Model Mathematics

\frac{d}{d t} (q_1) = - v_1 \frac{d}{d t} (q_2) = (- 2) v_1 \frac{d}{d t} (q_3) = v_2 \frac{d}{d t} (q_4) = 2 v_2 \frac{d}{d t} (q_e0) = v_2 - v_1 \frac{d}{d t} (q_e1) = v_1 - v_2 u_f_1 = u_1 + 2 u_2 + u_e0 u_r_1 = u_e1 u_f_2 = u_e1 u_r_2 = u_3 + 2 u_4 + u_e0 u_1 = RT \ln (K_q_1 q_1) u_2 = RT \ln (K_q_2 q_2) u_3 = RT \ln (K_q_3 q_3) u_4 = RT \ln (K_q_4 q_4) u_e0 = RT \ln (K_q_e0 q_e0) u_e1 = RT \ln (K_q_e1 q_e1) v_1 = kappa_1 (ⅇ^{\frac{u_f_1}{RT}} - (ⅇ^{\frac{u_r_1}{RT}})) v_2 = kappa_2 (ⅇ^{\frac{u_f_2}{RT}} - (ⅇ^{\frac{u_r_2}{RT}}))

q_e_tot = q_e0 + q_e1 A_f = \frac{q_e_tot K_q_e0 kappa_1 kappa_2}{kappa_1 + kappa_2} K_q_1 {K_q_2}^{2} A_r = \frac{q_e_tot K_q_e0 kappa_1 kappa_2}{kappa_1 + kappa_2} K_q_3 {K_q_4}^{2} B_f = \frac{\frac{K_q_e0}{K_q_e1} kappa_1}{kappa_1 + kappa_2} K_q_1 {K_q_2}^{2} B_r = \frac{\frac{K_q_e0}{K_q_e1} kappa_1}{kappa_1 + kappa_2} K_q_3 {K_q_4}^{2} v_SS = \frac{A_f q_1 {q_2}^{2} - (A_r q_3 {q_4}^{2})}{1 + B_f q_1 {q_2}^{2} + B_r q_3 {q_4}^{2}}