The results of certain calculations are given below. Round off each result to the appropriate number of significant figures:
A. 17.0141    B. 21.0239

Chemistry is an experimental science, and all quantitative measurements carry some degree of uncertainty. The concept of significant figures (also called significant digits or sig figs) provides a systematic way to communicate the precision of measurements and propagate uncertainty through calculations. Every measurement has a last certain digit and one uncertain (estimated) digit. All digits up to and including the first uncertain digit are significant. For example, a ruler graduated in millimetres allows you to measure to the nearest millimetre with certainty (say 15 mm) and estimate to the nearest 0.1 mm (say 15.3 mm) — the digit 3 is uncertain but significant. The result 15.3 mm has 3 significant figures. The key rules for counting significant figures: (1) All non-zero digits are always significant: 1, 2, 3, 4, 5, 6, 7, 8, 9. (2) Zeros between non-zero digits are always significant (these are "trapped" zeros): 1001 (4 sig figs), 20.05 (4 sig figs). (3) Leading zeros are never significant (they just establish the decimal point position): 0.0025 has only 2 sig figs (the 2 and the 5); the zeros before 25 are not significant. (4) Trailing zeros are significant only if there is a decimal point: 1700 is ambiguous (2, 3, or 4 sig figs possible), but 1700. has 4 sig figs; 1.700 has 4 sig figs; 1.70 has 3 sig figs; 1.7 has 2 sig figs. (5) Exact numbers (defined values like 1 inch = 2.54 cm exactly, or counted quantities like 5 students) have infinite sig figs and never limit the precision of calculations.

Every measurement in science has some associated uncertainty or error. Understanding error types and how they propagate through calculations is essential for assessing the quality and reliability of experimental results. Systematic errors (also called determinate errors) are reproducible inaccuracies that consistently bias measurements in the same direction. Sources: calibration errors (an uncalibrated balance, a thermometer that reads 1°C too high), method errors (a reaction that is not complete, an indicator that changes colour too early or too late), personal errors (parallax reading, incorrect technique). Systematic errors cannot be reduced by taking multiple measurements and averaging — they affect all measurements equally. They can only be detected and corrected by: using different measurement methods, comparing with primary standards, or performing a blank/control experiment. Random errors (also called indeterminate errors) arise from unpredictable, uncontrollable fluctuations in the measurement process — variations in reading the meniscus, fluctuations in room temperature during weighing, vibrations. Random errors are reduced by taking multiple measurements and calculating the mean (average). Statistical analysis: mean = Σx/n (arithmetic average); standard deviation σ = sqrt(Σ(xi-mean)²/(n-1)) (sample standard deviation) quantifies the spread of random errors; relative standard deviation (RSD) or coefficient of variation (CV) = σ/mean × 100% (normalised spread). Accuracy vs. precision: accuracy = closeness to the true value (affected by systematic errors). Precision = reproducibility of repeated measurements (affected by random errors).

The International System of Units (SI), adopted in 1960 and continuously updated, provides a coherent system of measurement units used worldwide in science and increasingly in everyday life. Seven base SI units: metre (m, length), kilogram (kg, mass), second (s, time), ampere (A, electric current), kelvin (K, thermodynamic temperature), mole (mol, amount of substance), candela (cd, luminous intensity). All other units are derived from these seven base units. Examples: Force = kg m s-2 = newton (N). Energy = kg m2 s-2 = joule (J). Pressure = kg m-1 s-2 = pascal (Pa). Electric charge = A s = coulomb (C). In chemistry, the most frequently used units: Concentration: mol/L (molarity, M) or mol/m3 (SI). Pressure: Pa or kPa (1 atm = 101.325 kPa = 760 mmHg). Temperature: K = °C + 273.15. Energy: J or kJ (1 cal = 4.184 J; 1 kcal = 4.184 kJ; 1 eV = 96.485 kJ/mol). Volume: m3 (SI), but mL and L used practically (1 L = 10-3 m3). Prefixes: kilo (k, 10^3), mega (M, 10^6), giga (G, 10^9), milli (m, 10^-3), micro (μ, 10^-6), nano (n, 10^-9), pico (p, 10^-12), femto (f, 10^-15). Unit conversion: use dimensional analysis (factor-label method): multiply by conversion factors (which equal 1, like 1000 mL/1 L) so that unwanted units cancel.

The mole concept is the bridge between the atomic/molecular world (individual atoms, molecules, ions) and the macroscopic, measurable world (grams, litres, particles that can be weighed and observed). One mole contains exactly N_A = 6.02214076 × 10^23 elementary entities (atoms, molecules, ions, electrons, etc.) — defined by the 2019 SI revision as an exact number (previously N_A was a measured constant). Molar mass (M): the mass in grams of exactly one mole of a substance = numerically equal to the relative atomic or molecular mass (previously called atomic weight or molecular weight). Atomic masses: H = 1.008, C = 12.011, N = 14.007, O = 15.999, F = 18.998, Na = 22.990, Mg = 24.305, Al = 26.982, S = 32.06, Cl = 35.45, K = 39.098, Ca = 40.078, Fe = 55.845, Cu = 63.546, Br = 79.904, Ag = 107.868, I = 126.904, Ba = 137.327, Pb = 207.2. Common molar mass calculations: NaCl: 22.99 + 35.45 = 58.44 g/mol. H2O: 2(1.008) + 15.999 = 18.015 g/mol. CO2: 12.011 + 2(15.999) = 44.009 g/mol. CaCO3: 40.078 + 12.011 + 3(15.999) = 100.086 g/mol. H2SO4: 2(1.008) + 32.06 + 4(15.999) = 98.079 g/mol.

The behaviour of gases at low pressure and high temperature is described by the ideal gas equation PV = nRT, which combines three empirical gas laws discovered in the 17th-19th centuries. Boyle's law (1662): at constant T and n, PV = constant. Graphically: P vs 1/V is linear; P vs V is a rectangular hyperbola. Charles' law (1787): at constant P and n, V/T = constant (volume proportional to absolute temperature). Avogadro's law (1811): at constant T and P, V/n = constant (equal volumes of gases at same T and P contain equal numbers of molecules). Combined: PV = nRT where R = 8.314 J mol-1 K-1 = 0.08206 L atm mol-1 K-1 = 8.314 Pa m3 mol-1 K-1. STP (Standard Temperature and Pressure): 0°C (273.15 K) and 1 bar (100 kPa) in current IUPAC definition. At STP, 1 mol ideal gas occupies 22.4 L (at old STP with P = 1 atm = 101.325 kPa; at new IUPAC STP with 1 bar, molar volume = 22.711 L). Real gases deviate from ideal behaviour at: high pressure (molecules are close together, intermolecular forces and excluded volume matter) and low temperature (near condensation, attractive forces important). Van der Waals equation: (P + an²/V²)(V - nb) = nRT, where a = correction for intermolecular attractions (Pa m6/mol2 = atm L2/mol2), b = correction for molecular volume (m3/mol = L/mol).

Concentration of a solution can be expressed in several different ways, each useful in different contexts. Molarity (M): moles of solute per litre of solution. M = n/V(L). Most commonly used in chemistry calculations. Note: molarity depends on temperature (volume changes with T). Molality (m): moles of solute per kilogram of solvent. m = n/w_solvent(kg). Used for colligative property calculations (boiling point elevation, freezing point depression) because molality is temperature-independent. Mole fraction (x): moles of component divided by total moles. x_solute = n_solute/(n_solute + n_solvent). Used in Raoult's law. Mass fraction (w): mass of solute / total mass of solution. Often expressed as percentage (% w/w). Volume fraction (φ): volume of component / total volume. Often expressed as % v/v. Normality (N): equivalents per litre = n-factor × molarity. Used in acid-base and redox titrations. ppm and ppb: mg/kg or mg/L (water quality measurements). Dilution calculation: C1V1 = C2V2 (for dilution, where no chemical reaction occurs and amount of solute is conserved). For example: to prepare 250 mL of 0.1 M HCl from 12 M HCl stock: V1 = C2V2/C1 = (0.1 × 250)/12 = 2.08 mL. Add 2.08 mL of 12 M HCl to enough water to make 250 mL total volume.