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Consider the following species:

$$ \underset{(\text{A})}{\ce{\overset{⊖}{O}H}} \quad \underset{(\text{B})}{\ce{CH3-\overset{⊖}{O}}} \quad \underset{(\text{C})}{\ce{\overset{⊖}{C}H3}} \quad \underset{(\text{D})}{\ce{\overset{⊖}{N}H2}} \quad $$

Arrange these species in their decreasing order of nucleophilicity.

(a) C > D > A > B
(b) B > A > C > D
(c) A > B > C > D
(d) C > A > B > D

My teacher taught in polar protic solvent nucleophilicity is inversely proportional to basic character. I assumed solvent to be $\ce{H2O}$ as solvent was not mentioned in the question. As acidity order is

$$\ce{H2O} > \ce{CH3OH} > \ce{NH3} > \ce{CH4},$$

so basicity order should be

$$\ce{CH3-} > \ce{NH2-} > \ce{CH3O-} > \ce{OH-},$$

therefore using above proportionality I arrived at the nucleophilicity order

$$\ce{OH-} > \ce{CH3O-} > \ce{NH2-} > \ce{CH3-},$$

but you can see the option is not even given. Why is it so?

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What you probably missed was that the order you mention is for the SAME GROUP elements. For example, down the group basicity of halide ions is-

F->Cl->Br->I-

But the the nucleophilicity order in polar protic solvents is-

I->Br->Cl->F-

This reverse in order is due to higher charge density of smaller halide ions and thus better interaction with polar protic solvents.

But along a period the charge density is similar and thus the interaction between solvent and nucleophile is almost similar. Thus, along a period the order follows basicity order.

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