The correct order of increasing solubility in water is:

Options

A. `KF lt NaF lt LiF`
B. `NaHCO_(3) lt KHCO_(3) lt RbHCO_(3)`
C. `K_(2)CO_(3) lt Na_(2)CO_(3) lt Li_(2)CO_(3)`
D. `LiNO_(3) lt NaNO_(3) lt KNO_(3)`

Explanation

To determine the correct order of increasing solubility in water among different salts of alkali metals, we need to analyze the solubility trends based on the electropositive character of the alkali metals and the nature of the anions involved.

### Step-by-Step Solution:

1. **Understand the Electropositive Character**:
- The electropositive character of alkali metals increases down the group from lithium (Li) to cesium (Cs). This means that cesium is the most electropositive, while lithium is the least.
- The order of increasing electropositive character is:
\[ \text{Li} < \text{Na} < \text{K} < \text{Rb} < \text{Cs} \]

2. **Identify the Anions**:
- The solubility of salts in water can depend on the size and charge of the anions. For smaller anions (like halides, carbonates, sulfates, and bicarbonates), solubility generally increases with increasing electropositive character of the cation.
- For larger anions (like nitrates), the trend is reversed; solubility decreases with increasing electropositive character.

3. **Evaluate Each Option**:
- **Option 1**: \( \text{K}_f < \text{NaF} < \text{LiF} \)
- This is incorrect because LiF has the least solubility among these due to its higher lattice energy.
- **Option 2**: \( \text{NaHCO}_3 < \text{KHCO}_3 < \text{RbHCO}_3 \)
- This is correct. As the electropositive character increases from Na to Rb, the solubility also increases.
- **Option 3**: \( \text{K}_2\text{CO}_3 < \text{Na}_2\text{CO}_3 < \text{Li}_2\text{CO}_3 \)
- This is incorrect. Li2CO3 has the least solubility due to its lower electropositive character.
- **Option 4**: \( \text{LiNO}_3 < \text{NaNO}_3 < \text{KNO}_3 \)
- This is incorrect. For nitrates, the solubility decreases with increasing electropositive character, so LiNO3 has the highest solubility.

4. **Conclusion**:
- The only correct order of increasing solubility in water is given by **Option 2**:
\[ \text{NaHCO}_3 < \text{KHCO}_3 < \text{RbHCO}_3 \]