The Brunauer-Emmett-Teller BET theory is a widely used method to calculate the specific surface area of a solid material. The BET equation is given by:N = N_m * C * P / [ 1 - P * 1 + C - 1 * P ]where:N = the quantity of gas adsorbed at a certain pressure moles N_m = the quantity of gas required for a monolayer coverage moles C = the BET constant, which is related to the heat of adsorptionP = the relative pressure P/P0 , where P0 is the saturation pressure of the adsorbateIn this problem, we are given N = 0.25 moles, the weight of the catalyst used is 0.5 g, and the monolayer coverage of the adsorbate is 0.2. We are not given the values of C and P, so we cannot directly calculate the specific surface area using the BET equation. However, we can still estimate the specific surface area using the monolayer coverage information.The specific surface area A can be calculated using the following equation:A = N_m * V_m * A_m / mwhere:A = specific surface area m/g N_m = the quantity of gas required for a monolayer coverage moles V_m = the molar volume of the gas L/mol A_m = the molecular area of the adsorbate m/mol m = the weight of the catalyst used g We are given N_m = 0.2, V_m = 22.4 L/mol, and m = 0.5 g. To calculate A_m, we can use the following equation:A_m = V_m * 10^-3 / N_Awhere:A_m = the molecular area of the adsorbate m/mol V_m = the molar volume of the gas L/mol N_A = Avogadro's number 6.022 x 10^23 molecules/mol A_m = 22.4 * 10^-3 / 6.022 x 10^23 A_m 3.72 x 10^-20 m/molNow, we can calculate the specific surface area A :A = 0.2 * 22.4 * 3.72 x 10^-20 / 0.5A 3.34 x 10^-19 m/gThe specific surface area of the catalyst is approximately 3.34 x 10^-19 m/g. Please note that this value is an estimation, as we do not have the values of C and P to directly apply the BET equation.