Hi, I am new here. In my chemistry book, I came across a derivation of the Boyle's law using calculus which I did not understand. Well, actually, it is a small part that I did not understand. My question is not purely based on chemistry, it is mostly mathematics, and so I am posting here. Anyway, here is how it goes:
There is an imaginary cylinder of base A and slant height ct, where c is the molecular speed and t, a small interval of time. Θ is the angle between the axis of the cylinder and a perpendicular from the wall. And Φ is the angle on the surface of the wall. Molecules in it are moving parallel to the axis, and hence have a perpendicular component to the wall c cosΘ. So, the momentum imparted on the wall by one such molecule will be 2mc cosΘ (where m is the mass of the molecule).
The number of molecules that move parallel to the axis = (Act cosΘ) × (N/V) × (dΦ sinΘ dΘ/4π) [where N is the total number of molecules and V is the total volume, and π is pi though it does not quite look like it].
(dΦ sinΘ dΘ/4π) is found by dividing (r2 sinΘ dΘ dΦ) by the total surface area of the sphere (4πr2). This is what I did not understand, how did they obtain (r2 sinΘ dΘ dΦ)?