Concentration Terms and Their Conversion
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Direct answer
Molarity, molality, mole fraction, normality and mass percent describe one solution from different angles, and every conversion runs through the same two quantities — moles of solute and mass of solvent — with density as the bridge between mass and volume. The formula worth memorising: molality = (1000 × M)/(1000d − M × M2), where d is density in gram per millilitre and M2 the solute's molar mass. Mole fraction is safest built on a 1 kg solvent basis: x(solute) = m/(m + 1000/M1). Normality is just molarity times the n-factor. And temperature sorts the terms: molarity and normality change with heating because volume does; molality, mole fraction and mass percent do not — the property colligative chapters rely on.
What you must remember
- Definitions in one line each: M = moles of solute per litre of solution; m = moles per kilogram of solvent; x = moles of one component over total moles; N = equivalents per litre; mass percent = (mass solute/mass solution) × 100; ppm = mass fraction × 10⁶.
- The bridge formula: m = (1000 M)/(1000d − M × M2); with d in g/mL it converts any molarity to molality in one substitution.
- Mole fraction from molality: x(solute) = m/(m + 1000/M1); for water, 1000/18 = 55.5 mol per kg is the number to hold.
- Normality link: N = M × n-factor (2 for H2SO4, 5 for acidified KMnO4); titration arithmetic N1V1 = N2V2 rides on it.
- Mass percent to molarity: M = (10 × percent w/w × d)/M2, again with d in g/mL.
- Temperature behaviour: M and N vary with temperature, m, x and mass percent do not — a one-mark assertion nearly every session.
- Dilution: M1V1 = M2V2 for molarity; the molality counterpart needs the added solvent mass, not volume.
One solution converted into every term
Take 3.0 M sodium chloride with density 1.25 g/mL, and walk it through every unit. Per litre of solution: 3 mol of NaCl, mass of solute 3 × 58.5 = 175.5 g, mass of one litre = 1250 g, so the solvent weighs 1250 − 175.5 = 1074.5 g. Molality = 3/1.0745 = 2.79 mol per kg — or straight through the bridge formula, 3000/(1250 − 175.5) = 2.79. Mole fraction: moles of water = 1074.5/18 = 59.7; x(NaCl) = 3/(3 + 59.7) = 0.048. Mass percent = 175.5/1250 × 100 = 14.0%. Every term from the same two quantities, exactly as the definition says.
Reverse the flow for practice. A 20% w/w sugar solution of density 1.08 g/mL: molarity = 10 × 20 × 1.08/342 = 0.63 M. The habit to build is sketching the litre of solution with its solute moles and solvent mass first; the formulas then become descriptions rather than memory.
Solvent versus solution slips
The solvent-versus-solution confusion generates most wrong answers: molality divides by solvent mass, molarity by solution volume, and the two differ by the solute's own mass — about 14% in the worked example above. The bridge formula's minus sign is the second casualty: 1000d minus M × M2, because the solute mass must leave before water is counted; writing 1000d + M M2 or plain 1000d puts the molality high. Third, density in kilogram per litre fed where gram per millilitre belongs — a factor of 1000 slips silently. Fourth, normality's n-factor: 1 M sulphuric acid is 2 N only when both protons react; the question's context sets it. And when a problem supplies no density, it usually wants mole fraction or molality built from masses alone — no volume-to-mass step is needed, and inventing one wastes time.
Frequently asked questions
Which concentration terms survive a temperature change?
Molality, mole fraction and mass percent — they are mass-based; molarity and normality shift because solution volume expands on heating.
What is the molality of 3 M NaCl with density 1.25 g/mL?
m = 1000 × 3/(1000 × 1.25 − 3 × 58.5) = 3000/1074.5 ≈ 2.8 mol per kg of water.
How does mass percent convert to molarity?
Through M = (10 × percent w/w × d)/M2 with density in g/mL — 20% sugar at 1.08 g/mL is about 0.63 M.
Why is density indispensable in molarity-molality conversion?
Because molarity lives in volume and molality in mass, and density is the only bridge between a solution's volume and its mass.
When is normality the more convenient unit?
In titration arithmetic — N1V1 = N2V2 holds for any stoichiometry because equivalents react one to one.