To calculate the surface area of the activated carbon sample using the BET theory, we need to determine the amount of gas adsorbed per unit mass of the sample and then use the BET equation to find the surface area. First, let's find the amount of gas adsorbed per unit mass of the sample:Amount of nitrogen gas adsorbed = 2.5 molesSample weight = 3 gramsMolecular weight of nitrogen = 28 g/molAmount of gas adsorbed per gram of sample = 2.5 moles / 3 grams = 0.833 moles/gramNow, we can use the BET equation to find the surface area. The BET equation is:V = V_m * C * P / [ P_0 - P * 1 + C - 1 * P / P_0 ]Where:V = volume of gas adsorbed at pressure PV_m = volume of gas adsorbed at monolayer coverageC = BET constantP = pressure of the gasP_0 = saturation pressure of the gasIn this problem, we are given the pressure P as 1 atm and the temperature T as 77 K. We need to find the saturation pressure P_0 of nitrogen gas at 77 K. The saturation pressure can be found using the Antoine equation:log P_0 = A - B / T + C For nitrogen, the Antoine constants are:A = 6.89272B = 840.0C = 237.0Let's find P_0:log P_0 = 6.89272 - 840.0 / 77 + 237 P_0 = 10^6.89272 - 840.0 / 314 P_0 0.835 atmNow, we can use the BET equation to find V_m and C. However, we are not given the values of V_m and C in the problem. In practice, these values are determined experimentally by measuring the adsorption isotherm and fitting the BET equation to the data. Without these values, we cannot directly calculate the surface area using the BET equation.If we had the values of V_m and C, we could calculate the surface area using the following equation:Surface area = V_m * N_A * A_cross / molecular weight of nitrogen Where:N_A = Avogadro's number 6.022 x 10^23 molecules/mol A_cross = cross-sectional area of a nitrogen molecule 0.162 nm Unfortunately, without the values of V_m and C, we cannot proceed further in calculating the surface area of the activated carbon sample using the BET theory.