Final answer:
The N-O bond lengths from shortest to longest are predicted to be: NO+ (shortest), NO₃⁻, and NO₂⁻ (longest), based on their bond orders and resonance structures.
Step-by-step explanation:
The predict the ordering of N-O bond lengths from shortest to longest in NO+, NO₂⁻, and NO₃⁻, we must consider the bond orders and resonance structures. In NO+, there is a bond order of 2.5 which suggests a shorter bond length compared to a single bond. In NO₂⁻ (nitrite ion), the actual structure is an average of the two resonance structures, resulting in equivalent N-O bonds that have a bond length between that of a single and a double bond. And in NO₃⁻ (nitrate ion), due to the presence of three resonance structures, all N-O bonds are equivalent and the bond lengths are expected to be shorter than a single N-O bond but longer than those in NO+.
Therefore, the predicted order from shortest to longest N-O bond length is: NO+ < NO₃⁻ < NO₂⁻. This prediction is based upon the bond order and resonance stabilization which lends oneself to shorter bond lengths in ions with higher bond orders and more delocalization of electrons.