Ratios and proportional reasoning Knowledge-Based Study

Study notes

1. What is a Ratio?

   In chemistry, quantitative relationships govern how substances interact, combine, and change. A ratio is a mathematical comparison of two quantities that indicates how much of one entity is present relative to another.

   Ratios can be expressed in several formats:

   • Using a colon, such as 2:1
   • As a fraction, such as $\frac{2}{1}$
   • In words, such as "2 to 1"

   For example, in a molecule of water, there are always 2 hydrogen atoms for every 1 oxygen atom. This constant relationship represents a fixed atomic ratio of 2:1.

   Understanding Proportions

   A proportion is a mathematical statement showing that two ratios are equal:

   $\frac{a}{b} = \frac{c}{d}$

   In chemistry, many physical properties and chemical processes exhibit direct proportionality. When two quantities are directly proportional, increasing one quantity by a certain factor causes the other quantity to increase by the same factor, keeping their ratio constant.

   Consider density, which is defined as the ratio of mass to volume:

   $\text{Density} = \frac{\text{mass}}{\text{volume}}$

   Because density is an intensive property of a pure substance at a given temperature, doubling the mass of a sample also doubles its volume, maintaining a constant ratio.

   Using Cross-Multiplication

   When three terms of a proportion are known, the unknown fourth term can be calculated using cross-multiplication.

   Suppose a 5.0 mL sample of an unknown liquid has a mass of 4.0 g. To determine the mass of a 25.0 mL sample of the same liquid, set up a proportion based on constant density:

   $\frac{4.0\text{ g}}{5.0\text{ mL}} = \frac{x}{25.0\text{ mL}}$

   Cross-multiplying gives:

   $5.0 \cdot x = 4.0 \cdot 25.0$

   $5.0 \cdot x = 100.0$

   $x = 20.0\text{ g}$

   Proportional reasoning allows chemists to scale quantities systematically in laboratory preparations, solution dilutions, and stoichiometric predictions.