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Firstly, dissolution is the process by which a solute forms a solution when added in a solvent. A common example would be salt added to water. When salt, the solute, is added to water, the solvent, the salt particles seem to disappear in the water. This happens because salt, NaCl dissociate into its ions, Na+ and Cl-, because of the attraction of water with them.
The rate of dissolution simply tells us how fast or slow a given solute will dissolve in a particular solvent. There are four main factors that affect this rate: temperature, stirring, area of contact, and saturation. (We'll focus on stirring).
In order for a solute to dissolve, the water molecules (or the solvent molecules, in general) must come into contact with the solute particles. For example, in the salt example, the water molecules must be able to interact with each ions. Stirring exposes more of the solute, and makes more of the solute come in contact with the solvent, thus increasing the rate of dissolution. Put in another way, stirring brings "fresh parts" of the solute into contact with the solvent.
If you simply leave a chunk of salt in the middle of a glass full of water, the dissolution will be slow as initially, water can only interact with the particles that are on the surface of the block of salt. If you stir the system, however, you are exposing not only those in the surface, but also the other particles in the bulk of the salt, and hence water can come into contact with them and dissolve the particles simultaneously.
Stirring increases the rate of dissolution. When you stir a solute, more surface area of the substance is exposed to the solvent. The higher the surface area in relation to volume is, the faster a solute will dissolve.
When you want to make Kool-Aid, you are advised to stir until all the sugar crystals are dissolved. By stirring, each sugar crystal is exposed to the water, giving it a chance to dissolve more rapidly than if it was just at the bottom of the pitcher.
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