Question

In the electrolysis of a sodium chloride solution, what volume of \(\mathrm{H}_{2}(g)\) is produced in the same time it takes to produce \(257 \mathrm{~L}\) \(\mathrm{Cl}_{2}(g)\), with both volumes measured at \(50 .{ }^{\circ} \mathrm{C}\) and \(2.50 \mathrm{~atm} ?\)

Short answer

The volume of \(H_{2}(g)\) produced in the same time it takes to produce 257 L of \(Cl_{2}(g)\) is about 257 L, both measured at 50°C and 2.50 atm.

Step-by-step solution

Write the balanced equation for the electrolysis of sodium chloride

The electrolysis of sodium chloride produces hydrogen gas, chlorine gas, and sodium hydroxide. The balanced chemical equation for this process is: \(2 NaCl (aq) + 2 H_{2}O (l) \rightarrow 2 NaOH (aq) + H_{2} (g) + Cl_{2} (g)\)

Determine the stoichiometric ratio of hydrogen to chlorine

From the balanced equation, we can determine the stoichiometric ratio of hydrogen gas to chlorine gas: 1 mole of \(H_{2}(g)\) is produced for every 1 mole of \(Cl_{2}(g)\).

Calculate the moles of chlorine produced

We can use the ideal gas law to calculate the moles of \(Cl_{2}(g)\) produced. The ideal gas law is given by the equation: \(PV = nRT\) Where: P = pressure in atm V = volume in liters n = moles of gas R = universal gas constant = 0.0821 L atm / K mol T = temperature in Kelvin Given: P = 2.50 atm V = 257 L (volume of \(Cl_{2}(g)\)) T = 50°C + 273.15 = 323.15 K Rearrange the ideal gas law equation to solve for n: \(n = \frac{PV}{RT}\) Plug in the values and calculate the moles of \(Cl_{2}(g)\): \(n_{Cl_2} = \frac{(2.50~atm)(257~L)}{(0.0821~\frac{L~atm}{K~mol})(323.15~K)} \approx 25.69~moles\)

Calculate the moles of hydrogen produced

Using the stoichiometric ratio determined in Step 2, we can calculate the moles of \(H_{2}(g)\) produced: 1 mole of \(H_{2}\) is produced for every 1 mole of \(Cl_{2}\), therefore, the moles of \(H_{2}(g)\) produced are equal to the moles of \(Cl_{2}(g)\) produced: \(n_{H_2} = n_{Cl_2} = 25.69~moles\)

Calculate the volume of hydrogen produced

Now, we'll use the ideal gas law equation again to find the volume of \(H_{2}(g)\) produced: \(PV = nRT\) We need to calculate V for hydrogen gas. Rearrange the equation to solve for V: \(V = \frac{nRT}{P}\) Given: n = 25.69 moles (\(H_{2}(g)\)) R = 0.0821 L atm / K mol T = 323.15 K (converted from 50°C) P = 2.50 atm Plug in the values and calculate the volume of \(H_{2}(g)\): \(V_{H_2} = \frac{(25.69~moles)(0.0821~\frac{L~atm}{K~mol})(323.15~K)}{(2.50~atm)} \approx 257~L\) Thus, the volume of \(H_{2}(g)\) produced in the same time it takes to produce 257 L of \(Cl_{2}(g)\) is about 257 L, both measured at 50°C and 2.50 atm.

Key concepts

Stoichiometry in Chemical Reactions

Stoichiometry is like a mathematical recipe for chemistry. It's the study of the quantitative relationships between the reactants and products in a chemical reaction, based on the balanced chemical equation. When we balance an equation, we ensure the same number of atoms of each element is present on both the reactant and product sides of the equation. This principle of conservation of mass is fundamental to all chemical reactions.

For example, in the electrolysis of sodium chloride solution, the balanced equation tells us exactly how much hydrogen gas and chlorine gas is produced from a given amount of sodium chloride. The key to solving stoichiometry problems is always starting with a balanced chemical equation and then using the mole ratios to convert between substances. In the exercise, we harnessed the understanding that for every mole of chlorine gas produced, there's also a mole of hydrogen gas produced. This ratio (1:1) is what allows us to quickly find the volume of hydrogen gas from the given volume of chlorine gas.

Ideal Gas Law and Its Application

The ideal gas law is a crucial equation in chemistry and physics. It provides a clear relationship between pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. Represented by the formula PV = nRT, where 'R' stands for the universal gas constant, this law supports calculations involving the conditions of a gaseous system.

The usefulness of the ideal gas law lies in its ability to predict the behavior of a gas when subject to changes in temperature, pressure, or volume. In this scenario involving the electrolysis of sodium chloride, we utilize the ideal gas law to convert the volume of chlorine gas measured at certain conditions to moles and vice versa for hydrogen. By doing so, students can see how the law enables us to work with physical conditions to find quantities that connect to the stoichiometry of the reaction.

Balancing Chemical Equations for Reaction Analysis

Balancing chemical equations is paramount in chemistry because it ensures the law of conservation of mass is upheld. When we have a balanced equation, we can precisely determine the stoichiometry of the reaction—how much of each reactant is needed and how much of each product is formed.

In the context of the sodium chloride electrolysis problem, the balanced equation is the starting point that dictates the subsequent calculations. It tells us the stoichiometric relationship between the reactants and products. This relationship is fundamental for solving stoichiometry problems because it tells us, quantitatively, how the reactants will combine and what quantities of products will form. By understanding and applying the art of balancing chemical equations, students gain the skill to approach and solve a myriad of chemistry problems.