Calculate pressure with the ideal gas law or $\rho gh$?

Question

In a container the pressure on the walls due to a gas can be calculated using the ideal gas law $\bigl(pV=NRT\bigr)$. However, for a column of water, or the pressure at earth's surface due to a column of air, the pressure can be calculated as $P=F/A=\rho gh$.

When does one use which model or is it two completely different principles?

For a liquid bridge between two spheres, would the water pressure be a function of the kinetic behaviour (ideal gas law) or the weight ($\rho gh$)?

Answer 1

Are you interested in the relative pressure difference between two points or the absolute pressure?

In a container the pressure on the walls due to a gas can be calculated using the ideal gas law (pV=NRT). However, for a column of water, or the pressure at earth's surface due to a column of air, the pressure can be calculated as P=F/A=ρgh.

The first example is for absolute pressure, and the second example is for relative pressure. If I have a glass of water in front of me, the pressure difference from bottom to top is about 1/100th of the total absolute pressure.

Liquids also have a state function, similar to air. You can calculate the absolute pressure of a liquid from its temperature and density, it's just not a very accurate calculation since it's a sharply sloping function.

Likewise, you can use an analog to ρgh for gases. For constant gravity, I would write:

$$ \Delta P = \mu g \\ \mu = \int \rho dh $$

This would be (mostly) valid for calculating the relative difference in air pressure between two cities at different altitudes. If density is constant, this is the same as your ρgh approximation.

Answer 2

Here it is an incomplete answer, but just to think about some issues.

An ideal gas is assumed to have particles non-interacting and with no extension (point-like particles). In particular for an ideal gas you discard gravity. While the Stevin's law (the relation $\rho g h$) is a direct consequence of the fact that you have a fluid in a gravitational field.

Answer 3

The ideal gas law is derived without consideration of the gravitational field. The kinetic behavior of a gas is, by far, more complex than the very simple example of the ideal gas law.

So, answering to your two questions:

1. Yes, the range of application of each equation is different.
2. Use the hydrostatic equation and not the ideal gas one. Anyway, kinetic theory os also correct here, but more complicated than the ideal gas law.