Step 1: Understanding the Topic
This question is about the Langmuir adsorption isotherm, a fundamental model in surface chemistry. An adsorption isotherm describes the equilibrium of adsorption of a substance on a solid surface at a constant temperature. The Langmuir model is based on specific assumptions about the nature of the surface and the adsorption process.
Step 2: Key Assumptions and Equation of the Langmuir Isotherm
The Langmuir theory makes the following key assumptions:
Adsorption occurs on specific, fixed sites on the surface.
The adsorption is limited to a monolayer (a single layer of molecules).
All adsorption sites are energetically identical (homogeneous surface).
There are no interactions between adsorbed molecules on adjacent sites.
Based on these assumptions, the mathematical relationship between the fractional surface coverage ($\theta$) and the gas pressure ($P$) is derived as:
\[
\theta = \frac{KP}{1+KP}
\]
where K is the adsorption equilibrium constant.
Step 3: Evaluating the Options
(A) Multilayer adsorption: This is incorrect. Langmuir theory explicitly assumes monolayer adsorption. Multilayer adsorption is described by the BET (Brunauer–Emmett–Teller) isotherm.
(B) and (C): These options describe specific conditions of temperature and graphical representation but do not define the fundamental equation or principle of the Langmuir isotherm itself.
(D) Reciprocal form gives \( \theta = \dfrac{KP}{1+KP} \): This option presents the correct and defining mathematical expression for the Langmuir isotherm. (Note: The option text is slightly confusing; it should just state that the equation is $\theta = \frac{KP}{1+KP}$. The reciprocal form is actually used for linear plotting, i.e., $\frac{1}{\theta} = \frac{1}{KP} + 1$). However, this option is the only one that provides the correct mathematical model.
Step 4: Final Answer
Option (D) correctly identifies the mathematical formula that defines the Langmuir adsorption isotherm.