Following up from our previous equation, it is necessary to consider the possible variation in  before the equation can be integrated. While we already know what compressible gases are, it is equally important to note that since specific weights of gases are comparatively small compared to liquids, it follows from

that the pressure gradient in the vertical direction is correspondingly small, or even over distances of several hundred feet the pressure will remain essentially constant for a gas. This implies that we can neglect the effect of elevation changes on the pressure in gases in tanks, pipes, in which the distances involved are small.

Should the variations in heights be large, on the order of thousands of feet, attention must be given to the specific weight. Our way around this is to use the equation of state for an ideal gas.

where p is the absolute pressure, R is the gas constant, and T is the absolute temperature. Combing these equations, we get

and by separating variables

where g and R are assumed to be constant over the elevation change from  and . Changes in g are usually small and so we assumed it to be constant at some average value for the range of elevation involved.

We still have one more condition to set. Notice that temperature T may also vary due to the change in height. Hence, we assume the temperature to be a constant  over the range  and . This is called an isothermal condition, quite often used in analysis of liquids. By integrating, we get

This equation provides the desired pressure-elevation relationship for an isothermal layer.