91Ó°ÊÓ

Various amounts of activated carbon were added to a fixed amount of raw cane sugar solution \((48 \mathrm{wt} \%\) sucrose in water) at \(80^{\circ} \mathrm{C} .\) A colorimeter was used to measure the color of the solutions, \(R,\) which is proportional to the concentration of trace unknown impurities in the solution. The following data were obtained (adapted from the reference in Footnote \(20,\) p. 652 ):$$\begin{array}{|l|r|r|r|r|r|r|} \hline \text { kg carbon/kg dry sucrose } & 0 & 0.005 & 0.010 & 0.015 & 0.020 & 0.030 \\ \hline R \text { (color units/kg sucrose) } & 20.0 & 10.6 & 6.0 & 3.4 & 2.0 & 1.0 \\ \hline\end{array}.$$ The reduction in color units is a measure of the mass of impurities (the adsorbate) adsorbed on the carbon (the adsorbent).(a) The general form of the Freundlich isotherm is $$X_{i}^{*}=K_{\mathrm{F}} c_{i}^{\beta}$$ where \(X_{i}^{*}\) is the mass of \(i\) adsorbed/mass of adsorbent and \(c_{i}\) is the concentration of \(i\) in solution. Demonstrate that the Freundlich isotherm may be formulated for the system described above as $$\vartheta=K_{\mathrm{F}}^{\prime} R^{\beta}$$ where \(\vartheta\) is the \(\%\) removal of color/[mass of carbon/mass of dissolved sucrose]. Then determine \(K_{\mathrm{F}}^{\prime}\) and \(\beta\) by fitting this expression to the given data, using one of the methods in Section \(2.7 .\) (b) Calculate the amount of carbon that would have to be added to a vat containing \(1000 \mathrm{kg}\) of the 48 wt\% sugar solution at \(80^{\circ} \mathrm{C}\) for a reduction in color content to \(2.5 \%\) of the original value.

Step by step solution

01

Derive the Freundlich Isotherm

First, it's important to establish the Freundlich isotherm specific for this system. This means relating \(X_{i}^{*}\), the mass of impurities adsorbed per mass of carbon, to \(c_{i}\), the concentration of impurities in the solution, using the Freundlich isotherm, \(X_{i}^{*}=K_{\mathrm{F}} c_{i}^{\beta}\). Here, the concentration of impurities is represented by \(R\), the color of the solution. Similarly, \(\%\) removal of color per [mass of carbon/mass of dissolved sucrose] is equal to the mass of \(i\) adsorbed per mass of carbon (\(X_{i}^{*}\)). Therefore, for this specific system, the Freundlich isotherm is: \(\vartheta=K_{\mathrm{F}}^{\prime} R^{\beta}\).

02

Determine the Freundlich constants

The Freundlich constants \(K_{\mathrm{F}}^{\prime}\) and \(\beta\), are obtained by fitting this derived isotherm to the provided data (kg carbon/kg dry sucrose and R), using logarithmic plotting or numerical analysis methods. This equation is in the form of a power function, so it can be converted to linear form by taking the logarithm of both sides: \(\log \vartheta=\log K_{\mathrm{F}}^{\prime} + \beta \log R\). This linear form can be plotted using the provided data, with the resulting slope equal to \(\beta\) and the intercept equal to \(\log K_{\mathrm{F}}^{\prime}\). Therefore, the constants \(K_{\mathrm{F}}^{\prime}\) and \(\beta\) can be determined using this line.

03

Calculate the required activated carbon

Upon finding the constants \(K_{\mathrm{F}}^{\prime}\) and \(\beta\), use them to calculate the required amount of activated carbon necessary to achieve a desired reduction in color content to \(2.5 \%\) of the original value. This is done using the derived Freundlich isotherm, solving for \(\vartheta\) (since it's equal to activated carbon required/kg dry sucrose) when \(R\) is \(2.5\%\) of the original value, and the mass of dry sucrose is known (in this case, considering the 48 wt% sugar solution).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Freundlich Isotherm

The Freundlich Isotherm is an empirical equation describing how solutes interact with surfaces in adsorption processes. It is particularly useful for heterogeneous surface energies, which occur in real-life scenarios. In our exercise, the isotherm takes the form \(X_{i}^{*}=K_{\mathrm{F}} c_{i}^{\beta}\), where \(X_{i}^{*}\) is the mass of the impurities adsorbed per mass of adsorbent (carbon), and \(c_{i}\) is the concentration in the solution.
By manipulating this, we get \(\vartheta=K_{\mathrm{F}}^{\prime} R^{\beta}\), which relates to the percent color removal per mass of carbon per mass of dissolved sucrose. Here, \(\vartheta\) represents a redefined term for our system.
This transformation allows us to link impurity concentration, indicated by color (\(R\)), to how much gets adsorbed on carbon. By fitting data to this model, constants \(K_{\mathrm{F}}^{\prime}\) and \(\beta\) can be identified, essential for predicting adsorption dynamics in various scenarios.

Adsorbent

An adsorbent is a material tasked with bonding to atoms, ions, or molecules (known collectively as adsorbates) during a process called adsorption. In this context, activated carbon plays the role of the adsorbent. It’s a highly porous substance with a large surface area, which makes it effective in capturing impurities.
The active sites on the carbon interact with various contaminants in the sugar solution, including trace impurities that impact color. Activated carbon's efficiency makes it extensively used in industries for water purification, air filtration, and sugar refining.
The amount of activated carbon utilized can be directly linked to how much of the impurities—or color—in the solution are removed. This relationship is critical when applying the Freundlich isotherm to understand the system's behavior.

Trace Impurities

In processes involving solutions like our sugar mixture, trace impurities are undesired tiny particles or substances that affect quality. Despite being in low concentrations, these impurities can significantly impact the overall appearance or properties of a solution.
Here, trace impurities affect the sugar solution’s color. The use of activated carbon serves to reduce these impurities by adsorbing them onto its surface, thereby enhancing the quality and clarity of the sugar solution.
Understanding and managing trace impurities is crucial in industries such as food processing, where the purity of ingredients is vital for product safety and consumer satisfaction.

Colorimetry

Colorimetry is a technique used to determine the concentration of colored compounds in solution by measuring the light absorption's intensity. In this exercise, colorimetry assists in quantifying trace impurities in the sugar solution by examining its color.
The measurement obtained, denoted as \(R\), reflects the amount of impurities because the more color present, the higher the concentration of these impurities. This measurement provides insights into how effective the adsorbent—activated carbon—has been in cleansing the solution.
Using colorimetry helps in accurately gauging the achievement of desired purity levels in solutions, making the process critical in quality control and assurance across many industries.