A few months ago, I made a post where I derived the Maxwell-Boltzmann distribution formula. Now I'll show you how to derive a not-so- esoteric law: PV=vRT
You should read the article from July where I introduced the multiplicity Ω.

       The multiplicity is proportional to the disorder degree. Think of a particle in an empty box. There are more posible positions of the particle if the volume V is bigger. So ,Ω must be proportional to V. What is more, the momentum of the particle matters as well, therefore a relation between the momentum of the particle, which has components on the three axes, and the multiplicity ,must be found.
       The relations and formulas I found in books are very complicated( it uses the area of an N-dimensional sphere, etc). However, I found by myself a simpler way( which is not as accurate,but works just fine for this purpose). 
        First of all, I'll introduce the  bi dimensional phase space. It is actually simple, it's a graph that shows the momentum depending on the position of the body.

This graph is just for a linear movement( it shows hoe px varies with x coordinate). We can make a six dimensional phase space to represent a tri dimensional movement. The axes will be px, py, pz ,x, y, z( it is not  very difficult to imagine).
  We have to finf out how many states( positions) of the particle are possoble in the 6 dimensional phase space. Let's note the dimensions of the box X, Y, Z and the components of the particle momentum Px, Py, Pz. Thus, the total phase volume is XYZPxPyPz. But how can we find the number of positions? Remember the Heisenberg incertitude principle  

So, Ω=C X*Y*Z*Px*Py*Pz/h^3, where C is a constant

Therefore, S=klnV+kln PxPyPz+ C2, k is Boltzmann constant

The energy of a particle in the ideal gas U= Px^2+Py^2+Pz^2/  2m

over long periods of time: <Px^2>=<Py^2>=<Pz^2> , so U=3Px^2/ 2= 3Py^2/ 2=.....=>

PxPyPz=C3 U^1.5

so,
 S=klnV+1.5klnU+ C2

remember this is for one particle

for N particles we can sum the entrophies

so S=Nkln V+ 1.5NklnU+ C


Finnaly, I managed to write the entrophy as a function of volume and internal energy of an ideal, monoatomic gas.

Next, I'll show you  how temperature and pressure are defined.

1) temperature

Imagine two bodies that can ineteract and change only energy, so that the first one has an entrophy Sa and internal energy Ua and the second, Sb and Ub. Ua+Ub=Utoatal= constant

The graph from above shows how Sa(blue), Sb( red) and Stoatal=Sa + Sb varies when Ua changes

The bodies continue to exchange energy,until dStotal/dU=0, or when dSa/dUa+ dSb/dUa=0

which means( regarding that dUa=-dUb)

dSa/ dUa  = dSb/dUb in that moment. 

but, as you know at, the termal equilibrium betweeen 2 bodies Ta= Tb( temperatures)

also, it is logical that a hotter body( with a bigger T) varies it's entrophy harder than a cooler body for the same amount of energy gain.

so, the definition of temperature is:( E is energy)
(remember that volume and number of particles of the body remain constant)

Pressure

Something similar happens when 2 bodies can exchange only volume

so that dSa/dVa= dSb/dSb at equilibrium

thus, pressure defines as

P= T dS/ dV, the term T has been added in order to respect the physical unit measure(Pa)

if we use this final equation

and  S=NklnV+1.5NklnU+ kC2

we will really obtain that P=NkT/V=> PV=NRT/Na=>PV=vRT