(a) Which ion is smaller, \(\mathrm{Co}^{3+}\) or \(\mathrm{Co}^{4+} ?(\mathbf{b})\) In a lithium-ion battery that is discharging to power a device, for every \(\mathrm{Li}^{+}\) that inserts into the lithium cobalt oxide electrode, a \(\mathrm{Co}^{4+}\) ion must be reduced to a \(\mathrm{Co}^{3+}\) ion to balance charge. Using the CRC Handbook of Chemistry and Physics or other standard reference, find the ionic radii of \(\mathrm{Li}^{+}, \mathrm{Co}^{3+},\) and \(\mathrm{Co}^{4+} .\) Order these ions from smallest to largest. (c) Will the lithium cobalt oxide cathode expand or contract as lithium ions are inserted? (d) Lithium is not nearly as abundant as sodium. If sodium ion batteries were developed that function in the same manner as lithium ion batteries, do you think "sodium cobalt oxide" would still work as the electrode material? Explain. (e) If you don’t think cobalt would work as the redox-active partner ion in the sodium version of the electrode, suggest an alternative metal ion and explain your reasoning.

Step by step solution

01

(a) Comparing Co³⁺ and Co⁴⁺ ion sizes

When comparing ions from the same element, each additional positive charge will attract the electrons stronger towards the nucleus, resulting in a smaller ionic radius. Therefore, the ion with a higher charge will be smaller: Co⁴⁺ is smaller than Co³⁺.

02

(b) Finding ionic radii and ordering ions

Using the CRC Handbook of Chemistry and Physics or another standard reference, we find the following ionic radii: - Li⁺: 76 pm - Co³⁺: 54.5 pm - Co⁴⁺: 51 pm Now we can order these ions from smallest to largest: Co⁴⁺ < Co³⁺ < Li⁺.

03

(c) Expansion or contraction of lithium cobalt oxide cathode

When lithium ions are inserted into the lithium cobalt oxide electrode, they are larger than the cobalt ions they replace (Li⁺ > Co³⁺). As a result, the cathode will expand to accommodate the larger lithium ions.

04

(d) Feasibility of sodium cobalt oxide as an electrode material

Sodium ions are larger than lithium ions. While it is possible that a sodium-ion battery may work similarly to a lithium-ion battery, the larger size of sodium ions may lead to a significant expansion in the sodium cobalt oxide electrode during charge/discharge cycles. This could result in decreased stability and performance over time. Therefore, sodium cobalt oxide might not be the ideal choice for an electrode material in sodium-ion batteries.

05

(e) Suggesting an alternative metal ion for the sodium version of the electrode

Considering that using cobalt as the redox-active partner ion with sodium might lead to stability issues, we could search for a metal ion with similar redox properties but a larger ionic radius, more compatible with the size of sodium ions. One possible candidate could be the manganese ion (Mn³⁺/Mn⁴⁺). This ion has a larger ionic radius than cobalt ions, and its stable oxidation states are compatible with the redox process needed for a sodium-ion battery's functionality. Additionally, manganese is more abundant and less toxic than cobalt, which could be advantageous for large-scale battery production.

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