Problem 80 - Chemistry The Central Science In Si Units

Question

Li-ion batteries used in automobiles typically use a \(\mathrm{LiMn}_{2} \mathrm{O}_{4}\) cathode in place of the \(\mathrm{LiCoO}_{2}\) cathode found in most Li-ion batteries. (a) Calculate the mass percent lithium in each electrode material. (b) Which material has a higher percentage of lithium? Does this help to explain why batteries made with \(\mathrm{LiMn}_{2} \mathrm{O}_{4}\) cathodes deliver less power on discharging? (c) In a battery that uses a \(\mathrm{LiCoO}_{2}\) cathode, approximately \(50 \%\) of the lithium migrates from the cathode to the anode on charging. In a battery that uses a \(\operatorname{LiMn}_{2} \mathrm{O}_{4}\) cathode, what fraction of the lithium in \(\operatorname{LiMn}_{2} \mathrm{O}_{4}\) would need to migrate out of the cathode to deliver the same amount of lithium to the graphite anode?

Step-by-Step Solution

Answer

(a) LiCoO2: 7.09%, LiMn2O4: 3.84%. (b) LiCoO2 has more Li. (c) 50% should migrate in LiMn2O4.

Step 1: Calculate Molar Masses

First, calculate the molar mass of each compound. The molar mass of \(\mathrm{LiCoO}_2\) is calculated as follows:- Lithium (Li): 6.94 g/mol- Cobalt (Co): 58.93 g/mol- Oxygen (O): 16 g/mol (each)Thus, the molar mass of \(\mathrm{LiCoO}_2\) is \(6.94 + 58.93 + 2 \times 16 = 97.87\) g/mol.Now, calculate the molar mass of \(\mathrm{LiMn}_2\mathrm{O}_4\):- Lithium (Li): 6.94 g/mol- Manganese (Mn): 54.94 g/mol (each)- Oxygen (O): 16 g/mol (each)Thus, the molar mass of \(\mathrm{LiMn}_2\mathrm{O}_4\) is \(6.94 + 2 \times 54.94 + 4 \times 16 = 180.81\) g/mol.

Step 2: Calculate Mass Percent of Lithium in LiCoO2

The mass percent of lithium in \(\mathrm{LiCoO}_2\) is calculated using the formula:\[\text{Mass percent of Li} = \left(\frac{\text{molar mass of Li}}{\text{molar mass of } \mathrm{LiCoO}_2}\right) \times 100\%\]Substitute the values:\[\text{Mass percent of Li} = \left(\frac{6.94}{97.87}\right) \times 100\% = 7.09\%\]

Step 3: Calculate Mass Percent of Lithium in LiMn2O4

Similarly, calculate the mass percent of lithium in \(\mathrm{LiMn}_2\mathrm{O}_4\):\[\text{Mass percent of Li} = \left(\frac{\text{molar mass of Li}}{\text{molar mass of } \mathrm{LiMn}_2\mathrm{O}_4}\right) \times 100\%\]Substitute the values:\[\text{Mass percent of Li} = \left(\frac{6.94}{180.81}\right) \times 100\% = 3.84\%\]

Step 4: Compare Mass Percentages

Compare the mass percentages calculated. \(\mathrm{LiCoO}_2\) has a mass percent of 7.09% lithium, whereas \(\mathrm{LiMn}_2\mathrm{O}_4\) has 3.84% lithium. Thus, \(\mathrm{LiCoO}_2\) has a higher percentage of lithium.

Step 5: Discharge Power Explanation

The higher lithium content in \(\mathrm{LiCoO}_2\) suggests it can contribute more lithium ions compared to \(\mathrm{LiMn}_2\mathrm{O}_4\), possibly delivering more power on discharging. This could explain why \(\mathrm{LiMn}_2\mathrm{O}_4\) batteries deliver less power.

Step 6: Calculate Fraction of Lithium Migration

We need to calculate what fraction of the lithium in \(\mathrm{LiMn}_2\mathrm{O}_4\) must migrate to deliver the same amount to the anode as \(\mathrm{LiCoO}_2\), where 50% migrates.Assume each material starts with 1 mole:For \(\mathrm{LiCoO}_2\): 50% of 1 mole of Li migrates, which is 0.5 moles.For \(\mathrm{LiMn}_2\mathrm{O}_4\): To deliver 0.5 moles, solve \(x \times 1 \text{ mole}\) = 0.5 moles, where \(x\) is the fraction migrating.Thus, \(x = 0.5\), meaning 50% of the lithium should migrate.

Key Concepts

Cathode Materials in Batteries

Lithium-ion batteries, commonly used in various devices and electric vehicles, have a crucial component called the cathode. The performance and efficiency of a battery largely depend on the materials used in the cathode. Two popular materials in Li-ion batteries are

• Lithium Cobalt Oxide (\(LiCoO_2\))

and

• Lithium Manganese Oxide (\(LiMn_2O_4\)).

\(LiCoO_2\) is widely used due to its high energy density, which translates to longer battery life in applications such as smartphones. In contrast, \(LiMn_2O_4\) offers excellent thermal stability and safety features, making it a suitable choice for electric vehicles. However, it also exhibits a lower specific energy compared to \(LiCoO_2\), which can influence the battery's power delivery.
The choice of cathode material is a trade-off between energy density and safety, determining the battery's efficiency and application suitability.

Lithium Migration

Lithium migration is a critical process in rechargeable lithium-ion batteries. During the charging process, lithium ions move from the cathode to the anode. For a battery with a \(LiCoO_2\) cathode, around 50% of the lithium migrates towards the anode. This migration is essential for storing energy that can power a device.

Lithium Migration in \(LiMn_2O_4\) Batteries

In batteries with a \(LiMn_2O_4\) cathode, the same principle applies. To release the same amount of lithium ions to the anode as \(LiCoO_2\), 50% of the lithium in \(LiMn_2O_4\) must migrate. This migration influences the battery's capacity and discharge power. A better understanding of lithium migration can help improve battery design and boost the efficiency of energy storage systems.

Molar Mass Calculation

Molar mass calculation is an essential step in understanding battery chemistry. It allows us to calculate the mass percent of elements in a compound. For

• \(LiCoO_2\)

, the molar mass is calculated by summing the atomic masses of Li (6.94 g/mol), Co (58.93 g/mol), and twice O (2 x 16 g/mol), resulting in 97.87 g/mol.
Similarly, for

• \(LiMn_2O_4\)

, it is: Li (6.94 g/mol), twice Mn (2 x 54.94 g/mol), and four times O (4 x 16 g/mol), totaling 180.81 g/mol.
From this information, we can determine that

• The mass percent of lithium in \(LiCoO_2\)

is \(7.09\)%, and

• in \(LiMn_2O_4\)

it is \(3.84\)%. Molar mass calculations are vital for evaluating and comparing cathode materials and understanding their effect on the battery's performance.

Battery Discharge Power

The discharge power of a battery is an integral factor when evaluating its performance. Discharge power refers to the energy that a battery releases over time. A higher mass percent of lithium in the cathode material, like in \(LiCoO_2\), often results in a greater discharge power.

The Role of Lithium Concentration

This is because higher lithium content contributes more ions during discharge, leading to enhanced power delivery. When considering \(LiMn_2O_4\), its lower lithium percentage compared to \(LiCoO_2\) means it typically delivers less power. However, technologies utilizing \(LiMn_2O_4\) benefit from improved safety and thermal stability, which are critical for specific applications, such as electric vehicles.
In optimizing battery design, balancing discharge power with safety factors significantly impacts the choice of cathode materials.