Question

A solution containing a metal ion \(\left(M^{2+}(a q)\right)\) is electrolyzed by a current of \(7.8\) A. After \(15.5\) minutes, \(2.39 \mathrm{~g}\) of the metal is plated out. (a) How many coulombs are supplied by the battery? (b) What is the metal? (Assume \(100 \%\) efficiency.)

Short answer

Answer: The metal plated out during electrolysis is copper (Cu).

Step-by-step solution

Calculate the Total Charge in Coulombs

Use the current (I) and time (t) given in the problem to calculate the total charge (Q) supplied by the battery: Q = I × t Where: - Q is the total charge in coulombs (C) - I is the current in amperes (A) - t is the time in seconds (s) To convert the given time from minutes to seconds, multiply by 60: t = 15.5 minutes × 60 (s/minute) = 930 seconds Now, calculate the total charge Q: Q = 7.8 A × 930 s = 7254 C

Use Faraday's Law of Electrolysis to Find the Number of Moles

Faraday's Law states that the amount of substance (in moles) deposited at an electrode is directly proportional to the charge. To find the number of mo_goals of the metal ion (n), divide the total charge (Q) by Faraday's constant (F): n = Q / (z*F) Where: - n is the number of moles of the deposited metal - z is the charge of the metal ion (+2 in this case, as indicated by the problem: \(M^{2+}\) ) - F is the Faraday's constant = 96,485 C/mol.e n = 7254 C / (2 * 96,485 C/mol.e)= 0.03756 mol

Determine the Molar Mass of the Metal

To find the molar mass of the metal (M), divide the mass deposited (m) by the number of moles deposited (n): M = m / n Where: - M is the molar mass of the metal - m is the mass of the deposited metal (2.39 g as given in the problem) - n is the number of moles of the metal (0.03756 mol) M = 2.39 g / 0.03756 mol = 63.67 g/mol

Identify the Metal

Now that we know the molar mass of the metal (63.67 g/mol), we can match it to the closest atomic mass of a known metal found in the periodic table: The molar mass is close to that of copper (Cu) with an atomic mass of 63.55 g/mol. So, the metal plated out during electrolysis is copper (Cu).

Key concepts

Faraday's Law

Faraday's Law is fundamental in understanding the process of electrolysis. It states that the mass of a substance altered at an electrode during electrolysis is directly proportional to the amount of electricity that passes through the solution. This means the more electricity used, the more substance is deposited or dissolved.

In mathematical terms, this concept connects the electric charge passing through a solution (measured in coulombs) to the amount of substance changed (in moles). The relationship is governed by the equation:

• \[ n = \frac{Q}{z imes F} \]
• Where \( n \) is the number of moles, \( Q \) is the total charge in coulombs, \( z \) is the number of electrons transferred per ion (also known as valency), and \( F \) is Faraday's constant \( (96,485 \text{ C/mol}) \).

To fully understand Faraday's Law, it is necessary to grasp how the amount of electricity (Q) relates to the chemical reaction happening at the electrodes. This understanding can help in calculating the number of moles of a substance involved in an electrochemical reaction.

Coulombs

A coulomb is a unit of electric charge that is part of the International System of Units (SI). It is named after Charles-Augustin de Coulomb, a French physicist.

One coulomb is defined as the quantity of electricity transported by a current of one ampere in one second. This means that a current of one amp (A) flowing through a wire for one second results in the passage of one coulomb of charge.

• Formula: \[ Q = I \times t \]
• Where \( Q \) is the total charge in coulombs, \( I \) is the current in amps, and \( t \) is the time in seconds.

In electrolysis, the total charge transferred determines how much material is deposited or dissolved across the electrodes according to Faraday's Law. This makes understanding and calculating coulombs a crucial step in solving electrochemical problems.

Metal Ion

Metal ions play a central role in the process of electrolysis. In a solution, metal ions possess a positive charge due to losing electrons. For example, \( M^{2+} \) denotes a metal ion with a double positive charge, indicating it has lost two electrons.

During electrolysis, these metal ions are attracted to the cathode (negative electrode) because of their positive charge. Once they reach the cathode, they gain electrons to form metal atoms. This process causes the metal to ‘plate out’, or deposit on the cathode. Understanding the type of metal ion and its charge is essential for calculating the number of moles of metal being deposited using Faraday's Law.

• Key Points:
  • Metal ions are positively charged.
  • The charge number (like \( 2+ \)) shows how many electrons are lost.
  • The ionic charge affects how many moles of electrons (\( z \)) are needed for reduction during electrolysis.

This understanding is vital for effectively applying Faraday's Law and determining the amounts of substances involved in electrolysis.

Molar Mass

Molar mass is the mass of one mole of a substance, measured in grams per mole (g/mol). It is a fundamental property used to convert between the mass of a substance and the number of moles.

When you electrolyze a solution, you can determine the molar mass of the deposited metal through an equation:

• \[ M = \frac{m}{n} \]
• Where \( M \) is the molar mass, \( m \) is the mass of the deposited metal, and \( n \) is the number of moles.

Having found the molar mass, often you can identify the metal by comparing it to known values from the periodic table. In electrolysis problems, knowing both the deposited metal's mass and the calculated moles allows for the determination of molar mass, assisting in confirming the metal's identity, as seen when recognizing \( 63.67 \text{ g/mol} \) as close to copper's molar mass.