Do metal springs really store sig work as potential?

[vanesch]

This is an interesting discussion, and the OP raised an interesting point that BTW never occurred to me, but I think he's right!

Indeed, the internal energy of an ideal gas is function only of temperature:
u = u(T), and not of the other thermodynamical variable (pressure, entropy, density, whatever).

As pointed out before, u is the macroscopic version of the hamiltonian, and given that an ideal gas has no potential interaction energy, it is purely kinetic energy, which is purely determined by the temperature.

So, what if we use a gas bottle as a spring ? We can do that in two ways: we can do it relatively quickly (like, in looking at the oscillations of a weight on a piston), or we can do it slowly (fill a tank, and come back a few days later).
The first one is adiabatic, the second one is isothermal.

In the adiabatic case, the potential energy of the gas-spring is stored in the gas, and this is simply done by the increase in temperature. So this is the simple "storage of energy in the medium as 'potential' energy". Only, it is not really microscopically potential energy, but rather internal energy in this case ; nevertheless, macroscopically, we can call this the "potential energy of the gas spring" as long as we don't "look inside".

However, the isothermal case is more interesting. As pointed out by the OP (and I never realized this until reading his post ), the fact that there is still "pressure to do work" in the tank after an isothermal compression (with heat loss to the environment exactly equal to the amount of work done on the "spring") is a very peculiar property of gasses, and is a thermodynamic effect. The gas works indeed as a heat engine but we don't realize it!

If the expansion (the "work done by the gas spring"), after being at room temperature again, is adiabatic, then the gas TAKES energy from its own energy content, lowers its temperature to expand. Clearly, this can only be done if the gas is not at 0K! (that's why there are no ideal gasses at 0K). So we have the gas acting both as a heat engine, and as a heat reservoir.

If the expansion is slow, and isothermal, then the gas acts as a heat engine, but the environment acts as the heat reservoir.

But in both cases, the tank with compressed gas is a heat engine, which transforms heat into work. It is not the restoration of stored "potential" energy from the compression, as in a conservative force field.

Now, as to gravitational effects: the gravitational source is the internal energy, and will hence be given by u(T). That means that, during adiabatic compression, the gas heats, has more internal energy u, and will have hence a (minuscule) increase in gravitational mass, relativistically speaking. This is because the molecules are moving faster in the COG of the tank, and hence the relativistic mass will increase slightly.
When the gas cools, its weight will decrease (very very tiny effect in reality!).
When the gas will expand adiabatically, it will cool down below room temperature, and have even less weight (relativistically speaking).

So, indeed, in agreement with the OP, there is NO storage of energy in a compressed tank by the pressure. There is only a decrease in entropy, which allows a "one-shot" thermal engine to extract heat from the environment and to do work with it.

EDIT: I see that bgwowk said about the same (left the editor open on my computer and forgot to submit the text above... and went to a meeting).

 

[vanesch]

On the other hand, if you consider stretching apart the plates of a charged capacitor, this case clearly does alter the potential energy (rather than having a thermal effect). At first glance the metallic spring looks more similar to this latter case than to the ideal elastic, but (“once bitten, twice shy”) I have to suspect it could be shown otherwise.

I guess that if the "constant of elasticity" (in Hooke's law) is the same for slow (isothermal) and for fast (adiabatic) motion, then one can say that there is some form of local storage of the work done (which could be called, in broad terms, "potential energy" although as we saw with the adiabatic ideal gas spring, was in fact microscopic kinetic energy).

 

[Cyrus]

Please do enlighten us.

You put mass into the bottle when you filled it. It has to weigh more than the empty tank, and it has to have more pressure than the outside. Thats the whole point of storing it in a pressure vessel. You did work to fill this bottle in the form of flow work. It took work to push the air against the already pressurized air inside the botttle.

The pressure is an energy density per unit volume. If the tank is 1L and you increased the pressure ten fold, then that energy is sitting there inside that bottle waiting to do work. It never went away. I get the feeling steve is saying that the energy inside the bottle goes away but the order remains and that it is this order that 'unorders' itself when you open the bottle, thus outputting work. As I said, its the energy in the form of pressure, which never left the bottle, which is doing that.

 

[vanesch]

You put mass into the bottle when you filled it. It has to weigh more than the empty tank, and it has to have more pressure than the outside. Thats the whole point of storing it in a pressure vessel. You did work to fill this bottle in the form of flow work. It took work to push the air against the already pressurized air inside the botttle.

I think you misunderstood what the weight increase was about: it was about a relativistic effect (which is, for practical pressure vessels, unmeasurable).

Consider a vessel with a piston. The amount of gas doesn't change when pushing on the piston. Put it on a superduper balance. Now, when the gas is relaxed, it has (bottle + gas + piston) a certain mass (found by the weight) M0.

Now, compress the gas using the piston. This will first heat the gas, as the work you've done on the piston goes into internal energy of the gas, which is u(T). If you weight it now, it will have total mass M = M0 + dM, where dM is the work you've done divided by c^2 (this is not measurable in practice). The energy is now in the increased thermal energy of the gas molecules, which go slightly faster, and hence have a slightly greater relativistic mass (not rest mass).

Now, let the vessel (piston down) cool down to ambient temperature. Guess what ? If you weight it now, the mass will be M0 again. dM is gone, and is in fact dissipated in the environment.

Now, let the piston expand and do some work (lift a weight or something). If you weight the vessel now, the mass will be M0 - dM'. This is because the gas cooled itself down in order to expand. It worked as a thermal engine.

Let the vessel now heat up again to ambient temperature. The mass will again be M0.

The pressure is an energy density per unit volume. If the tank is 1L and you increased the pressure ten fold, then that energy is sitting there inside that bottle waiting to do work. It never went away.

Then how come that the gas COOLS (looses thermal energy) when it expands adiabatically ? If the "pressure energy" was there all along, it shouldn't need to pick heat energy to do its work, right ?

No, the pressure gives you the thermodynamic possibility to convert thermal energy (u(T)) into work.

 

[Cyrus]

Consider a vessel with a piston. The amount of gas doesn't change when pushing on the piston. Put it on a superduper balance. Now, when the gas is relaxed, it has (bottle + gas + piston) a certain mass (found by the weight) M0.

Ok, I see what your saying. But I was referring to a scuba tank. In a scuba tank, it is not physically represented by a closed amount of mass being compressed. There is a definite, finite, measurable amount of increase in mass to the system.

I am not arguing with what your saying,...Im saying I am talking about a totally different system. I do agree with what you are saying about YOUR system though.

I was responding to Steves comment:

But the work you do to get the gas into the tank all goes into heat that goes away long before you want to use the gas for work, so it's not located anymore in the tank, so it's not stored in the tank as energy, in ANY sense.

I am saying that in THAT case the energy is inside the tank in the form of pressure. Yes, no, maybe so?

 

[bgwowk]

Isn't the potential for the compressed gas to do work dependent on the difference in pressure between it and, say, the atmosphere ?

Once a gas expands so that its pressure is equal to ambient pressure, it can do no more useful work. But for air in a scuba tank that starts off at 100 times atmospheric pressure, whether expansion is stopped at one atmosphere or zero atmospheres (vacuum) makes little difference in total work done.

 

[bgwowk]

I was responding to Steves comment:

"But the work you do to get the gas into the tank all goes into heat that goes away long before you want to use the gas for work, so it's not located anymore in the tank, so it's not stored in the tank as energy, in ANY sense."

I am saying that in THAT case the energy is inside the tank in the form of pressure. Yes, no, maybe so?

Negative. The energy content of an ideal gas depends only on temperature, not pressure. Here's what happens: As you compress gas into a scuba tank, the work done on the gas appears as heat in the gas. As long as the gas remains hot, it still contains the energy you put into it by compressing it. But as the gas cools back down to room temperature, ALL the energy you put into the gas by compressing it leaves as heat. All of it.

By being compressed, the gas contains no extra energy. But its low entropy makes it ideally positioned to do useful work. The energy of any work it does during expansion comes at expense of cooling the gas or the environment around it. Although the tremendous ability of compressed gas to do work SEEMS like stored energy, it really isn't. Put compressed gas to work doing work, and it will cool the room around it. It is physically impossible to do any process with compressed gas that will heat a room because there's no actual excess energy in the gas.

 

[Cyrus]

So what about the notion of pressure being the energy density per unit volume?

 

[vanesch]

So what about the notion of pressure being the energy density per unit volume?

Consider a tank of water. Now, push on a tiny piston, do a tiny bit of work, and raise the pressure in the big tank to, say, 100 bars. The volume of the water almost didn't change, and the amount of work you did was ridiculously low. Nevertheless, you have the same volume and pressure as with the gas tank.

 

[vanesch]

I suppose that a good way to view the "potential work a tank under pressure" can do, is similar to the potential work 100 liters of hot water and 100 liters of icy-cold water can do.
The total energy of the hot and cold water is the same as the energy of 200 liters of lukewarm water, but the first system has the thermodynamic potential to do work, while the second doesn't.
So we're slowly drifting to Helmholtz free energy as our notion for "potential energy" :-)

 

[bgwowk]

So what about the notion of pressure being the energy density per unit volume?

For an ideal gas, pressure is indeed proportional to energy density per unit volume. That's because at constant volume, pressure is proportional to temperature which is proportional to energy. And at constant temperature, pressure is proportional to number density which is proportional to energy density.

Note though that increasing the pressure of a gas at constant temperature doesn't increase the total energy content of the gas. That's because although the energy density increases with pressure, the volume decreases by the same amount, so volume * energy density remains constant.

vanesch is right that with a tank of compressed gas at ambient temperature, we are dealing with a system with a thermodynamic potential to do work, not stored energy of compression. You could say that the compressed gas is storing its thermal energy in a smaller space, and that by being more dense the energy is more "available" to do mechanical work. The mathematical manifestation of that is decreased entropy, and increased Free Energy.