Problem 32 - Chemistry and Chemical Reactivity

Question

Silane, \(\operatorname{SiH}_{4}\), reacts with \(\mathrm{O}_{2}\) to give silicon dioxide and water: $$ \mathrm{SiH}_{4}(\mathrm{g})+2 \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{SiO}_{2}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(\ell) $$ A 5.20 -L sample of \(\mathrm{SiH}_{4}\) gas at \(356 \mathrm{mm}\) Hg pressure and \(25^{\circ} \mathrm{C}\) is allowed to react with \(\mathrm{O}_{2}\) gas. What volume of \(\mathrm{O}_{2}\) gas, in liters, is required for complete reaction if the oxygen has a pressure of \(425 \mathrm{mm} \mathrm{Hg}\) at \(25^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

To complete the reaction, approximately 8.06 L of \(\text{O}_2\) is needed.

Step 1: Identify the Reaction Requirements

We first identify that the complete reaction of silane, \(\text{SiH}_4\), with oxygen, \(\text{O}_2\), involves a 1:2 molar ratio according to the given balanced chemical equation. For every 1 mole of \(\text{SiH}_4\), 2 moles of \(\text{O}_2\) are required.

Step 2: Convert Conditions to Standard Units

Convert all given conditions to appropriate units for using the ideal gas law. We have the initial conditions of silane gas: \(\text{P}_{\text{SiH}_4} = 356\, \text{mm Hg}\), \(\text{V}_{\text{SiH}_4} = 5.20\, \text{L}\), \(T = 25^{\circ} \text{C} = 298\, \text{K}\). We will also need to use these to find the required volume of \(\text{O}_2\) under its given pressure \(\text{P}_{\text{O}_2} = 425\, \text{mm Hg}\).

Step 3: Apply the Ideal Gas Law to Silane

Use the ideal gas law \( PV = nRT \) to find the moles of \(\text{SiH}_4\). \[ n_{\text{SiH}_4} = \frac{P_{\text{SiH}_4} \cdot V_{\text{SiH}_4}}{R \cdot T} \]Where \(R = 0.0821\, \text{L atm mol}^{-1} \text{ K}^{-1}\), and pressure needs to be in atm: \[P_{\text{SiH}_4} = \frac{356}{760} \approx 0.4684\, \text{atm}\]Calculate \(n_{\text{SiH}_4}\) with the converted pressures.

Step 4: Calculate the Required Moles of Oxygen

Using the stoichiometry from the balanced equation, compute the moles of \(\text{O}_2\) needed. Since the reaction ratio from \(\text{SiH}_4\) to \(\text{O}_2\) is 1:2, \[ n_{\text{O}_2} = 2 \cdot n_{\text{SiH}_4} \]

Step 5: Apply the Ideal Gas Law to Oxygen

Use the ideal gas law again \( PV = nRT \) with the calculated moles of \(\text{O}_2\) and solve for its volume, \(V_{\text{O}_2}\) at its conditions:\[ n_{\text{O}_2} = \frac{P_{\text{O}_2} \cdot V_{\text{O}_2}}{R \cdot T} \]Rearrange to find \( V_{\text{O}_2} \):\[ V_{\text{O}_2} = \frac{n_{\text{O}_2} \cdot R \cdot T}{P_{\text{O}_2}} \]Ensure \(P_{\text{O}_2}\) is also in atm using \(0.5592\, \text{atm} \approx \frac{425}{760}\).

Step 6: Calculate and Provide the Final Volume of Oxygen

Insert all known values into the above formula to determine \(V_{\text{O}_2}\). Make sure to maintain consistent units (L, atm, and K). This will provide the required volume of \(\text{O}_2\) in liters for the complete reaction.

Key Concepts

Stoichiometry

Stoichiometry is a core concept in chemistry that involves the quantitative relationship between reactants and products in a chemical reaction. For the reaction of silane (\( \text{SiH}_4 \)) with oxygen (\( \text{O}_2 \)), stoichiometry helps us understand how much of each reactant is needed.

• The balanced chemical equation shows the stoichiometry of the reaction: \( \text{SiH}_4( ext{g}) + 2 \text{O}_2( ext{g}) \rightarrow \text{SiO}_2( ext{s}) + 2 \text{H}_2\text{O}( ext{l}) \).
• The coefficients in the equation, such as "1" for \( \text{SiH}_4 \) and "2" for \( \text{O}_2 \), indicate their molar ratios.
• This means that for every mole of \( \text{SiH}_4 \), two moles of \( \text{O}_2 \) are required.

Using stoichiometry in this scenario helps us determine the amount of \( \text{O}_2 \) needed to completely react with a given volume of \( \text{SiH}_4 \). This balance ensures that there are no leftover reactants and that the products are formed correctly.

Chemical Reactions

Chemical reactions describe how substances interact to form new products. These reactions, like the one involving \( \text{SiH}_4 \) and \( \text{O}_2 \), rearrange atomic structure and are governed by certain rules..

• Reactants are substances that start a reaction. For this chemical equation, \( \text{SiH}_4 \) and \( \text{O}_2 \) are reactants.
• Products are the result of the reactants' interactions. Here, the products are \( \text{SiO}_2 \) and water (\( \text{H}_2\text{O} \)).

The balanced chemical equation is key because it shows that mass is conserved; the number of each type of atom remains constant before and after the reaction. Understanding the reaction specifics allows us to predict how much of each substance we need and how much will be produced.

Gas Laws

Gas laws help understand the behavior and properties of gases under different conditions. In our problem, we use the ideal gas law, \( PV = nRT \), which relates pressure (\( P \)), volume (\( V \)), moles of gas (\( n \)), and temperature (\( T \)).

• The ideal gas constant \( R \) is generally given as 0.0821 L atm mol\(^{-1} \text{ K}^{-1}\).
• Converting pressures from mm Hg to atm is crucial for unit consistency. For example, \( 356 \, \text{mm Hg} \) becomes approximately \( 0.4684 \, \text{atm} \).
• Temperature should be in Kelvin, which is \( 298 \text{ K} \) for \( 25^{\circ} \mathrm{C} \).
• This equation lets us find the moles of a gas if we know the other three parameters, which helps in figuring out reactant quantities.

With the ideal gas law, we assess how much oxygen is needed based on the reaction conditions, ensuring we have exact measurements for a complete chemical transformation.