Question

How many grams of \(\mathrm{Cr}\) are deposited in the electrolysis of solution of \(\mathrm{Cr}\left(\mathrm{NO}_{3}\right)_{3}\) in the same time that it takes to deposit \(0.54 \mathrm{~g}\) of \(\mathrm{Ag}\) in a silver coulometer arranged in series with the \(\mathrm{Cr}\left(\mathrm{NO}_{3}\right)_{3}\) cell? \(\quad\) (Atomic weight : \(\mathrm{Cr}=52.0 ; \mathrm{Ag}=108\) ) (a) \(0.0866\) (b) \(0.0288\) (c) \(0.173\) (d) \(0.220\)

Short answer

0.173 g of Cr are deposited.

Step-by-step solution

Determine the charge required to deposit the given mass of Ag

First, calculate the amount of charge required to deposit 0.54 g of Ag using the equation:\[ q = \frac{{m \times F}}{{A \times n}} \] where \( m \) is the mass of Ag deposited, \( F \) is Faraday's constant (approximately 96500 C/mol), \( A \) is the atomic mass of Ag, and \( n \) is the number of electrons transferred per atom (for Ag, \( n = 1 \) because Ag deposits as Ag+).

Calculate the amount of Cr deposited using the same charge

Use the charge calculated in step 1 to find the mass of Cr deposited using the equation:\[ m_{\mathrm{Cr}} = \frac{{q \times A_{\mathrm{Cr}} \times n_{\mathrm{Cr}}}}{{F}} \] where \( A_{\mathrm{Cr}} \) is the atomic mass of Cr, and \( n_{\mathrm{Cr}} \) is the number of electrons transferred per atom of Cr (for Cr, \( n = 3 \) because Cr deposits as Cr^3+).

Perform the calculations

Substitute the known values into the equations and solve for the mass of Cr.

Key concepts

Faraday's Laws of Electrolysis

Faraday's laws of electrolysis are crucial for understanding how electrolytic processes work, including the deposition of metals. These laws enable us to calculate the amount of substance that will be deposited at an electrode during electrochemical reactions.

The first law states that the amount of chemical change (or mass of the substance deposited) at an electrode during electrolysis is directly proportional to the quantity of electricity that passes through the electrolyte. This quantity of electricity is measured in coulombs, which is the product of current in amperes and time in seconds.

The second law of Faraday states that the amounts of different substances liberated by the same quantity of electricity passing through the electrolytic solution are proportional to their equivalent weights. The equivalent weight of a substance is its molar mass divided by the number of electrons required to reduce or oxidize one mole of the substance.

In the context of our exercise, these laws allow us to relate the charge needed to deposit chromium to the charge used to deposit silver. Since the two processes are arranged in series, the same quantity of electricity will pass through both cells, leading to the deposition of both metals.

Mole Concept in Chemistry

The mole concept in chemistry provides a bridge between the microscopic world of atoms and ions and the macroscopic world we can measure. One mole of any substance contains Avogadro's number of particles (atoms, ions, or molecules), which is approximately \(6.022 \times 10^{23}\) entities.

To connect the mole concept to electrochemistry, we recall that a mole of electrons has a charge equal to Faraday's constant (\(F \text{, which is approximately } 96,500 \text{ C/mol}\) ). This is essential when we consider the deposition of metals: One mole of metal ions will require one or more moles of electrons to be neutralized and deposited according to the ion's charge. In the case of silver (\text{Ag}), each ion requires one electron, while chromium (\text{Cr}^3+) requires three electrons.

Applying this concept, we can relate the mass of a metal deposited to the moles of electrons transferred, and hence to the basic unit of electrochemical quantity, the faraday. So, in our problem, the calculation of mass of chromium deposited begins with understanding the relationship between the mass of silver deposited, the moles of electrons transferred, and the stoichiometry of the chromium ions in solution.

Stoichiometry

Stoichiometry is the quantitative relationship in which reactants combine to form products in a chemical reaction, it's similarly used in electrochemistry for the reactions at electrodes. In the context of electrolysis, stoichiometry involves the calculation of masses of reactants and products based on the balanced chemical equations for the redox reactions occurring at the electrodes.

In practical terms, stoichiometry helps us predict the outcome of chemical processes – in our case, how much of a metal will be deposited from its ions in solution when a specific amount of charge is passed through the electrolyte. The stoichiometric calculation involves understanding the valency of the metal ions involved (how many electrons one ion of the metal will accept), the mass of the ions, and the relationship between electrons transferred and metal ions reduced or oxidized.

By incorporating stoichiometry into our calculations for the exercise, we can determine the exact mass of chromium that would be deposited given a particular charge that has also deposited a known mass of silver. We use the valencies of silver and chromium to find the relation between the moles of electrons and the equivalent mass that is deposited.