The increasing order of `pK_b` value of the following is :
(I) `HC -= C^(Ө)`
`H^(Ө)`
`overset (Ө) NH_2`
(IV) `C overset (Ө) H_3`.

Options

A. `IV lt III lt II lt I`
B. `I lt II lt III lt IV`
C. `IV lt II lt III lt I`
D. `I lt III lt II lt IV`

Explanation

To determine the increasing order of \( pK_b \) values for the given compounds, we need to analyze their basicity. The lower the \( pK_b \) value, the stronger the base. Thus, we will first assess the basicity of each compound.

### Step 1: Identify the compounds and their basicity
The compounds given are:
1. \( HC \equiv C^- \) (acetylide ion)
2. \( H^- \) (hydride ion)
3. \( NH_2^- \) (amide ion)
4. \( CH_3^- \) (methyl ion)

### Step 2: Determine the strength of each base
- **Acetylide ion (\( HC \equiv C^- \))**: This ion is derived from a terminal alkyne and has a significant amount of \( s \)-character (50% \( s \)-character). This means it is a relatively weak base.

- **Hydride ion (\( H^- \))**: This is a very strong base as it is derived from hydrogen, which has no \( s \)-character.

- **Amide ion (\( NH_2^- \))**: This ion is a stronger base than the acetylide ion but weaker than the hydride ion. It has about 33% \( s \)-character.

- **Methyl ion (\( CH_3^- \))**: This ion is a strong base as well, stronger than the amide ion but weaker than the hydride ion. It has about 25% \( s \)-character.

### Step 3: Rank the bases from strongest to weakest
Based on the analysis:
1. \( H^- \) (strongest base)
2. \( CH_3^- \) (second strongest)
3. \( NH_2^- \) (third strongest)
4. \( HC \equiv C^- \) (weakest base)

### Step 4: Determine the \( pK_b \) values
Since the lower the \( pK_b \), the stronger the base, we can now write the order of \( pK_b \) values:
- \( pK_b(H^-) < pK_b(CH_3^-) < pK_b(NH_2^-) < pK_b(HC \equiv C^-) \)

### Step 5: Write the increasing order of \( pK_b \) values
Thus, the increasing order of \( pK_b \) values is:
\[ pK_b(HC \equiv C^-) < pK_b(NH_2^-) < pK_b(CH_3^-) < pK_b(H^-) \]

### Final Answer
The increasing order of \( pK_b \) values is:
\[ HC \equiv C^- < NH_2^- < CH_3^- < H^- \]