Problem 11 - Chemistry in 30 days

Question

Freundlich adsorption isotherm gives a straight line on plotting (a) \(x / m\) vs \(P\) (b) \(\log x / m\) vs \(P\) (c) \(\log x / m\) vs \(\log P\) (d) \(x / m\) vs \(l / P\)

Step-by-Step Solution

Answer

The correct option is (c) \( \log x / m \) vs \( \log P \).

Step 1: Understand the Freundlich isotherm equation

The Freundlich adsorption isotherm is described by the equation \( \frac{x}{m} = K P^{1/n} \), where \( \frac{x}{m} \) is the amount adsorbed per unit mass of adsorbent, \( P \) is the pressure, \( K \) is the adsorption capacity constant, and \( 1/n \) is the adsorption intensity.

Step 2: Transform the Freundlich isotherm for linearization

Taking the logarithm of both sides of the Freundlich equation gives \( \log \left(\frac{x}{m}\right) = \log K + \frac{1}{n} \log P \). This is a linear equation of the form \( y = mx + c \), where \( y = \log \left(\frac{x}{m}\right) \), \( m = \frac{1}{n} \), \( x = \log P \), and \( c = \log K \).

Step 3: Identify the correct plotting variables

To obtain a straight line, the variables in the linear form of the equation must be plotted. Thus, \( \log \left(\frac{x}{m}\right) \) should be plotted against \( \log P \) to get a straight line.

Step 4: Select the correct multiple choice answer

Given the transformed linear equation \( \log \left(\frac{x}{m}\right) = \log K + \frac{1}{n} \log P \), option (c) \( \log x / m \) vs. \( \log P \) matches the setup needed for a straight line plot.

Key Concepts

Adsorption Intensity

The term adsorption intensity, represented by \( 1/n \) in the Freundlich adsorption isotherm equation, indicates how strongly an adsorbate adheres to the surface of an adsorbent. In the equation \( \frac{x}{m} = K P^{1/n} \), \( \frac{x}{m} \) is the amount of adsorbate per unit mass of adsorbent, \( P \) is the pressure, and \( K \) is a constant related to adsorption capacity.
When \( 1/n \) is closer to zero, adsorption is more intense. The adsorbate molecules bond more tightly due to strong interactions between the adsorbent and adsorbate. Conversely, a \( 1/n \) value closer to one suggests weaker adsorption intensity, indicating that the adsorbate does not bond as strongly.

• A low value of \( 1/n \) implies stronger adsorption and suggests a highly heterogeneous surface.
• A high \( 1/n \) value points to weak adsorption and a more uniform surface.

Understanding the concept of adsorption intensity helps in assessing the effectiveness of adsorbent materials in various practical applications, such as water purification or gas capture.

Linearization of Isotherms

Linearization is a mathematical technique used to reformulate non-linear equations into a linear form. This is crucial for interpreting adsorption data through the Freundlich adsorption isotherm. The original Freundlich equation \( \frac{x}{m} = K P^{1/n} \) is non-linear due to the exponential relationship between pressure \( P \) and adsorption \( \frac{x}{m} \).
By taking the logarithm of both sides, the equation transforms into a linear format: \[\log \left(\frac{x}{m}\right) = \log K + \frac{1}{n} \log P.\]

• The linear form resembles \( y = mx + c \), making it easy to plot and analyze.
• It allows determination of the constants \( K \) and \( 1/n \) by interpreting the y-intercept and slope of the line, respectively.

Linearizing the isotherm simplifies visualization, helping researchers understand the interactions between adsorbent and adsorbate better.

Pressure in Adsorption

Pressure plays a significant role in the adsorption process outlined by the Freundlich isotherm.
It determines how much adsorbate a fixed amount of adsorbent can hold. When the pressure \( P \) increases, the number of adsorbate molecules available for binding to the adsorbent surface also increases. This, in turn, enhances the adsorption process until saturation occurs.

• Higher pressure results in more adsorbent binding sites being occupied.
• Low-pressure environments are indicative of less robust adsorption processes.

Understanding pressure effects in adsorption informs the design of systems for practical applications, ensuring efficiency at varying pressures and preventing overloading the adsorbent material.

Plotting Adsorption Data

Plotting adsorption data accurately is essential to interpret the Freundlich isotherm.
Through graphical representation, researchers can visually assess adsorption characteristics, such as adsorption intensity and capacity.To obtain a straight line using the Freundlich isotherm, plot \( \log \left(\frac{x}{m}\right) \) against \( \log P \).

• This plot allows for easy determination of \( 1/n \) (slope) and \( \log K \) (y-intercept).
• It simplifies comparison between different adsorption systems and conditions by offering a straightforward way to visualize differences.

Effectively plotting and interpreting these graphs ensures that the adsorption characteristics are correctly understood, guiding the optimal design and application of adsorbent materials in various settings.