Say I have a sphere that has a volume of 0.330 liters, contains 0.075 moles of an ideal gas and has a temperature of 293 K. I need to calculate...

In this question, 

V = 0.330 l

T = 293 K

and n = 0.075 moles

As you rightly stated, we can use the ideal gas law (PV = nRT) to determine the total pressure exerted by the gas on the container.

And hence, P = nRT/V = (0.075 x 0.0821 x 293)/(0.330) = 5.47 atm. (about the same as you have mentioned).

To determine the pressure exerted by each atom, we need to determine the total number of atoms in the sphere. 

1 mole of any substance has an Avogadro's number of atoms. Here we have 0.075 moles of gas. Thus, the total number of atoms are

= =  atoms.

Since the total pressure exerted by the gas is 5.47 atm, the pressure exerted by each atom can be calculated as:

Pressure by each atom = total pressure/number of atoms

=   atm/atom

 =  atm/atom.

Thus, each atom exerts about   atm pressure.

For a sphere of volume 0.33 l, we can calculate its radius.

Volume =   =0.33 l = 330 cm^3

solving this equation, we get, r = 4.29 cm

The surface area of the sphere is   = 231.27 cm^2

Area per atom is 231.27 cm^2 /4.52 x 10^22 atoms = 5.12 x 10^-21 cm^2/atom = 5.12 x 10^-25 m^2/atom. 

Since, force is the product of pressure and area, the force exerted by each atom can be calculated as:

Force = pressure x area = 

=   N/atom

Here a very simplifying assumption has been made that each atom is on the surface of the container and is applying force only on the container surface and not each other. 

Hope this helps.