Quantitative Electrolysis
The Foundation: Electricity and Chemical Change
Electrolysis is a process that uses a direct electric current (DC) to drive a non-spontaneous chemical reaction. It occurs in an electrolytic cell, which contains an electrolyte (a substance that conducts electricity when molten or dissolved in water) and two electrodes (conductors that provide the interface with the electrolyte). The key to quantitative electrolysis is understanding that the flow of electric current is actually the flow of tiny, charged particles called electrons. The amount of substance that is oxidized or reduced at an electrode is directly proportional to the quantity of electric charge that passes through the cell.
Understanding Faraday's Laws of Electrolysis
The relationship between electricity and chemical change was formally established by the scientist Michael Faraday in the 1830s. His findings are summarized in two fundamental laws.
Faraday's First Law: The mass of a substance altered at an electrode during electrolysis is directly proportional to the quantity of electric charge passed through the electrolyte.
This can be written as: $m \propto Q$, where $m$ is the mass deposited or liberated and $Q$ is the total electric charge.
Faraday's Second Law: When the same quantity of electricity is passed through different electrolytes, the masses of different substances produced at the electrodes are proportional to their equivalent masses[1].
This law allows us to compare the amounts of different elements produced by the same current over the same time.
The Key Players: Charge, Current, Time, and the Faraday Constant
To perform calculations, we need to define and connect a few important quantities.
| Quantity | Symbol & Unit | Description |
|---|---|---|
| Electric Charge | $Q$ (Coulombs, C) | The total amount of electricity passed. It is calculated from current and time: $Q = I \times t$. |
| Electric Current | $I$ (Amperes, A) | The rate of flow of electric charge. |
| Time | $t$ (seconds, s) | The duration for which the current flows. |
| Faraday Constant | $F$ (C mol$^{-1}$) | The charge of one mole of electrons. $F \approx 96,500$ C mol$^{-1}$. |
The most crucial link is the Faraday Constant (F). It tells us that one mole of electrons carries a charge of approximately 96,500 Coulombs. This number is the bridge between the microscopic world of atoms and moles and the macroscopic world of amps and seconds that we can measure.
The Universal Calculation Method
We can combine all these concepts into a single, powerful formula for calculating the mass of a substance produced during electrolysis:
$m = \frac{(I \times t)}{F} \times \frac{M}{z}$
Where:
- $m$ = mass of substance produced (in grams)
- $I$ = current (in Amperes, A)
- $t$ = time (in seconds, s)
- $F$ = Faraday constant (96,500 C mol$^{-1}$)
- $M$ = molar mass of the substance (in g mol$^{-1}$)
- $z$ = the number of electrons transferred per ion (e.g., $z=1$ for $Ag^+$, $z=2$ for $Cu^{2+}$, $z=2$ for $Cl^-$ to form $Cl_2$)
The term $\frac{(I \times t)}{F}$ gives us the number of moles of electrons that have flowed. The term $\frac{M}{z}$ is the mass of substance deposited by one mole of electrons, which is its equivalent mass.
This formula is your key to solving almost any quantitative electrolysis problem involving mass.
A Section with the Theme of Practical Application or Concrete Example
Let's apply our knowledge to real-world scenarios. Electroplating and the industrial production of chemicals are two major applications of quantitative electrolysis.
Example 1: Electroplating a Spoon with Silver
Imagine you want to coat a spoon with a layer of silver. The spoon is made the cathode in an electrolytic cell containing silver ions ($Ag^+$). The half-reaction at the cathode is: $Ag^+ + e^- \to Ag(s)$.
Problem: A current of 2.5 A is passed through the cell for 30 minutes. What mass of silver will be deposited on the spoon? (Molar mass of Ag = 107.9 g mol$^{-1}$)
Solution:
- Identify the variables from the formula $m = \frac{(I \times t) \times M}{F \times z}$.
- $I = 2.5$ A
- $t = 30$ minutes $ \times 60 = 1800$ s
- $M = 107.9$ g mol$^{-1}$
- $F = 96500$ C mol$^{-1}$
- $z = 1$ (because only 1 electron is needed per $Ag^+$ ion)
- Substitute into the formula: $m = \frac{(2.5 \times 1800) \times 107.9}{96500 \times 1}$
- Calculate step-by-step: $Q = I \times t = 4500$ C. Then $m = \frac{4500 \times 107.9}{96500} = \frac{485550}{96500} \approx 5.03$ g.
So, approximately 5.03 grams of silver will be plated onto the spoon.
Example 2: Calculating the Volume of a Gas Liberated
In the electrolysis of concentrated sodium chloride solution (brine), chlorine gas ($Cl_2$) is produced at the anode: $2Cl^- \to Cl_2 + 2e^-$.
Problem: What volume of chlorine gas, measured at Standard Temperature and Pressure (STP)[2], is produced when a current of 10 A flows for 1 hour? (Molar volume at STP = 22.4 dm$^3$ mol$^{-1}$)
Solution:
- First, find the moles of electrons: $n(e^-) = \frac{I \times t}{F} = \frac{10 \times (60 \times 60)}{96500} = \frac{36000}{96500} \approx 0.373$ mol.
- From the equation $2Cl^- \to Cl_2 + 2e^-$, we see that 2 moles of electrons produce 1 mole of $Cl_2$ gas. So, moles of $Cl_2$ produced = $\frac{0.373}{2} = 0.1865$ mol.
- Volume at STP = moles $ \times $ molar volume = $0.1865 \times 22.4 \approx 4.18$ dm$^3$.
Therefore, about 4.18 liters of chlorine gas are liberated.
Important Questions
Cathode: $2H_2O + 2e^- \to H_2 + 2OH^-$ (z=2 for producing one mole of $H_2$)
Anode: $2H_2O \to O_2 + 4H^+ + 4e^-$ (z=4 for producing one mole of $O_2$)
You can use the same calculation method to find the mass or volume of hydrogen and oxygen gas produced from a given amount of electricity.
Quantitative electrolysis provides a powerful and predictable link between the abstract concept of electric current and the tangible results of chemical reactions. By mastering the relationship defined by Faraday's Laws and utilizing the universal formula $m = \frac{(I \times t) \times M}{F \times z}$, we can accurately determine the mass of a metal plated in an electroplating bath, the volume of a gas produced in an industrial plant, or the time required to refine a batch of copper. This understanding is fundamental not only to academic chemistry but also to countless technological and industrial processes that shape our modern world.
Footnote
[1] Equivalent Mass: The mass of a substance that will combine with or displace 1 gram of hydrogen or 8 grams of oxygen. In electrolysis, it is calculated as Molar Mass ($M$) divided by the charge on the ion ($z$). For example, the equivalent mass of aluminum ($Al^{3+}$) is $27 / 3 = 9$ g mol$^{-1}$.
[2] STP (Standard Temperature and Pressure): A standard set of conditions for measuring gases, defined as a temperature of 0$^\circ$C (273 K) and a pressure of 1 atmosphere. At STP, one mole of any ideal gas occupies 22.4 liters (22.4 dm$^3$).