Both statements given above have implications relating to the phenomena of capillarity, which involves the interaction between a liquid and a solid (in this case, a capillary tube). Let's break down each statement for clarity.

Statement I: When a capillary tube is dipped into a liquid and the liquid neither rises nor falls in the capillary, it suggests a scenario where the adhesive forces (between the liquid and the solid) and the cohesive forces (among the liquid molecules) are in perfect balance. The contact angle, which is the angle formed by the tangent to the liquid surface at the point of contact with the wall of the tube, plays a crucial role here. A contact angle of $$0^{\circ}$$ implies complete wetting, meaning the liquid spreads out to maximize contact with the solid. However, the statement that the liquid neither rises nor falls specifically with a contact angle of $$0^{\circ}$$ seems inaccurate. In reality, when the contact angle is $$0^{\circ}$$, it denotes perfect wetting, and the liquid tends to rise in the capillary tube. Thus, the precision of Statement I could be disputed based on the typical behavior of liquids in the context of capillarity and contact angles.

Statement II: The contact angle really is a property of both the solid and the liquid. It depends on the nature of the solid surface (whether it's hydrophilic or hydrophobic) and the type of liquid. This is because the contact angle reflects the degree of interaction between the liquid and solid surfaces, which is influenced by characteristics such as surface tension of the liquid and the surface energies of both the liquid and solid. Thus, Statement II accurately describes the nature of the contact angle as being dependent on both the material of the solid and the liquid.

Considering the analysis above:

• Statement I suggests a specific scenario that does not align accurately with the principles of capillarity and wettability, particularly at a contact angle of $$0^{\circ}$$, which typically would result in the liquid rising in the capillary, not staying neutral.
• Statement II accurately reflects the factors influencing the contact angle between a liquid and a solid surface.

Therefore, the correct option based on the given statements and their analysis would be:

Option B: Statement I is false but Statement II is true.