Notes on Avogadro's Number, Molar Mass, Percent Composition, and Formulas

2.7 The Mole and Avogadro's Number

• Avogadro's number: N_A = 6.02214076 \times 10^{23} particles per mole.
• The mole is the SI base unit for measuring an amount of substance.
• One mole of a substance contains the same number of particles as one mole of any other substance: 1\text{ mol} = N_A \text{ particles}
• The mole concept enables counting particles (atoms, molecules, ions) by weighing macroscopic samples.
• Avogadro's number establishes the link between the macroscopic mass and the number of elementary entities.
• Molar mass: the mass in grams of one mole of particles of a substance. It is the mass of 1 mole of that substance in g/mol.
• An element's molar mass is numerically equal to its atomic weight: M \;(\text{g/mol}) \approx Ar where $Ar$ is the atomic weight in amu.
• The molar mass concept is the cornerstone of quantitative chemistry.
• Mass–moles–molar mass conversions:
  • Mass to moles: n = \frac{m}{M}
  • Moles to mass: m = nM
• The molar masses of the elements are typically known to four or more significant figures.
• The formula of a compound indicates the types of atoms or ions present and the relative numbers of each.
• Subscripts indicate either the number of atoms in a molecule or the number of moles of that element in the substance (depending on context).
• The Mole is a Counting Unit that bridges microscopic particles and macroscopic mass.
• Core ideas:
  • 1\text{ mol} = N_A particles, regardless of the substance.
  • Molar mass $M$ is the mass per mole in g/mol.
  • Mass, moles, and molar mass form a set of interchangeable units for quantitative analysis.

2.8 Chemical Analysis and Percent/Molecular Composition

• Chemical Analysis: the determination of the amounts or identities of the components of a mixture.

• Percent Composition: a central principle is that any sample of a pure compound consists of the same elements in the same proportion by mass.

  • This allows expressing composition as a percent by mass.

• Molecular Composition: can be expressed as a percent (mass of an element in a 100.0 g sample).

• Empirical and Molecular Formulas from Percent Composition
  1) Know the identity of the elements in the sample.
  2) Determine the mass of each element in a given mass of compound by chemical analysis.
  3) Calculate the relative amount (moles) of each element, which is the relative number of atoms of each element in the formula of the compound.

• Steps to obtain empirical formula from mass percent (illustrative template):
  Step 1: Convert mass percent to mass.
  Step 2: Convert mass to amount (moles).
  Step 3: Find the mole ratio by dividing by the smallest number of moles.
  Step 4: Determine the smallest whole-number ratio for the elements to obtain the empirical formula.

• Step 1: For mass percent data, assume a 100.0 g sample. Then the mass of each element equals its percent value in grams:

  • If element A is pA %, then mA = p_A\,\text{g} (for a 100.0 g sample).
  • If element B is pB %, then mB = p_B\,\text{g}.
  • In general, mi = pi\;\text{g} for a 100.0 g sample.

• Step 2: Convert each element’s mass to moles:
  ni = \frac{mi}{Mi} where $Mi$ is the molar mass of element $i$.

• Step 3: Compute mole ratios by dividing by the smallest $n_i$ among the elements.

• Step 4: Convert these ratios to the smallest whole numbers to obtain the empirical formula: if the ratios are $nA : nB : \dots$, then empirical formula is $A{x}B{y}\dots$ with $x,y$ being the rounded whole-number ratios.

• Empirical formulas:

  • Represent the simplest whole-number ratio of atoms in a compound.
  • Use at least 3 significant figures when reporting the mole ratios.
  • When determining atom ratios, always divide the larger number of moles by the smaller one.

• Molecular vs Empirical Formulas:

  • Empirical formula shows the simplest whole-number ratio of atoms.
  • Molecular formula shows the actual total number of each type of atom in a molecule.
  • For ionic compounds, there is no distinct molecular formula; only the empirical formula is meaningful.
  • To obtain the molecular formula from the empirical formula, you must know the molar mass of the compound:
    M = n \times M{emp} where $M{emp}$ is the molar mass of the empirical formula and $n$ is a whole-number multiplier.

• Determining the molecular formula from empirical data requires the compound’s molar mass (experimentally determined or known).

• Empirical vs Molecular Formulas:

  • Empirical formula: simplest ratio of atoms.
  • Molecular formula: total number of atoms in the actual molecule; may be a multiple of the empirical formula.
  • Ionic compounds: molecular formula is not defined in the same way; empirical formula is used to represent the simplest ratio.

Determining a Formula from Mass Data

• The mass-percent composition of a compound gives the mass of each element in a 100.0 g sample.

• In the laboratory, additional information about composition can be used to determine formulas of compounds.

• Method: combining known masses of elements to yield a sample of known mass:

  • Element masses can be converted to moles.
  • The ratio of these amounts gives the combining ratio of atoms, i.e., the empirical formula.

• Alternative method: decomposing a known mass of an unknown compound into pieces of known composition:

  • If the masses of the pieces are known, the ratio of moles of the pieces yields the formula.

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