Problem 65 - Chemistry An Atoms First Approach

Question

Consider the following reaction: $$4 \mathrm{Al}(s)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{Al}_{2} \mathrm{O}_{3}(s)$$ It takes \(2.00 \mathrm{L}\) of pure oxygen gas at \(STP\) to react completely with a certain sample of aluminum. What is the mass of aluminum reacted?

Step-by-Step Solution

Answer

The mass of aluminum reacted can be calculated using the following steps: 1. Find the number of moles of oxygen gas: \( \frac{2.00 \, \text{L}}{22.4 \, \text{L/mol}}\) 2. Find the number of moles of aluminum reacted: \( (\text{Moles of O}_2) \times \frac{4 \, \text{moles Al}}{3 \, \text{moles O}_2}\) 3. Calculate the mass of aluminum reacted: \( (\text{Moles of Al}) \times (26.98 \, \text{g/mol})\)

Step 1: Find the number of moles of oxygen gas

We are given that the volume of oxygen gas at STP is 2.00 L. To find the number of moles, we will use the ideal gas law, which at STP can be simplified to: n = PV/RT At STP, P = 1 atm, V = 2.00 L, R = 0.08206 L atm/mol K, and T = 273.15 K. However, we can simplify the problem since we know 1 mole of any gas occupies 22.4 L at STP. Therefore: Number of moles of oxygen gas = (2.00 L) / (22.4 L/mol)

Step 2: Find the number of moles of aluminum reacted

Now that we have the number of moles of oxygen gas, we can use the stoichiometry of the balanced chemical equation to find the number of moles of aluminum reacted. From the balanced equation, we see that 4 moles of Al reacts with 3 moles of O₂. Hence, we can set up a proportion to determine the moles of aluminum: Moles of Al = (Moles of O₂) × (4 moles Al / 3 moles O₂)

Step 3: Calculate the mass of aluminum reacted

Now that we have the number of moles of aluminum reacted, we can calculate the mass using the molar mass of aluminum, which is approximately 26.98 g/mol. Mass of Al = (Moles of Al) × (26.98 g/mol)

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle used to relate the pressure, volume, and temperature of an ideal gas. The law is expressed in the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.
At Standard Temperature and Pressure (STP), which is 1 atm and 273.15 K, the equation can be simplified to find that 1 mole of an ideal gas occupies 22.4 L.
This simplification is useful for quickly finding the number of moles when dealing with gases at STP without needing detailed calculations. To solve problems involving gases at STP:

• Use 22.4 L/mol directly to find moles from volume.
• This avoids complex calculation using the full Ideal Gas Law formula.

Balanced Chemical Equation

A balanced chemical equation ensures that the number of atoms for each element is the same on both sides of the equation. This equality maintains the Law of Conservation of Mass. In the equation \( 4 \mathrm{Al}(s) + 3 \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{Al}_{2} \mathrm{O}_{3}(s) \), the coefficients in front of each compound or element show the molar ratio.
This balanced equation indicates:

• 4 moles of aluminum react with 3 moles of oxygen to produce 2 moles of aluminum oxide.
• Understanding these mole ratios is crucial in stoichiometry because they dictate how much of each reactant is needed or how much product is formed.

The balanced equation helps calculate the exact amount of a substance needed or produced in a chemical reaction.

Molar Mass

Molar mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). For example, the molar mass of aluminum is approximately 26.98 g/mol.
This value is used to convert moles to grams, which is necessary for determining the mass of a substance when the number of moles is known. Steps for calculation:

• Determine the number of moles using stoichiometry from the balanced equation.
• Multiply the moles by the molar mass to find the mass in grams.

Molar mass allows you to translate between the count of atoms/molecules and their mass in lab measurements.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) are the conditions where the temperature is 273.15 K (0°C) and the pressure is 1 atm. These conditions are commonly used as a reference point in chemistry. At STP:

• One mole of any ideal gas occupies a volume of 22.4 liters.
• These standard conditions make it easier to compare gas measurements and perform calculations using the Ideal Gas Law.

STP is particularly helpful when conducting experiments or solving problems involving gases, as it provides a consistent basis for measurement and calculation.