The removal or recovery of volatile organic compounds (VOCs) from exhaust gas streams is an important process in environmental engineering. Activated carbon has long been used as an adsorbent in this process, but the presence of moisture in the stream reduces its effectiveness. M.-S. Chou and J.-H. Chiou (J. Envir. Engrg. ASCE, \(123,437(1997)\) ) have studied the effect of moisture content on the adsorption capacities of granular activated carbon (GAC) for normal hexane and cyclohexane in air streams. From their data for dry streams containing cyclohexane, shown in the table below, they conclude that GAC obeys a Langmuir type model in which \(\mathrm{q}_{\mathrm{VCORH}=0}=\mathrm{abc}_{\mathrm{VOC}}(1+\) \(\left.\mathrm{bc}_{\mathrm{VOC}}\right)\), where \(\mathrm{q}=\mathrm{m}_{\mathrm{VOC}} / \mathrm{m}_{\mathrm{GAC}}\), RH denotes relative humidity, a the maximum adsorption capacity, \(\mathrm{b}\) is an affinity parameter, and \(\mathrm{c}\) is the abundance in parts per million (ppm). The following table gives values of \(\mathrm{q}_{\mathrm{voc}, \mathrm{RH}=0}\) for cyclohexane: $$ \begin{array}{llllll} \text { c/ppm } & 33.6^{\circ} \mathrm{C} & 41.5^{\circ} \mathrm{C} & 57.4^{\circ} \mathrm{C} & 76.4^{\circ} \mathrm{C} & 99^{\circ} \mathrm{C} \\ 200 & 0.080 & 0.069 & 0.052 & 0.042 & 0.027 \\ 500 & 0.093 & 0.083 & 0.072 & 0.056 & 0.042 \\ 1000 & 0.101 & 0.088 & 0.076 & 0.063 & 0.045 \\ 2000 & 0.105 & 0.092 & 0.083 & 0.068 & 0.052 \\ 3000 & 0.112 & 0.102 & 0.087 & 0.072 & 0.058 \end{array} $$ (a) By linear regression of \(1 / \mathrm{q}_{\mathrm{voc}}, \mathrm{RH}=0\) against \(1 / \mathrm{c}_{\mathrm{VOC}}\), test the goodness of fit and determine values of a and b. (b) The parameters a and b can be related to \(\Delta_{\mathrm{ads}} \mathrm{H}\), the enthalpy of adsorption, and \(\Delta_{\mathrm{b}} \mathrm{H}\), the difference in activation energy for adsorption and desorption of the VOC molecules, through Arrhenius type equations of the form \(\mathrm{a}==\mathrm{k}_{\mathrm{a}} \exp \left(-\Delta_{\mathrm{ads}} \mathrm{H} / \mathrm{RT}\right)\) and \(\mathrm{b}=\mathrm{k}_{\mathrm{b}} \exp (-\) \(\left.\mathrm{A}_{\mathrm{b}} \mathrm{H} / \mathrm{RT}\right)\). Test the goodness of fit of the data to these equations and obtain values for \(\mathrm{k}_{\mathrm{a}}, \mathrm{k}_{\mathrm{b}}, \Delta_{\mathrm{ads}} \mathrm{H}\), and \(\Delta_{\mathrm{b}} \mathrm{H.}\) (c) What interpretation might you give to \(\mathrm{k}_{\mathrm{a}}\) and \(\mathrm{k}_{\mathrm{b}}\) ?

Step by step solution

01

Understanding Langmuir Adsorption Model

The Langmuir adsorption model is given by the equation: \[ q_{\mathrm{VCORH}=0} = \frac{abc_{\mathrm{VOC}}}{1 + bc_{\mathrm{VOC}}} \] where \(q = \frac{m_{\mathrm{VOC}}}{m_{\mathrm{GAC}}}\) represents the adsorption capacity (mass of VOC per mass of GAC), \(a\) is the maximum adsorption capacity, \(b\) is an affinity parameter reflecting the strength of the adsorption bond, and \(c_{\mathrm{VOC}}\) is the concentration of VOCs in ppm. The model assumes monolayer adsorption on a surface with a finite number of identical sites.

02

Linear Regression of Langmuir Model

The Langmuir equation can be linearized by taking the reciprocal of both sides, resulting in: \[ \frac{1}{q_{\mathrm{voc}, \mathrm{RH}=0}} = \frac{1}{a} + \frac{1}{abc_{\mathrm{VOC}}} = \frac{1}{a} + \frac{b}{a} \times \frac{1}{c_{\mathrm{VOC}}} \] This form suggests a linear relationship between \(\frac{1}{q_{\mathrm{voc}, \mathrm{RH}=0}}\) and \(\frac{1}{c_{\mathrm{VOC}}}\). By plotting these reciprocal values and performing linear regression, the slope will give \(\frac{b}{a}\) and the y-intercept will give \(\frac{1}{a}\), from which the values of \(a\) and \(b\) can be determined.

03

Goodness of Fit

The goodness of fit for the linear regression can be assessed using the coefficient of determination, \(R^2\), which reflects how well the data fits the linear model. A value of \(R^2\) close to 1 indicates a good fit. The p-value from the regression analysis can also be used to determine the statistical significance of the fit.

04

Evaluating Arrhenius Type Equations

To evaluate the Arrhenius type equations given by: \[ a = k_a \exp\left(-\frac{\Delta_{\mathrm{ads}}H}{RT}\right) \] and \[ b = k_b \exp\left(-\frac{\Delta_bH}{RT}\right) \] plot \( \ln(a) \) versus \( \frac{1}{RT} \) and \( \ln(b) \) versus \( \frac{1}{RT} \) respectively. The slopes of these plots will give \(-\Delta_{\mathrm{ads}}H\) and \(-\Delta_bH\), and the y-intercepts will give \(\ln(k_a)\) and \(\ln(k_b)\), from which the respective pre-exponential factors and enthalpy changes can be calculated.

05

Interpretation of Pre-Exponential Factors

The pre-exponential factors \(k_a\) and \(k_b\) from the Arrhenius equations represent the frequency of collisions leading to adsorption and desorption, respectively, independent of the activation energy barriers. They can be related to the physical properties of the system, such as the surface area of the adsorbent and the kinetic energy of the adsorbate molecules.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Understanding Adsorption Capacity

Adsorption capacity, denoted as q in the Langmuir adsorption model, is at the core of understanding how efficiently an adsorbent like granular activated carbon (GAC) can remove contaminants from a system. It refers to the mass of the adsorbate (in this case, VOCs like cyclohexane) that can be held per unit mass of the adsorbent (GAC).

The maximum adsorption capacity, represented by the variable a in the Langmuir model, indicates the total amount of the adsorbate that can be adsorbed onto the adsorbent surface to form a monolayer. It assumes that once the adsorbent surface is covered in a monolayer of adsorbate molecules, no more adsorption can occur, as there are a limited number of binding sites.

In practical terms, a high adsorption capacity signifies that a lesser amount of adsorbent is required to remove a given mass of contaminant from a stream, making the process more efficient and cost-effective. This parameter is crucial in the design and operation of filtration systems, where the choice of adsorbent and its capacity directly impacts the effectiveness of contaminant removal.

Enthalpy of Adsorption

Significance of Enthalpy of Adsorption

The enthalpy of adsorption, Δ_adsH, relates to the amount of heat released or absorbed when an adsorbate molecule attaches itself to the adsorbent surface. In the context of the Langmuir model, the enthalpy of adsorption is considered a vital thermodynamic parameter as it indicates the strength of the interaction between the adsorbate and the adsorbent.

A negative value of Δ_adsH (exothermic process) implies that heat is released during adsorption. The more negative the value, the stronger the bond between the adsorbate and adsorbent, which typically results in higher adsorption capacity at lower temperatures. Conversely, a positive value (endothermic process) would require heat input for adsorption to occur.

Understanding this enthalpic change is crucial for engineers and scientists as it affects the choice of adsorbent, the adsorption process design, and the overall efficiency of the system under various temperature conditions. In environmental engineering applications, such as VOC removal, knowing the Δ_adsH can help in optimizing the energy consumption of the adsorption system.

Activation Energy in Adsorption

Importance of Activation Energy

Activation energy is a fundamental concept in the adsorption process that explains the energy barrier that must be overcome for adsorption (and desorption) to occur. The difference in activation energy for adsorption and desorption, indicated by Δ_bH in the exercise, influences how easily the adsorbate can transition between being free in the stream and being bound to the adsorbent.

The lower the activation energy for adsorption, the easier it is for the molecules to adhere to the adsorbent surface. In the case of desorption, a higher activation energy suggests that more energy is required to release the molecule from the surface, implying stronger adsorption.

Activation energy plays a critical role in the rate of adsorption and desorption, impacting not just the efficacy of the adsorption process but also its reversibility. This makes it especially important when selecting adsorbents for applications where regeneration (recovery of the adsorbent) is necessary. In the study of VOC removal using GAC, understanding the activation energy can assist in determining the operational parameters necessary to achieve optimal performance in dynamic environmental conditions.