Solubility is the ability of a solute to dissolve into a solvent. It is calculated by creating an equilibrium equation and solving for the concentration of dissolved ions.

For this question, we will need to set up equilibrium constant equations for each compound.

Note that the solid compounds are not included in the equilibrium expressions. Now we can work on finding the solubilities. For each mole of calcium hydroxide dissolved, one mole of calcium ions and two moles of hydroxide ions are released. For each mole of lithium hydroxide dissolved, one mole of lithium ions and one mole of hydroxide ions are released. For each mole of potassium hydroxide dissolved, one mole of potassium ions and one mole of hydroxide ions are released. Mathematically, we can equate these ratios to the ion concentrations in the equilibrium calculations.

In these calculations, is the solubility. Given the solubility constants, we can solve for .

Calcium hydroxide has the greatest solubility.

Solubility is the ability of a solute to dissolve into a solvent. It is calculated by creating an equilibrium equation and solving for the concentration of dissolved ions.

For this question, we will need to set up the equilibrium constant equation for calcium hydroxide.

Note that the solid compound is not included in the equilibrium expression. Now we can work on finding the solubility. For each mole of calcium hydroxide dissolved, one mole of calcium ions and two moles of hydroxide ions are released. Mathematically, we can equate this ratio to the ion concentrations in the equilibrium calculation.

In this calculation, is the solubility. Given the solubility constant, we can solve for .

Recall that the solubility product constant is given by the equation below, for a reaction in the following format.

Using our balanced reaction, we can find the solubility product equation for the dissociation of lead (II) fluoride.

For each molecule of lead (II) fluoride that dissolves, it produces one lead ion and two fluoride ions. We can conclude that the concentration of fluoride ions in solution will be twice the concentration of lead ions.

Use these variables in the solubility product equation, along with the given value from the question.

Now we can solve for the value of .

Remember that this value is equal to the concentration of lead ions in solution and half the concentration of fluoride ions in solution.

To determine the solubility product constant, we only need the concentrations and coefficients of the ions. The effective concentration of any pure substance (solid, liquid, or gas) is equal to one by definition, so does not influence the value of . The equation for the solubility product constant of this reaction is:

The units of the solubility product constant will depend on the coefficients of the products. will be a constant for the reaction, and will not change as more solid dissolves or as the reaction progresses.

Increasing the temperature of a gas solute in a liquid solvent will decrease solubility. While dissolved, the gas is in equilibrium with the liquid. Adding heat will push the equilibrium in favor of the gas, causing it to precipitate from solution in the form of bubbles.

Increasing the pressure of a gas solute in a liquid solvent, decreasing the temperature of a gas solute in a liquid solvent, and increasing the temperature of a solid solute in a liquid solvent will all increase the solubility. 

Knowledge of basic solubility rules is enough to answer this question. Nitrates and salts of alkali metals are always soluble, as are salts of ammonium. Acetates are also always soluble. Acid-base reactions, such as with hydrofluoric acid, produce a soluble salt and water. Sulfates are commonly soluble, with certain exceptions (mostly alkaline earth metals).

The only reaction that produces an insoluble product is that between silver nitrate and potassium chloride. Silver chloride is a common precipitate obtained in double replacement reactions.

A colloid is a suspension of particles larger than molecules, but too small to be individually distinguished by the naked eye. Only colloids disperse light by the Tyndall effect. Milk is an example of a colloid.

Knowing some of the general solubility trends will greatly facilitate your understanding of solubility in chemistry. Here are a couple guidelines you can follow in order to predict which compounds are soluble.

1. Compounds containing alkali metals, ammonium cations, or nitrate anions are soluble.

2. Compounds containing halogen anions are soluble. Key exceptions are halogens attached to silver, mercury, and lead.

3. Sulfates are soluble, except when attached to heavier alkaline metals, like barium (thus the correct answer).

4. Carbonates, phosphates, and hydroxides are generally insoluble; however, if they are attached to one of the ions mentioned in one of the above points, they are soluble.

These solubility rules should be known for the MCAT and are presented as a hierarchy.

1. All group 1 salts and ammonium salts are soluble

2. All nitrates, perchlorates, and acetates are soluble

3. All mercury, lead, and silver salts are NOT soluble

Understanding these solubility rules as a hierarchy, we can understand why Mg(NO₃)₂ and AgNO₃ are soluble and why AgCl is not soluble. 

Molecules are most soluble in solvents that possess similar properties and intermolecular forces to the solute.

Pentane is a non-polar compound, which interacts predominately through london dispersion forces. Ammonia, silver chloride, and phosphorus trichloride are all polar molecules. Ammonia has a lone pair on the nitrogen; silver is an ionic compound; phosphorus trichloride has lone pairs on both the phosphorus and chlorine atoms.

In contrast, carbon dioxide has lone pairs only on the oxygen atoms and has a linear geometry. This creates a line of symmetry in the molecule such that the net polarities of the carbon-oxygen bonds cancel each other. Even though carbon dioxide has polar bonds, it is not a polar molecule and would likely dissolve in pentane.