Question

Atomic masses can be determined by electrolysis. In one hour, a current of 0.600 A deposits \(2.42 \mathrm{~g}\) of a certain metal, \(\mathrm{M},\) which is present in solution as \(\mathrm{M}^{+}\) ions. What is the atomic mass of the metal?

Short answer

Question: Calculate the atomic mass of a metal whose 2.42 grams are deposited when a current of 0.600 A is passed for 1 hour using electrolysis. The metal ion has an oxidation state of +1. Answer: The atomic mass of the metal is 108 g/mol.

Step-by-step solution

Determine the moles of electrons transferred in the process

We can use the formula \(Q = It\) to calculate the total charge transferred during the electrolysis process, where \(Q\) is the total charge, \(I\) is the current, and \(t\) is the time. We have: - Current, \(I = 0.600 \ \text{A}\) - Time, \(t = 1 \ \text{hour} = 3600 \ \text{s}\) Now, we can find the total charge transferred, \(Q\): $$Q = It = (0.600 \ \text{A}) \times (3600 \ \text{s}) = 2160 \ \text{C}$$ We can convert this charge into moles of electrons using the Faraday constant, \(F = 96,485 \ \frac{\text{C}}{\text{mol}}\): $$\text{moles of electrons} = \frac{Q}{F} = \frac{2160 \ \text{C}}{96,485 \ \frac{\text{C}}{\text{mol}}} = 0.0224 \ \text{mol}$$

Determine the moles of the metal ion

Since the oxidation state of the metal ion is +1, the number of moles of metal ions is equal to the moles of electrons transferred: $$\text{moles of metal ions} = 0.0224 \ \text{mol}$$

Determine the atomic mass of the metal

Using the mass of metal deposited and the moles of metal ions, we can calculate the atomic mass: $$\text{Atomic mass} = \frac{\text{Mass of metal}}{\text{Moles of metal ions}} = \frac{2.42 \ \text{g}}{0.0224 \ \text{mol}} = 108 \ \frac{\text{g}}{\text{mol}}$$ The atomic mass of the metal is \(\boxed{108 \ \frac{\text{g}}{\text{mol}}}\).

Key concepts

Electrolysis

Electrolysis is a fascinating electrochemical process where electrical energy is used to drive a non-spontaneous chemical reaction. It involves passing an electric current through a substance that is either molten or dissolved in a suitable solvent to cause a chemical change. The substance is often separated into its component ions, with positively charged ions (cations) moving towards the negatively charged electrode (cathode) and negatively charged ions (anions) moving towards the positively charged electrode (anode).

For instance, when electrolysis is applied to a solution of a metal salt, the metal ions move towards the cathode where they gain electrons (reduce) to form the metal. This is exactly the principle used to determine the atomic mass of a metal in our textbook example—a known quantity of the metal is deposited on the cathode, and by calculating the mass of the metal and the charge passed, we can deduce its atomic mass.

Faraday's Laws of Electrolysis

Faraday's laws of electrolysis are critical to understanding how the quantity of substance affected during electrolysis relates to the quantity of electricity used in the process. The first law states that the mass of a substance altered at an electrode during electrolysis is directly proportional to the quantity of electricity that passes through the circuit. The second law provides a deeper layer by informing us that for the same quantity of electricity, the amounts of different substances deposited or dissolved are proportional to their equivalent weights.

Expressed mathematically, the quantity of a substance (in moles) deposited at an electrode is equal to the total charge passed through the substance divided by the Faraday constant (F). This constant, approximately 96,485 coulombs per mole of electrons, is pivotal in calculating the moles of electrons involved during electrolysis—as demonstrated in our solution.

Mole Concept

The mole concept is an indispensable part of chemistry that deals with counting atoms, molecules, ions, or other entities using the unit 'mole'. One mole is equivalent to Avogadro's number, approximately 6.022 x 10^23, of these entities. The application of the mole concept allows chemists to work with macroscopic amounts of substances even as they think about atomic-scale interactions.

In electrochemistry, such as during electrolysis, the mole concept allows us to translate the mass of a substance deposited on an electrode into moles, which can then be related to the number of electrons or Faraday's constant to find essential properties, like the atomic mass of a metal. With the mole concept, we were able to calculate the atomic mass of the deposited metal by dividing the mass of the metal by the moles of metal ions, which we had found earlier.

Electrochemistry

Electrochemistry sits at the crossroads of electrical science and chemical reactions, studying how these two seemingly diverse fields can influence each other. It deals with the processes that convert electrical energy into chemical energy and vice versa. One of the core applications of electrochemistry is in batteries, where chemical energy is converted into electrical energy to provide power.

Electrolysis, as considered in our exercise, is another key application of electrochemistry. By utilizing it, we can not only determine the atomic mass of an element, but also refine metals, manufacture chemicals, and coat objects with a thin layer of a precious metal (electroplating). Understanding electrochemistry is essential for grasping concepts like redox reactions, electrode potentials, and the energy changes associated with electrical work done by chemical systems.