Values of dissociation constant, $$K_a$$ are given as follows :
Acid             $$K_a$$
HCN             $$6.2 \times 10^{-10}$$
HF               $$7.2 \times 10^{-4}$$
HNO$$_2$$          $$4.0 \times 10^{-4}$$
Correct order of increasing base strength of $$CN^-$$, $$F^-$$ and $$NO_2^-$$ will be :

We are given the dissociation constants $$K_a$$ for three acids: HCN, HF, and HNO$$_2$$. The values are:

• HCN: $$K_a = 6.2 \times 10^{-10}$$
• HF: $$K_a = 7.2 \times 10^{-4}$$
• HNO$$_2$$: $$K_a = 4.0 \times 10^{-4}$$

We need to find the order of increasing base strength for their conjugate bases: CN$$^-$$, F$$^-$$, and NO$$_2^-$$.

The strength of a conjugate base is inversely related to the strength of its conjugate acid. A stronger acid has a larger $$K_a$$ value and a weaker conjugate base, while a weaker acid has a smaller $$K_a$$ value and a stronger conjugate base. Therefore, to compare the base strengths, we compare the $$K_a$$ values of their conjugate acids.

Let's list the $$K_a$$ values:

• HCN: $$K_a = 6.2 \times 10^{-10}$$
• HF: $$K_a = 7.2 \times 10^{-4}$$
• HNO$$_2$$: $$K_a = 4.0 \times 10^{-4}$$

Comparing these values:

• The $$K_a$$ for HCN is $$6.2 \times 10^{-10}$$, which is the smallest because the exponent $$-10$$ is more negative than $$-4$$. This means HCN is the weakest acid.
• Between HF and HNO$$_2$$, both have exponents of $$-4$$, so we compare the coefficients: HF has $$7.2$$ and HNO$$_2$$ has $$4.0$$. Since $$4.0 < 7.2$$, $$4.0 \times 10^{-4} < 7.2 \times 10^{-4}$$, so HNO$$_2$$ is weaker than HF.

Ordering the acids from weakest to strongest:

1. HCN ($$K_a = 6.2 \times 10^{-10}$$)
2. HNO$$_2$$ ($$K_a = 4.0 \times 10^{-4}$$)
3. HF ($$K_a = 7.2 \times 10^{-4}$$)

Since conjugate base strength is inversely proportional to acid strength:

• The weakest acid HCN has the strongest conjugate base CN$$^-$$.
• The next weakest acid HNO$$_2$$ has the next strongest conjugate base NO$$_2^-$$.
• The strongest acid HF has the weakest conjugate base F$$^-$$.

Therefore, the order of increasing base strength (from weakest to strongest) is F$$^-$$ < NO$$_2^-$$ < CN$$^-$$.

Now, comparing with the options:

• Option A: F$$^-$$ < CN$$^-$$ < NO$$_2^-$$ → Incorrect
• Option B: NO$$_2^-$$ < CN$$^-$$ < F$$^-$$ → Incorrect
• Option C: F$$^-$$ < NO$$_2^-$$ < CN$$^-$$ → Correct
• Option D: NO$$_2^-$$ < F$$^-$$ < CN$$^-$$ → Incorrect

Hence, the correct answer is Option C.