Hydrostatic equation giving the relation between height and pressure at a given temperature :
\(\frac{dp}{dz} = -\rho g\)
Consider a thin horizontal slab of air of thickness \(dz\) and unit area. Its weight is \(\rho g\,dz\), where \(\rho\) is air density and \(g\) is gravitational acceleration.
In hydrostatic balance the pressure at the bottom of the slab exceeds that at the top by exactly this weight: \(p - (p + dp) = \rho g\,dz\).
Rearranging gives \(\frac{dp}{dz} = -\rho g\); the minus sign shows that pressure decreases with height.
The other forms are dimensionally wrong: \(\rho T\) and \(\rho / T\) have units of density times or divided by temperature, and \(\rho / g\) does not have units of pressure per length. Temperature enters only through \(\rho = p/(RT)\), which leads to the hypsometric equation.
Therefore the hydrostatic equation is pressure gradient equals minus density times gravity.
Hence, \(\frac{dp}{dz} = -\rho g\).
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