Question

Suppose the chemical reactions and corresponding redox potentials in a battery are given by [137]: \(\mathrm{Li} \rightarrow \mathrm{Li}^{+}+e^{-} \quad V_{r p}=3.04 \mathrm{~V}\) \(\mathrm{S}+2 e^{-} \rightarrow \mathrm{S}^{2-} \quad V_{r p}=-0.57 \mathrm{~V}\) (a) Find the overall theoretical specific capacity of the battery in \(\frac{\mathrm{C}}{\mathrm{g}}\). (b) Find the overall theoretical specific energy of the battery in \(\frac{\mathrm{J}}{\mathrm{g}}\). (c) Which material, lithium or sulfur, gets oxidized, and which material gets reduced?

Short answer

(a) 13900 C/g. (b) 50279 J/g. (c) Lithium is oxidized; sulfur is reduced.

Step-by-step solution

Determine the Oxidation and Reduction Reactions

To find which material is oxidized and which is reduced, identify the changes in oxidation states. The reaction \( \mathrm{Li} \rightarrow \mathrm{Li}^{+} + e^{-} \) represents lithium losing an electron, indicating oxidation. Conversely, \( \mathrm{S} + 2 e^{-} \rightarrow \mathrm{S}^{2-} \) represents sulfur gaining electrons, indicating reduction.

Calculate the Theoretical Specific Capacity

Specific capacity in \( \frac{\text{C}}{\text{g}} \) is calculated using the formula \( \text{Specific Capacity} = \frac{n \cdot F}{M} \), where \( n \) is the number of moles of electrons transferred, \( F \) is Faraday's constant (~96485 C/mol), and \( M \) is the molar mass. Since 1 mole of \( \mathrm{Li} \) transfers 1 electron and 1 mole of \( \mathrm{S} \) receives 2 electrons, the reaction transfers 2 electrons. Use lithium's molar mass, 6.94 g/mol, for calculation because it is often the limiting factor.

Perform Specific Capacity Calculation

Plug in the values: \( \text{Specific Capacity} = \frac{1 \cdot 96485}{6.94} = 13900 \frac{\text{C}}{\text{g}}. \) This calculation uses lithium because it provides the practical limit of the battery's specific capacity, being the lighter element involved in the reaction.

Calculate the Theoretical Specific Energy

Specific energy in \( \frac{\text{J}}{\text{g}} \) is calculated using \( \text{Specific Energy} = \text{Specific Capacity} \times \text{Cell Voltage} \). The cell voltage \( E \) is computed by summing the standard reduction potentials: \( V_r = (3.04 - (-0.57)) = 3.61 \text{ V}. \)

Perform Specific Energy Calculation

Once you have the specific capacity (13900 \( \frac{\text{C}}{\text{g}} \)) and the cell voltage (3.61 V), compute the specific energy: \( \text{Specific Energy} = 13900 \times 3.61 = 50279 \frac{\text{J}}{\text{g}}.\)

Conclusion and Answer Statements

Based on the calculations: (a) The theoretical specific capacity is 13900 \( \frac{\text{C}}{\text{g}} \), (b) the theoretical specific energy is approximately 50279 \( \frac{\text{J}}{\text{g}} \), and (c) lithium is oxidized while sulfur is reduced.

Key concepts

Redox Reactions

In electrochemistry, redox reactions stand at the center of how batteries function. The term "redox" is short for reduction-oxidation, describing the simultaneous chemical processes. A redox reaction involves the exchange of electrons between substances, where one substance loses electrons (oxidation) and another gains them (reduction). In our exercise, the reaction \( \mathrm{Li} \rightarrow \mathrm{Li}^{+} + e^{-} \) signifies lithium undergoing oxidation as it loses an electron. Simultaneously, sulfur undergoes reduction in the reaction \( \mathrm{S} + 2 e^{-} \rightarrow \mathrm{S}^{2-} \), where it gains two electrons. Understanding these processes is vital since they dictate the flow of electrons, ultimately powering batteries.

Battery Chemistry

Battery chemistry involves the intricate chemical reactions that occur within a battery to produce electrical energy. In the case of lithium-sulfur batteries, the cathode and anode reactions define the battery's operation. Lithium, with a higher oxidation potential of 3.04 V, serves as the anode, losing electrons as it oxidizes. This process is complemented by sulfur functioning as the cathode, with a reduction potential of -0.57 V, thereby gaining electrons. The total cell voltage, crucial for the battery's performance, is determined by the difference in these potential values, calculated to be 3.61 V. Understanding battery chemistry is crucial for developing efficient energy storage solutions and is foundational in evaluating battery capacities and energy densities.

Specific Capacity Calculation

Specific capacity is an essential parameter in battery technology, defining the charge stored per gram of active material, measured in \( \frac{\text{Coulombs}}{\text{gram}} \). This concept helps us understand how much charge a battery can hold relative to its mass. Calculating specific capacity involves the formula \( \text{Specific Capacity} = \frac{n \cdot F}{M} \), where \( n \) denotes the number of moles of electrons transferred, \( F \) is Faraday's constant (approximately 96485 C/mol), and \( M \) is the molar mass of the material. In our scenario, the specific capacity centers around lithium due to its limiting influence, as it participates by providing electrons and has a molar mass of 6.94 g/mol. By applying these values, lithium's specific capacity is computed as 13900 \( \frac{\text{C}}{\text{g}} \), guiding us toward understanding the battery's potential charge storage volume.

Specific Energy Calculation

Specific energy is another pivotal concept in evaluating a battery's potential, representing the energy supplied per gram, and is denoted in \( \frac{\text{Joules}}{\text{gram}} \). This metric indicates how much work a battery can perform in relation to its mass. Specific energy is calculated using \( \text{Specific Energy} = \text{Specific Capacity} \times \text{Cell Voltage} \). With a specific capacity already obtained (13900 \( \frac{\text{C}}{\text{g}} \)) and a cell voltage of 3.61 V (derived from the sum of reduction potentials), the specific energy results in 50279 \( \frac{\text{J}}{\text{g}} \). This value signifies the power efficiency of the battery, showcasing its ability to deliver energy efficiently relative to its weight. Efficient specific energy ensures that a battery not only stores charge effectively but also delivers ample energy when needed.