Question

It took 2.30 min using a current of 2.00 A to plate out all the silver from 0.250 L of a solution containing Ag \(^{+} .\) What was the original concentration of \(\mathrm{Ag}^{+}\) in the solution?

Short answer

The original concentration of Ag+ ions in the solution was approximately 0.01144 mol/L.

Step-by-step solution

Calculate the Charge (Coulombs) Passed Through the Cell

We are given that a current of 2.00 A flowed for 2.30 minutes. The total charge in Coulombs can be calculated using the formula: Charge (Q) = Current (I) × Time (t) Now we need to convert the time from minutes to seconds, as the unit of current (Ampere) is in terms of seconds. 1 minute = 60 seconds, so 2.30 minutes = 2.30 × 60 seconds = 138 seconds Now, we can find the total charge: Q = I × t Q = 2.00 A × 138 seconds Q = 276 Coulombs

Calculate the Moles of Ag+ ions Deposited

Next, we will use Faraday's laws of electrolysis to find the moles of Ag+ ions deposited. The relationship between the charge, the number of moles of substance, and the Faraday's constant is: Charge (Q) = Number of moles (n) × Faraday's constant (F) F is the Faraday constant, equal to 96485 C/mol In this case, the deposited substance is silver (Ag), which is a monovalent ion (Ag+), so the Faraday's law for Ag+ can be represented as: Q = n × F Now, rearrange the formula to find the number of moles: Number of moles (n) = Charge (Q) / Faraday's constant (F) n = 276 C / 96485 C/mol n ≈ 0.00286 mol

Calculate the Original Concentration of Ag+ ions

Finally, to find the original concentration of Ag+ ions in the solution, we will use the formula: Concentration (c) = Number of moles (n) / Volume (V) We are given that the volume of the solution is 0.250 L. Using the moles of Ag+ ions calculated in Step 2, we can find the concentration: c = 0.00286 mol / 0.250 L c ≈ 0.01144 mol/L So, the original concentration of Ag+ ions in the solution was approximately 0.01144 mol/L.

Key concepts

Faraday's laws of electrolysis

Faraday's laws of electrolysis are fundamental principles that relate the amount of substance deposited at an electrode to the electric charge passed through the electrolyte. This involves two main ideas:
1. **First Law**: The mass of the substance deposited or dissolved at the electrode is directly proportional to the amount of electric charge passed through the electrolyte.
2. **Second Law**: For a given amount of electric charge, the mass of different substances deposited or dissolved is proportional to their equivalent weights.
In our exercise, we see this law applied to the deposition of silver. Since silver dissolves or deposits as Ag^{+} ions, the quantity of silver deposited can be computed using an understanding of these laws and precisely knowing the charge passed. The knowledge that one mole of electrons corresponds to the Faraday constant (96,485 C/mol) allows for converting electrical measurements to chemical results.

Charge calculation

Charge calculation is crucial in electrochemical processes where the amount of electricity used directly affects the reaction's outcome.
To determine the charge ( Q ) in the exercise, we use the formula: Q = I imes t , where I is the current (in amperes) and t is the time (in seconds).
It's essential to remember to convert time into the correct units to align with the current measured in seconds. In the problem, 2.30 minutes was converted to seconds by multiplying by 60, resulting in 138 seconds. With a current of 2.00 A, the total charge calculated was 276 Coulombs.
This value represents the total amount of electric charge that contributed to the silver plating process.

Concentration determination

Concentration determination allows chemists to find out how diluted or concentrated a solution is.
To determine concentration, we need the number of moles of a solute and the volume of the solution. The formula used is c = rac{n}{V} , where c is the concentration, n is the number of moles, and V is the volume (in liters).
In our exercise, after calculating 0.00286 moles of silver ions were deposited, using a solution volume of 0.250 liters allowed us to find the original concentration of 0.01144 mol/L.
This calculation is essential for understanding the solution's initial conditions and the efficiency of the electrolysis process, as well as ensuring that the desired reaction takes place correctly.

Silver deposition

Silver deposition refers to the process of depositing silver metal onto a surface from a solution.
In electrochemistry, this process involves moving silver ions ( Ag^{+} ) from the solution to an electrode where they gain electrons and transform back into solid silver.
During the electrolysis of a silver solution, applying an electric current enables the ions to move towards a negatively charged electrode (cathode), facilitating the deposition of silver metal.
This is a practical use of Faraday's laws and is critical in various industries, including electronics and jewelry-making, where a thin layer of silver plating can enhance appearance or conductivity.

Electrolysis process

The electrolysis process is a technique that uses an electric current to drive a non-spontaneous chemical reaction.
In this context, it involves the breakdown of chemical compounds in a solution to deposit a material (like silver) onto an electrode.
By applying certain voltages, ions in the solution migrate to electrodes based on charge: positive ions (cations) migrate to the cathode to gain electrons, while negative ions (anions) move to the anode to lose electrons.
This movement of ions is crucial for the deposition of materials like through electroplating, or in chemical synthesis and resource extraction.
Applying this process properly requires understanding the relationship between current, time, and the nature of the ions involved, as described by Faraday's laws of electrolysis.