Problem 94 - Chemistry and Chemical Reactivity

Question

An old method of measuring the current flowing in a circuit was to use a "silver coulometer." The current passed first through a solution of \(\mathrm{Ag}^{+}(\mathrm{aq})\) and then into another solution containing an electroactive species. The amount of silver metal deposited at the cathode was weighed. From the mass of silver, the number of atoms of silver was calculated. since the reduction of a silver ion requires one electron, this value equaled the number of electrons passing through the circuit. If the time was noted, the average current could be calculated. If, in such an experiment, \(0.052 \mathrm{g}\) of \(\mathrm{Ag}\) is deposited during \(450 \mathrm{s},\) what was the current flowing in the circuit?

Step-by-Step Solution

Answer

The current was approximately 0.104 A.

Step 1: Determine the Moles of Silver Deposited

To find the number of moles of silver (\(\mathrm{Ag}\)) deposited, use the molar mass of silver, which is approximately \(107.87 \, \mathrm{g/mol}\). Calculate the moles of silver by dividing the mass of silver deposited by its molar mass:\[\text{Moles of } \mathrm{Ag} = \frac{0.052 \, \mathrm{g}}{107.87 \, \mathrm{g/mol}} \approx 0.000482 \, \text{mol}\]

Step 2: Convert Moles to Number of Electrons

Since the reduction of one silver ion requires one electron, the number of moles of electrons is equal to the moles of silver deposited. Thus, we have \(0.000482 \, \text{mol}\) of electrons.

Step 3: Calculate Charge Passed through the Circuit

Use Faraday's constant to convert moles of electrons to charge. Faraday's constant is \(96485 \, \mathrm{C/mol}\). The total charge \(Q\) is calculated as:\[Q = 0.000482 \, \text{mol} \times 96485 \, \mathrm{C/mol} = 46.51402 \, \mathrm{C}\]

Step 4: Calculate the Current

Current \(I\) is the charge passed per unit time. Given that the time \(t\) is \(450\, \text{s}\), the current can be calculated using:\[I = \frac{Q}{t} = \frac{46.51402 \, \mathrm{C}}{450 \, \text{s}} \approx 0.1036 \, \mathrm{A}\]

Key Concepts

Understanding Current Calculation

Current is defined as the rate of flow of electric charge through a conductor. In electrochemistry, calculating current involves determining how many electrons flow through a system over a certain period. In our exercise, silver ions (\(\text{Ag}^{+}\)) become reduced to silver metal (\(\text{Ag}\)) by gaining electrons. The number of electrons is directly related to the amount of silver deposited. To find the current, you first need to determine how many moles of silver are deposited, which can be done using the given mass and the molar mass of silver. Next, use Faraday's constant to find the total charge. Finally, calculate the current by dividing the total charge by time. Current is measured in amperes and represents how fast electrons are flowing.

Faraday's Constant in Electrochemistry

Faraday's constant is essential in electrochemistry for converting moles of electrons into coulombs. The constant, approximately \(96485 \, \text{C/mol}\), represents the electric charge per mole of electrons. It is derived from Avogadro's number and the elementary charge of an electron.

When you need to find out how much charge flows through a system, you multiply the moles of electrons by Faraday's constant. This is crucial in determining how much electric charge has been transferred in reactions involving the movement of electrons. For example, in the exercise, this constant was used to calculate the total charge from the moles of silver ions reduced to silver metal.

What is a Coulometer?

A coulometer is a device used to measure the quantity of electricity (electric charge) passed in an electrochemical cell. It is particularly useful for determining the amount of a substance deposited at an electrode during electrolysis. Coulometry is based on the principle that the amount of chemical change in a system is proportional to the total charge passed.

• The silver coulometer, as mentioned in the exercise, uses silver ions because they deposit readily at the cathode, thus allowing a measurable mass to be quickly obtained.
• This deposited mass can then be used to calculate the number of moles of electrons, giving insight into the current flowing through the circuit at any given time.

A coulometer can either be potentiometric, measuring voltage to deduce charge, or gravimetric, weighing a deposited material, such as silver, to compute the transferred charge.

Concept of Silver Deposition

Silver deposition is a process where silver ions in solution are reduced to form solid silver on an electrode. This occurs when electrons are supplied to the electrode, causing the silver ions (\(\text{Ag}^{+}\)) to gain electrons and transform into metallic silver (\(\text{Ag}\)).
A key aspect of silver deposition is its direct relationship with the amount of electric current. The more electrons that flow into the system, the more silver ions can be reduced to form solid silver, which can then be weighed to measure the reaction's progress. This relationship is why silver deposition is useful in coulometry to measure current.

• The weight of deposited silver helps in understanding how many moles of silver, and thus electrons, have been transferred.
• The stoichiometry of the reaction is simple, as one mole of silver ion requires exactly one mole of electrons for deposition.

This makes calculations straightforward when estimating the electric current that has passed through the electrode during electrolysis.