Question

In the first extraction, the amount of the substance left unextracted is given by the formula \((K\) is distribution coefficient, \(V \mathrm{ml}\) of the aqueous solution contain \(A\) grams of an organic substance) (a) \(x_{1}=A \frac{K V}{K V+v}\) (b) \(x_{1}=A \frac{K V}{K V+v^{2}}\) (c) \(x_{1}=A^{2} \frac{K V}{K V+v}\) (d) \(x_{1}=A \frac{K^{2} V}{K V+v}\)

Short answer

The correct equation for the amount of substance left unextracted \(x_{1}\) is \(x_{1}=A \frac{K V}{K V+v}\), which is represented by option (a).

Step-by-step solution

Understanding the Variables

A represents the amount of the organic substance in grams. K is the distribution coefficient which shows the distribution of a substance between two immiscible phases at equilibrium, in this case, the organic substance between the aqueous solution and the secondary phase. V stands for the volume of the aqueous solution (in milliliters), and v represents the volume of the solvent used for extraction.

Understanding the Extraction Process

During the extraction process, a portion of the substance is transferred from the aqueous phase to the organic phase. The amount of substance left in the aqueous phase after the first extraction depends on the distribution coefficient, and the volumes of the aqueous solution (V) and the solvent (v).

Identifying the Correct Equation

Based on the principles of extraction, the amount of substance left unextracted (x1) can be expressed as \(x_{1}=A \frac{K V}{K V+v}\). This equation expresses that the amount of substance left unextracted is proportional to the initial total amount of substance (A), depends on the distribution coefficient (K) and the volume of the aqueous solution (V), and inversely proportional to the sum of the product of the distribution coefficient and the aqueous solution (KV) and the volume of the solvent (v). Thus, the correct equation is (a) \(x_{1}=A \frac{K V}{K V+v}\).

Key concepts

Chemical Extraction

Chemical extraction is a fundamental laboratory and industrial technique utilized to separate compounds from a mixture. It is based on the fundamental principle that different substances have different solubilities in different solvents. In practical terms, it involves dissolving a specific compound of interest from a solid or liquid mixture into a solvent in which the compound is more soluble.

During this process, the mixture is usually introduced to a solvent in which the desired compound is highly soluble, while impurities are not. The result is a two-phase system where the target compound migrates into the chosen solvent. This separation can be enhanced by using various techniques, such as shaking or stirring, which help distribute the compounds between the two phases more efficiently.

Chemical extraction is a powerful tool because it allows for the purification, isolation, and concentration of specific compounds from complex mixtures, making it essential in fields like pharmaceuticals, environmental testing, and chemical analysis.

Solvent Volume

The role of solvent volume in the context of chemical extraction cannot be overstated. Solvent volume, represented by the variable 'v' in our initial formula, is a critical factor that directly influences the efficiency and the extent of extraction. Choosing the right volume of solvent is all about finding the balance between maximizing the extraction of the desired compound and minimizing the extraction of undesired impurities.

When you increase the volume of the solvent, the ability of the solvent to dissolve more of the substance increases—potentially leading to better extraction results. However, an excessive volume may not be practical or cost-effective, and can also dilute the compound of interest, making it harder to recover afterwards.

On the other hand, too little solvent may not adequately extract the target compound, requiring multiple extractions and leading to reduced efficiency. This makes the optimization of solvent volume a crucial step for effective chemical extraction processes.

Equilibrium in Extraction Processes

Equilibrium is a key concept in understanding extraction processes, and it refers to a state where the rates of transfer of a substance in two opposing directions are equal, resulting in a constant ratio of the concentration of the substance in both phases, given no other changes occur in the system. This condition is described by the distribution coefficient (K).

The distribution coefficient is a measure of how a particular substance is partitioned between the two immiscible liquids, typically an aqueous phase and an organic solvent. It is defined by the ratio of the concentrations of the solute in each phase at equilibrium. If you have a high distribution coefficient, the substance prefers the organic solvent; if it is low, the substance is more comfortable in the aqueous phase.

During the extraction process, achieving equilibrium is essential because it ensures that the maximum amount of substance is extracted under the given conditions. Shaking or stirring can help attain equilibrium faster, as it allows for increased contact between the two phases. Understanding this equilibrium is crucial when trying to determine the amount of substance that remains in each phase after extraction, as clarified in the provided equation \(x_{1}=A \frac{K V}{K V+v}\).