Concentration Units & Calculations

Definition of Concentration

• Concentration = amount of solute dissolved in a solvent.
• Expressed in multiple standardized units adapted to context.
  • Liquors: volume percent (% v/v); “proof” = 2 × volume percent.
  • Fruit-juice sweetness: degrees Brix (°Bx) = mass percent (g glucose / g solution × 100 %).

Units of Concentration Tested on the MCAT

• Percent composition by mass
• Mole fraction
• Molarity (M)
• Molality (m)
• Normality (N)
• Dilution relationships

Percent Composition by Mass (Mass %)

• Formula: \text{Mass %}=\frac{\text{mass of solute}}{\text{mass of solution}}\times100\%
• Applies to aqueous and non-aqueous mixtures (e.g., metal alloys).
• Example
  • “100 g solution contains 20 g NaCl.”
  • $\frac{20\,g}{100\,g}\times100\%=20\%$ NaCl by mass.
• Significance
  • Simple experimental measurement (weigh only).
  • Useful when densities vary or solvents are solids.

Mole Fraction (χ)

• Formula for component A: $\chi<em>A=\frac{n</em>A}{\sum n<em>i}$ where $\sum \chi</em>i =1$.
• Typical Uses
  • Raoult’s law → vapor-pressure depression.
  • Partial pressures of gases (Chapter 8 topics).
• Example: 184 g glycerol (C₃H₈O₃) + 180 g H₂O
  • Molar masses given: H₂O = 18 g mol⁻¹ ; glycerol = 92 g mol⁻¹.
  • $n<em>{\text{H}</em>2\text{O}}=\frac{180\,g}{18\,g\,mol^{-1}}=10\,mol$
  • $n_{\text{glycerol}}=\frac{184\,g}{92\,g\,mol^{-1}}=2\,mol$
  • Total moles = 12.
  • $\chi<em>{\text{H}</em>2\text{O}}=\frac{10}{12}=0.83$
  • $\chi_{\text{glycerol}}=\frac{2}{12}=0.17$
• Notes
  • Unitless, temperature-independent.
  • Essential in colligative-property calculations.

Molarity (M)

• Definition: $M=\frac{\text{moles of solute}}{\text{liters of solution}}$
• Square-bracket notation: $[\text{Na}^+],\,[H^+],\,[\text{A}]$ always implies molarity unless stated otherwise.
• Denominator = volume of final solution (not solvent alone).
• Key Applications
  • Rate laws, equilibrium (law of mass action), pH/pOH, osmotic pressure, Nernst equation.
• Example: 11 g CaCl₂ diluted to 100 mL
  • Molar mass CaCl₂ ≈ 111.1 g mol⁻¹.
  • $n=\frac{11\,g}{111.1\,g\,mol^{-1}}=0.10\,mol$
  • Volume = 0.100 L.
  • $M=\frac{0.10\,mol}{0.100\,L}=1.0\,M$
• Practical Insight
  • Temperature sensitive (volume expands/contracts).

Molality (m)

• Definition: $m=\frac{\text{moles of solute}}{\text{kilograms of solvent}}$
• For dilute aqueous solutions at $25\,^{\circ}!\text{C}$, $m\approx M$ because $\rho<em>{\text{H}</em>2\text{O}}\approx1\,kg\,L^{-1}$.
• Divergence occurs at high solute concentration since solute >> H₂O molar mass → density ↑.
• Primary Uses
  • Boiling-point elevation.
  • Freezing-point depression.
• Example: 10 g NaOH in 500 g H₂O
  • Molar mass NaOH = 40 g mol⁻¹.
  • $n=\frac{10\,g}{40\,g\,mol^{-1}}=0.25\,mol$
  • Solvent mass = 0.500 kg.
  • $m=\frac{0.25\,mol}{0.500\,kg}=0.50\,m$

Normality (N)

• Definition: $N=\frac{\text{equivalents of species of interest}}{\text{liter of solution}}$
• “Equivalent” = amount that reacts or supplies one mole of the species of interest (H⁺, e⁻, etc.).
• Reaction-dependent: same solute can have different N in different contexts.
  • Acid-base: count H⁺ or OH⁻ equivalents.
  • Redox: count e⁻ equivalents.
• Example scenarios with permanganate $\text{MnO}_4^-$
  • Acidic medium: 1 mol accepts 5 e⁻ → 1 M = 5 N.
  • Alkaline medium: 1 mol accepts 1 e⁻ → 1 M = 1 N.
• Connections
  • Tied to gram-equivalent weight (Chapter 4).
  • Useful in titration calculations where stoichiometry ≠1:1.

Dilution Calculations

• When solvent is added, moles remain constant: $M<em>iV</em>i = M<em>fV</em>f$.
• Works with any consistent volume units (mL or L).
• Example: Prepare 300 mL of 1.1 M NaOH from 5.5 M stock
  • $V<em>i = \frac{M</em>fV<em>f}{M</em>i} = \frac{1.1\,M \times 300\,mL}{5.5\,M}=60\,mL$ stock needed.
  • Add water until total volume = 300 mL.

Comparative & Practical Insights

• Choice of unit depends on property measured, experimental convenience, and sensitivity to temperature or density.
  • Mass % and molality are temperature independent (mass doesn’t change with T).
  • Molarity and normality vary with temperature (dependent on solution volume).
• Real-world relevance
  • Quality control (proof of alcohol, °Bx for juice sweetness).
  • Clinical labs & pharmaceuticals often use normality for acid/base reagents.
  • Environmental chemistry uses mole fraction for gas mixtures.
• Ethical/Practical implications
  • Proper labeling of concentration critical for safety (lab reagents, beverages).
  • Misinterpretation of units can lead to dosing errors or experimental failure.