Question

An unknown metal \(\mathrm{M}\) is electrolyzed. It took \(74.1 \mathrm{~s}\) for a current of \(2.00 \mathrm{~A}\) to plate out \(0.107 \mathrm{~g}\) of the metal from a solution containing \(\mathrm{M}\left(\mathrm{NO}_{3}\right)_{3}\). Identify the metal.

Short answer

The metal with a molar mass of approximately 208.98 g/mol, determined using Faraday's law of electrolysis, is Thallium (Tl).

Step-by-step solution

Given data

Time taken, t = 74.1 s; Current, I = 2.00 A; Mass of the metal deposited, m = 0.107 g; and the solution containing the metal ion is \(\mathrm{M(NO_3)_3}\).

Calculate the charge passed using the given time and current

To calculate the charge passed (Q) during the electrolysis, we can use the formula: Q = I*t Where Q is the charge, I is the current, and t is the time. Plugging in the values: Q = (2.00 A) * (74.1 s) = 148.2 C

Use Faraday's law to find the moles of electrons involved in the reaction

According to Faraday's law, moles of electrons (n) are related to the charge passed (Q) as: n = Q/F Where F is the Faraday constant, approximately 96485 C/mol. Plugging in the values: n = (148.2 C) / (96485 C/mol) = 0.001536 mol of electrons

Determine the moles of metal deposited

The reaction of the metal ion during electrolysis is: \(\mathrm{M^{3+} + 3e^- -> M}\) As the metal has a +3 charge, 3 moles of electrons will deposit 1 mole of metal. To find the moles of metal (m_M) deposited: m_M = n/3 m_M = 0.001536 mol of electrons / 3 = 0.000512 mol of metal

Calculate the molar mass of the metal

To find the molar mass of the metal, we can use the formula: Molar mass (M) = mass of metal (m) / moles of metal (m_M) Plugging in the values: M = 0.107 g / 0.000512 mol = 208.98 g/mol

Identify the metal

The molar mass of the metal is found to be approximately 208.98 g/mol. Checking the periodic table, the metal with a molar mass close to this is thallium (Tl) with a molar mass of 204.38 g/mol. Thus, the unknown metal is Thallium (Tl).

Key concepts

Faraday's Law of Electrolysis

Faraday's law of electrolysis is crucial in understanding how electric current can cause a chemical change during an electrolysis process. This principle tells us how much chemical change occurs at an electrode in an electrochemical cell, related to the amount of electrical charge passed through the system.

According to Faraday's first law, the amount of substance that undergoes oxidation or reduction at each electrode during electrolysis is directly proportional to the total electric charge passed through the solution. In simple terms, if you double the charge, you double the amount of substance electrolyzed.

Here's how you can calculate moles of electrons using Faraday's law in a step-by-step manner:

• First, you calculate the total charge that passed through the electrolyte using the equation: Q = I * t
• Then, you calculate the number of moles of electrons transferred using the relation:n = Q / F

where I is the current in amperes (A), t is the time in seconds (s), Q is the total electrical charge in coulombs (C), and F is Faraday's constant approximately equal to 96485 C/mol.

Understanding this law is fundamental for solving electrolysis problems as it provides a bridge between the physical flow of current and the chemical processes occurring in the solution.

Calculating Molar Mass

Calculating molar mass is a vital skill in chemistry, especially when dealing with stoichiometry and electrolysis reactions. Molar mass is essentially the mass of one mole of a substance, and it is expressed in grams per mole (g/mol).

To calculate the molar mass of an element, you look up the atomic weight of that element on the periodic table. However, when dealing with an unknown element, like in our exercise, you find it indirectly by using experimental data from an electrolysis reaction.

Here are the general steps to calculate molar mass from electrolysis data:

• Determine the mass of the deposited metal after electrolysis.
• Calculate the number of moles of metal deposited using the stoichiometry of the reaction and the moles of electrons involved.
• Finally, use the formula:Molar mass (M) = mass of the metal (m) / moles of metal (m_M)

By applying these steps, you can determine the molar mass of the electrolyzed metal, which helps to identify the substance in cases such as the given exercise.

Stoichiometry of Electrolysis Reactions

Understanding the stoichiometry of electrolysis reactions is critical as it defines the quantitative relationship between reactants and products in a chemical reaction occurring during the electrolysis process. In the context of our exercise, the stoichiometry helps us link the moles of electrons to the moles of metal deposited.

For our given metal ion \( \mathrm{M^{3+}} \) that undergoes reduction to the metal \( \mathrm{M} \), the stoichiometric equation is:\[\mathrm{M^{3+} + 3e^− → M}\]This indicates that three moles of electrons (\( \mathrm{e^{-}} \) ) reduce one mole of the metal ion. When tackling stoichiometry problems:

• Write the balanced chemical equation for the reaction.
• Use stoichiometric coefficients to determine the ratio between reactants and products.
• Relate these ratios to the data provided, like the number of moles of electrons, to calculate the number of moles of the substance produced or consumed.

For successful problem-solving in electrolysis, combining Faraday's law to determine the charge and moles of electrons with stoichiometry will yield the moles of metal deposited, thereby enabling us to find the molar mass and identify the metal, as shown in the textbook exercise.