To calculate the surface area of the activated carbon sample, we can use the Brunauer-Emmett-Teller BET theory, which relates the amount of gas adsorbed to the surface area of the adsorbent. The BET equation is:n = n_m * C * P / [ P_0 - P * 1 + C - 1 * P / P_0 ]where:n = amount of gas adsorbed in moles n_m = monolayer capacity in moles C = BET constant dimensionless P = partial pressure of the gas in atm P_0 = saturation pressure of the gas in atm At 77 K and 1 atm pressure, nitrogen gas behaves as an ideal gas, so we can assume that the partial pressure of nitrogen P is equal to the atmospheric pressure 1 atm . The saturation pressure of nitrogen P_0 is also 1 atm. Since the activated carbon adsorbed 2.5 moles of nitrogen gas, we can assume that the monolayer capacity n_m is equal to 2.5 moles. The BET constant C is typically in the range of 20-200 for activated carbon. However, the exact value of C is not provided, so we cannot determine the surface area using the BET equation.Instead, we can use the Langmuir isotherm, which is a simpler model that assumes a monolayer adsorption:n = n_m * K * P / 1 + K * P where:n = amount of gas adsorbed in moles n_m = monolayer capacity in moles K = Langmuir constant in atm^-1 P = partial pressure of the gas in atm Since we know that n = 2.5 moles and P = 1 atm, we can rewrite the equation as:2.5 = n_m * K / 1 + K However, we still do not have enough information to solve for n_m or K.Assuming that the entire 2.5 moles of nitrogen gas form a monolayer on the activated carbon surface, we can estimate the surface area using the cross-sectional area of nitrogen gas 16.2 square angstroms .Surface area = n * N_A * where:n = amount of gas adsorbed in moles N_A = Avogadro's number 6.022 x 10^23 molecules/mol = cross-sectional area of nitrogen gas 16.2 square angstroms Surface area = 2.5 moles * 6.022 x 10^23 molecules/mol * 16.2 square angstroms Surface area 2.45 x 10^25 square angstromsThe estimated surface area of the activated carbon sample is approximately 2.45 x 10^25 square angstroms. However, this is a rough estimate, as we made several assumptions and did not have all the necessary parameters to use the BET or Langmuir equations.