Problem 58 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Electrolysis of an acetate solution produces ethane according to the reaction: $$ 2 \mathrm{CH}_{3} \mathrm{COO}^{-} \rightarrow \mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{~g})+2 \mathrm{CO}_{2}(\mathrm{~g})+2 \mathrm{e}^{-} $$ What total volume of ethane and \(\mathrm{CO}_{2}\) would be produced at \(0^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\), if a current of \(0.5 \mathrm{~A}\) is passed through the solution for \(482.5\) min? Assume current efficiency \(80 \%\). (a) \(1.344 \mathrm{~L}\) (b) \(2.688 \mathrm{~L}\) (c) \(4.032 \mathrm{~L}\) (d) \(1.792 \mathrm{~L}\)

Step-by-Step Solution

Answer

The total volume of ethane and \(CO_2\) produced would be 1.344 L.

Step 1: Calculate the Charge Passed

Determine the total amount of charge (in coulombs) that passes through the solution using the formula: charge (Q) = current (I) × time (t). Convert the time from minutes to seconds.

Step 2: Reactions at the Electrode

Understand the electrolysis process. For each mole of \(C_2H_6\) and \(CO_2\), 2 moles of electrons are needed. Relate the charge passed to the amount of \(C_2H_6\) and \(CO_2\) formed using Faraday's first law of electrolysis. Note that the number of moles of electrons (n) is given by Q/F (where F is the Faraday constant, 96485 C/mol).

Step 3: Calculate the Amount of Gas Produced

Incorporate the current efficiency into the calculation. Calculate the actual number of moles using the efficiency. Then, determine the number of moles of \(C_2H_6\) and \(CO_2\) produced.

Step 4: Use the Ideal Gas Law

Use the ideal gas law to find the total volume of the gases produced. The ideal gas law is PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. Use the conditions provided: \(P = 1 \text{atm}\), \(T = 273 K\), and \(R = 0.0821 \text{L}\cdot\text{atm}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}\).

Step 5: Compute the Total Volume of Gas

Calculate the volume for each gas separately (using the number of moles from Step 3) and then add these volumes to get the total volume.

Key Concepts

Faraday's Laws of Electrolysis

Understanding Faraday's laws of electrolysis is essential for solving stoichiometry problems involving electrochemical processes. These laws allow us to relate electrical charge to chemical change.

When an electric current passes through an electrolyte, a chemical reaction occurs at the electrodes, resulting in the liberation or deposition of substances. Faraday's first law states that the amount of substance liberated at an electrode is directly proportional to the quantity of electricity (charge) that passes through the electrolyte. Mathematically, it's expressed as \(m = (Q/F) \times M\), where \(m\) is the mass of the substance, \(Q\) is the total charge, \(F\) is Faraday's constant (\(96485 \text{C/mol}\)), and \(M\) is the molar mass of the substance.

Faraday's second law states that when the same amount of electric current is passed through different electrolytes, the quantity of substances produced is proportional to their chemical equivalents (the mass of a substance that combines with or replaces one mole of hydrogen).

For solving problems, translating the charge passed through the electrolyte into moles of electrons is fundamental, as this will be subsequently used to calculate the amounts of substances produced at the electrodes.

Ideal Gas Law Calculations

The ideal gas law is a pivotal concept used to calculate the properties of gases under various conditions. It is represented by the equation \(PV = nRT\), where \(P\) stands for pressure, \(V\) is the volume, \(n\) represents the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

To solve our electrochemistry problem involving gas production, we apply the ideal gas law to find the volume of gases produced. With our given conditions of standard temperature (\(0^{\text{°}}C\), or \(273 K\)) and pressure (\(1 atm\)), and knowing the number of moles calculated from the amount of charge passed and Faraday's laws, we can determine the volume of the gaseous products produced during electrolysis.

Example Calculation

If we have calculated the number of moles of \(C_2H_6\), we can find its volume at the given temperature and pressure by rearranging the ideal gas law to \(V = \frac{nRT}{P}\) and substituting the values. In a similar fashion, the volume of \(CO_2\) is calculated, allowing us to sum their volumes to find the total volume of gas produced.

Current Efficiency in Electrolysis

Current efficiency is a measure that indicates how effectively the electric current is being used to produce the desired product in an electrochemical cell. It is expressed as a percentage of the theoretical amount of product expected based on the amount of charge passed through the electrolyte.

Not all the current used in an electrolysis process goes into producing the targeted chemical compounds; some of it is lost or used in side reactions. Current efficiency accounts for these losses. If a problem states that the current efficiency is 80%, it means that only 80% of the electrical current contributes to the reaction you're interested in.

For calculations involving current efficiency, we adjust the number of moles of product expected from the charge passed (calculated using Faraday's first law) by multiplying it by the current efficiency (in decimal form). Therefore, if you calculated that \(1 mole\) of gas should be produced, but the current efficiency is \(80%\), then effectively, only \(0.8 moles\) of gas would be produced. This consideration is crucial for accurate stoichiometry calculations in electrolysis problems.