# Why does the enthalpy equation include work (PV term) twice?

1. Mar 11, 2013

### JeweliaHeart

Hello. I am a thermodynamics novice trying to gain a better understanding of state functions, particularly enthalpy.

I understand that enthalpy is defined as

"A measure of the total energy of a thermodynamic system, including internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure."

The equation:

ΔH=ΔU(internal energy) + ΔPV

confuses me b/c

ΔU= q(heat added) - w(work done by system on environment)

so

ΔH really means:

ΔH=q - w + ΔPV

There are two terms of work (w and ΔPV) and b/c of the opposite sign, they cancel out, leaving only q. This means ΔH= q which is at odds with the accepted definition of enthalpy. Where did I mess up?

2. Mar 12, 2013

### DrDu

That not at odds with the usual definition. Mostly enthalpy is determined by measuring heat.
However $\Delta(PV)=P\Delta V=w$ only if p is constant and volume work is the only kind of work the system is performing.

3. Mar 12, 2013

### JeweliaHeart

Okay,
so you are saying that enthalpy is only equivalent to heat if pressure is held constant?

Meaning,

ΔH=q - PΔV + PΔV= q (only when pressure is constant and only PV work is being exerted)

Otherwise, when pressure is not constant the equation should like this, perhaps?:

ΔH=q - w + PΔV

And the work defined by the 'w' above includes all forms of work, whether PV or mechanical, etc?

If so, that makes a little more sense. It's just that all the example problems I've encountered with ΔU only use PV work and no other form.

4. Mar 12, 2013

### DrDu

In general, $d(PV)=PdV+VdP$. The second term will not vanish when P is not constant while the first term gives the volume work done in an infinitesimal step.
Hence $\Delta H=q-w+\int PdV +\int V dP$ in general.
If there is no work done other than volume work, this reduces to
$\Delta H=q+\int V dP$.
An example of non-volume work is e.g. the work done when stirring a viscous fluid.