Consider 15 mL of an aqueous solution containing 2.0 g of an organic solute. If the distribution coefficient for the solute between ether and water is 20, calculate the percent of the compound that can be recovered with a single extraction with 20.4 mL of ether.

Question

NOTE: Enter your numerical answer in the space below to two significant figures.
Your Answer:

Answer 1

To solve this problem, we first need to understand the distribution coefficient (K) and how it relates to the concentrations of the solute in two different phases (aqueous and ether).

Given:

• Volume of water (V_water) = 15 mL
• Mass of solute = 2.0 g
• Distribution coefficient (K) = 20
• Volume of ether (V_ether) = 20.4 mL

The distribution coefficient, K, is defined as:

\[ K = \frac{C_{ether}}{C_{water}} \]

where \(C_{ether}\) and \(C_{water}\) are the concentrations of the solute in ether and water, respectively.

Let \(x\) be the amount of solute (in grams) that partitions into the ether layer after extraction, and \(2.0 - x\) will be the amount remaining in the water layer.

Step 1: Write the concentrations

The concentration of the solute in the ether after extraction is:

\[ C_{ether} = \frac{x}{20.4 \text{ mL}} \]

The concentration of the solute remaining in the water is:

\[ C_{water} = \frac{2.0 - x}{15 \text{ mL}} \]

Step 2: Use the distribution coefficient

Using the distribution coefficient in terms of concentrations:

\[ K = \frac{C_{ether}}{C_{water}} = \frac{\frac{x}{20.4}}{\frac{2.0 - x}{15}} \]

Substituting \(K = 20\):

\[ 20 = \frac{\frac{x}{20.4}}{\frac{2.0 - x}{15}} \]

Step 3: Solve for \(x\)

Cross multiply to solve for \(x\):

\[ 20 \cdot (2.0 - x) = \frac{x \cdot 15}{20.4} \]

Reorganizing:

\[ 20(2.0 - x) = \frac{15x}{20.4} \]

Multiply both sides by 20.4 to eliminate the fraction:

\[ 20.4 \cdot 20(2.0 - x) = 15x \]

Expanding:

\[ 408.0 - 20.4 \cdot 20x = 15x \]

\[ 408.0 = 15x + 408x \rightarrow 408.0 = 15x + 408.0x \]

Combine like terms:

\[ 408.0 = 423.0x \]

Now solve for \(x\):

\[ x = \frac{408.0}{423.0} \approx 0.9646\text{ g} \]

Step 4: Calculate percent recovery

The percent recovery is the amount recovered divided by the initial amount, multiplied by 100:

\[ \text{Percent recovery} = \left(\frac{x}{2.0}\right) \times 100 \]

\[ \text{Percent recovery} = \left(\frac{0.9646}{2.0}\right) \times 100 \approx 48.2% \]

Conclusion

After rounding to two significant figures, the percent of the compound that can be recovered with a single extraction with 20.4 mL of ether is:

\[ \boxed{48} \]