Question

Li-ion batteries used in automobiles typically use a \(\operatorname{LiMn}_{2} \mathrm{O}_{4}\) 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 \(\operatorname{LiMn}_{2} \mathrm{O}_{4}\) 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 \(\mathrm{LiMn}_{2} \mathrm{O}_{4}\) would need to migrate out of the cathode to deliver the same amount of lithium to the graphite anode?

Short answer

The mass percentage of lithium in LiCoO₂ is 7.09%, while in LiMn₂O₄ it is 3.84%. LiCoO₂ has a higher percentage of lithium, which helps explain why batteries made with LiMn₂O₄ cathodes deliver less power on discharging. To deliver the same amount of lithium to the graphite anode, approximately 92.4% of the lithium in the LiMn₂O₄ cathode needs to migrate out of the cathode.

Step-by-step solution

Calculate the molar mass of LiCoO₂ and LiMn₂O₄

In order to find the mass percentage of lithium in each electrode material, we should first calculate the molar mass of each compound. For LiCoO₂, its molar mass is calculated as follows: Molar mass = 1 Li + 1 Co + 2 O = (1 × 6.939) + (1 × 58.933) + (2 × 16.00) = 97.872 g/mol For LiMn₂O₄, its molar mass is calculated as follows: Molar mass = 1 Li + 2 Mn + 4 O = (1 × 6.939) + (2 × 54.938) + (4 × 16.00) = 180.815 g/mol

Calculate the mass percent of lithium in each electrode material

Now, we can find the mass percent of lithium in each material using the formula: Mass percent = (Mass of Li in the formula / Molar mass of the compound) × 100 For LiCoO₂: Mass percent = (6.939 / 97.872) × 100 = 0.0709 × 100 = 7.09% For LiMn₂O₄: Mass percent = (6.939 / 180.815) × 100 = 0.0384 × 100 = 3.84%

Compare the materials and discuss power delivery

From Step 2, we can see that LiCoO₂ has a higher mass percentage of lithium (7.09%) compared to LiMn₂O₄ (3.84%). This indicates that batteries made with LiMn₂O₄ deliver less power on discharging compared to batteries made with LiCoO₂ since they have a lower lithium content.

Calculate the fraction of lithium migration for LiMn₂O₄

It is given that in a LiCoO₂ battery, approximately 50% of the lithium migrates from the cathode to the anode on charging. We need to find what fraction of lithium in LiMn₂O₄ needs to migrate out of the cathode to deliver the same amount of lithium to the graphite anode. Let x be the fraction of lithium that needs to migrate from LiMn₂O₄. Since both batteries have to deliver the same amount of lithium, we can use the following equation: Li migration (LiCoO₂) = Li migration (LiMn₂O₄) 0.5 (Mass percent of Li in LiCoO₂) = x (Mass percent of Li in LiMn₂O₄) x = (0.5 × 7.09%) / 3.84% = 0.924 So, around 92.4% of the lithium in the LiMn₂O₄ cathode needs to migrate out of the cathode to deliver the same amount of lithium to the graphite anode.

Key concepts

Lithium cathode materials

Lithium-ion (Li-ion) batteries are a major player in the world of rechargeable batteries, especially for portable electronic devices and electric vehicles. One crucial component of these batteries is the cathode material, which significantly affects the battery's performance. Two common cathode materials are

• **LiCoO₂** (Lithium Cobalt Oxide)
• **LiMn₂O₄** (Lithium Manganese Oxide)

These materials are used because they allow lithium ions to flow between the anode and cathode, which is essential for the charging and discharging processes.
The choice of a specific cathode material is critical, as it influences the battery's capacity (the amount of energy it can store) and lifespan. LiCoO₂ is known for its higher energy density, which means it can store more energy than LiMn₂O₄. However, LiMn₂O₄ is often chosen for applications requiring high power output and reduced costs. Although each has distinct advantages, the capacity to deliver power effectively and its overall safety are what often determine which material is utilized.
Understanding these materials helps explain some of the intricacies involved in battery design and how they affect performance, crucial for students undertaking studies in materials science and battery technology.

Mass percent calculations

Mass percent calculations are vital in the analysis and design of compounds like those used in Li-ion battery cathodes. Mass percent is a way of expressing the concentration of an element in a compound, essential for understanding the contribution of each component to the overall properties of the material.
To calculate the mass percent of an element in a compound, you use the formula: \[ \text{Mass percent} = \frac{\text{Mass of the element in the formula}}{\text{Molar mass of the compound}} \times 100 \]For example, consider the calculation for Lithium in LiCoO₂:

• The molar mass of Li is approximately 6.939 g/mol.
• The total molar mass for LiCoO₂ was found to be approximately 97.872 g/mol.
• The mass percent of lithium in LiCoO₂ is thus \[ \left( \frac{6.939}{97.872} \right) \times 100 = 7.09\% \]

Similarly, for LiMn₂O₄, the molar mass is 180.815 g/mol, resulting in a lithium mass percent of 3.84%.
Grasping these calculations helps measure how much of a specific element, like lithium, is present in different cathodes and gives insight into their potential performance and applications in batteries.

Electrode composition

Understanding electrode composition is essential in appreciating the efficiency and capability of Li-ion batteries. The composition of the electrode affects both the battery's capacity and its ability to release energy efficiently.
Electrodes are composed of active materials, current collectors, binders, and other additives. In the context of our Li-ion battery discussion, the "active material" is where electrochemical reactions occur to store and release energy.

• For LiCoO₂, an electrode with a high lithium content (7.09%) results in high energy density.
• LiMn₂O₄, on the other hand, has a lithium content of 3.84%, which reflects lower energy storage capacity but offers stability and is often cheaper.

The choice of materials must balance energy needs, cost, safety, and cycle life (how many times a battery can be charged and discharged). While LiCoO₂ may offer more energy, LiMn₂O₄'s composition allows it to deliver power safely and is generally more environmentally friendly.
These factors must be considered in designing battery systems, making understanding electrode composition foundational to optimizing their use in various applications.

Battery discharge

Battery discharge refers to the process of delivering stored energy within a battery, which powers electronic devices or vehicles. The discharge rate and efficiency depend significantly on the battery's internal materials and design.
Li-ion batteries, like those with LiCoO₂ or LiMn₂O₄ cathodes, discharge by transferring lithium ions back to their original electrode, generating electricity. The efficiency of this process can be measured by the fraction of lithium that migrates during charging or discharging.

• LiCoO₂ experiences about 50% lithium migration during charging, allowing for significant energy release on discharge.
• With a lower lithium content, LiMn₂O₄ requires a higher fraction, around 92.4%, of lithium migration to deliver equivalent energy levels.

A high migration rate in LiMn₂O₄ batteries can strain the material over time, affecting the battery's lifespan and efficiency. Discharge characteristics are part of why LiMn₂O₄ batteries may deliver less power compared to their LiCoO₂ counterparts.
Thus, analyzing discharge processes helps identify what fraction of lithium must migrate in different materials to meet energy demands, influencing how batteries are used in practical applications.