Question

If cobalt metal is mixed with excess sulfur and heated strongly, a sulfide is produced that contains $55.06\mathrm{\%}$ cobalt by mass. Calculate the empirical formula of the sulfide.

Short answer

The empirical formula of the cobalt sulfide compound is ${\text{Co}}_2{\text{S}}_3$.

Step-by-step solution

Convert mass percentage to grams

Since we are given that the compound contains $55.06\mathrm{\%}$ of cobalt by mass, we can assume that we have a $100\text{g}$ sample. Thus, there would be $55.06\text{g}$ of cobalt and the remaining $44.94\text{g}$ would be sulfur.

Calculate the moles of cobalt and sulfur

Next, we need to determine the moles of cobalt and sulfur in the compound. To do this, divide the masses from Step 1 by the respective molar masses. For cobalt: $$n_{Co}=\frac{55.06\text{g}}{58.93\text{g/mol}}=0.934\text{mol}$$ For sulfur: $$n_S=\frac{44.94\text{g}}{32.06\text{g/mol}}=1.402\text{mol}$$

Determine the mole ratio

Now, divide each of the mole values from Step 2 by the smallest value to get the mole ratio. Mole ratio of cobalt: $$\frac{n_{Co}}{\text{min}(n_{Co},n_S)}=\frac{0.934\text{mol}}{0.934\text{mol}}=1$$ Mole ratio of sulfur: $$\frac{n_S}{\text{min}(n_{Co},n_S)}=\frac{1.402\text{mol}}{0.934\text{mol}}\approx 1.5$$

Simplify the mole ratio to a whole number

The mole ratio from Step 3 is approximately $1:1.5$. The value $1.5$ is not a whole number, but it is very close to $\frac{3}{2}$. So, we can multiply the mole ratio by $2$ to obtain a ratio of whole numbers. This will provide us with a ratio of: Cobalt : Sulfur = $1\times 2$ : $1.5\times 2$ = $2:3$

Write the empirical formula

Based on the simplified mole ratio from Step 4, the empirical formula of the cobalt sulfide compound is ${\text{Co}}_2{\text{S}}_3$.

Key concepts

Chemistry

Chemistry is the study of matter and how it changes. When we talk about empirical formulas in chemistry, we are discussing the simplest ratio of the elements in a compound. This exercise provides us with percent composition data, which is a practical aspect of chemistry analysis.

Percent composition helps chemists determine the formula of a compound, which then allows them to understand its properties and reactions. In this problem, we determine the empirical formula by analyzing a cobalt sulfide compound. The empirical formula gives us the lowest whole number ratio of cobalt to sulfur in this compound.

To calculate empirical formulas, understanding how matter behaves on an atomic scale is key. States of matter, chemical reactions, and atomic theory all play a role in these calculations.

• Empirical formulas represent the simplest whole-number ratio of elements in a compound.
• They aid in identifying substances based on their elemental composition.
• This contributes to broader applications in identifying unknown compounds and understanding complex biochemical processes.

Mole Ratio

The mole ratio is a fundamental concept in chemistry that shows the proportions of elements in a compound. It is derived from the number of moles, which are units that measure the amount of substance.

To find the mole ratio, the given mass percentages are converted to moles, using the molar mass of the elements involved. In this exercise, we calculated the moles of cobalt and sulfur based on their masses. This step is crucial, as moles allow chemists to compare quantities of different elements directly, despite their different atomic weights.

A mole ratio reflects the proportion of cobalt to sulfur in the compound. By dividing the number of moles of each element by the smallest number of moles present, we obtain a simple whole-number ratio. In this case, the final ratio of cobalt to sulfur was found to be 2:3, giving us the empirical formula Co₂S₃.

• Mole ratios provide insights into the stoichiometry of chemical compounds.
• This method standardizes comparisons among different elements and simplifies complex reactions.
• Understanding mole ratios is also key in converting between grams and moles, a common need in laboratory settings.

Percent Composition

Percent composition indicates the percentage by mass of each element in a compound. It is a stepping stone for calculating empirical formulas. This concept is crucial for chemists when analyzing the purity of a substance or when formulating compounds.

In this problem, we had a cobalt sulfide compound with 55.06% cobalt by mass. To calculate empirical formulas, we assume a 100 g sample, which simplifies computations, breaking things down to mention that there is 55.06 g of cobalt and the rest is sulfur.

Understanding percent composition allows us to understand a compound's properties and helps us to manufacture chemicals to certain specifications. It is widely used in pharmaceutical, biochemical, and industrial processes.

• Percent composition is calculated using the formula: $\text{Percentage of element}=\frac{\text{mass of element in 1 mole of compound}}{\text{molar mass of compound}}\times 100$
• It provides information about the elemental composition, important for deriving empirical formulas.
• Knowing the percent composition helps in ensuring that the reactant mixtures are correct in precise formulations.