Question

what factors might control the maximum amount of solute that will dissolve in a given amount of solvent?

Answer 1

Temperature, the nature of the reactants and the presence of other ions affect solubility and the rate of dissolving.

Solubility of a substance is its maximum amount that can be dissolved in a specified amount of solvent. It depends upon the nature of solute and solvent as well as temperature and pressure.
Let us consider the effect of these factors in a solution of a solid or a gas in a liquid. When a solid solute is added to the solvent, some solute dissolves and its concentration increases in solution. This process is known as dissolution. Some solute particles in solution collide with the solid solute particles and get separated out of solution. This process is known as crystallisation.
A stage is reached when the two processes occur at the same rate. Under such conditions, the number of solute particles going into solution will be equal to the solute particles separating out and a state of dynamic equilibrium is reached. At this stage, the concentration of solute in solution will remain constant under the given conditions, i.e., temperature and pressure.
Similar process is followed when gases are dissolved in liquid solvents.
Such a solution in which no more solute can be dissolved at the same temperature and pressure is called a saturated solution. An unsaturated solution is one in which more solute can be dissolved at the same temperature.
The maximum amount of solute that dissolves completely in a given amount of solvent at constant temperature is called solubility of the solute.

Solubility is defined as the upper limit of solute that can be dissolved in a given amount of solvent at equilibrium. In such an equilibrium, Le Chatelier's principle can be used to explain most of the main factors that affect solubility. Le Châtelier's principle  dictates that the effect of a stress upon a system in chemical equilibrium can be predicted in that the system tends to shift in such a way as to alleviate that stress.

Solute-Solvent Interactions Affect Solubility

The relation between the solute and solvent is very important in determining solubility. Strong solute-solvent attractions equate to greater solubility while weak solute-solvent attractions equate to lesser solubility. In turn, polar solutes tend to dissolve best in polar solvents while non-polar solutes tend to dissolve best in non-polar solvents. In the case of a polar solute and non-polar solvent (or vice versa), it tends to be insoluble or only soluble to a miniscule degree. A general rule to remember is, "Like dissolves like."

COMMON ION EFFECT

The presence of other ions in the solution can affect solubility, for example acids will increase the solubility of carbonates. The presence of common ions in the solution can decrease the solubility.

Pressure Affects Solubility of Gases

The effects of pressure are only significant in affecting the solubility of gases in liquids.

Solids & Liquids: The effects of pressure changes on the solubility of solids and liquids are negligible.

Gases: The effects of pressure on the solubility of gases in liquids can best be described through a combination of Henry's law and Le Châtelier principle. Henry's law dictates that when temperature is constant, the solubility of the gas corresponds to it's partial pressure. Consider the following formula of Henry's law:

P=KhC
where:

p is the partial pressure of the gas above the liquid,
kh is Henry's law constant, and
c is the concentrate of the gas in the liquid.
This formula indicates that (at a constant temperature) when the partial pressure decreases, the concentration of gas in the liquid decreases as well, and consequently the solubility also decreases. Conversely, when the partial pressure increases in such a situation, the concentration of gas in the liquid will increase as well; the solubility also increases. Extending the implications from Henry's law, the usefulness of Le Châtelier's principle is enhanced in predicting the effects of pressure on the solubility of gases.

Consider a system consisting of a gas that is partially dissolved in liquid. An increase in pressure would result in greater partial pressure (because the gas is being further compressed). This increased partial pressure means that more gas particles will enter the liquid (there is therefore less gas above the liquid, so the partial pressure decreases) in order to alleviate the stress created by the increase in pressure, resulting in greater solubility.The converse case in such a system is also true, as a decrease in pressure equates to more gas particles escaping the liquid to compensate.

Temperature Affects Solubility

Temperature changes affect the solubility of solids, liquids and gases differently. However, those effects are finitely determined only for solids and gases.

Solids
The effects of temperature on the solubility of solids differ depending on whether the reaction is endothermic or exothermic. Using Le Chatelier's principle, the effects of temperature in both scenarios can be determined.

First, consider an endothermic reaction ( ΔHsolvation>0ΔHsolvation>0 ): Increasing the temperature results in a stress on the reactants side from the additional heat. Le Châtelier's principle predicts that the system shifts toward the product side in order to alleviate this stress. By shifting towards the product side, more of the solid is dissociated when equilibrium is again established, resulting in increased solubility.
Second, consider an exothermic reaction (( ΔHsolvation<0ΔHsolvation<0 ): Increasing the temperature results in a stress on the products side from the additional heat. Le Châtelier's principle predicts that the system shifts toward the reactant side in order to alleviate this stress. By shifting towards the reactant's side, less of the solid is dissociated when equilibrium is again established, resulting in decreased solubility.
Liquids
In the case of liquids, there is no defined trends for the effects of temperature on the solubility of liquids.

Gases
In understanding the effects of temperature on the solubility of gases, it is first important to remember that temperature is a measure of the average kinetic energy. As temperature increases, kinetic energy increases. The greater kinetic energy results in greater molecular motion of the gas particles. As a result, the gas particles dissolved in the liquid are more likely to escape to the gas phase and the existing gas particles are less likely to be dissolved. The converse is true as well. The trend is thus as follows: increased temperatures mean lesser solubility and decreased temperatures mean higher solubility.

Le Chatelier's principle allows better conceptualization of these trends. First, note that the process of dissolving gas in liquid is usually exothermic. As such, increasing temperatures result in stress on the product side (because heat is on the product side). In turn, Le Chatelier's principle predicts that the system shifts towards the reactant side in order to alleviate this new stress. Consequently, the equilibrium concentration of the gas particles in gaseous phase increases, resulting in lowered solubility.

Conversely, decreasing temperatures result in stress on the reactant side (because heat is on the product side). In turn, Le Châtelier's principle predicts that the system shifts toward the product side in order to compensate for this new stress. Consequently, the equilibrium concentration of the gas particles in gaseous phase would decrease, resulting in greater solubility.

Molecular size

The larger the molecules of the solute are, the larger is their molecular weight and their size. It is more difficult it is for solvent molecules to surround bigger molecules. If all of the above mentioned factors ale excluded, a general rule can be found that larger particles are generally less soluble. If the pressure, and temperature are the same than out of two solutes of the same polarity, the one with smaller particles is usually more soluble.

Stirring increases the speed of dissolving

Stirring does not have an affect on solubility of a substance, but everyone knows that if he puts sugar in his tea and does not stir, it will not dissolve. Actually, if we left the tea to stand for a long enough time, the sugar would dissolve. Stirring only increases the speed of the process - it increases move of the solvent what exposes solute to fresh portions of it, thus enabling solubility. As molecules in liquid substances are in constant move, the process would take place anyway, but it would take more time.