Question

What is the value of intercept on y-axis when $\log_{10} \frac{x}{m}$ is plotted against $\log_{10} P$ in Freundlich isotherm?

• A. $\frac{1}{n}$
• B. $\log_{10} k$
• C. $n$
• D. $k$

Hint:

In physical chemistry linear log graphs, multiplication operations turn into additions. Since the constant $k$ was multiplied by the pressure term in the original non-log equation ($\frac{x}{m} = k \cdot P^{1/n}$), taking logs turns it into an additive constant ($\log k$), which sets the starting baseline position or $y$-intercept of the line.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical value of the vertical $y$-axis intercept when the logarithmic form of the Freundlich adsorption isotherm equation is plotted as a straight-line graph.

Step 2: Key Formula or Approach:
The mathematical empirical formula for the Freundlich adsorption isotherm is: $$\frac{x}{m} = k \cdot P^{1/n}$$ To convert this exponential relationship into a linear form, take the logarithm (base 10) on both sides of the expression and match it to the standard straight-line equation: $$y = m \cdot x + c$$ where $m$ is the slope and $c$ is the $y$-axis intercept.

Step 3: Detailed Explanation:
1. Start with the core Freundlich equation: $$\frac{x}{m} = k \cdot P^{1/n}$$ 2. Take log base 10 on both sides: $$\log_{10}\left(\frac{x}{m}\right) = \log_{10}\left(k \cdot P^{1/n}\right)$$ 3. Apply logarithmic product rules ($\log(a \cdot b) = \log a + \log b$): $$\log_{10}\left(\frac{x}{m}\right) = \log_{10} k + \log_{10}\left(P^{1/n}\right)$$ 4. Apply logarithmic power rules ($\log a^b = b \log a$): $$\log_{10}\left(\frac{x}{m}\right) = \log_{10} k + \frac{1}{n}\log_{10} P$$ 5. Rearrange the terms to align with the linear slope-intercept format ($y = m \cdot x + c$): $$\underbrace{\log_{10}\left(\frac{x}{m}\right)}_{y} = \underbrace{\left(\frac{1}{n}\right)}_{slope} \cdot \underbrace{\log_{10} P}_{x} + \underbrace{\log_{10} k}_{intercept}$$ By inspection, plotting $\log_{10} \frac{x}{m}$ on the $y$-axis versus $\log_{10} P$ on the $x$-axis produces a straight line where: The slope of the line is equal to $\frac{1}{n}$. The intercept on the $y$-axis is equal to $\log_{10} k$.

Step 4: Final Answer:
The value of the intercept on the $y$-axis is $\log_{10} k$, which corresponds to option (B).