Suppose the pressure at height h is p and that at `h+dh is p+dp. Then `
`dp=-rhog dh. …(i)`
Now considering any small volume `DeltaV` if air of mass
`Deltam, `
`pDeltaV=nRt=(Deltam)/(M)RT`
`p=(Deltam)/(DeltaV) (RT)/(M)=(rhoRT)/(M)`
`or, =rho=(M)/(RT)p.`
Putting in (i),
`dp=-(M)/(RT)pg dh`ltbr.`or, `int_(p_(0))^(p)(dp)/(p)=int_(0)^(h)-(M)/(RT)g dh`
`or, In(p)/(p_(0))=-(Mgh)/(RT)`
where p_(0) is the pressure at h=0.`
Thus, `p=p_(0) e^(-(Mgh)/(RT)).`