[Shadow]

Gibbs energy is defined as the amount of enthalpy which can be converted to work: dG=dH-tdV. Isn't Gibbs energy also: dG=dH-dU(internal energy)=pdV?

[mjc123]

No. dG = dH - TdS

[Shadow]

But enthalpy is defined as:
dH=dU+tdV where tdV is the amount of heat which is converted to work. Gibbs energy is defined as the amount of enthalpy that can be converted to work, so why isn't dG=tdV=dH-dU?

[Corribus]

You can tell your equation (dH = dU + tdV) is incorrect just by units. tdV does not have units of energy. The correct equation is H = U + pV.

[Shadow]

That's a typo, I meant pdV. How is now my conclusion incorrect?

[Corribus]

As mjc123 already pointed out, G = H - TS, not G = H - U, nor G = H - pV.  If G = H - TS, then G = U + pV - TS.

[Shadow]

Okay, I see. One more question: pV-TS should be the energy which can be converted to work, but by definition of Gibbs energy, it is only TS, how?

[Corribus]

Maybe it helps to explain what these values mean and where they come from. What follows is my best understanding. I know these concepts are confusing -indeed I find myself reformulating my understanding as the years go by.

Let's say you want to create some system. U is the amount energy it takes to make the system, assuming that you don't change either the volume the system occupies or its temperature. We call this the internal energy because it's just a characteristic of the system itself, without accounting for the changes you need to make to the environment in order to create it.

In reality, systems occupy space, and there is usually external pressure that makes it difficult to create that space. The enthalpy is an expression of the amount of energy it takes to create the system PLUS the amount of energy it takes to create the space the system will occupy. 

Furthermore, although we have to expend energy to create our system, if the environment is at a different temperature than the system, heat from the environment will be transferred to the system (or vice-versa), which will modulate the amount of energy required to create the system. If heat flows into the system from the environment, then the amount of energy needed to create the system will be slightly less than it would otherwise, because spontaneous heat flow will provide some of the energy needed. The amount of work you have to do to create the system would be less in this case. Of course, the amount of heat energy that can be "borrowed" against the environment can be expressed in terms of the final temperature and the entropy.

The Gibbs energy, then, is an expression of the total amount of energy it takes to create the system, accounting for the amount of energy you have to "waste" to create the space for the system, plus a "tweak" that accounts for the heat that will flow into the system or out of the system after it's creation. (If heat flows out, then you have to add a little extra energy to account for it; if heat flows in, this provides a little bit of the energy needed to create the system.)

Now, it's obvious we never actually create a system from nothing. We take one system and change it into some other system. As a result, although we often speak in terms of absolute quantities - in that they are expressed compared to a commonly accepted, idealized reference point - in reality what we want to know is how much energy it takes to change the system from some starting point to some final point.

For example, the change in internal energy ΔU is how much energy it takes to change the system, without accounting for whether or not you have to do work to create space to accomodate the change, or whether there will be some amount of heat flow from the environment. The enthalpy change, ΔH, is the amount of energy it takes to change the system, AS WELL AS accounting for whether or not you have to create space to accomodate the change. (The change in internal energy plus the pressure-volume work.) The Gibbs energy change ΔG is a reflection of how much energy it takes to change the system itself, plus how much energy it costs you to accomodate the extra space needed to effect the change, plus any additional cost/payback (positive or negative) to account for spontaneous heat flow due to a temperature differential, expressed as the product of the temperature and the difference in entropy that occurs during the process.

In chemistry, we usually consider ΔG to be an expression of the spontaneity of a chemical process. It's worth thinking about why this is the case. A reaction is spontaneous if it happens "naturally", i.e., without needing any extra energy applied to make it happen. From the description above, any chemical process involves a change in the internal energy of the sytem (that is, the energy to actually change the system) plus a change in the amount of space the system occupies before and after the process has occurred. The Gibbs energy is an expression of the amount of energy needed to effect the change plus the amount of energy needed to make space for the change plus any heat transferred into our out of the system as a result of the change. Such a process will happen spontaneously IF enough heat will enter the system to drive the process without any external energy being applied. When the Gibbs energy change is negative, more than enough heat is transferred into the system to mmake up for the amount of energy it takes to make the system change. If the Gibbs energy change is positive, the spontaneous transfer of heat due to the second law of thermodynamics is not enough to equal the amount of energy it costs to change the system PLUS the amount of energy it costs to make space for the new system, and therefore the change won't happen unless you put extra energy into the system (raising the temperature of the system, say).

This is why ΔG = ΔH - TΔS, or ΔG = ΔU + pΔV - TΔS. And it's why we say a reaction is spontaneous if ΔG < 0.

It has also been said that the Gibbs energy is the largest amount of non-expansion work that can be done by a closed system (under constant pressure conditions). Keep in mind that the Gibbs energy is a thermodynamic potential. The energy available to do work can come in the form of the amount of energy stored internally, or from the entropy, which is a way of quantifying the amount of heat energy that may be transferred into or out of the system during a process change. Although the Gibbs energy incorporates the amount of work that must be done to make the space needed to create the system, in some sense this energy is "wasted" - it has gone into creating the system's environment and is not available to do work elsewhere. Anything left over after creating the space is useful work that can be expended toward useful processes... although in practice the Gibbs energy is a maximum potential value that assumes a fully reversible thermodynamic process.

[Shadow]

So ΔU-TΔS is an additional amount of energy which can be converted to work. Which work? The pΔV is against the external pressure, what about the ΔU-TΔS work?

[Corribus]

You can think of the system as a means of storing energy for use at a later time. The Gibbs energy tells you how much energy you need to expend to make the system, but looked at another way, it also tells you what is the maximum amount of energy you can get back out of the system to do work on another system somewhere else. Assuming 100% efficiency (no energy is lost at any time), you should be able to get the same amount out of the system as you put into it in the first place. Realize, these thermodynamic potentials are really only useful in comparisons with the potentials of other possible states of the system - so as to determine whether a process is likely to happen without additional energy expenditure.

Other than that, I'm not really clear on what you're asking. It seems like you're very hung up on the equations. The pΔV tells you how much of the stored energy needs to be expended toward changing the volume of the system to accomodate the process. Everything else can be used - in principle - to change the system's internal state. If there's not enough energy available, it won't happen spontaneously under the conditions specified (i.e., at that temperature).

[Shadow]

You can think of the system as a means of storing energy for use at a later time. The Gibbs energy tells you how much energy you need to expend to make the system, but looked at another way, it also tells you what is the maximum amount of energy you can get back out of the system to do work on another system somewhere else.

What kind of work?

[Corribus]

Chemical work.