To calculate the exciton binding energy you must just follow the procedure described by Sturge, Phys. Rev. 127, 768 (1962). The theory of optical absorption of semiconductor near the band edge has been worked out in the effective mass approximation by R.J. Elliot. The absorption can be expressed by a simple equation 9equation 1 in this paper) in terms of photon energy, the energy gap, the matrix element , the effectife mass and the refractive index. The exciton binding energy is also given in terms of the the electron and hole masses and the dielectric constant. The energy gap and the excition binding energy thus calculated have been given in Table I of the same paper. I do not know why do you invoke Arrhenius equation?