Question

Sulfur trioxide, \(\mathrm{SO}_{3},\) is produced in enormous quantities each year for use in the synthesis of sulfuric acid. $$ \begin{array}{c} \mathrm{S}(s)+\mathrm{O}_{2}(g) \rightarrow \mathrm{SO}_{2}(g) \\ 2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{SO}_{3}(g) \end{array} $$ What volume of \(\mathrm{O}_{2}(g)\) at \(350 .^{\circ} \mathrm{C}\) and a pressure of 5.25 atm is needed to completely convert \(5.00 \mathrm{g}\) of sulfur to sulfur trioxide?

Short answer

\(3.12 \: L\) of \(\mathrm{O}_{2}(g)\) is needed to completely convert \(5.00 \: g\) of sulfur to sulfur trioxide under the given conditions.

Step-by-step solution

Calculate the moles of S(s)

To calculate the number of moles, we will use the mass and molar mass of sulfur: Mass of S(s) = 5.00 g Molar Mass of S = 32.07 g/mol (from the periodic table) Moles of S(s) = (mass of S) / (molar mass of S) Moles of S(s) = 5.00 g / 32.07 g/mol = 0.1558 mol

Calculate the moles of O₂(g) needed

Using the balanced chemical equations, we can determine the moles of O₂(g) needed: 1 mol of S(s) needs 1 mol + 1 mol = 2 mol of O₂(g) in total (1 mol for the 1st equation, and 1 mol for the 2nd equation) So, moles of O₂(g) needed = 2 × moles of S(s) Moles of O₂(g) needed = 2 × 0.1558 mol = 0.3116 mol

Calculate the volume of O₂(g) using the Ideal Gas Law

The Ideal Gas Law is given by: PV = nRT, where P = Pressure (in atm) V = Volume (in L) n = Moles of the gas R = Ideal Gas Constant (0.0821 L atm/mol K) T = Temperature (in K) Temperature in Kelvin = 350°C + 273.15 = 623.15 K Pressure = 5.25 atm Moles of O₂(g) = 0.3116 mol We will now solve for the volume, V: V = nRT/P V = (0.3116 mol) × (0.0821 L atm/mol K) × (623.15 K) / (5.25 atm) V = 3.1166 L Hence, 3.12 L of O₂(g) is needed to completely convert 5.00 g of sulfur to sulfur trioxide under the given conditions.

Key concepts

Stoichiometry Calculations

Stoichiometry is essentially the math behind chemistry. It allows chemists to make predictions about the outcomes of chemical reactions. When tackling stoichiometry calculations, one must first understand the relationship between the reactants and products as expressed in a balanced chemical equation.

The balanced equation tells us the proportion of molecules and moles that are consumed and produced. For instance, in the sulfur trioxide production process, the balanced equations reveal that one mole of sulfur reacts with one mole of oxygen to form one mole of sulfur dioxide which then reacts with half a mole of oxygen to produce sulfur trioxide. To solve the problem, first determine the number of moles of the given substance, in this case, sulfur. Then, use the mole ratios from the balanced equation to find out how many moles of another reactant, here oxygen, are needed.

Once you understand the stoichiometry of a reaction, you can calculate masses, volumes, and even energy changes, as long as you have enough information and a balanced chemical equation to work from.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that relates the pressure, volume, number of moles, and temperature of an ideal gas. It's expressed as PV = nRT, where P stands for pressure in atmospheres (atm), V for volume in liters (L), n for the number of moles, R for the ideal gas constant (0.0821 L atm/mol K), and T for temperature in kelvins (K).

This law allows us to calculate any one of these four variables if the other three are known. In our sulfur trioxide formation example, after determining the moles of oxygen gas needed stoichiometrically, we can apply the Ideal Gas Law to find the required volume of this reactant under specified conditions (pressure and temperature).

Applying this law in practical scenarios, like the synthesis of sulfuric acid, provides crucial insights for chemical engineering and industrial applications, emphasizing its importance in real-world contexts beyond the classroom.

Chemical Reactions

Chemical reactions are processes where reactant molecules are transformed into product molecules. These transformations involve breaking the original bonds and forming new ones, resulting in new substances with different properties. There are various types of chemical reactions, such as synthesis, decomposition, single replacement, double replacement, and combustion.

In the given example, we deal with a synthesis reaction, where sulfur and oxygen combine to form sulfur dioxide, and then sulfur dioxide reacts further with oxygen to produce sulfur trioxide. The sulfur and oxygen atoms are rearranged during the process, creating new chemical structures and substances.

Understanding chemical reactions is crucial, as it not only helps in predicting the products but also in ensuring that processes are carried out safely and efficiently, especially in industrial settings where vast quantities of substances are involved. Knowing how substances react enables us to control and optimize conditions to achieve the desired outcome, such as creating a high yield of sulfur trioxide for sulfuric acid production.