Problem 94 - Chemistry: An Atoms-Focused Approach

Question

Miners' Lamps Before the development of reliable batterics, miners' lamps burned acetylene produced by the reaction of calcium carbide with water: $$\mathrm{CaC}_{2}(s)+\mathrm{H}_{2} \mathrm{O}(\ell) \rightarrow \mathrm{C}_{2} \mathrm{H}_{2}(g)+\mathrm{CaO}(s)$$ Suppose a lamp uses \(4.8 \mathrm{L}\) of acetylene per hour at 1.02 atm pressure and \(25^{\circ} \mathrm{C}\) a. How many moles of \(\mathrm{C}_{2} \mathrm{H}_{2}\) are used per hour? b. How many grams of calcium carbide are consumed for a four-hour shift?

Step-by-Step Solution

Answer

Answer: Approximately 51.28 grams of calcium carbide would be consumed for a four-hour shift.

Step 1: Convert temperature to Kelvin

Given temperature is \(25^{\circ} \mathrm{C}\). To convert the temperature to Kelvin, add 273.15 to the given temperature in Celsius. \(25^{\circ} \mathrm{C} + 273.15 = 298.15 K\) (approximately equal to 298 K)

Step 2: Applying the Ideal Gas Law

We are given P=1.02 atm, V=4.8 L, R=0.0821 atm·L/mol·K, and T=298 K. We can now solve for the moles (n) of C2H2 by plugging those values into the Ideal Gas Law formula. \(PV = nRT\) \((1.02\,\text{atm})(4.8\,\text{L}) = n(0.0821\,\text{atm·L/mol·K})(298\,\text{K})\) Solving for n, we get \(n \approx 0.2\,\text{moles}\) of C2H2 used per hour.

Step 3: Calculate moles of CaC2 consumed in a four-hour shift

In the balanced chemical equation, 1 mole of CaC2 reacts to form 1 mole of C2H2. Since we found out that 0.2 moles of C2H2 are used per hour, the same number of moles of CaC2 will be consumed. For a four-hour shift, we need to multiply the number of moles of C2H2 per hour by 4: \((0.2\,\text{moles }\mathrm{CaC}_{2}/\text{hour}) × 4\,\text{hours} = 0.8\,\text{moles}\,\underline{\phantom{xxx}}\mathrm{CaC}_{2}\)

Step 4: Convert moles of CaC2 to grams

To calculate how many grams of CaC2 are consumed for a four-hour shift, multiply the number of moles of CaC2 by the molar mass of CaC2 (64.1 g/mol). \(0.8\,\text{moles}\,\underline{\phantom{xxx}}\mathrm{CaC}_{2} \times 64.1\, \text{g/mol}\,\mathrm{CaC}_{2} = 51.28\, \text{g}\,\mathrm{CaC}_{2}\) So, for a four-hour shift, approximately 51.28 grams of calcium carbide are consumed.

Key Concepts

Chemical Reactions

Chemical reactions form the foundation of chemistry, involving the transformation of reactants into products. In the case of the miners' lamps, calcium carbide (\(\text{CaC}_2\)) reacts with water (\(\text{H}_2\text{O}\)) to produce acetylene gas (\(\text{C}_2\text{H}_2\)) and calcium oxide (\(\text{CaO}\)). This particular equation exemplifies a typical chemical reaction.

• Reactants: \(\text{CaC}_2\) and \(\text{H}_2\text{O}\)
• Products: \(\text{C}_2\text{H}_2\) and \(\text{CaO}\)

Each balanced equation indicates the number of moles of each substance involved. In this context, one mole of calcium carbide reacts with one mole of water, producing one mole of acetylene. Understanding the stoichiometry of reactions enables us to calculate how much of each reactant is needed and how much product is formed. This is crucial in both industrial applications and laboratory settings. Chemical reactions can be classified in various ways, such as synthesis, decomposition, single displacement, and double displacement reactions. The miners' lamp reaction fits the category of a synthesis reaction, where multiple reactants combine to form a single product. Recognizing these patterns helps us predict reaction outcomes and balance equations effectively.

Gas Stoichiometry

Gas stoichiometry involves calculating the volumes of gases involved in chemical reactions based on the laws of chemistry. With the Ideal Gas Law, which is represented as \(PV = nRT\), we can relate the pressure (P), volume (V), temperature (T), and moles (n) of a gas. The Ideal Gas Constant (R) is typically 0.0821 atm·L/mol·K.

In the miners' lamp scenario, acetylene gas's production rate is crucial as it determines how long the gas can sustain a lamp. The Ideal Gas Law helps us calculate moles of acetylene gas consumed per hour by using:- \(P = 1.02 \, \text{atm}\)- \(V = 4.8 \, \text{L}\)- \(T = 298 \, \text{K}\)By substituting these values into the equation, we find \(n \approx 0.2\) moles of acetylene gas are used each hour. This calculation ensures that miners have a clear idea of how much produce that is available for operation over a given time period. Understanding gas stoichiometry is essential for optimizing resource use in processes involving gases, such as combustion, fermentation, and air purification.

Calculations with Moles

To understand reactions at a molecular level, calculations with moles are indispensable. A mole represents a specific number of molecules or atoms, precisely Avogadro's number, which is \(6.022 \times 10^{23}\). When conducting mole calculations, chemists can deduce the relationships between quantities calculated from balanced chemical equations.

In our example, we determine the moles of reactants and products by using stoichiometric relationships. For every mole of \(\text{CaC}_2\) used, one mole of \(\text{C}_2\text{H}_2\) is produced, as dictated by the balanced chemical reaction. Over a four-hour shift using 0.2 moles of \(\text{C}_2\text{H}_2\) each hour, we consume 0.8 moles of calcium carbide.

Furthermore, to find how many grams of calcium carbide are used, the number of moles is multiplied by the molar mass:- \(\text{CaC}_2\) molar mass: \(64.1 \, \text{g/mol}\)Hence, \(0.8 \, \text{moles} \times 64.1 \, \text{g/mol} = 51.28 \, \text{g}\). Calculating with moles allows for precise measurements in chemical reactions, ensuring efficiency and accuracy whether in labs or industrial applications.