Question

Two different compounds of elements \(\mathrm{A}\) and \(\mathrm{B}\) were found to have the following composition: first compound, \(1.188 \mathrm{~g}\) of \(\mathrm{A}\) combined with \(0.711 \mathrm{~g}\) of \(\mathrm{B} ;\) second compound, \(0.396 \mathrm{~g}\) of \(\mathrm{A}\) combined with \(0.474 \mathrm{~g}\) of \(\mathrm{B}\). (a) Show that these data are in accord with the law of multiple proportions, (b) If the formula for the first compound is \(\mathrm{AB}_{2}\), what is the formula for the second?

Short answer

The mass ratios of A and B in the two compounds are found to be approximately in a simple whole number ratio, confirming the law of multiple proportions. Given the formula for the first compound is AB\(_{2}\), the formula for the second compound is \(\mathrm{A}_{2}\mathrm{B}\).

Step-by-step solution

Determine mass ratios of A and B for each compound

To determine the mass ratios, we must divide the mass of A by the mass of B for each compound. First compound: Ratio of A to B = \(\frac{1.188}{0.711}\) Second compound: Ratio of A to B = \(\frac{0.396}{0.474}\)

Find the simplest whole number ratio between the two ratios

It might be easier to handle if we take a ratio of two ratios: \(\frac{\frac{1.188}{0.711}}{\frac{0.396}{0.474}} = \frac{1.188 \times 0.474}{0.711 \times 0.396}\) Now, perform the calculation: \( \frac{1.188 \times 0.474}{0.711 \times 0.396} \approx \frac{0.563}{0.281} \approx 2\) The ratio is approximately 2, which is a small whole number. This confirms that the law of multiple proportions is satisfied.

Determine the formula for the second compound

Given the formula for the first compound is AB\(_{2}\), we know that for every one A, there are 2 B atoms. Since the ratio between both compounds is approximately 2, this means that the ratio between the atoms in the second compound should be doubled. For the second compound, considering the ratio of 2 (from the previous step), we can write a new mass ratio of A to B as: \(\frac{2\times{0.396}}{0.474}\) = \(\frac{1.188}{0.474}\) Since we have doubled the mass of A compared to that of the first compound, it means that there are now 2 atoms of A for each atom of B in the second compound. Hence, the formula for the second compound is \(\mathrm{A}_{2}\mathrm{B}\).

Key concepts

Chemical Compounds

Understanding chemical compounds is essential in the field of chemistry. These are substances formed by two or more different types of atoms bonded together in fixed proportions. Each compound has its unique properties differing from the elements it is composed of. For example, water ( H_2O ) is a compound made of hydrogen and oxygen; despite hydrogen and oxygen being gases, water is a liquid under normal conditions. Compounds can be broken down into simpler substances by chemical reactions, unlike elements, which are the basic building blocks of matter and cannot be broken down any further.

The diversity of chemical compounds is vast, and their formation follows specific laws and principles, one of which is the law of multiple proportions. This law is essential when it comes to predicting and understanding the composition of compounds in chemical reactions.

Mass Ratio

The mass ratio is a fundamental concept in stoichiometry, involving the relative proportions of elements in compounds. It represents the mass of one element compared to the mass of another within a compound. This ratio is constant for a given compound and is related to the relative numbers of atoms of each element involved, according to their atomic weights.

In the context of our exercise, the mass ratio is used to directly compare the quantities of elements A and B in two different compounds. The calculation of mass ratios provides a way to apply the law of multiple proportions and demonstrates whether or not the same elements combine in small, whole-number ratios, which is a cornerstone of Dalton's atomic theory.

Stoichiometry

Stoichiometry is the study of the quantitative relationships, or ratios, of the reactants and products in chemical reactions. It is like the recipe of chemistry, dictating the exact amounts of each substance involved to ensure a reaction goes to completion. Stoichiometry is used to predict the amounts of products that will form in a given reaction, based on the mass ratios of the reactants and the balanced chemical equation.

In our textbook exercise, stoichiometry appears when determining the formulas of compounds from their mass ratios. By knowing the mass of each element in a compound, we can calculate the stoichiometric coefficients, which indicate the number of atoms of each element that combine to form one unit of the compound.

Empirical Formula

An empirical formula represents the simplest whole-number ratio of the atoms of each element present in a compound. Unlike the molecular formula, which shows the exact number of atoms, the empirical formula might not be the same as the molecular formula. It is derived from the mass ratio of the elements in the compound and gives us an idea of the composition of the compound without detailing the exact quantity of atoms.

In the given exercise, once the mass ratios are determined, they are then used to deduce the empirical formula of the compound. If we know that one compound is AB_2 , and we have established the mass ratio for another compound with the same elements, we use stoichiometry to deduce an empirical formula that fits this ratio. This process illustrates how empirical formulas are obtained and used in chemistry to express the identities of compounds in a simplified manner.