MadSci Network: Physics Query:

### Re: Why does it get colder as you go up a mountian?

Date: Fri Aug 28 10:35:50 1998
Posted By: Jason Goodman, Graduate Student, Massachusetts Institute of Technology
Area of science: Physics
ID: 903653176.Ph
Message:

The air at the ground is at a higher pressure than air higher up; this is because the air at the ground is compressed by the weight of all the air above it. Air weighs quite a lot: A square column of air one meter across weighs ten tonnes! The air pressure decreases steadily as you go up, since there's less air pressing down from above, until you reach zero pressure, which is outer space.

If you've ever opened a container of compressed gas, like a bottle of carbon dioxide or nitrogen, you'll notice that the air coming out of it is very cold. (Even a bottle of soda does this a little; it's the cold that makes it "steam" briefly when opened.) Reducing the pressure makes most gases colder.

In what follows, I'll talk a lot about "air parcels". You should imagine a "chunk" of air, identical to the air around it, but identified by drawing an imaginary boundary around it, or by marking each of the molecules in it with a little tattoo.

Now, the air in the lower atmosphere (the "troposphere") is constantly mixed by convection, which results from sunlight heating the ground. So air parcels are being constantly lifted from the ground to high altitude and back again. As a chunk of air rises, its pressure decreases, and so its temperature drops. Descending parcels are warmed as they're compressed. This is what makes the air cooler higher up: the process is called adiabatic expansion/compression.

But why do gases cool when they're compressed? I'll give a verbal and a mathematical explanation. Imagine a parcel rising from the ground high into the atmosphere without gaining any net energy. As it rises, it is gaining gravitational potential energy. This energy must come from somewhere: the only source of energy is the internal thermal energy of the molecules in the parcel -- the energy in the shaking and jiggling of the molecules. If the molecules shake and jiggle less, it means they must have a lower temperature.

The mathemtical explanation: you may have heard of the ideal gas law:

``` PV = nRT					(1)
```
which says that gas pressure (P) times volume (V) is proportional to the temperature (T). (n is the number of moles of gas molecules in the parcel, and R is a fundamental constant; both remain the same in this problem). So we see that when P goes down (on top of the mountain), T also goes down. But wait! The volume V of the gas goes up at the same time, which would increase the temperature! Which of P and V wins?

To find the answer, we need the "adiabatic expansion" equation, which says that if no energy is gained or lost by an air parcel,

```P Vgamma = constant = P0 V0gamma	(2)
```
where P0 is the pressure and V0 is the volume before we expanded or compressed the gas. Putting equation 1 into equation 2, we can eliminate the volume V:
```
P/P0 = (V/V0)(-gamma)
V = V0 (P/P0)(-1/gamma)
PV = P V0(P/P0)(-1/gamma)

(n R)
P(1-1/gamma) = -----  P0(-1/gamma) T
V0
```
Since n, R, P0, and V0 are all constants, this is an equation relating pressure P to temperature T for a parcel. As long as (1-1/gamma) > 0, T will decrease whenever P decreases (as we go up in the atmosphere). gamma = 1.4 for air, giving (1-1/gamma) = .28 > 0. Therefore, the atmosphere gets colder as you go up and pressure decreases.

Why does the ideal gas law hold? Where does the adiabatic expansion law come from? Why is gamma = 1.4 for air? These are much harder questions, dealt with in college thermodynamics courses, and beyond the scope of this question.

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