I don't think that is really appropriate for this site, but whatever. Right now I cannot find a single written discussion encompassing the whole derivation, but there are basically three steps: (1) derivation of the ideal gas law, (2) derivation of the equation for an ideal gas in an adiabatic process, and lastly (3) modeling longitudinal sound waves as the limiting process of a series of springs which expand and contract according to the adiabatic process.

(1) If you start by considering a gas as a collection of discrete particles, then set up the system using Hamiltonian mechanics, and assuming ergodicity, then we can use the equipartition theorem to derive the ideal gas law. You can find this derivation in any statistical mechanics book.

(2) To derive the equation for an ideal gas undergoing an adiabatic process, one considers the rotational and translational degrees of freedom of the gas molecule (vibrational modes are ignored). Notice that, in this derivation, we DO NOT assume that the gas is made up of spheres; the molecule structure is important. You can find this derivation in any thermodynamics book.

(3) Now we think of sound waves as a series of springs with a variable spring constant given by the adiabatic expansion and contraction of the air. You can find this derivation in a number of places. When I use Google, I find a number of sources.

The primary resources I used for the above were "Thermal Physics" (Kittel, Kroemer), and "Waves" (Frank Crawford).