Question

At T(K), adsorption of a gas on surface of a solid follows Freundlich adsorption isotherm. A graph of \( \log \left( \frac{x}{m} \right) \) (on y-axis) and \( \log p \) (on x-axis) for this gas gave a straight line with slope = 2 and intercept = 1.5. What is the value of n ?

• A. 2.0
• B. 1.5
• C. 0.5
• D. 0.666

Hint:

Always remember the logarithmic form of the Freundlich adsorption isotherm: $\log \left( \frac{x}{m} \right) = \log k + \frac{1}{n} \log p$. This equation is analogous to $y = c + mx$, where the slope is $\frac{1}{n}$ and the y-intercept is $\log k$. The question specifically asks for $n$, which is derived directly from the slope.

Answer 1

The Correct Option is C

Step 1: State the Freundlich Adsorption Isotherm Equation
The Freundlich adsorption isotherm describes the adsorption of gases on solid surfaces. The equation is: \[ \frac{x}{m} = k p^{1/n} \] Where: \begin{itemize} \item $\frac{x}{m}$ is the mass of the gas adsorbed per unit mass of the adsorbent. \item $k$ is a constant related to the extent of adsorption. \item $p$ is the pressure of the gas. \item $n$ is a constant, where $n>1$. \end{itemize} Step 2: Convert the Freundlich Isotherm to its Logarithmic Form
To obtain a straight line graph, take the logarithm on both sides of the Freundlich equation: \[ \log \left( \frac{x}{m} \right) = \log (k p^{1/n}) \] Using logarithm properties ($\log(ab) = \log a + \log b$ and $\log(a^b) = b \log a$): \[ \log \left( \frac{x}{m} \right) = \log k + \log (p^{1/n}) \] \[ \log \left( \frac{x}{m} \right) = \log k + \frac{1}{n} \log p \quad \cdots (1) \] Step 3: Relate the Logarithmic Form to the Equation of a Straight Line
The equation of a straight line is typically written as $y = mx + c$, where: \begin{itemize} \item $y$ is the variable on the y-axis. \item $m$ is the slope. \item $x$ is the variable on the x-axis. \item $c$ is the y-intercept. \end{itemize} Comparing equation (1) with $y = mx + c$: \begin{itemize} \item $y = \log \left( \frac{x}{m} \right)$ \item $x = \log p$ \item Slope ($m_{\text{graph}}$) = $\frac{1}{n}$ \item Intercept ($c_{\text{graph}}$) = $\log k$ \end{itemize} Step 4: Use the Given Slope to Find the Value of n
The problem states that the graph of \( \log \left( \frac{x}{m} \right) \) (on y-axis) and \( \log p \) (on x-axis) gave a straight line with slope = 2. So, we have: \[ \text{Slope} = \frac{1}{n} \] Given Slope = 2: \[ 2 = \frac{1}{n} \] Solve for $n$: \[ n = \frac{1}{2} \] \[ n = 0.5 \] The intercept given (1.5) corresponds to $\log k = 1.5$, which means $k = 10^{1.5}$, but this information is not needed to find $n$. Step 5: Analyze Options
\begin{itemize} \item Option (1): 2.0. Incorrect. \item Option (2): 1.5. Incorrect. \item Option (3): 0.5. Correct, as it matches our calculated value of $n$. \item Option (4): 0.666. Incorrect. \end{itemize}