICS

\[\Delta U = q + w\]

Enthalpy: \(H = U + PV\). For reactions: \(\Delta H = \Delta U + \Delta n_g RT\).

SECOND LAW & ENTROPY

Entropy change (general): \[\Delta S = nC_v \ln \frac{T_2}{T_1} + nR \ln \frac{V_2}{V_1}\]

For reversible process: \(\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr} = 0\).

GIBBS FREE ENERGY

\[\Delta G = \Delta H - T\Delta S\]

• \(\Delta G < 0\): Spontaneous.
• \(\Delta G = 0\): Equilibrium.
• \(\Delta G > 0\): Non-spontaneous.

THERMOCHEMISTRY

• Bond Enthalpy: \(\Delta_r H = \sum BE_{reactants} - \sum BE_{products}\).
• Neutralization: \(\Delta H_{neut} = -13.7\text{ kCal/eq} = -57.3\text{ kJ/eq}\) (for SA+SB).

CHAPTER 5: Chemical Equilibrium

For reaction \(aA + bB \rightleftharpoons cC + dD\):

\[K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]

\[K_p = K_c(RT)^{\Delta n_g}\]

Relation with Gibbs Energy: \(\Delta G^{\circ} = -RT \ln K_c\).

Le-Chatelier's Principle

Equilibrium shifts to counteract external changes in concentration, pressure, or temperature.

CHAPTER 6: Ionic Equilibrium

• Ostwald’s Dilution Law: \(\alpha = \sqrt{\frac{K_a}{C}}\).
• pH Calculation: \(pH = -\log[H^+]\). At 298 K, \(pH + pOH = 14\).

Salt Hydrolysis Table

Salt Type	Nature	pH Formula
Weak Acid + Strong Base	Alkaline	\(pH = 7 + \frac{1}{2}(pK_a + \log C)\)
Strong Acid + Weak Base	Acidic	\(pH = 7 - \frac{1}{2}(pK_b + \log C)\)
Weak Acid + Weak Base	Neutral/Variable	\(pH = 7 + \frac{1}{2}(pK_a - pK_b)\)

Buffer Solutions

Henderson Equation (Acidic Buffer): \[pH = pK_a + \log \frac{[Salt]}{[Acid]}\]

Henderson Equation (Basic Buffer): \[pOH = pK_b + \log \frac{[Salt]}{[Base]}\]

Solubility Product (\(K_{sp}\))

For \(A_xB_y \rightleftharpoons xA^{y+} + yB^{x-}\): \[K_{sp} = (xS)^x(yS)^y\]

CHAPTER 7: Solid State

Density of Unit Cell: \[d = \frac{Z \times M}{N_A \times a^3}\]

Unit Cell Comparisons

Type	Z	Relation (a & r)	Efficiency
SCC	1	\(a = 2r\)	52.4%
BCC	2	\(a = \frac{4r}{\sqrt{3}}\)	68%
FCC	4	\(a = 2\sqrt{2}r\)	74%

Defects

• Schottky: Equal cations/anions missing. Density decreases.
• Frenkel: Ion shifts to interstitial site. Density remains same.

CHAPTER 8: Solutions

Raoult's Law: \(P_A = P_A^{\circ} X_A\).

Colligative Properties

1. Relative lowering of V.P.: \(\frac{P^{\circ} - P_s}{P^{\circ}} = i X_B\)
2. Elevation in B.P.: \(\Delta T_b = i K_b m\)
3. Depression in F.P.: \(\Delta T_f = i K_f m\)
4. Osmotic Pressure: \(\pi = iCRT\)

Van't Hoff Factor (\(i\)): For dissociation, \(i = 1 + (n-1)\alpha\); for association, \(i = 1 + (1/n - 1)\alpha\).

CHAPTER 9: Electrochemistry

Gibbs Energy: \(\Delta G = -nFE_{cell}\).

Nernst Equation (298 K): \[E = E^{\circ} - \frac{0.0591}{n} \log Q\]

Faraday's Law: \(W = \frac{EIt}{96500}\).

Molar Conductance: \(\Lambda_m = \frac{\kappa \times 1000}{M}\).

CHAPTER 10: Chemical Kinetics

• Zero Order: \([A]_0 - [A] = kt; t_{1/2} = \frac{[A]_0}{2k}\)
• First Order: \(kt = \ln \frac{[A]_0}{[A]}; t_{1/2} = \frac{0.693}{k}\)
• Arrhenius Eq: \(k = Ae^{-E_a/RT}\)

ORGANIC CHEMISTRY (Chapters 12-24)

Stability & Reactivity

• Carbocation stability: \(3^{\circ} > 2^{\circ} > 1^{\circ} > CH_3^+\).
• Electronic effects: Mesomeric > Hyperconjugation > Inductive.
• Acidic strength: \(RSO_3H > RCOOH > Phenol > ROH > Alkyne\).

Name Reactions Summary

• Aldol: Needs \(\alpha-H\) + dil. NaOH.
• Cannizzaro: No \(\alpha-H\) + conc. NaOH.
• Reimer-Tiemann: Phenol + \(CHCl_3/KOH \rightarrow\) Salicylaldehyde.
• Hoffmann Bromamide: Amide \(\rightarrow\) Primary amine (1C less).

Distinction Tests

• Lucas Test: \(3^{\circ}\) Alcohol (Immediate), \(2^{\circ}\) (5 min), \(1^{\circ}\) (Heating).
• Tollen's Reagent: Aldehydes (Silver mirror), Ketones (-ve).
• Iodoform Test: Methyl ketones/alcohols (Yellow ppt).

INORGANIC CHEMISTRY (Chapters 25-32)

Periodic Trends

• Atomic radius: Decreases along period, increases down group.
• IE/EN: Increases along period, decreases down group.

Chemical Bonding

Hybridization: \(sp\) (Linear), \(sp^2\) (Trigonal), \(sp^3\) (Tetrahedral), \(sp^3d\) (TBP), \(sp^3d^2\) (Octahedral).

Bond Order: \(BO = \frac{1}{2}(N_b - N_a)\).

Salt Analysis

• Group I: \(Ag^+, Pb^{2+}, Hg_2^{2+}\) (as Chlorides).
• Group II: \(Cu^{2+}, Cd^{2+}, Bi^{3+}, As^{3+}\) (as Sulphides in acidic medium).
• Group III: \(Fe^{3+}, Al^{3+}, Cr^{3+}\) (as Hydroxides).
• Tests: Chromyl chloride test (for \(Cl^-\)), Brown ring test (for \(NO_3^-\)).