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 charge passed through the electrolyte is calculated using \(Q = I \times t\), where Q is the charge, I is the current, and t is the time. The charge is found to be \(148.2 C\). Using Faraday's law, \(Q = nF\), the moles of electrons transferred are found to be approximately \(0.001536 \ \text{mol}\). Since the metal ion is trivalent, moles of the unknown metal are calculated as \(0.000512 \ \text{mol}\). The molar mass of the metal is calculated using \(\frac{\text{Mass of M}}{\text{Moles of M}} = 209 \ \frac{\text{g}}{\text{mol}}\). The metal is identified as Thallium (Tl) based on its molar mass from the periodic table.

Step-by-step solution

1. Calculate the charge passed through the electrolyte

To calculate the charge, we'll use the given current of \(2.00 A\) and the given time of \(74.1 s\). We can find the charge (Q) using the formula: \[Q = I \times t\] where Q is the charge, I is the current, and t is the time. Using the given values, we have: \[Q = (2.00 A) (74.1 s) = 148.2 C\] So, the charge passed through the electrolyte is \(148.2 C\).

2. Determine the moles of electrons transferred

To find the moles of electrons transferred from the unknown metal ions in the electrolyte to the metal, we will use Faraday's law: \[Q = nF\] where Q is the charge, n is the number of mole of electrons, and F is the Faraday constant, which is approximately 96485 C/mol. We can rearrange the equation to solve for n: \[n = \frac{Q}{F}\] Now, we substitute the given values: \[n = \frac{148.2 C}{96485 C/mol} \approx 0.001536 \ \text{mol}\] So, \(0.001536 \ \text{mol}\) of electrons were transferred.

3. Calculate the molar mass of the metal and identify it

Since we are given that the unknown metal comes from a compound with the formula \(\mathrm{M(NO_{3})_{3}}\), it means that the metal ion is trivalent, which means that 3 moles of electrons are needed for each mole of the metal to form the metal from the ion. So, we can find the moles of the unknown metal M formed from the given electrons as: Moles of M = \(\frac{\text{Moles of electrons transferred}}{3} = \frac{0.001536 \ \text{mol}}{3} = 0.000512 \ \text{mol}\) Now, we will calculate the molar mass by dividing the mass of the metal plated out (0.107 g) by the moles of the metal: Molar mass of M = \(\frac{\text{Mass of M}}{\text{Moles of M}} = \frac{0.107 \ \text{g}}{0.000512 \ \text{mol}} = 209 \ \frac{\text{g}}{\text{mol}}\) Now, since the molar mass of M is approximately 209 g/mol, we can identify the metal as Thallium (Tl) based on its molar mass from the periodic table.

Key concepts

Faraday's Law

Faraday's Law is essential when dealing with electrolysis. It helps us understand the relationship between the electricity used in the course of an electrochemical reaction and the amount of substance produced in that reaction. The law states:

• The amount of substance produced at an electrode during electrolysis is proportional to the quantity of electricity that is passed through the electrolyte.
• This relationship can be used to calculate the moles of electrons transferred, which is critical in identifying the amount of metal deposited during electrolysis.

In this specific problem, Faraday's Law enables us to calculate the number of moles of electrons (n) using the formula \[ n = \frac{Q}{F} \]where \( Q \) is the total electric charge passed through the electrolyte, and \( F \) is Faraday's constant, approximately 96485 C/mol. Knowing these principles allows us to break down complex electrolysis reactions and makes it easier to solve them step-by-step.
Faraday's Law empowers students to understand electrochemical reactions quantitatively and establish a direct link between charge and moles of substances involved.

Molar Mass

The concept of molar mass is a key aspect in chemistry, especially when trying to identify unknown substances based on reactions in electrolysis. Molar mass refers to the mass of a given substance (chemical element or chemical compound) divided by the amount of substance, expressed in moles. It is usually measured in grams per mole (g/mol).

• In the electrolysis problem, we used the known mass of the deposited metal and the number of moles determined by Faraday's Law to calculate the molar mass.
• By dividing the mass of the plated-out metal (0.107 g) by the calculated moles (0.000512 mol), we determined the molar mass of the unknown metal to be approximately 209 g/mol.

Calculating the molar mass was crucial for identifying the unknown metal as thallium since it has a known molar mass close to our result. Understanding how to find and use molar mass aids greatly in chemical identification and reaction quantification.
For students, mastering this calculation is essential for linking the theoretical with the practical aspects of chemistry.

Trivalent Metal Ions

Understanding trivalent metal ions is fundamental when examining reactions involving metal electrolytes. A trivalent metal ion is one that has a valency of three, meaning it can donate or share three electrons per ion. This concept is crucial for solving problems related to electrolysis when the metal comes from compounds like \( \text{M(NO}_{3})_{3} \).

• In the given exercise, the unknown metal M formed a trivalent ion, which dictated that 3 moles of electrons were needed for the reduction of each mole of metal ions to metallic form.
• The concept determines how many moles of the metal are plated out based on the moles of electrons transferred in the electrolytic reaction.

By knowing that the metal ion is trivalent, we understand that each mole of M requires three moles of electrons to convert it from an ionic to a metallic state. This knowledge was crucial to calculating the exact amount of metal deposited and, subsequently, its molar mass. Understanding the nature of trivalent ions is vital for solving electrochemical reaction problems and predicting how metal ions behave during operations like electrolysis.