The correct order of increasing solubility in water is:

Options

A. `KFltNaFltLiF`
B. `NaHCO_(3)ltKHCO_(3)ltRbHCO_(3)`
C. `K_(2)CO_(3)ltNa_(2)CO_(3)ltLi_(2)CO_(3)`
D. `LiNO_(3)ltNaNO_(3)`

Explanation

To determine the correct order of increasing solubility in water for the given compounds, we need to analyze the concepts of lattice energy and hydration energy.

### Step-by-Step Solution:

1. **Understand Lattice Energy**:
- Lattice energy is the energy required to separate one mole of an ionic solid into its gaseous ions. It is influenced by the charges of the ions and the distance between them (size of the ions).
- The formula for lattice energy is given by:
\[ U \propto \frac{Q_1 \times Q_2}{R} \]
where \(Q_1\) and \(Q_2\) are the charges of the ions, and \(R\) is the distance between the ions (ionic radii).

2. **Analyze Ionic Sizes**:
- For alkali metal ions (Li\(^+\), Na\(^+\), K\(^+\), Rb\(^+\)), as the size of the ion increases, the lattice energy decreases. This is because larger ions have a greater distance between them, leading to a weaker attraction.

3. **Determine Solubility Order**:
- Since solubility is inversely related to lattice energy, the order of solubility will be the reverse of the order of lattice energy.
- The order of lattice energy for alkali metal ions is:
\[ \text{Li}^+ > \text{Na}^+ > \text{K}^+ > \text{Rb}^+ \]
- Therefore, the order of increasing solubility in water will be:
\[ \text{Rb}^+ < \text{K}^+ < \text{Na}^+ < \text{Li}^+ \]

4. **Consider Hydration Energy**:
- Hydration energy is the energy released when ions are surrounded by water molecules. It is directly proportional to the charge density of the ions.
- Smaller ions with higher charge density (like Li\(^+\)) will have higher hydration energy and thus higher solubility.

5. **Final Order of Solubility**:
- Based on the analysis above, the correct order of increasing solubility in water for the given compounds is:
\[ \text{Rb}^+ < \text{K}^+ < \text{Na}^+ < \text{Li}^+ \]

### Conclusion:
The correct answer for the order of increasing solubility in water is:
- Rb\(^+\) < K\(^+\) < Na\(^+\) < Li\(^+\)