# What is the relationship between enthalpy and internal energy? Where does #q = DeltaH# come from?

##### 1 Answer

**IMPLICIT ASSUMPTIONS**

Okay, then let's look at that formula. You say that:

#q_p = DeltaE + PDeltaV = DeltaH#

Here's how your formula turns out:

#q_p = DeltaE + PDeltaV = DeltaH#

#q_p = q + w + PDeltaV = DeltaH#

#q_p = q - cancel(PDeltaV) + cancel(PDeltaV) = DeltaH#

#color(green)(q_p = DeltaH)#

Here, you have implicitly assumed ahead of time that *is* correct.

** But**, it

*obscures*the assumption that you may not have realized you

*already*made:

**that the pressure is constant.**

**ENTHALPY VS. HEAT FLOW**

To see what that means, let's compare it to a formula that I know to be correct (and is where I would start an enthalpy derivation of this kind), as well as two related equations:

#\mathbf(DeltaH = DeltaE + Delta(PV))#

#DeltaE = q + w#

#w = -PDeltaV#

*The bolded equation shows the full relationship before making any assumptions at all, and is thus more helpful when determining relationships between internal energy and enthalpy in multiple different conditions.*

Next, you should notice that

#Delta(PV) = PDeltaV + VDeltaP + DeltaPDeltaV#

Thus, the relationship for enthalpy vs. heat flow is:

#DeltaH = q + w + PDeltaV + VDeltaP + DeltaPDeltaV#

#= q - cancel(PDeltaV) + cancel(PDeltaV) + VDeltaP + DeltaPDeltaV#

#color(blue)(DeltaH = q + VDeltaP + DeltaPDeltaV)#

So enthalpy is heat flow plus any pressure changes at a specific initial volume, plus the effects from changes in both simultaneously.

At this point, it is obvious that a constant pressure yields

**ENTHALPY VS. INTERNAL ENERGY**

Now, we can make comparisons between enthalpy and internal energy (recall the first law of thermodynamics for

#color(blue)(DeltaH = q + VDeltaP + DeltaPDeltaV)#

#color(blue)(DeltaE = q - PDeltaV)#

From this, here's what we can figure out:

You should notice that **enthalpy at a constant pressure is entirely dependent on the heat flow.**

The other thing you should notice is that **internal energy at a constant volume is entirely dependent on the heat flow.**

Finally, based on the bolded equation, we should see that:

*Enthalpy is internal energy plus any work required to change the pressure or volume without worrying about keeping the other constant.*