Question

What is the hourly production rate of chlorine gas (in $\mathrm{kg})$ from an electrolytic cell using aqueous $\mathrm{NaCl}$ electrolyte and carrying a current of $1.500\times {10}^3\mathrm{A}?$ The anode efficiency for the oxidation of ${\mathrm{Cl}}^{-}$ is 93.0 percent.

Short answer

The hourly production rate of chlorine gas is approximately 1.846 kg/h.

Step-by-step solution

Understand the Reaction

The reaction at the anode in an electrolytic cell containing $\mathrm{NaCl}$ is the oxidation of chloride ions (${\mathrm{Cl}}^{-}$) to chlorine gas (${\mathrm{Cl}}_2$). The balanced chemical equation for the oxidation is: $$2{\mathrm{Cl}}^{-}\to {\mathrm{Cl}}_2+2e^{-}$$ This means that 2 moles of electrons produce 1 mole of chlorine gas.

Calculate the Theoretical Charge Required

Using Faraday's constant ($F=96485\text{C/mol}$), calculate the charge needed to produce 1 mole of ${\mathrm{Cl}}_2$ gas: 2 moles of electrons are needed to produce 1 mole of ${\mathrm{Cl}}_2$.1 mole of electrons carries a charge of $96485\text{C}$, so 2 moles would carry $2\times 96485=192970\text{C}$. This charge produces 1 mole of chlorine gas (${\mathrm{Cl}}_2$).

Find the Theoretical Mass of Chlorine Produced per Second

We are given a current of $1.500\times {10}^3\text{A}$. The current (I) is the rate of flow of charge $(Q/t)$, so the total charge per second is $1.500\times {10}^3\text{C/s}$. The production of ${\mathrm{Cl}}_2$ can then be calculated by considering the mass per coulomb: $$m=\left(\frac{molar\mathrm{mass}\mathrm{of}{\mathrm{Cl}}_2}{2\times 96485}\right)\times 1.500\times {10}^3=\frac{70.9}{192970}\times 1.500\times {10}^3$$ $$\approx 0.552\mathrm{g}/\mathrm{s}$$, where 70.9 $\mathrm{g}/\mathrm{mol}$ is the molar mass of ${\mathrm{Cl}}_2$.

Adjust for Anode Efficiency

The anode efficiency is 93%, which means that only 93% of the chlorine predicted by the theoretical calculations is actually formed. Multiply the theoretical production mass per second by the anode efficiency: $$\approx 0.552\mathrm{g}/\mathrm{s}\times 0.93=0.513\mathrm{g}/\mathrm{s}$$

Convert to Hourly Production Rate

The question requires the rate in $\mathrm{kg}\mathrm{per}\mathrm{hour}$. Convert from $\mathrm{g}/\mathrm{s}$ to $\mathrm{kg}/\mathrm{h}$: $$\approx 0.513\mathrm{g}/\mathrm{s}\times 3600\mathrm{seconds}/\mathrm{hour}=1846\mathrm{g}/\mathrm{h}$$ Convert grams to kilograms: $$1846\mathrm{g}/\mathrm{h}\approx 1.846\mathrm{kg}/\mathrm{h}$$

Key concepts

Chlorine Gas Production

Electrolytic cells are devices that use electrical energy to drive chemical reactions. In the case of chlorine gas production, an electrolytic cell is used to oxidize chloride ions (${\mathrm{Cl}}^{-}$) into chlorine gas (${\mathrm{Cl}}_2$). The reaction occurs at the anode, the positive electrode of the cell. The process can be described by the balanced chemical equation:$$2{\mathrm{Cl}}^{-}\to {\mathrm{Cl}}_2+2e^{-}$$In this reaction, two moles of electrons are required to produce one mole of chlorine gas. This balanced equation provides the basis for calculating how much chlorine gas is produced given a certain amount of electricity flowing through the cell. Chlorine gas is important in various industrial applications, including water purification and the production of compounds like polyvinyl chloride (PVC). By harnessing the principles of electrolysis, industries can produce chlorine gas efficiently on a large scale.

Faraday's Constant

Faraday's constant ($F$) is a fundamental value in electrochemistry that relates the amount of electric charge carried by one mole of electrons. It is defined as 96485 coulombs per mole (C/mol). This constant is crucial for calculating the charge required to produce a specific amount of a substance in an electrolytic process.In the context of chlorine gas production, Faraday's constant is used to determine how much electric charge is needed to generate a particular quantity of ${\mathrm{Cl}}_2$. Since two moles of electrons are necessary to form one mole of chlorine gas, the total charge for this reaction can be calculated by:$$Charge=2\times F=2\times 96485=192970\text{Coulombs}$$Using Faraday's constant allows us to transform theoretical chemical equations into practical calculations for real-world applications, ensuring accurate predictions of product quantities based on the electrical input.

Anode Efficiency

Anode efficiency is a measure of how effectively an anode in an electrolytic cell converts its input into the desired product. It is expressed as a percentage, indicating the proportion of the theoretical yield that is actually obtained. In the case of chlorine gas production, the anode efficiency refers to the efficiency of converting chloride ions into ${\mathrm{Cl}}_2$ gas.For example, if the anode efficiency is 93%, it means that 93% of the calculated theoretical chlorine production is realized in practice. Several factors can affect anode efficiency, including the nature of the electrolyte and the condition of the anode surface. Here, the calculation incorporates anode efficiency by adjusting the theoretical mass of chlorine produced per second from 0.552 g/s to 0.513 g/s, acknowledging the real-world limitations of the electrolysis process.

Current and Charge Relationship

In electrochemical processes, the relationship between electric current and charge is a critical aspect. Current ($I$) is the flow of electric charge per unit time, measured in amperes (A). Mathematically, it is described by the equation:$$I=\frac{Q}{t}$$where $Q$ is the charge in coulombs and $t$ is the time in seconds. This relationship is pivotal in understanding how the flow of electric current over a period translates into a specific amount of charge that drives the electrolytic reactions.
In our scenario of producing chlorine gas, the given current is $1.500\times {10}^3\text{A}$, indicating the rate at which charge is supplied to the system. By calculating the charge delivered per second, we can accurately predict the mass of chlorine gas produced:$$Q=1.500\times {10}^3\text{C/s}$$Using this relationship, we can transition from theoretical to actual production rates, understanding that the electricity supply directly dictates how much of the chemical reaction occurs.