Problem 95 - Chemistry and Chemical Reactivity

Question

A "silver coulometer" (Study Question 94) was used in the past to measure the current flowing in an electrochemical cell. Suppose you found that the current flowing through an electrolysis cell deposited \(0.089 \mathrm{g}\) of Ag metal at the cathode after exactly 10 min. If this same current then passed through a cell containing gold(III) ion in the form of \(\left[\mathrm{AuCl}_{4}\right]^{-}\), how much gold was deposited at the cathode in that electrolysis cell?

Step-by-Step Solution

Answer

Approximately 0.0542 g of gold is deposited.

Step 1: Calculate Moles of Silver Deposited

First, we need to determine the number of moles of silver (Ag) deposited. The molar mass of silver is approximately 107.87 g/mol. Using the mass provided (0.089 g), use the formula: \[ \text{Moles of Ag} = \frac{\text{mass of Ag}}{\text{molar mass of Ag}} = \frac{0.089}{107.87} \approx 0.000825 \text{ mol} \]

Step 2: Calculate Electrons Required for Silver Deposition

The deposition of silver involves a one-electron reduction process: \[ \text{Ag}^{+} + e^{-} \rightarrow \text{Ag} \] Since the reaction involves one electron per mole of silver, 0.000825 moles of electrons were used to deposit silver.

Step 3: Calculate Current Flow

Using Faraday's Law of Electrolysis, we can relate the moles of electrons to current: \[ Q = n \cdot F \] Where \(Q\) is the total charge, \(n\) is moles of electrons (0.000825 mol), and \(F\) is Faraday's constant (~96485 C/mol e\(^-\)). Calculate total charge: \[ Q = 0.000825 \cdot 96485 \approx 79.58 \text{ C} \] The current \(I\) is then found by: \[ I = \frac{Q}{t} = \frac{79.58}{600} \approx 0.1326 \text{ A} \] where the time \(t\) is 600 seconds (10 minutes).

Step 4: Calculate Moles of Gold Deposited

Gold(III) ions are reduced involving the equation: \[ \left[\text{AuCl}_4\right]^{-} + 3e^{-} \rightarrow \text{Au} \] Since three moles of electrons deposit one mole of gold, calculate moles of gold deposited: \[ \text{Total electrons in 10 min} = I \cdot t = 0.1326 \cdot 600 = 79.56 \text{ C} \] \[ \text{Moles of electrons} = \frac{79.56}{96485} \approx 0.000825 \text{ mol} \] Thus, moles of gold deposited: \[ \text{Moles of Au} = \frac{0.000825}{3} \approx 0.000275 \text{ mol} \]

Step 5: Calculate Mass of Gold Deposited

Use the moles of gold to find its mass. The molar mass of gold is approximately 196.97 g/mol: \[ \text{Mass of Au} = \text{Moles of Au} \cdot \text{Molar mass of Au} = 0.000275 \cdot 196.97 \approx 0.0542 \text{ g} \]

Key Concepts

Silver Coulometer

A silver coulometer is a device traditionally used to measure electric current in solutions through the process of electrolysis. It operates on the principle of depositing a known mass of silver when a specific amount of electric charge passes through it. This method is based on the fundamental law of electrolysis, which dictates that the mass of a substance altered at an electrode during electrolysis is directly proportional to the total electric charge passed through the substance.

• Silver coulometer is often used for accurate calibration in electrochemical measurements.
• Relies on the predictable deposition of silver, which involves a simple one-electron exchange process.
• Historically important for determining the strength of currents used in various electrical devices.

The silver deposited on the cathode provides a measurable way to determine the amount of electric charge that has passed through the cell. This measurement helps calculate the actual current flowing during the reaction if the time is also known.

Current Calculation

To calculate the current flowing through an electrochemical cell during electrolysis, you first need to determine the total charge passed through the cell. Faraday's laws provide a guideline, where the total charge (Q) is the product of the moles of electrons (n) and Faraday's constant (F). The formula is given by:
\[ Q = n \cdot F \]

• Faraday's constant (F) is approximately 96485 Coulombs per mole of electrons.
• The moles of electrons are calculated based on the deposited mass and molar mass of the involved substance.

With this charge value, the current (I) is determined using the relationship:
\[ I = \frac{Q}{t} \]
where \( t \) is the time in seconds during which the charge flowed. This method allows for monitoring and calculation of current in electrolysis setups over defined time periods.

Electrodeposition

Electrodeposition is the process by which a metal is deposited onto a conductive surface from a solution containing ions of that metal, using electrolysis. It is a critical procedure in various industries for coating, purifying, or building up thick layers of metals like silver, gold, copper, and nickel.

• Involves reduction at the cathode, where metal ions gain electrons and form solid metal.
• Useful in applications such as plating and manufacturing electronic components.
• The thickness of the deposited layer can be precisely controlled through current and time adjustments.

Understanding the principles of electrodeposition allows for fine-tuning processes to achieve desired metal layer thicknesses and qualities, critical in areas such as electronics and jewelry.

Gold Deposition

Gold deposition via electrolysis involves reducing gold ions, often in the form of \( \left[ \text{AuCl}_4 \right]^{-} \), to solid gold on the cathode. This process is important in various applications, including electronics, where gold's excellent conductive properties are valuable.

• Gold deposition involves a three-electron reduction process, which differs from the one-electron process in silver deposition.
• This process calculates how many moles of gold can be deposited based on the current passed through the solution.
• By knowing the molar mass and moles of deposited gold, the mass of gold can be accurately determined.

Gold's distinctive properties and its conductive efficiency make it a popular choice for electrodeposition, wherein maintaining precise control over deposition is crucial for achieving optimal product quality.