Question

A certain type of straw (with 15 wt\% moisture content) is submerged in water at a mass ratio of water to straw of 4 . The straw initially contains \(0.8 \mathrm{wt} \% \mathrm{Cl}\) (ar basis). When taken out of the water, the straw is very wet with a moisture content of \(80 \mathrm{wt} \%\) but a \(\mathrm{Cl}\) content of only \(0.03 \mathrm{wt} \%\). What is the partitioning coefficient of \(\mathrm{Cl}\) when this is defined as the mass fraction of \(\mathrm{Cl}\) in the water phase divided by the \(\mathrm{Cl}\) mass fraction in the wet biomass before submerging? What do you assume?

Short answer

Answer: The partitioning coefficient of Cl in this system is approximately 0.345.

Step-by-step solution

Understand the problem and define the variables

Let's denote the mass of the straw as \(m_s\) and the mass of the water as \(m_w\). We are given that the mass ratio of water to straw is 4, meaning that \(m_w = 4m_s\). We are also given the initial mole fractions of water in the straw (\(\mathrm{15\ wt\%}\)), initial \(\mathrm{Cl}\) content in the straw (\(\mathrm{0.8\ wt\%}\)), and the final moisture (\(\mathrm{80\ wt\%}\)) and \(\mathrm{Cl}\) content in the straw after submersion (\(\mathrm{0.03\ wt\%}\)). The partitioning coefficient of \(\mathrm{Cl}\), denoted as \(K_\mathrm{Cl}\), is the mass fraction of \(\mathrm{Cl}\) in the water phase divided by the mass fraction of \(\mathrm{Cl}\) in the wet biomass before submerging.

Calculate the mass fraction of \(\mathrm{Cl}\) in the wet biomass before submerging

In order to calculate the mass fraction of \(\mathrm{Cl}\) in the wet biomass before submerging, we will use the initial \(\mathrm{Cl}\) content in the straw (\(\mathrm{0.8\ wt\%}\)), which can be written as: Initial \(\mathrm{Cl}\) mass fraction in straw = \(\dfrac{0.008m_s}{m_s}\).

Calculate the mass fraction of \(\mathrm{Cl}\) in the water phase after submerging

In order to calculate the mass fraction of \(\mathrm{Cl}\) in the water phase after submerging, we will first need to calculate the total mass of \(\mathrm{Cl}\) transferred from the straw to the water. The initial mass of \(\mathrm{Cl}\) in the straw is \(0.008m_s\), and the final mass of \(\mathrm{Cl}\) in the straw is \(0.0003(m_s + 0.8m_w)\) based on the final moisture (\(\mathrm{80\ wt\%}\)) and \(\mathrm{Cl}\) content in the straw (\(\mathrm{0.03\ wt\%}\)). Since we know that \(m_w = 4m_s\), we can rewrite the final mass of \(\mathrm{Cl}\) in the straw as \(0.0003(m_s + 3.2m_s)\). Therefore, the mass of \(\mathrm{Cl}\) transferred to the water is: \(\mathrm{Mass\ of\ Cl\ transferred} = 0.008m_s - 0.0003(4.2m_s)\). Now, we can find the mass fraction of \(\mathrm{Cl}\) in the water phase after submerging: \(\mathrm{Cl\ mass\ fraction\ in\ water\ phase} = \dfrac{0.008m_s - 0.0003(4.2m_s)}{m_w}\).

Calculate the partitioning coefficient of \(\mathrm{Cl}\)

Now that we have the mass fractions of \(\mathrm{Cl}\) in the water phase after submerging and in the wet biomass before submerging, we can calculate the partitioning coefficient \(K_\mathrm{Cl}\): \(K_\mathrm{Cl} = \dfrac{\mathrm{Cl\ mass\ fraction\ in\ water\ phase}}{\mathrm{Initial\ Cl\ mass\ fraction\ in\ straw}}\). Substituting the expressions we found in Steps 2 and 3: \(K_\mathrm{Cl} = \dfrac{\dfrac{0.008m_s - 0.0003(4.2m_s)}{m_w}}{\dfrac{0.008m_s}{m_s}}\). Using the relationship \(m_w = 4m_s\), the partitioning coefficient simplifies to: \(K_\mathrm{Cl} = \dfrac{0.008 - 0.0003(4.2)}{4 \times 0.008}\). Finally, we can calculate the value of \(K_\mathrm{Cl}\): \(K_\mathrm{Cl} \approx 0.345\). So, the partitioning coefficient of \(\mathrm{Cl}\) is approximately \(0.345\). This result assumes that the system is at equilibrium and that water is the only molecule absorbed in straw.

Key concepts

Chemical Partitioning

Chemical partitioning is a process that describes how a substance distributes itself between two different phases or media at equilibrium. In the context of our exercise, the partitioning coefficient of Chlorine (\text{Cl}) is fundamental to understanding how it divides itself between the 'water phase' and the 'wet biomass phase' after the straw is submerged into the water. This coefficient, denoted as \(K_\text{Cl}\), is calculated as the ratio of the mass fraction of \(\text{Cl}\) in the water to the mass fraction in the straw prior to submerging.

The concept of partitioning is used extensively in environmental science, particularly to assess the fate and transport of contaminants in different environmental compartments such as water, air, soil, and biota. It is also a critical concept in areas such as pharmacokinetics and chemical engineering. Understanding the factors that influence chemical partitioning, like temperature, pH levels, the presence of other chemicals, can help predict the movement and impact of a substance in an environment.

Mass Fraction Calculation

When dealing with mixtures or solutions, mass fraction calculation allows us to express the concentration of a component relative to the total mass of the solution or mixture. In the exercise, we calculated the initial and final mass fractions of \(\text{Cl}\) in the straw, considering the water weight percentage as a defining factor.

Mass fraction is dimensionless and often expressed as a percentage. Its calculation plays a vital role in areas like food science, material science, and chemistry, where it helps to determine the composition of a mixture and to understand reaction yields. Calculating the initial mass fraction of \(\text{Cl}\) in the straw (\(0.8\ wt\%\)) and the mass fraction after the straw was removed from the water (\(0.03\ wt\%\)) provided essential inputs for determining the partitioning coefficient, a crucial step in quantifying the extent of \(\text{Cl}\)'s migration into the water.

Equilibrium Concentration

Equilibrium concentration refers to the steady-state concentration of a solute in a solvent when the rates of dissolution and precipitation (or absorption and desorption) are equal. This concept is pivotal in understanding the final distribution of the substance between two phases. In our case, we assume equilibrium has been reached between the \(\text{Cl}\) ions in straw and in water when the straw is soaked and subsequently removed.

The idea of equilibrium is a foundational principle in thermodynamics and is applicable to various fields, such as chemistry, environmental science, and pharmacology. Information about equilibrium concentrations can help in predicting the outcome of chemical reactions, the behavior of pollutants in the environment, or the body's response to medication. For the exercise solution, we computed the equilibrium concentration of \(\text{Cl}\) to deduce the amount that had transferred from the straw to the water, ultimately allowing us to calculate the partitioning coefficient.