Problem 60 - Chemistry: The Central Science

Question

Acetylene gas, \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\), can be prepared by the reaction of calcium carbide with water: $$ \mathrm{CaC}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q)+\mathrm{C}_{2} \mathrm{H}_{2}(g) $$ Calculate the volume of \(\mathrm{C}_{2} \mathrm{H}_{2}\) that is collected over water at \(23{ }^{\circ} \mathrm{C}\) by reaction of \(1.524 \mathrm{~g}\) of \(\mathrm{CaC}_{2}\) if the total pressure of the gas is 753 torr. (The vapor pressure of water is tabulated in Appendix B.)

Step-by-Step Solution

Answer

The volume of acetylene gas (\(C_2H_2\)) collected over water at 23°C and a total pressure of 753 torr, when \(1.524~g\) of calcium carbide reacts with water, is approximately 0.596 liters.

Step 1: Find the moles of calcium carbide and acetylene

Given the mass of calcium carbide: \(1.524~g\). The molar mass of calcium carbide, \(CaC_2\), is \(40.08 + 2(12.01) = 64.1~g/mol\). To find the moles of \(CaC_2\), divide the given mass by the molar mass: \(\frac{1.524~g}{64.1~g/mol}=0.0238~mol\) According to the balanced chemical equation, 1 mole of \(CaC_2\) reacts to produce 1 mole of \(C_2H_2\). Thus, the number of moles of \(C_2H_2\) produced is also \(0.0238~mol\).

Step 2: Calculating the pressure of acetylene gas

We are given the total pressure as 753 torr, and the temperature is 23°C (296 K). We need to use the vapor pressure of water at 23°C from Appendix B to calculate the partial pressure of acetylene. Let's assume the vapor pressure of water at 23°C is 21.1 torr. According to Dalton's law of partial pressures: \(P_{C_2H_2} = P_{total} - P_{H_2O} = 753~torr - 21.1~torr = 731.9~torr\)

Step 3: Applying the Ideal Gas Law to calculate the volume

The Ideal Gas Law is given as: \(PV=nRT\) Where: - P is the pressure of the gas (in atm) - V is the volume of the gas (in L) - n is the number of moles of the gas - R is the ideal gas constant (\(0.0821 L\cdot atm/(mol\cdot K)\)) - T is the temperature of the gas (in K) First, we need to convert the pressure of acetylene in torr to atm: \(P_{C_2H_2} = \frac{731.9~torr}{760 \frac{torr}{atm}} = 0.963~atm\) Now, we can use the Ideal Gas Law to find the volume of acetylene gas (we will leave the units of volume as L to keep it simple): \(V = \frac{nRT}{P} = \frac{0.0238~mol \cdot 0.0821\frac{L\cdot atm}{mol\cdot K} \cdot 296~K}{0.963~atm} = 0.596~L\)

Step 4: The volume of acetylene gas

The volume of acetylene gas (\(C_2H_2\)) that is collected over water at 23°C and a total pressure of 753 torr, when \(1.524~g\) of calcium carbide reacts with water, is approximately 0.596 liters.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of an ideal gas. The formula for the law is expressed as \(PV=nRT\), where:

• \(P\) stands for the pressure of the gas, typically measured in atmospheres (atm) or torr.
• \(V\) represents the volume of the gas, which is typically in liters (L).
• \(n\) is the number of moles of the gas, a measure of the quantity of gas particles.
• \(R\) is the ideal gas constant, which has a value of \(0.0821 L\cdot atm/(mol\cdot K)\).
• \(T\) is the temperature of the gas in Kelvin (K).

To solve problems using the Ideal Gas Law, one must first make sure all units are consistent. Pressure must be in atm, volume in L, temperature in K, and the gas constant \(R\) must be compatible with these units. It allows for the calculation of one of the four properties of a gas if the other three are known. In the context of the exercise, the volume of acetylene gas was calculated by rearranging the Ideal Gas Law formula to solve for \(V\), using the moles of gas, temperature, and pressure provided.

Dalton's Law of Partial Pressures

When dealing with gas mixtures, Dalton's Law of Partial Pressures becomes particularly significant. This law states that the total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases. The partial pressure of any single gas is the pressure it would exert if it occupied the entire volume alone at the same temperature.

How to Apply Dalton's Law

In practical terms, to find the pressure of a specific gas in a mixture, you subtract the pressures of other gases from the total pressure. For example, as seen in the exercise, when acetylene gas is collected over water, the total pressure inside the collection container is the sum of the pressure of acetylene and the vapor pressure of water. To get the pressure of just the acetylene, you would subtract the water’s vapor pressure from the total pressure (using the value found in a reference, such as Appendix B). This process is crucial when using the Ideal Gas Law to calculate the volume of a specific gas collected in a reaction.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantifiable relationships of the reactants and products in a chemical reaction. In the context of our exercise, stoichiometry is used to calculate the moles of acetylene gas produced from the reaction of a known mass of calcium carbide with water.

Stoichiometric Calculations

To perform these calculations, one must understand the balanced chemical equation of the reaction. Each coefficient in the reaction denotes the ratio in which compounds react and are produced. For instance, the balanced equation given in the exercise implies that one mole of calcium carbide yields one mole of acetylene gas. By calculating the moles of reactant (calcium carbide in this case), stoichiometry allows for the determination of moles of product (acetylene), which can then be used alongside the Ideal Gas Law to find the gas's volume at a certain temperature and pressure. This is particularly important in lab settings, where precise amounts of materials must be used to achieve the desired chemical reaction and yield.