If the fraction of surface sites occupied by an adsorbate is given by \(\frac{V}{V_m}\), the equation for adsorption isotherm is :
\(\frac{1}{V} = \frac{1}{PbV_m} + \frac{1}{V_m}\)
The Langmuir adsorption isotherm describes monolayer adsorption on a surface of identical, independent sites. Writing the fractional coverage as \(\theta = \frac{V}{V_m}\), where \(V_m\) is the volume needed for a complete monolayer, the isotherm is
\(\theta = \frac{bP}{1 + bP}\), so \(\frac{V}{V_m} = \frac{bP}{1 + bP}\).
Rearrange to the linear form. Invert both sides:
\(\frac{V_m}{V} = \frac{1 + bP}{bP} = \frac{1}{bP} + 1\).
Divide through by \(V_m\):
\(\frac{1}{V} = \frac{1}{PbV_m} + \frac{1}{V_m}\).
This is the useful working form: plotting \(\frac{1}{V}\) against \(\frac{1}{P}\) gives a straight line of slope \(\frac{1}{bV_m}\) and intercept \(\frac{1}{V_m}\), from which both the monolayer capacity and the adsorption equilibrium constant are obtained. The monolayer capacity in turn gives the surface area of the adsorbent.
Checking the limits confirms the expression. At low pressure, \(bP \ll 1\) and \(\theta \approx bP\), so coverage is proportional to pressure. At high pressure, \(bP \gg 1\) and \(\theta \to 1\) — the surface saturates at one monolayer, which is the defining assumption of the model.
The other options are not rearrangements of the isotherm at all; only one has \(\frac{1}{V}\) isolated with the correct two terms on the right.
Hence the isotherm is \(\frac{1}{V} = \frac{1}{PbV_m} + \frac{1}{V_m}\).
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