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 \(257 \mathrm{~L}\) \(\mathrm{Cl}_{2}(\mathrm{~g})\), with both volumes measured at \(50 .{ }^{\circ} \mathrm{C}\) and \(2.50 \mathrm{~atm} ?\)

Short answer

In the electrolysis of a sodium chloride solution, 257 L of H₂ (g) is produced in the same time it takes to produce 257 L of Cl₂(g), with both volumes measured at 50.°C and 2.50 atm.

Step-by-step solution

Write the balanced chemical equation for the electrolysis of sodium chloride solution.

The balanced equation for the electrolysis of sodium chloride solution is as follows: \[2 \mathrm{NaCl} (aq) + 2 \mathrm{H}_{2}O (l) \rightarrow 2 \mathrm{NaOH} (aq) + \mathrm{Cl}_{2} (g) + \mathrm{H}_{2} (g)\] The stoichiometry from the balanced equation shows that one mole of Cl₂ gas is produced for every one mole of H₂ gas.

Find the amount of moles of Cl₂ gas produced.

We will use the Ideal Gas Law formula \(PV = nRT\) to find the moles of Cl₂ gas produced, where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is temperature in Kelvin. Given: Volume of Cl₂ = 257 L Pressure = 2.50 atm Temperature = 50 °C = 323 K (To convert from Celsius to Kelvin, add 273) Universal Gas Constant R = 0.0821 L atm / (mol K) Plugging these values into the Ideal Gas Law formula: \(2.50 \cdot 257 = n \cdot 0.0821 \cdot 323\) Now, we will solve the equation for the number of moles (n) of Cl₂.

Calculate the moles of Cl₂ produced.

Solving for the number of moles (n) of Cl₂ produced: \(n_{Cl2} = \frac{2.50 \cdot 257}{0.0821 \cdot 323} = 25.92~\text{moles}\) We find that 25.92 moles of Cl₂ gas are produced during the electrolysis of the sodium chloride solution.

Find the moles of H₂ gas produced.

In the balanced equation, for every mole of Cl₂ produced, one mole of H₂ is also produced: \(n_{H2} = n_{Cl2}\) \(n_{H2} = 25.92~\text{moles}\) So, 25.92 moles of H₂ gas are produced.

Calculate the volume of H₂ gas produced.

Now that we have the moles of H₂ (n), we can use the Ideal Gas Law formula to find the volume of H₂ gas produced under the given conditions: \(PV = nRT\) \(V_{H2} = \frac{n_{H2} \cdot R \cdot T} {P}\) Plugging the values into the formula: \(V_{H2} = \frac {25.92 \cdot 0.0821 \cdot 323}{2.50} = 257~\text{L}\) Thus, the volume of H₂ gas produced under these conditions is 257 L.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fascinating and vital concept in chemistry and physics. It helps us understand the behavior of gases under different conditions by relating pressure, volume, temperature, and the number of moles. The formula for the Ideal Gas Law is \( PV = nRT \), where:

• \(P\) stands for pressure (in atm).
• \(V\) is the volume (in L).
• \(n\) is the number of moles of gas.
• \(R\) is the Universal Gas Constant, approximately 0.0821 L atm / (mol K).
• \(T\) is the temperature in Kelvin, which you get by adding 273 to the Celsius temperature.

In our example, the Ideal Gas Law helps calculate the moles of \( \mathrm{Cl}_2 \) gas, which in turn allows us to find the volume of \( \mathrm{H}_2 \) gas produced. This shows the interconnection between these variables and how changing one affects the others.

Chemical Equation Balancing

Chemical equation balancing is crucial for accurately representing chemical reactions. Each side of the equation must have the same number of atoms for each element. This balance indicates the conservation of mass, meaning matter cannot be created or destroyed. In our example, the chemical equation for the electrolysis of sodium chloride solution is:\[ 2 \mathrm{NaCl} (aq) + 2 \mathrm{H}_{2}O (l) \rightarrow 2 \mathrm{NaOH} (aq) + \mathrm{Cl}_{2} (g) + \mathrm{H}_{2} (g) \]Notice how the number of each type of atom on the reactant side equals the number on the product side. Balancing this equation is essential for further calculations, like determining the ratio of products formed or the reactants consumed. Compounds like \( \mathrm{NaCl} \) and \( \mathrm{H}_2O \) decompose and rearrange to create new chemicals, keeping the atom count intact.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It uses the coefficients from balanced chemical equations to predict how much product will form, or how much reactant is needed, when the reaction occurs. In the context of our problem, stoichiometry involves understanding that one mole of \( \mathrm{Cl}_2 \) gas produced corresponds to one mole of \( \mathrm{H}_2 \) gas due to the balanced equation. Hence, if \(25.92\) moles of \( \mathrm{Cl}_2 \) are formed, \(25.92\) moles of \( \mathrm{H}_2 \) are also generated. Stoichiometry forms the backbone of precisely predicting quantities in chemical reactions, ensuring accuracy in laboratory and industrial settings.

Gas Laws

Gas laws describe the behavior of gases and are fundamental to understanding how changes in pressure, volume, and temperature affect gases. Several laws culminate in the Ideal Gas Law, such as:

• Boyle's Law: At constant temperature, the pressure of a gas is inversely proportional to its volume.
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin.
• Avogadro's Law: Equal volumes of gases at the same temperature and pressure have the same number of molecules (or moles).

The understanding of these laws allows us to anticipate how a gas will behave when any one of these conditions changes. In our example, these principles allow us to calculate the volumes of gases produced during electrolysis by relating the conditions of temperature and pressure with the types and quantities of gases concerned. This insight into gas behavior is essential for both academic studies and practical applications in engineering and environmental sciences.