The image presents a derivation of some formulae used in meteorology, specifically related to the characteristic gas equation, pressure and height relationship, and adiabatic relations. It shows how pressure changes with height and temperature, and how these relationships are derived. We are asked to derive the equation for the height interval between two pressure levels p1 and p2, assuming constant temperature, based on the given equations.

1. Problem Description

The image presents a derivation of some formulae used in meteorology, specifically related to the characteristic gas equation, pressure and height relationship, and adiabatic relations. It shows how pressure changes with height and temperature, and how these relationships are derived. We are asked to derive the equation for the height interval between two pressure levels p1 and p2, assuming constant temperature, based on the given equations.

2. Solution Steps

First, let's identify the relevant equations.

The characteristic gas equation is given as:

$pv = RT$ (1)

Where $p$ is pressure, $v$ is volume, $R$ is a gas constant and $T$ is absolute temperature.

Also,

$p = \rho R T$ (2)

Where $\rho$ is density.

We are given that

$g\rho dz = -dp$ (A)

And, eliminating the density $\rho$ by means of equation (2), we obtain:

$dz = -\frac{RT}{gp} dp$ (3)

The height interval between two pressure levels $p_1$ and $p_2$ is obtained by integrating equation (3). Since the temperature is constant, we can write:

$\int_{z_1}^{z_2} dz = -\int_{p_1}^{p_2} \frac{RT}{gp} dp$

$h_2 - h_1 = -\frac{RT}{g} \int_{p_1}^{p_2} \frac{1}{p} dp$

Where $h_1 = z_1$ and $h_2 = z_2$

The integral of $\frac{1}{p}$ with respect to $p$ is $ln(p)$:

$h_2 - h_1 = -\frac{RT}{g} [ln(p)]_{p_1}^{p_2}$

$h_2 - h_1 = -\frac{RT}{g} (ln(p_2) - ln(p_1))$

$h_2 - h_1 = \frac{RT}{g} (ln(p_1) - ln(p_2))$

$h_2 - h_1 = \frac{RT}{g} ln(\frac{p_1}{p_2})$

If we are using units such that R/g is again 96, while if the logarithms are changed from base e to base 10, the conversion factor 2.303 must be introduced. This gives

$h_2 - h_1 = 221.1 T(log p_1 - log p_2)$ feet

$[h_2 - h_1] = 67.47 T(log p_1 - log p_2)$ metres.

3. Final Answer

The height interval is given by:

$h_2 - h_1 = \frac{RT}{g} ln(\frac{p_1}{p_2})$

Or,

$h_2 - h_1 = 221.1 T(log p_1 - log p_2)$ feet

$[h_2 - h_1] = 67.47 T(log p_1 - log p_2)$ metres