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According to this classification, in S isotherms the more molecules of adsorbate are already adsorbed, the easier it is for additional amounts of adsorbate to become fixed to the surface of the adsorbent.
It is assumed that the adsorption sites have equal affinities for molecules of adsorbate and that the presence of adsorbed molecules at one site will not affect the adsorption of molecules at an adjacent site.
The Langmuir isotherm model assumes monolayer adsorption and neglects interactions between the molecules of adsorbate [34, 35].
This can be partially explained by the replacement of a number of water molecules from the surface, by molecules of adsorbate.
Molecules of adsorbate that show an S-shape isotherm (as in the case of phenol), are considered monofunctional, where the attraction towards the surface of the adsorbent arises from its OH group, giving an arrangement of adsorbed molecules oriented perpendicularly to the surface.
For 1/n = 1, the partition between the two phases is independent of the concentration; the situation 1/n < 1 is the most common and corresponds to a normal L-type Langmuir isotherm, while 1/n > 1 is indicative of a cooperative adsorption which involves strong interaction between the molecules of adsorbate.
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The average free energy, E, of a molecule of adsorbent expresses the energy released when a molecule of adsorbate passes from the solution into the adsorbent (Kilic et al. 2011).
D R isotherm is also used to determine the thermodynamic nature of the adsorption by estimating the mean free energy of the sorption per molecule of adsorbate, E (kJ mol−1), which is expressed as E = frac{1}{{sqrt {2k_{text{DR}} } }}.
The approach was usually applied to distinguish the physical and chemical adsorption of metal ions with its mean adsorption free energy, E per molecule of adsorbate (for removing a molecule from its location in the adsorption space to the infinity) can be computed by the relationship (Bello et al. 2014): E = - left[ {frac{1}{{sqrt {2A_{text{DKR}}} }}} right].
Therefore, as dosage of adsorbent increases, the competition for bonding sites between molecules of the adsorbate and adsorbent decreases.
Freundlich model is based on the sorption of solute on a heterogeneous surface and involves the interaction between the molecules of the adsorbate.
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