The problem is how the specific equation should be derived by using boundary condition. It consists of the final equation (which is based on Langmuir isotherm) and the boundary condition is as well known so just want to know to reach the final equation by using boundary condition? Step by step. The model is on the assumptions that:
i) In a As-HA binary system, sorbent binding sites form complexes represented as SA when arsenic binding occurs and complexes denoted SH when HA is bound and
ii) Equilibrium can be expressed as S+A? SA, where K_{1} = [SA]/[S][A], S + H ? SH, where K_{2} = [SH]/[S][H] and A + H ? AH where K_{3} = [AH]/[A][H]. In this model, considering an interactions resulting in SAH complexes, the equilibrium can be expressed as SA + H? SAH, where K_{1,2} = [SAH]/[SA][H], SH+A ? SHA, where K_{2,1} = [SHA]/[SH][A] and S+AH ? SAH where K_{3,3} = [SAH]/[S][AH]. Then one can assume S+A+H? SAH and S+H+A? SHA, where K= K_{1}K_{1,2} = K_{2}K_{2,1} = K_{3}K_{3,3}.
Assuming that the sorption system is in equilibrium (there are no net changes of [SA], [SH] and [SAH] with respect to time), the following can be written: d[SA]/dt = 0 , d[SH]/dt = 0 and d[SAH]/dt = 0 and [S0] = [S]+[SA]+[SH]+[SAH]. In such case, the model can be presented to the form: qAs = (qmax Ce [As]{1+(K_{1}/K )Ce [HA] })/({K_{1}+Ce [As]+(K_{1}/K_{2}) Ce [HA]+2(K_{1}/K) Ce [As] Ce [HA]}).
So, describe how to reach the above final equation by using the mentioned assumptions.