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A gas stream is placed into contact with an adsorbent material at temperature T. Sites are available within the material to adsorb up to nmax moles of gas, but the pressure P of the gas stream is such that, at equilibrium, half the adsorption sites in the material are occupied and half of them are empty. Heat (specifically in the form of isosteric heat of adsorption) is released during the adsorption process, although it can be assumed that such heat is conducted to the surroundings sufficiently quickly that any temperature rise is negligible.
(a) Assume that the molar concentration C of the gas stream can be obtained from the ideal gas law. Assume also that the adsorbed phase follows a Langmuir isotherm, but with a temperature-sensitive Langmuir parameter a. Under these circumstances, determine an expression for the heat released during the adsorption process (that takes the system from no moles adsorbed to half of the adsorption sites being occupied). Simplify the expression so that it is written in terms of T, nmax, a and da/dT. [12 marks]