Stoichiometric rule

Core concept and mathematical expression

The stoichiometric rule is derived from the strict proportionality between the amounts of reacting substances and their coefficients in a chemical equation. It is used for calculations involving reactions with any coefficients.

Mathematically, the rule is expressed as follows: the ratio of the amount of substance ($n_i$, in moles) to its stoichiometric coefficient ($\nu_i$) yields the reduced amount of substance ($n'$).

A rigorous formulation of the rule relates the change in the amount of substance ($\Delta n_i$) to the stoichiometric coefficient and the extent of the reaction ($\Delta n'$). For initial reactants, the amount decreases ($\Delta n < 0$), while for reaction products, it increases ($\Delta n > 0$). The value of $n'$ itself can be interpreted as the number of theoretical "reaction cycles" that the given portion of the reactant will suffice for.

Matrix calculation algorithm

To simplify calculations, a matrix layout is used, placed directly beneath the reaction equation. This eliminates the need to write out intermediate formulas on paper.

The matrix structure includes four main rows (from top to bottom):

1. $M$, g/mol — molar mass.
2. $m$, g — mass of the substance (initial or formed).
3. $n$, mol — actual amount of substance (calculated as $m/M$).
4. $n'$, mol — reduced amount of substance (calculated as $n / \nu$).

Solution workflow resembles the Latin letter "U":

• First, move down ($\downarrow$) from the known mass to $n$, and then to $n'$.
• Next, transfer the value of $n'$ horizontally ($\rightarrow$ or $\leftarrow$) to the target substance, since the ratio is stoichiometrically equivalent.
• Finally, move up ($\uparrow$): multiply $n'$ by the coefficient to obtain $n$, and then multiply $n$ by the molar mass to obtain the target mass $m$.

Calculations involving excess and deficiency

In practice, starting materials are rarely taken in exact stoichiometric ratios. This may be due to technological necessity (complete conversion of one reactant requires an excess of another), neutralization of toxic agents (reagents are taken in deliberate excess), or accidental mixing during accidents.

If the masses of both reactants are given, it is necessary to determine the limiting factor:

• Calculate $n'$ for each starting material.
• The substance with the smallest $n'$ value is in deficiency. It is consumed completely and determines the amount of products formed.
• The substance with the larger $n'$ value is in excess (part of it will remain after the reaction is complete).

To verify the accuracy of the calculations, the law of conservation of mass is used: the sum of the masses of the reactants before the reaction must equal the sum of the masses of the substances (including excess residues and precipitates) after the reaction.

Application to elemental composition

The stoichiometric rule applies not only to reaction equations but also to molecular formulas. The ratio of the amount of a specific element to its subscript in the compound's formula is identical for all elements in that molecule.

This property allows you to:

• Calculate the mass of a specific atom within a given mass of a complex compound (e.g., calculating the mass of aluminum in aluminum sulfide).
• Determine the empirical formulas of unknown substances knowing the mass ratios of the elements. To do this, the mass ratio is divided by the atomic masses of the elements, the molar ratio is found, and it is reduced to the simplest whole-number subscripts, while checking the result against valency (e.g., when deriving the formula of the freon $CF_2Cl_2$).

Class and exam questions

What else to study on this topic

• Calculation of the reaction matrix with changes in the amount of substance (Δn) and material balance.
• Determining the formula of a salt based on thermal decomposition data (e.g., calcination of KClO_x).
• Biomedical problems: calculating the mass of protein macromolecules via equivalents.
• Problems on metal displacement from salt solutions taking into account excess and deficiency.

Other topics in «Chapter 2. Chemical Reactions»

Prepared by WhiteCoat Pro faculty in accordance with the medical school curriculum. For educational purposes only.