Question

Sulfur trioxide, ${\mathrm{SO}}_3,$ is produced in enormous quantities each year for use in the synthesis of sulfuric acid. $$\mathrm{S}(s)+{\mathrm{O}}_2(g)⟶{\mathrm{SO}}_2(g)$$ $$2{\mathrm{SO}}_2(g)+{\mathrm{O}}_2(g)⟶2{\mathrm{SO}}_3(g)$$ 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 $\mathrm{g}$ sulfur to sulfur trioxide?

Short answer

First, convert 5.00 g of sulfur to moles: $MolesofS=\frac{5.00g}{32.07g\cdot mo{l}^{-1}}$. Next, find the moles of O2 needed: $Molesof{O}_2=MolesofS\times 1.5mol{O}_2\cdot mo{l}_S^{-1}$. Finally, use the ideal gas law to find the volume of O2 gas needed: $V=\frac{(Molesof{O}_2)(0.0821L\cdot atm\cdot K^{-1}\cdot mo{l}^{-1})(623K)}{5.25atm}$. Calculate V to obtain the volume of O2 gas required.

Step-by-step solution

Convert sulfur mass to moles

First, we need to convert the mass of sulfur (S) given in the problem (5.00 g) to moles. The molar mass of S is approximately 32.07 g/mol, so we have: Moles of S = $\frac{5.00g}{32.07g\cdot mo{l}^{-1}}$

Determine moles of O2 needed for the reaction

From the balanced chemical equation, we see that: - 1 mole of S reacts with 1 mole of O2 gas in the first reaction to produce SO2 (S + O2 → SO2). - In the second reaction (2 SO2 + O2 → 2 SO3), 2 moles of SO2 react with 1 mole of O2 gas to produce 2 moles of SO3. Because every mole of S reacts with a mole of O2 and we need one more mole of O2 for two moles of SO2, we will need 1.5 moles of O2 for every mole of S. Now we can find the moles of O2 needed: Moles of O2 = Moles of S × $1.5mol{O}_2\cdot mo{l}_S^{-1}$

Calculate the volume of O2 gas using the ideal gas law

Now that we have the moles of O2, we can use the ideal gas law to find the volume of O2 gas needed. The ideal gas law is expressed as: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. We are given the pressure (P = 5.25 atm) and temperature (T = 350°C = 623 K). We'll convert the given information to the proper units to use the Ideal Gas Law with the gas constant R = 0.0821 L·atm/mol·K. So, we can rearrange the ideal gas law equation to solve for V: V = $\frac{nRT}{P}$ Now substitute the known values: V = $\frac{(Molesof{O}_2)(0.0821L\cdot atm\cdot K^{-1}\cdot mo{l}^{-1})(623K)}{5.25atm}$ Calculate the volume (V) with the obtained values, and you'll find the volume of O2 gas needed to completely convert 5.00 g of sulfur to sulfur trioxide.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry that describes the behavior of gases under different conditions. This law is expressed with the formula $PV=nRT$, where:

• $P$ stands for pressure, measured in units like atmospheres (atm) or pascals (Pa).
• $V$ is the volume of the gas, typically in liters (L).
• $n$ represents the number of moles, which is a measure of the amount of substance.
• $R$ is the ideal gas constant, with values like 0.0821 L·atm/mol·K when using atmospheres for pressure and liters for volume.
• $T$ is the temperature, which must be in Kelvin (K) for the formula to work correctly.

To find the volume of a gas (in this exercise, $O_2$), given moles, temperature, and pressure, rearrange the equation to $V=\frac{nRT}{P}$. Plug in the given values of moles, the ideal gas constant, temperature (converted to Kelvin), and pressure to solve for the volume. Don't forget to maintain the correct units for consistency.

Chemical Reactions

Chemical reactions involve the transformation of substances through the reorganization of atoms. In these reactions, reactants are converted into products with different properties. Consider the sequence provided where sulfur and oxygen gases react to form sulfur dioxide, which then further reacts with oxygen to create sulfur trioxide. The equations are balanced, meaning the number of each type of atom on one side of the equation is equal to that on the other side.The steps are:

• $\mathrm{S}(s)+{\mathrm{O}}_2(g)\to {\mathrm{SO}}_2(g)$
• $2{\mathrm{SO}}_2(g)+{\mathrm{O}}_2(g)\to 2{\mathrm{SO}}_3(g)$

In the first reaction, sulfur atoms react with oxygen molecules to form sulfur dioxide. In the second, sulfur dioxide molecules react with more oxygen molecules to produce sulfur trioxide. This sequence needs careful stoichiometric consideration, ensuring that for every mole of sulfur, corresponding moles of oxygen are supplied as per reaction requirements. This balance is crucial to determine how much of each reactant you need.

Mole Conversion

Mole conversion is an essential step in stoichiometry, enabling chemists to convert between mass, number of molecules, or volume (for gases at standard conditions). The concept hinges on the fact that a mole represents Avogadro's number $6.022\times {10}^{23}$ particles and is tied to the molar mass of a substance.For instance, in the exercise, you begin with sulfur's mass, which you convert to moles using its molar mass ($32.07g/mol$). This conversion allows you to link the mass of sulfur to its chemical role in the reaction:Moles of $S=\frac{5.00g}{32.07g/mol}$.Next, using stoichiometry, you determine how many moles of $O_2$ are needed by relating them to moles of sulfur. In the given reaction series, 1 mole of sulfur implies 1.5 moles of $O_2$ are needed. Mole conversion helps in the calculation not only of the quantities of reactants required but also the expected amount of products formed, providing a comprehensive understanding of the entire reaction process.