Question

Consider the hypothetical reaction $$ 2 \mathrm{R}_{2} \mathrm{X}_{5}+2 \mathrm{Z}_{8} \longrightarrow 5 \mathrm{X}_{2}+4 \mathrm{RZ}_{4} $$ When \(25.00 \mathrm{~g}\) of \(\mathrm{Z}_{8}(\mathrm{MM}=197.4 \mathrm{~g} / \mathrm{mol})\) react with an excess of \(\mathrm{R}_{2} \mathrm{X}_{5}, 21.72 \mathrm{~g}\) of \(\mathrm{X}_{2}\) are produced. (a) How many moles of \(\mathrm{X}_{2}\) are produced? (b) What is the molar mass of \(\mathrm{X}_{2}\) ?

Short answer

Answer: \(0.3165\) moles of \(X_2\) are produced and its molar mass is \(68.6\) g/mol.

Step-by-step solution

Calculate moles of \(Z_8\)

To find the moles of \(Z_8\), use the given mass of \(Z_8\) and its molar mass using the formula: moles = mass / molar mass moles of \(Z_8 = \dfrac{25.00~g}{197.4~g/mol} = 0.1266~mol\)

Use stoichiometry to determine moles of \(X_2\)

Using the stoichiometry of the balanced equation, we can find the moles of \(X_2\) produced. For each 2 moles of \(Z_8\) reacting, 5 moles of \(X_2\) are produced. Therefore: mol \(X_2 = (\dfrac{5~mol~X_2}{2~mol~Z_8}) \times 0.1266~mol~Z_8 = 0.3165~mol~X_2\) (a) So, \(0.3165\) moles of \(X_2\) are produced.

Calculate the molar mass of \(X_2\)

Now, we'll use the mass of \(X_2\) produced and the previously calculated moles of \(X_2\) to determine its molar mass. We can use the formula: molar mass = mass / moles molar mass of \(X_2 = \dfrac{21.72~g}{0.3165~mol} = 68.6~g/mol\) (b) The molar mass of \(X_2\) is \(68.6\) g/mol.

Key concepts

Molar Mass

Understanding the molar mass of a substance is vital in chemistry, particularly when dealing with chemical reactions. In essence, the molar mass is the weight of one mole of a particular substance, usually expressed in grams per mole (\(g/mol\)). It corresponds to the combined atomic masses of all the atoms in a molecule of that substance, as detailed on the periodic table.

The molar mass allows us to convert between the weight of a substance and the number of moles. This provides a bridge between the macroscopic world (grams) and the microscopic world (molecular or atomic scale), which is handy for stoichiometric calculations. For example, if you have 21.72 g of a compound \(X_2\), and you know its molar mass is 68.6 g/mol, you can quickly calculate the number of moles. The molar mass becomes particularly important when dealing with limiting reagents and theoretical yields in chemical reactions.

Chemical Reactions

Chemical reactions are processes where reactants transform into products through the breaking and forming of chemical bonds. Such processes are described by chemical equations that are balanced to obey the law of conservation of mass, indicating that atoms are neither created nor destroyed in a reaction.

A balanced chemical equation tells us the ratio in which substances react and are produced (\(stoichiometry\)). In the given hypothetical reaction between \(R_2X_5\) and \(Z_8\), leading to the production of \(X_2\) and \(RZ_4\), the coefficients represent this ratio—a cornerstone for stoichiometric calculations. For instance, 2 moles of \(Z_8\) will react to form 5 moles of \(X_2\). This ratio allows the calculation of how much product is formed from a given amount of reactant, given that the reaction goes to completion.

Mole Concept

The mole concept is a fundamental concept in chemistry that provides a method for translating between atoms, molecules, and ions to a scale that we can measure in the laboratory. One mole is defined as exactly 6.022 x 10²³ entities (Avogadro’s number) of the substance. It could be atoms, molecules, ions, or other chemical units.

The mole concept allows chemists to count these particles by weighing them. Instead of counting individual atoms, which is impractical, one can weigh a sample of a substance to find out how many moles it contains. For example, in the exercise, we determine the number of moles of \(Z_8\) by dividing its mass (25.00 g) by its molar mass (197.4 g/mol). Without the mole concept, stoichiometry would be incredibly complex, as there would be no efficient way to relate the mass of substances to the number of particles involved in a chemical reaction.

Stoichiometric Calculations

Stoichiometric calculations are the quantitative aspect of chemistry that involves calculating the amounts of reactants and products in a chemical reaction. These calculations rely on the balanced chemical equation and the use of conversion factors derived from that equation, known as mole ratios.

During stoichiometric calculations, you often deal with questions like, 'If I start with a certain amount of one reactant, how much of a product can I expect to get?'. To answer this, you apply mole ratios as shown in the exercise, where the mole ratio of \(Z_8\) to \(X_2\) from the balanced equation (2 to 5) allowed the calculation of moles of \(X_2\) formed from a known amount of \(Z_8\). These types of calculations are the backbone of chemical reaction predictability and are essential for everything from synthesizing compounds to scaling up industrial chemical processes.