Question

Two elements, \(\mathrm{R}\) and \(\mathrm{Q}\), combine to form two binary compounds. In the first compound, \(14.0 \mathrm{~g}\) of \(\mathrm{R}\) combines with \(3.00 \mathrm{~g}\) of \(\mathrm{Q}\). In the second compound, \(7.00 \mathrm{~g}\) of \(\mathrm{R}\) combines with \(4.50 \mathrm{~g}\) of \(\mathrm{Q}\). Show that these data are in accord with the law of multiple proportions. If the formula of the second compound is RQ, what is the formula of the first compound?

Short answer

The formula of the first compound is R₃Q, and the data are in accord with the Law of Multiple Proportions.

Step-by-step solution

Determine the ratio of weights in the first compound

Divide the weight of the element R in the first compound by the weight of element Q in the first compound: Ratio of weights in the first compound = \(\frac{14.0\,\mathrm{g\,of\,R}}{3.00\,\mathrm{g\,of\,Q}} = \frac{14}{3}\)

Determine the ratio of weights in the second compound

Divide the weight of the element R in the second compound by the weight of element Q in the second compound: Ratio of weights in the second compound = \(\frac{7.00\,\mathrm{g\,of\,R}}{4.50\,\mathrm{g\,of\,Q}} = \frac{7}{4.5}\)

Calculate the ratio of both weight ratios

Now, we will find the ratio of the two weight ratios. If this ratio is a simple whole number, the data follow the Law of Multiple Proportions: $$\frac{\frac{14.0\,\mathrm{g\,of\,R}}{3.00\,\mathrm{g\,of\,Q}}}{\frac{7.00\,\mathrm{g\,of\,R}}{4.50\,\mathrm{g\,of\,Q}}} = \frac{\frac{14}{3}}{\frac{7}{4.5}}$$

Simplify the ratio

Simplify the ratio by cross-multiplying and reducing it to a whole number: $$\frac{\frac{14}{3}}{\frac{7}{4.5}} = \frac{14 \times 4.5}{3 \times 7} = \frac{63}{21}$$ The simplified ratio is \(\frac{63}{21} = 3\), which is a simple whole number.

Determine the formula of the first compound

Since the ratio of weight ratios turned out to be 3, the proportion of R and Q in the first compound would be 3 times that in the second compound. Since the second compound's formula is RQ, the first compound's formula would be R₃Q (3 times Q for each R). Therefore, the formula of the first compound is R₃Q, and the data are in accord with the Law of Multiple Proportions.

Key concepts

Binary Compounds

A binary compound is a chemical compound with exactly two different elements. In our exercise, elements \(R\) and \(Q\) form two separate binary compounds. Binary compounds are foundational in studying chemistry because they help us understand how different elements interact and combine with each other.
They often follow simple stoichiometric patterns which are guided by specific laws and rules, such as the law of multiple proportions. This law plays a crucial role when examining reactions and combinations in binary compounds, as it defines how one element can combine with another in multiple ways to form different compounds.

Chemical Formulas

Chemical formulas represent the composition of compounds using chemical symbols and numbers to show the ratio of atoms involved. In the problem, the chemical formula for the second compound is given as \(\text{RQ}\), indicating a 1:1 ratio of atoms of \(R\) to \(Q\).
Understanding chemical formulas allows chemists to determine the specific ratios of various elements in a compound. In binary compounds, these formulas are simple yet informative and are derived directly from the masses of each component. In our case, the first compound, determined by the mass ratios, is \(\text{R}_3\text{Q}\), which reveals a different stoichiometric relationship compared to the compound \(\text{RQ}\).

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It provides quantitative relationships between the amounts of elements and compounds. In the exercise, stoichiometry helps determine the ratio and formula of the compounds formed by elements \(R\) and \(Q\).
Through stoichiometry, we understand that 14.0 grams of \(R\) combining with 3.0 grams of \(Q\) forms the first compound, while 7.0 grams of \(R\) combining with 4.5 grams of \(Q\) forms the second. By comparing these ratios, stoichiometry illuminates how multiple compounds are formed by the same elements adhering to the law of multiple proportions.

Chemical Ratio

Chemical ratios describe the proportion of elements in compounds. In this exercise, ratios allow us to contrast the two binary compounds made from elements \(R\) and \(Q\). They provide a numerical expression of how atoms of each element combine.
By computing the ratio of mass of \(R\) to \(Q\) in each compound, we derive the formulas \(\text{R}_3\text{Q}\) and \(\text{RQ}\). This comparison demonstrates the law of multiple proportions—where the ratio \(\frac{14/3}{7/4.5} = 3/1\)—illustrating the law’s principle that the elements combine in simple whole number ratios. Understanding these chemical ratios is vital in predicting how different elements would react with one another and form compounds.