homework and exercises - Pressure and altitude

I am going to ask a simple question, for sure.

The pressure with respect to the altitude is given by this formula

Where

• sea level standard atmospheric pressure p0 = 101325 Pa


• sea level standard temperature T0 = 288.15 K


• Earth-surface gravitational acceleration g = 9.80665 m/s2.


• temperature lapse rate L = 0.0065 K/m


• universal gas constant R = 8.31447 J/(mol·K)


• molar mass of dry air M = 0.0289644 kg/mol

( from Wikipedia )

In addition to this, we have $L = \frac{g}{C_p}$ where $g = 9.80665\ m/s^2$ and $C_p$ is the constant pressure specific heat $= 1007\ J/(Kg\ K)$

Understood and thence the above formula can be written in this simple way:

$$\boxed{P = P_0\left(1 - \alpha h\right)^{\beta}}$$

where

$\alpha = \frac{g}{C_p\ T_0} \approx 3.3796\cdot 10^{-5}\ m^{-1}$

$\beta = \frac{C_p\ M}{R} \approx 3.5081971$

On the other side we learnt from the elementary physics that the formula for the pressure is also given by

$$\boxed{P = P_0 + \rho g h}$$

Where $\rho$ is the air density $(1.23\ kg/m^3)$.

The question

First thing first: I assumed that

$$h = h_1 - h_0$$

is that correct? I mean it's the difference between two heights (maybe from a table and the ground, just to say).

Since the two formulae seems quite different, I tried with a numerical example in calculating the pressure in two points, with a difference of height about $0.18$ m and I got a really similar result.

Since the first formula is more technique, I think it's the correct formula but I would like to understand if one could pass from the first to the second or vice versa somehow.

Also I would like to know if there are cases in which I can use only the second formula or only the first formula!