Question

The silver oxide-zinc battery used in watches delivers a voltage of \(1.60 \mathrm{~V}\). Calculate the free-energy change (in kilojoules) for the cell reaction $$ \mathrm{Zn}(s)+\mathrm{Ag}_{2} \mathrm{O}(s) \longrightarrow \mathrm{ZnO}(s)+2 \mathrm{Ag}(s) $$

Short answer

The free-energy change is \(-308.752\text{ kJ}\).

Step-by-step solution

Identify Formula for Free Energy Change

The free-energy change (\( \Delta G \)) for a reaction can be calculated using the formula \( \Delta G = -nFE \), where \( n \) is the number of moles of electrons transferred, \( F \) is the Faraday constant (approximately 96485 C/mol), and \( E \) is the cell potential (here it is 1.60 V).

Determine Number of Electrons Transferred

For the given reaction, \( \mathrm{Zn}(s) + \mathrm{Ag}_{2} \mathrm{O}(s) \rightarrow \mathrm{ZnO}(s) + 2 \mathrm{Ag}(s) \), one mole of \( \mathrm{Ag}_{2} \mathrm{O} \) produces 2 moles of \( \mathrm{Ag} \). Therefore, 2 moles of electrons are transferred (one for each silver atom changed from \( \mathrm{Ag}^{+} \) to \( \mathrm{Ag} \)). Thus, \( n = 2 \).

Calculate Free Energy Change

Plug the values into the formula: \( \Delta G = -nFE = -(2 \text{ mol})(96485 \text{ C/mol})(1.60 \text{ V}) \). This calculation gives \( \Delta G = -308,752 \text{ J} \), which is \( -308.752 \text{ kJ} \) when converted to kilojoules.

Key concepts

Silver Oxide-Zinc Battery

The silver oxide-zinc battery is a type of electrochemical cell commonly used in small devices, like watches and calculators. It is valued for its ability to provide a stable voltage and long shelf life.

This battery operates based on the chemical reaction between zinc and silver oxide. Here is the reaction equation:

• Zn(s) + Ag₂O(s) → ZnO(s) + 2Ag(s)

This reaction involves the transfer of electrons, powering the device it is contained within. Silver oxide-zinc batteries are preferred over some other types for their consistent output and compact size, making them ideal for precision electronics.

Cell Potential

Cell potential, denoted as \( E \), is a critical measure in electrochemistry. It represents the voltage or electrical potential difference of a battery when no current is flowing. The higher the cell potential, the greater the driving force of the electron flow.

For the silver oxide-zinc battery, the cell potential is calculated to be 1.60 volts. This value determines how much work can be done by the electrons released during the reaction between zinc and silver oxide. The cell potential is directly used in formulas to calculate other important terms, such as the free-energy change of the reaction.

Faraday Constant

The Faraday constant, \( F \), is a key concept in electrochemistry and represents the electric charge carried by one mole of electrons. It has a standard value of approximately 96485 coulombs per mole (C/mol).

The constant is used in calculations involving electrochemical reactions, like the one in a silver oxide-zinc battery, to determine the amount of charge involved in the reaction. By knowing the Faraday constant, you can convert between the amount of substance involved in the reaction and the charge moved, essential for precise calculations of energy changes.

Moles of Electrons

The term "moles of electrons" refers to the number of moles of electrons that participate in a particular chemical or electrochemical reaction.

• In the given reaction: Zn(s) + Ag₂O(s) → ZnO(s) + 2Ag(s),

two moles of electrons are transferred. Each silver ion ( Ag^+ ) is reduced to a silver atom ( Ag ), contributing one electron per ion, summing up to 2 electrons for two Silver ions.

Understanding the number of moles of electrons transferred is critical for applying calculations using the Faraday constant. This aspect helps in the determination of free-energy changes and efficiency of electrochemical cells.