# Colder with increasing altitude.

by chingel
Tags: altitude, colder, increasing
P: 752
 Quote by chingel How is the energy not lost if the temperature drops? Isn't temperature average kinetic energy per molecule, and the number of molecules surely stays the same.
Energy is not lost. Its conserved. Though you can say that when temperature of a body drops , its total internal energy in transit , which it possesses , is reduced. But total energy in an isolated system is again conserved.
 Homework Sci Advisor HW Helper Thanks P: 9,651 If a parcel of air expands a little, for whatever reason, it expands the total volume of the atmosphere that much. This means the average height of the molecules in it has increased. So work is done adding to the potential energy. This is quite different from a closed box, and the reason it does not warm another parcel of air. I think a lot of the confusion over this topic arises because people think of this process as causing the air to be colder higher up. If you could stop the convection, by inserting baffles all the way up, the temperature gradient would be far steeper. The tendency of air to cool when lofted to a lower pressure altitude inhibits such movements, and they only occur when the temperature gradient is steep enough to overcome it. So the expansion doesn't cause it to be colder higher up, it just limits convection's ability to prevent its being so. At the tropopause, the gradient is no longer steep enough and convection largely ceases.
 P: 474 A cube shaped block of air whose volume is one cubic kilometers expands with force 100 Giga Newtons. That is the force of air pressure on one side of the cube. If the volume increases 20 %, the energy of expansion will be 200 meters * 100 Giga Newtons = 20 Tera Joules. That's 20 Kilo Joules per one cubic meter, which has mass of about 1 kg, which cools about 20 degrees when 20 Kilo Joules of energy leaves it. Now we want to know where the energy goes. Well it goes everywhere at speed of sound. Because the expansion causes a slight air pressure increase everywhere.
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P: 9,651
 Quote by jartsa the expansion causes a slight air pressure increase everywhere.
No. The atmosphere is not a closed container.
Average air pressure at Earth's surface must be total weight of atmosphere divided by area. Increasing the volume of some of the air does not increase its total mass. If anything, it will decrease total weight slightly because the average height of the air molecules increases.
The energy goes into the atmosphere's gravitational potential energy.
P: 474
 Quote by haruspex No. The atmosphere is not a closed container. Average air pressure at Earth's surface must be total weight of atmosphere divided by area. Increasing the volume of some of the air does not increase its total mass. If anything, it will decrease total weight slightly because the average height of the air molecules increases. The energy goes into the atmosphere's gravitational potential energy.
Sounds reasonable.

But the energy to lift the atmosphere travels around the atmosphere as pressure wave and at speed of sound.
P: 254
 Quote by haruspex No. The atmosphere is not a closed container. Average air pressure at Earth's surface must be total weight of atmosphere divided by area. Increasing the volume of some of the air does not increase its total mass. If anything, it will decrease total weight slightly because the average height of the air molecules increases. The energy goes into the atmosphere's gravitational potential energy.
As some amount of air rises and expands, other air must take its place. Wouldn't the overall potential energy of the atmosphere stay the same because the average height of the air molecules stays the same, they just swap places?
P: 614
 Quote by DaveC426913 No, it does not cool by conduction, which is what you're describing. It cools because, the same amount of thermal energy is distributed throughout a larger volume. This is called adiabatic cooling. If you put a gallon of air in an insulated container that had a piston, then operated the piston to increase the volume to 2 gallons, the air's temperature would drop. This is Charles' Gas Law. V1 / T1 = V2 / T2 So, if V1 doubles to V2 then T1 will be halved to T2
Dave, if

$\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}$

and V2 = 2V1

then

$\frac{V_{1}}{T_{1}}=\frac{2V_{1}}{T_{2}}$

∴ T2 = 2T1

from which we end up with Charles's Gas Law, which states that, at constant pressure, the volume of an ideal gas is directly proportional to its temperature.

http://en.wikipedia.org/wiki/Charles%27s_law
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P: 9,651
 Quote by chingel As some amount of air rises and expands, other air must take its place. Wouldn't the overall potential energy of the atmosphere stay the same because the average height of the air molecules stays the same, they just swap places?
Of course, but I was just looking at the immediate consequence of a parcel of air expanding in situ, e.g. by being warmed. The question was, where has the energy gone? The suggestion had been that it had gone into compressing the air around it. Readjustment by convection comes later.
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P: 9,651
 Quote by SHISHKABOB Dave, if $\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}$ and V2 = 2V1 then $\frac{V_{1}}{T_{1}}=\frac{2V_{1}}{T_{2}}$ ∴ T2 = 2T1 from which we end up with Charles's Gas Law, which states that, at constant pressure, the volume of an ideal gas is directly proportional to its temperature. http://en.wikipedia.org/wiki/Charles%27s_law
Well spotted. DaveC's description clearly does not meet the constant pressure requirement of Charles' Law. To meet that, heat would need to be added to maintain the pressure, and at a doubled volume the temperature would also double.
P: 254
 Quote by haruspex Of course, but I was just looking at the immediate consequence of a parcel of air expanding in situ, e.g. by being warmed. The question was, where has the energy gone? The suggestion had been that it had gone into compressing the air around it. Readjustment by convection comes later.
OK, but once the air is already warmed, what stops it from rising and warming the upper layers. If it rises and cools, mustn't the energy go to the other air around it and if it does, doesn't it succed in warming the upper layers (ie give off its energy at a higher altitude as heat to the air around it)? Why aren't the upper layers then warmed by rising hot air?
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P: 9,651
 Quote by chingel OK, but once the air is already warmed, what stops it from rising and warming the upper layers. If it rises and cools, mustn't the energy go to the other air around it and if it does, doesn't it succed in warming the upper layers (ie give off its energy at a higher altitude as heat to the air around it)? Why aren't the upper layers then warmed by rising hot air?
Having expanded due to being heated, it rises (convection) because it is less dense than the air around it. As it rises, it encounters a slightly lower pressure, so expands a little more. This further expansion does more work in the same way as before, cooling the air adiabatically. There will also be some mixing, but in a large parcel of air that's a second order effect.
This rising-expanding-getting dense cycle has diminishing returns, so settles out when the air is at the same temperature as surrounding air. (This is for dry air - moist air is more complicated.)
At the same time, some air must descend to occupy the space vacated. That air gets compressed adiabatically, warming.
Looking at the total picture, the air as a whole is a little warmer than before (we added heat to kick this off), so is a little less dense, so the atmosphere extends a little further into space and has acquired a little extra potential energy. Thus the heat energy lost by the ascending air has not quite all gone into heating the descending air.
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P: 11,941
 Quote by chingel But if on the other side of the piston you also had air whose volume would decrease by the same amount, wouldn't the energy lost one one side of the piston be transferred to the other side? And isn't it the same way in the case of expanding air, air does work on other air?
Yes, but it is Mechanical Energy (Work) that is transferred, not thermal energy. What happens inside the cylinder doesn't depend on what the work happens to act on.
Mentor
P: 22,243
 Quote by SHISHKABOB Dave, if $\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}$ and V2 = 2V1 then $\frac{V_{1}}{T_{1}}=\frac{2V_{1}}{T_{2}}$ ∴ T2 = 2T1 from which we end up with Charles's Gas Law, which states that, at constant pressure, the volume of an ideal gas is directly proportional to its temperature. http://en.wikipedia.org/wiki/Charles%27s_law

What you are describing is what happens at the surface: it is heated and expands at constant pressure. That's what causes it to rise.
Mentor
P: 22,243
 Quote by haruspex Well spotted. DaveC's description clearly does not meet the constant pressure requirement of Charles' Law. To meet that, heat would need to be added to maintain the pressure, and at a doubled volume the temperature would also double.
Which is why that's not what happens. Again, Dave said adiabatic cooling.

http://apollo.lsc.vsc.edu/classes/me...diab_cool.html
P: 614
 Quote by russ_watters Adiabatic cooling does not happen at constant pressure.
well, right, I was just pointing out the error in terminology

I was *only* pointing out that Charle's Gas Law, which Dave mentioned, requires constant pressure. It was offtopic, I guess, technically, because I wasn't trying to add anything to the discussion of why it gets colder with increasing altitude.
 Mentor P: 22,243 Yes, you're right, sorry. Dave used the right term for what was happening but applied the wrong equation. That's what I get for only reading half a post.
PF Gold
P: 11,941
 Quote by chingel I have read some threads on this topic, but I am still confused. Why does temperature drop as you go up a mountain? I have read that since pressure depends on the weight of the air on top of you and as you increase your altitude the amount of air over you decreases, therefore pressure decreases, gas expands and expansion makes its temperature drop. The rising hotter air gets continually cooled due to expansion (or would it start rising at all?). Gas loses internal energy when it has to expand against a force, i.e. do work. But isn't exactly the same amount of work the gas does received as internal energy of the lump of gas next to it that the work is being done upon?
This thread seems to have lost its way a bit (so what's new?)
I quote the original question and the simplest answer must explain the steady state, static condition, not involving convection / mixing / adiabatic cooling / heating by the ground etc..
Consider a simple column of a gas air, in equilibrium in an insulated cylinder. In an equilibrium situation the air at the bottom of the column will not rise if it is MORE DENSE than the air above it. This can occur even when it is 'warmer', as long as the pressure is greater. So a temperature / pressure profile can occur which will not allow convection but where the temperature at the bottom is higher than the temperature at the top.
The atmospheric pressure is approximately halved for every 5km increase in height (for constant temperature). Applying
P1V1/T1 = P2V2/T2
to this simple model of a column seems to indicate that a mass of air at sea level at a temperature 300K will occupy a smaller volume than the same mass of air at 5km height (half the pressure), if the temperature at 5km is greater than 150K. (Someone else please check this)
This result is extreme and not realistic but it makes the point, I think. There are many other factors at work but it does put to bed the notion that the Hot Air must rise up through the Cold Air.
Mentor
P: 15,066
 Quote by sophiecentaur Consider a simple column of a gas air, in equilibrium in an insulated cylinder. In an equilibrium situation the air at the bottom of the column will not rise if it is MORE DENSE than the air above it.
The equilibrium condition for your isolated column of air is hydrostatic equilibrium and a uniform temperature throughout. This is the condition that maximizes entropy. There is a non-equilibrium local max in entropy for your isolated column. This local max also is in hydrostatic equilibrium but has temperature falling with increased altitude at the adiabatic lapse rate.

This isolated column is not a good model of the Earth's atmosphere. The Earth's atmosphere is not an isolated system and it is far from thermal equilibrium. The atmosphere is primarily heated from below and radiates into space from above.

The atmosphere is typically closer to that local max (adiabatic lapse rate) than it is to the global max (constant temperature), so the atmosphere is typically driven toward that local max in which temperature decreases with altitude. But not always. Sometimes thermal inversion layers set up in the atmosphere. Rising air stops at the inversion layer -- until it finally punches through. That's when all kinds of havoc such as tornados can result.

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