In a titration, 25.0 mL of $0.05$ M $\text{K}_2\text{C}_2\text{O}_4$ solution is titrated against $\text{KMnO}_4$ solution in acidic medium. If 12.5 mL of $\text{KMnO}_4$ is required, the molarity of $\text{KMnO}_4$ is:

Permanganometry refers to volumetric analysis using potassium permanganate (KMnO4) as the titrant (oxidising agent). It is one of the most widely used titrimetric methods for determining reducing agents due to several advantageous properties of KMnO4: it is a powerful oxidising agent (E° = +1.51 V in acid, +0.59 V in neutral, +0.56 V in alkaline medium); it is deeply purple-coloured even in dilute solution, making it a self-indicator (no external indicator needed — the endpoint is the first persistent pink/purple colour of excess permanganate); it is relatively inexpensive; and it can determine a wide variety of reducing analytes. The main disadvantages: KMnO4 is not a primary standard (it gradually oxidises organic matter and decomposes, especially in light and in the presence of Mn2+ which catalyses decomposition), so its solutions must be standardised regularly against a primary standard such as sodium oxalate (Na2C2O4, primary standard certified for purity and stability) or iron wire (Fe). The reaction of KMnO4 with reducing agents in acid medium is: MnO4- + 8H+ + 5e- → Mn2+ + 4H2O (E° = +1.51 V). In neutral/alkaline medium: MnO4- + 2H2O + 3e- → MnO2 + 4OH-.

The concept of equivalents and equivalent weight, while increasingly replaced by the mole concept in modern chemistry, remains essential for understanding volumetric analysis calculations, particularly for redox titrations where the stoichiometry depends on the number of electrons transferred. The n-factor (equivalents per mole) of a substance in a redox reaction is defined as the number of electrons gained or lost per formula unit in the reaction. For KMnO4 in acid medium: Mn goes from +7 to +2 (change of 5), so n-factor = 5. For K2Cr2O7 in acid medium: Cr goes from +6 to +3 (change of 3 per Cr × 2 Cr per formula unit = change of 6), so n-factor = 6. For Na2C2O4: C goes from +3 to +4 (change of 1 per C × 2 C per formula unit = 2), so n-factor = 2. For FeSO4: Fe goes from +2 to +3 (change of 1), so n-factor = 1. The law of equivalents states that at the equivalence point of any titration: equivalents of titrant = equivalents of analyte. n1 × M1 × V1 = n2 × M2 × V2. This equation is of fundamental importance in all volumetric calculations and allows calculation of the molarity of an unknown solution from the volume of a standard solution used in a titration.

A primary standard is a highly pure, stable, chemically well-defined substance that can be used to prepare a standard solution of exactly known concentration by direct weighing and dissolution. Requirements for a primary standard: high purity (99.9%+), chemical stability (does not decompose or absorb moisture, CO2, or O2 from air), large molar mass (to minimise weighing errors), readily available in pure form. Common primary standards for acid-base: potassium hydrogen phthalate (KHP) for standardising NaOH; anhydrous Na2CO3 for standardising HCl. Primary standards for redox: Na2C2O4 (sodium oxalate, for KMnO4), As2O3 (arsenic trioxide, for KMnO4, KBrO3), pure iron wire (for K2Cr2O7, KMnO4), K2Cr2O7 itself (primary standard, can be used to standardise Na2S2O3 indirectly), KIO3 (potassium iodate, for Na2S2O3), KBrO3 (for various reducing agents). A secondary standard is a solution standardised by titration against a primary standard. KMnO4 is a secondary standard (standardised against Na2C2O4). KMnO4 cannot be used as a primary standard because: it oxidises organic matter in water, it decomposes slowly in light, its equivalent weight is difficult to define exactly because it reacts differently in acid, neutral, and alkaline media.

Two related but distinct types of titrimetric methods involve iodine as either the titrant or the product: Iodimetry: direct titration with standard I2 solution (I2 as oxidising agent). Analytes: As(III) (arsenite → arsenate), Sn2+ (stannous → stannic), S2O3^2- (thiosulfate → tetrathionate), ascorbic acid (vitamin C). Indicator: starch solution gives intensely blue colour with I2 (I2 forms a complex with starch chains). Iodometry: indirect method where a known excess of KI is added to the analyte (oxidising agent); the I2 liberated by oxidation of I- is then titrated with standard Na2S2O3 (sodium thiosulfate). Used for: Cu2+ (Cu2+ + I- → CuI + I2; I2 titrated with Na2S2O3). KMnO4, K2Cr2O7, Cl2, H2O2, O3 (all oxidise I- to I2; I2 titrated with thiosulfate). Key reaction for Na2S2O3 titration: I2 + 2S2O3^2- → 2I- + S4O6^2- (tetrathionate). Starch is added near the endpoint (not at beginning, to avoid starch-I2 complex being too strong to break at low I2 concentrations). Na2S2O3 is standardised against KIO3 or K2Cr2O7.

Acid-base titrations are among the oldest and most widely practised methods of quantitative chemical analysis. The theory is based on the neutralisation reaction: H+ + OH- → H2O (in aqueous solution) or, more precisely for weak acids and bases, the transfer of a proton from the acid to the base. Titration curve: the pH versus volume of titrant added curve shows a characteristic sigmoid (S-shaped) shape. For strong acid vs strong base: pH changes slowly in the flat regions before and after the equivalence point, but changes very steeply (typically 6-8 pH units) at the equivalence point, which occurs at exactly pH 7 (because the salt formed is neutral). For weak acid vs strong base: the equivalence point is at pH > 7 (because the conjugate base of the weak acid is a base that hydrolyses water to give OH-). For weak base vs strong acid: equivalence point at pH < 7. The choice of indicator is critical: the indicator must change colour within the steep portion of the titration curve (which ideally encompasses the equivalence point). Phenolphthalein (changes from colourless to pink, pH range 8.3-10.0): suitable for weak acid vs strong base titrations where equivalence point is alkaline. Methyl orange (changes from red to yellow, pH range 3.1-4.4): suitable for strong acid vs weak base or strong acid vs strong base. Litmus (pH 5-8): less commonly used for titrations due to imprecise colour change. Buffer region: in a weak acid-strong base titration, at the half-equivalence point, [HA] = [A-] and pH = pKa (Henderson-Hasselbalch equation: pH = pKa + log([A-]/[HA]) = pKa when [A-] = [HA]).

Complexometric titrations use metal-chelating agents, particularly EDTA (ethylenediaminetetraacetic acid, H4Y), as the titrant to determine metal ion concentrations. EDTA is a hexadentate ligand (2 N + 4 O donor atoms) that forms very stable 1:1 complexes with virtually all metal ions: M^n+ + Y^4- → [MY]^(n-4). The stability constant (formation constant) Kf = [MY^(n-4)] / ([M^n+][Y^4-]) is very large for most metal-EDTA complexes (e.g., Kf for Ca-EDTA = 5×10^10, Mg-EDTA = 5×10^8, Fe3+-EDTA = 10^25, Pb2+-EDTA = 10^18). Indicator for EDTA titrations: metallochromic indicators change colour depending on whether metal is free or complexed with indicator. Eriochrome Black T (EBT): forms red complex with metal ions (M-EBT), free indicator is blue. At equivalence point, EDTA displaces EBT from M-EBT complex → solution turns from red/pink to pure blue. Important application: determination of water hardness. Temporary hardness: due to dissolved Ca(HCO3)2 and Mg(HCO3)2 — removed by boiling (CaCO3 and MgCO3 precipitate) or by adding Ca(OH)2 (Clark's process). Permanent hardness: due to dissolved CaCl2, CaSO4, MgCl2, MgSO4 — not removed by boiling. Removed by: adding Na2CO3 (Soda process), zeolite water softening (ion exchange: Ca2+/Mg2+ exchanged for Na+), or passing through ion exchange resins. EDTA titration: both Ca2+ and Mg2+ titrated together with EDTA at pH 10 (buffered with NH4+/NH3) using EBT indicator; total hardness = moles EDTA used × molar mass of CaCO3 expressed as ppm (mg CaCO3 per litre water).