Layer Thickness from Mean Temperature (Hypsometric Equation)

Warm air makes a thick layer. Pressure falls as you go up, so two pressure surfaces such as 1000 hPa and 500 hPa sit a certain distance apart, and that distance depends on how dense the air between them is. Warm air is less dense, so the same drop in pressure takes more height and the layer is thicker. The hypsometric equation $\Delta z = \frac{R_d\,\overline{T}}{g}\,\ln\frac{p_1}{p_2}$ puts a number on that, where $R_d$ is the gas constant of dry air, $g$ is gravity, $\overline{T}$ is the mean temperature of the layer, $p_1$ is the pressure at the bottom and $p_2$ the lower pressure at the top.

Where it comes from, and which temperature to use. It is the hydrostatic balance combined with the ideal gas law, the same balance that gives pressure at a single height from the surface pressure, here applied between two pressure levels. Wallace and Hobbs set out the hypsometric equation in chapter 3 of Atmospheric Science (2006). The temperature to use is the mean through the layer, and for the best accuracy it is the mean virtual temperature, since moist air is slightly lighter than dry air. The mean that belongs here is averaged in log pressure: for the standard atmosphere, with 288.15 K at the surface falling 6.5 K per km, the layer from 1000 to 500 hPa has a mean of 269.28 K.

Checking it. With that 269.28 K, 1000 to 500 hPa gives 5,463 m, the thickness of the ICAO standard atmosphere, whose 1000 hPa surface is at 111 m and whose 500 hPa surface is at 5,574 m. The 1000 to 500 hPa thickness of about 5,400 m is the old rule-of-thumb dividing line between rain and snow. A second check uses the Navy aerographer's shortcut, which the manual calls accurate within 2 percent. For 1000 to 850 hPa at 32 °F it gives 1,297.6 m. The shortcut replaces $\ln(p_1/p_2)$ with $2(p_1 - p_2)/(p_1 + p_2)$, which suits thin layers but runs 4 % low across 1000 to 500 hPa, so the thin layer is the fair comparison.

Limits. Only the ratio of the two pressures matters, so any pressure unit works, but $p_1$ must be the higher. The answer is a thickness, the height of the upper surface above the lower, and it is only as good as the mean temperature. The equation can also be run backward for the mean temperature or for either pressure.