Answer:
First, we need to calculate the number of moles of nitrogen gas required to form a monolayer:
n = (pv) / (rt)
where p is the pressure, v is the volume, r is the ideal gas constant, and t is the temperature in Kelvin.
At standard temperature and pressure, we have:
p = 1 atm
v = 7.6×10^3 mm^3 = 7.6×10^-6 m^3
t = 273 K
r = 8.31 J/(mol K)
So, n = (1 atm x 7.6×10^-6 m^3) / (8.31 J/(mol K) x 273 K) = 3.13×10^-7 mol
Next, we can calculate the number of nitrogen molecules in this amount of gas:
N = n x Na
where Na is Avogadro's number (6.02×10^23 molecules/mol).
N = 3.13×10^-7 mol x 6.02×10^23 molecules/mol = 1.88×10^17 molecules
Finally, we can calculate the surface area of the catalyst covered by these molecules:
A = N x a
where a is the area covered by a nitrogen molecule (0.162 nm^2), converted to m^2.
a = 0.162 nm^2 x (10^-18 m^2/nm^2) = 1.62×10^-20 m^2
A = 1.88×10^17 molecules x 1.62×10^-20 m^2/molecule = 3.05×10^-3 m^2
Therefore, the surface area of the catalyst covered by the nitrogen molecules is approximately 3.05×10^-3 m^2.