Even if the equations corresponding to these equations can be justified from a physical viewpoint, they are mainly phenomenological as they apply to many (very) different systems. Therefore, it is most of the times always possible to fit the adsorption data with one or another equation, with rather low differences of correlation data. You can try to fit your data with different isotherms, as shown in the attached paper (see Table 4). You'll find some differences, some isotherms being better than others, but it is not easy to say more and to explain why.
Yes, Temkin isotherm is independent of Langmuir and Freundlich isotherms. However, a particular adsorbent can follow multiple models not just one. This enables one to calculate constants and to create as complete a picture as possible, for the adsorption process of a particular adsorbent and for a particular adsorbate at particular pH, temperature etc. I hope this helps.
Langmuir adsorption isotherm represents formation of mono layer adsorbate on the surface of the adsorbent (Only single layer is formed)
Freundlich Adsorption isotherm represents characteristics of heterogeneous surface.
Temkin adsorption isotherm represents adsorbate adsorbent interactions. It states that the heat of adsorption of all molecules in layer decrease linearly.
Debashree, If you try out Langmuir isotherm , it is likely that it may fit in the data because single layer adsorption is quite possible. (For multi layer we have BET isotherm)
But it is clarified that all above 3 isotherms are different and have concepts of explaining adsorption are also different. Temkin adsorption isotherm thus is different from the Langmuir, Freundlich and BET isotherm. If your scope is limited to Temkin, you need not go ahead for other adsorption isotherm. Better check up with your research guide