K is Freundlich or affinity constant, since large K means that the dye is strongly adsorbed on carbon, and if K increasing by temperature that means your adsorption is favorable upon heating, and you can take advantage from the temperature dependence by calculating dG = RTlnK, which can tell you about the spontaneity of the dye-carbon adsorption.

If possible find out the type of adsorption of the dye on charcoal. i.e whether is it physisorption or chemisorption. Also, try to find if the process is reversible or irreversible.

The linear dependence according to Van't Hoff's equation is correct. Considering the slope, you should be able to tell if the adsorption is endothermic (negative slope) or exothermic (positive slope).

Refer -

Relationship between the Freundlich adsorption constants K and 1/N hydrophobic adsorption

Ikuo Abe, Katsumi Hayashi, Tsuneaki Hirashima, and Mutsuo KitagawaJournal of the American Chemical Society 1982 104 (23), 6452-6453DOI: 10.1021/ja00387a052

Monk, Paul (2004). Physical Chemistry: Understanding our Chemical World. Wiley. p. 162. ISBN 978-0471491811.

Empirical relation between extent of adsorption with pressure, at a constant temperature: 𝑥 /𝑚 = 𝐾𝑃 1/n. where, x is the mass of the gas adsorbed on mass, m, of the solid adsorbent at a pressure P. K and n are constants for a given adsorbate-adsorbent pair at a particular temperature. Taking logarithm of equation (1), we get log ( 𝑥 /𝑚 ) = 𝑙𝑜𝑔 𝐾 + 1 𝑛 𝑙𝑜𝑔 P/Po. where P o is the unit pressure required to make the pressure term unitless whose logarithm can then be taken. The validity of Freundlich isotherm can be verified by plotting a graph between log ( 𝑥 𝑚 ) and log (P/Po ). A straight line with slope equal 1/n and, intercept log K, validates Freundlich adsorption isotherm. The two constants can be evaluated from these values. The value of ‘n’ is generally greater than one and thus the reciprocal, 1/n, can have values between 0-1. Based on this, Freundlich isotherm is able to explain the type I adsorption isotherm as follows: · When 1 𝑛 = 0, 𝑥 𝑚 = constant. This means that extent of adsorption is independent of pressure When 1 𝑛 = 1, 𝑥 𝑚 = 𝐾𝑃. 𝑇ℎ𝑖𝑠 means that extent of adsorption varies linearly with pressure · In the intermediate range, 1 𝑛 varies between 0-1. This accounts for the fact that the increase in adsorption is not as fast as the increase in pressure Although, Freundlich isotherm gave satisfactory results, it was associated with the following limitations: · It was an empirical relation; there was no theoretical foundation · It could explain type I adsorption isotherm only · It failed at high pressures as the experimental isotherm approached saturation at high pressures, which Freundlich isotherm could not explain · K and n are not true constants for a particular adsorbate-absorbent system. They show temperature dependence.