To calculate the surface area of the powdered 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. In this case, we are given the amount of nitrogen gas adsorbed and the cross-sectional area of a nitrogen molecule.First, we need to convert the volume of nitrogen gas adsorbed 25.6 mL/g to moles. We can do this using the ideal gas law:PV = nRTwhere P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.Assuming standard pressure 1 atm and the given temperature of -196C 77 K , we can solve for n: 1 atm * 25.6 mL/g = n * 0.0821 L atm/mol K * 77 K Rearranging and solving for n:n = 1 atm * 25.6 mL/g / 0.0821 L atm/mol K * 77 K = 0.0102 mol/gNow, we can calculate the surface area of the activated carbon sample using the cross-sectional area of a nitrogen molecule 16.2 and the number of moles of nitrogen adsorbed:Surface area = 0.0102 mol/g * 6.022 x 10^23 molecules/mol * 16.2 /molecule Surface area = 1.00 x 10^22 /gTo convert the surface area to a more common unit, such as m/g, we can use the conversion factor:1 = 1 x 10^-20 mSurface area = 1.00 x 10^22 /g * 1 x 10^-20 m/ = 1000 m/gSo, the surface area of the powdered activated carbon sample is approximately 1000 m/g.