Question

At high pressure, Langmuir adsorption isotherm takes the form:
A) $\dfrac{x}{m}=\dfrac{aP}{1+bP}$
B) $\dfrac{x}{m}=\dfrac{a}{b}$
C) $\dfrac{x}{m}=aP$
D) $\dfrac{x}{m}=\dfrac{b}{a}+\dfrac{1}{aP}$

Answer 1

Hint: To attempt this question initially write down Langmuir adsorption isotherm equation and then solve the equation by considering the high value of pressure. At very high pressures, bP is very much greater than unity, use this hint.


Complete step by step solution:
Langmuir adsorption isotherm describes the adsorption process by assuming that the adsorbent behaves as an ideal gas at isothermal conditions. It is also used in describing the equilibrium between the adsorbent system and the adsorbate; and the adsorption of the adsorbate is limited to only one molecular layer before or at a relative pressure is attained as unity. According to this model, desorption and adsorption processes are reversible in nature.
The mathematical representation of Langmuir adsorption isotherm is denoted as:
$\dfrac{x}{m}=\dfrac{aP}{1+bP}$
Considering the case of high pressure,
At high pressure, bP is very much greater than unity ($bP\gg \gg 1$)
Therefore, \[\left( 1+bP \right)=bP\], Placing this changed value in above equation, the equation at high becomes
$\dfrac{x}{m}=\dfrac{aP}{bP}$
Cancelling the P value in numerator with denominator we get,
$\dfrac{x}{m}=\dfrac{a}{b}$
Hence, Option B is the correct answer.

Additional Information:
Freundlich adsorption isotherm is another adsorption isotherm used to describe the relationship between the quantity of gas adsorbed by unit mass of adsorbent and pressure at a constant temperature. This relationship is expressed as
\[\dfrac{x}{m}=k.{{p}^{{}^{1}/{}_{n}}}[n>1]\]

Note: For Langmuir isotherm model, some assumptions are valid, which are:
-The surface consisting of the adsorbing sites is homogeneous and perfectly flat plane
-All the adsorbing sites are energetically equivalent
-Each site can hold only one molecule of adsorbate
-The adsorbate adsorbs in an immobile state
-No type of interactions among adjacent adsorbate molecules