Problem 79 - Chemistry The Central Science In Si Units

Question

In a Li-ion battery the composition of the cathode is \(\mathrm{LiCoO}_{2}\) when completely discharged. On charging, approximately \(50 \%\) of the \(\mathrm{Li}^{+}\) ions can be extracted from the cathode and transported to the graphite anode where they intercalate between the layers. (a) What is the composition of the cathode when the battery is fully charged? (b) If the \(\mathrm{LiCoO}_{2}\) cathode has a mass of \(10 \mathrm{~g}\) (when fully dis-charged), how many coulombs of electricity can be delivered on completely discharging a fully charged battery?

Step-by-Step Solution

Answer

(a) \(\mathrm{Li}_{0.5}\mathrm{CoO}_{2}\). (b) 4931 Coulombs.

Step 1: Determine Cathode Composition

When the battery is fully charged, half of the \(\mathrm{Li}^{+}\) ions are extracted from the \(\mathrm{LiCoO}_2\) cathode. The initial composition is \(\mathrm{LiCoO}_2\). Remove half of the lithium ions to get the charged composition \(\mathrm{Li}_{0.5}\mathrm{CoO}_{2}\).

Step 2: Calculate Molar Mass

Calculate the molar masses of \(\mathrm{LiCoO}_2\) as follows: \(\mathrm{Li} = 6.94\, \text{g/mol}\), \(\mathrm{Co} = 58.93\, \text{g/mol}\), \(\mathrm{O} = 16.00\, \text{g/mol}\). The total molar mass \(\mathrm{LiCoO}_2\) is \(6.94 + 58.93 + 2 \times 16.00 = 97.87\, \text{g/mol}\).

Step 3: Calculate Moles on Full Charge

Given the mass of \(\mathrm{LiCoO}_2\) is \(10\, \text{g}\), use the molar mass to find the moles: \[ \text{Moles of } \mathrm{LiCoO}_2 = \frac{10}{97.87} = 0.1022\, \text{moles} \]

Step 4: Determine Moles of \(\mathrm{Li}^{+}\) Ions Transferred

Since \(50\%\) of \(\mathrm{Li}^{+}\) ions are transferred and there is 1 mole of \(\mathrm{Li}\) per mole of \(\mathrm{LiCoO}_2\), the moles of \(\mathrm{Li}^{+}\) ions transferred are \(0.1022 \times 0.5 = 0.0511\, \text{moles}\).

Step 5: Calculate Charge in Coulombs

Each mole of electrons carries a charge of \(96485\, \text{C/mol}\). Therefore, the charge in coulombs for \(0.0511\, \text{moles}\) is: \(0.0511 \times 96485 = 4931\, \text{C}\).

Key Concepts

Cathode Composition

In a lithium-ion battery, the cathode is a critical component responsible for storing and releasing lithium ions during charge and discharge cycles. The typical cathode material is composed of lithium cobalt oxide, denoted as \( \mathrm{LiCoO}_2 \) when fully discharged. This compound provides the structural framework to accommodate lithium ions.
When the battery is fully charged, approximately 50% of the \( \mathrm{Li}^+ \) ions are extracted from the cathode. This extraction changes the composition from \( \mathrm{LiCoO}_2 \) to \( \mathrm{Li}_{0.5}\mathrm{CoO}_2 \). The removal of lithium ions alters the electrical balance but retains the structural integrity of the material, allowing it to function effectively during the next discharge cycle.

• The charged composition aids in maintaining battery efficiency over cycles.
• This concept is central to the operation of rechargeable lithium-ion batteries.

Coulombs of Electricity

The concept of coulombs of electricity relates to how electrical charge is moved within a battery. One coulomb is equivalent to the charge of approximately 6.242 x 10¹⁸ electrons, which is a unit of electric charge in the International System of Units (SI).
In the context of a lithium-ion battery, the charge transferred when electrons move from the anode to the cathode (or vice versa) can be quantified in coulombs. This is crucial for calculating how much energy a battery can deliver.

• Coulombs are fundamental in battery discharge calculations.
• Understanding the amount of electron flow helps in designing batteries with better capacity and efficiency.

Molar Mass Calculation

Molar mass calculation is essential to understanding the chemical composition and energy capacity of battery materials. In the instance of \( \mathrm{LiCoO}_2 \), determining its molar mass involves summing the individual atomic masses of lithium, cobalt, and oxygen.
The molar masses for the elements are:

• \( \text{Li} = 6.94 \text{ g/mol} \)
• \( \text{Co} = 58.93 \text{ g/mol} \)
• \( \text{O} = 16.00 \text{ g/mol} \)

Therefore, the molar mass of \( \mathrm{LiCoO}_2 \) is calculated as \( 6.94 + 58.93 + 2 \times 16.00 = 97.87 \text{ g/mol} \).
By computing the molar mass, you can determine how many moles of the compound are present in a given mass, which aids in further calculations, such as energy discharge or amounts of reactants and products in chemical reactions.

Lithium Ion Transfer

Lithium ion transfer is the movement of lithium ions between the cathode and anode within a battery during charging and discharging cycles. This movement is fundamental to the battery's ability to efficiently store and release energy.
During the charging process, lithium ions move from the cathode to the anode, where they are intercalated into the graphite layers. This process reverses during discharge, supplying power to electronic devices.

• Approximately 50% of the \( \mathrm{Li}^+ \) ions are transferred during a complete charge-discharge cycle.
• The efficiency of lithium ion transfer directly influences the battery's cycle life and overall performance.

Managing the transfer process efficiently ensures that lithium-ion batteries can provide the high energy density and long life needed in many modern portable electronics.