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After combining two solutions, a direction in a 1970s British chemistry book says to wash the new solution with a mixture of baking soda, table salt, and water. Then it says to dry it with anhydrous magnesium sulfate.

What does washing and drying mean? Do the chemicals get physically mixed together? Doesn't that contaminate the solution?

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This is standard for purifying substances.

To wash means to add your product solution to an aqueous solution (or just water, but frequently a saturated solution) to a separatory funnel. After shaking, you drain the lower layer (which is usually aqueous). This process removes water soluble impurities. This is frequently repeated.

Drying is accomplished by taking the organic solution of your product and mixing it with a desiccant (usually anhydrous sodium sulfate or anhydrous magnesium sulfate). This removes any water that was left from the washing in the above step. The desiccant may then be removed by filtration.

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The solvent of this experiment should be non polar, and may be a hydrocarbon, or a chlorinated hydrocarbon. This organic phase may contain as impurities some acids and other polar substances soluble in water, that should be eliminated. This liquid is then mixed with an aqueous solution of soda $\ce{NaHCO3}$. Aqueous solutions and non polar substances are not miscible. But the soda reacts with the acidic impurities, producing a salt which is soluble into water. So, by stirring the two liquids, the impurities of the organic phase (salt plus other polar substances) will pass into water, and are eliminated from the organic phase. This organic phase is said to have been "washed" by the soda solution.

The organic phase has not the same density as the aqueous solution. So it is easy to separate the aqueous phase and keep the organic liquid. But the previous treatment has also added a few percent water which has entered the organic phase. This water must be eliminated in a second phase, called "drying". It is done by adding a powder to the organic phase which is insoluble in the organic phase but reacts with water. This powder is magnesium sulfate $\ce{MgSO4}$ which has a strong tendency to react and fix all water molecules in their vicinity. It produces a solid hydrate like $\ce{MgSO4·H2O}$ or even $\ce{MgSO4·7 H2O}$. After stirring the organic phase has lost all its water. The organic phase is called "dry".

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The description you provided is essentially a shorthand for a standard workup procedure. After performing a reaction, one wants to isolate one’s product while removing the catalyst or other reagents and side products that one doesn’t need. For example, in many cross-coupling reactions an inorganic halide salt is an undesired side-product and a base such as sodium hydrogencarbonate may have been added to quench protons that form during the reaction. Likewise, a reaction may be catalysed by a small amount of acid (e.g. sulphuric acid or acetic acid).

The washing part of workup makes use of the fact that a lot of these side-products are hydrophilic while the organic products typically are not. By combining the organic phase with an aquaeous solution (they will not mix) in a separating funnel and shaking the funnel to ensure large contact areas, hydrophilic side-products can easily transfer into the water phase.

You mention that your method suggests using sodium hydrogencarbonate solution for washing; thus, I assume that your reaction was run in acidic conditions. For acidic conditions, I typically wash with sodiumhydrogencarbonate solution while for basic conditions I wash with ammonium chloride; both have the effect of neutralising the acid or base generating ionic compounds which are then more easily transferred into the water phase. Contrary to the other answer, I do not use saturated solutions for washing; rather, I add approximately equal parts of a saturated solution and deionised water because in my experience using saturated solutions can lead to a supersaturated aquaeous phase and subsequent precipitation of salts—not what I want.

Many a time the methods suggest extracting the aquaeous solution with the organic solvent. This is to ensure that the maximum amount of organic product is taken out of the aquaeous phase. For this, the organic phase of the first separation step is put aside, the aquaeous phase readded into the separation funnel and further organic solvent is added; rinse and repeat.

After the initial washing, I wash the combined organic phases with deionised water—this is to remove any remaining traces of acid or base that may have passed through the first step or ammonia that may have been generated by washing with ammonium chloride.

Now, I could immediately remove the organic solvent at the rotavap. However, the solution has been saturated with water; although water is only poorly soluble in most organic solvents poorly is distinct from insoluble; were I to remove the wet solvent at the rotavap most of the water would likely remain with the product. This is where drying comes in: the name essentially means removal of most of the water from the organic phase to ensure that only the organic product will remain in the flask at the rotavap.

Usually, drying is a two-step process. First the solution is washed once again with saturated (no dilution this time!) sodium chloride solution or brine. This highly concentrated solution serves to remove most of the dissolved water from the organic phase into the aquaeous phase. Second, a drying agent (usually anhydrous sodium sulphate or magnesium sulphate) is added to the organic solution. Neither of these salts will dissolve in the organic solvent but both are very hygroscopic and will draw moisture towards themselves. The resulting solution is considered ‘dry’ for workup purposes. The inorganic sulphate is filtrated off and the solution is ready for the rotavap. When performed correctly, no residues of water remain in your rotavap-dried product.

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