# Applying the ideal gas law in the Earth's atmosphere

• Alexander83
In summary, the conversation discusses the application of the ideal gas law in our planet's atmosphere and its relation to the hydrostatic equilibrium. The question at hand is whether pressure can directly cause changes in temperature in the atmosphere, or if temperature is the driving force for pressure and density. The ideal gas law (P = ρRT) and the hydrostatic equation (dP/dz = -ρg) are used to understand the dynamics of pressure, temperature, and density in the atmosphere. The conversation also mentions the role of meteorologists in modeling these changes and the use of the adiabatic lapse rate as a simplified model for these factors. Ultimately, to apply the hydrostatic equation, one must know the vertical temperature profile and
Alexander83
Hi there,
I'm considering how the ideal gas law applies in practice in our planet's atmosphere. In particular, I'm considering this form of the law:

P = ρRT (1)

where P is pressure, ρ is density, R is the gas constant and T is the temperature.

I also know that, to a good approximation, the atmosphere is in hydrostatic equilibrium in which case a second equation is:

dP/dz = -ρg (2)

which is basically a statement saying that the gravitational force on a parcel of air is balanced by the vertical pressure gradient force.

My question has to do with cause and effect changes in, say, the pressure and temperature at a point on the surface. In particular, what I'm trying to detangle in my mind is which of the three variables in (1) actually drives and manipulates the other 2. I want to say that temperature, which is set by other environmental factors such as surface temperature and insolation determines atmospheric pressure and pressure and temperature together determine the density.

For instance, I understand that, if the atmospheric temperature profile is known, one can substitute (1) into (2) and integrate the resulting differential equation over the height of the atmosphere to determine the surface pressure. In this way, it's clear that the surface pressure must depend on temperature, not just at the surface, but through the entire atmospheric column and so changes in temperature (as determined by changes in solar insolation, cloud cover etc...) can cause changes in pressure.

What I'm trying to detangle in my mind is whether pressure can ever directly change temperature. (By direct, I mean, considering just the change in pressure, not changes in things like cloud cover). For instance, if a low pressure centre enters the region and the air pressure drops, then applying the ideal gas law (1) suggests that either temperature or density must also change. In this scenario would the air temperature change or would it simply be the density that would change? I feel that density is the factor in (1) that is always dependent on the other two, but wanted to confirm whether my intuition is correct.

Alex.

Yes, the density follows from the pressure and temperature. And, as you are aware, there are local meteorological changes in both the pressure and the temperature profiles. The meteorologists include both dynamical and thermodynamic effects in modeling these changes. Under average conditions, at least in the troposphere, the temperature varies vertically (approximately) according to the adiabatic lapse rate, which provides a simplified model of these factors. Basically, to apply the hydrostatic equation, you need to know the vertical temperature profile and the pressure at the surface.

Chet

## 1. What is the ideal gas law?

The ideal gas law is a fundamental equation in thermodynamics and describes the behavior of an ideal gas. It states that the pressure of a gas is directly proportional to its temperature and the number of molecules present, and inversely proportional to its volume.

## 2. How is the ideal gas law applied in the Earth's atmosphere?

The ideal gas law is used to study the behavior of gases in the Earth's atmosphere, such as air. It helps scientists understand how changes in temperature, pressure, and volume affect the behavior of gases in the atmosphere.

## 3. What are the units of measurement used in the ideal gas law?

The units of measurement used in the ideal gas law are pressure (P) in Pascals (Pa), volume (V) in cubic meters (m^3), temperature (T) in Kelvin (K), and the number of moles (n) of gas particles.

## 4. Are there any limitations to using the ideal gas law in the Earth's atmosphere?

Yes, there are limitations to using the ideal gas law in the Earth's atmosphere. The ideal gas law assumes that the gas molecules have no volume and do not interact with each other, which is not the case in reality. Additionally, it is only accurate for gases at low pressures and high temperatures.

## 5. How is the ideal gas law used in weather forecasting?

The ideal gas law is used in weather forecasting to predict changes in atmospheric pressure and temperature, which in turn affect weather patterns. It is also used to understand the behavior of gases in the atmosphere, such as the movement of air masses and the formation of clouds.

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