Which of the following is the correct equation of enthalpy?

$$ \begin{align} \Delta H &= \Delta U + \Delta (pV) \tag{1} \\ \Delta H &= \Delta U + p\,\Delta V \tag{2} \\ \Delta H &= \Delta U + W \tag{3} \end{align} $$

I am facing issues understanding them. Also, when am I supposed to use them? Finally, does $\Delta H = \Delta U$ when only the volume is constant, or both the volume and the pressure are?


2 Answers 2


We define a state function enthalpy as $$H=U+PV$$

Since we cannot measure the absolute value of enthalpy, but the change in it, we modify the equation to $$\Delta H= \Delta U +\Delta(PV)$$

Which is the correct relation. Now, if you assume pressure is constant, then you can take the $P$ out to get $$\Delta H=\Delta U+P\Delta V$$

Now $PV$ work (or the expansion work) is defined as $W=-P\Delta V$, so we have from the first law of theromodynamics ($\Delta U= q+W$), $$\Delta U= q-P\Delta V$$

Therefore at constant pressure , we have $$\Delta H= q-P\Delta V+P\Delta V=q$$

Which can be stated in words as "At constant pressure, the change in enthalpy is equal to the heat flow".

and finally when is $$\Delta H=\Delta U$$

This happens whenever $\Delta(PV)=0$, which implies $P_1V_1=P_2V_2$ must hold.

  • $\begingroup$ For an ideal gas $\Delta (PV) = 0 \Rightarrow \Delta (nRT) = 0 \Rightarrow \Delta U = \Delta (\alpha nRT) = 0$ so $\Delta H = \Delta U \Rightarrow \Delta H = \Delta U = 0$ $\endgroup$
    – ManRow
    Commented Feb 21, 2021 at 9:24

Enthalpy is the heat released during a chemical process under conditions of constant pressure. Therefore all the equations you have presented are correct. Expression two is the most specific definition of enthalpy.

There's no reason P can't be a constant in the first equation (although it's appearance inside the bracket admittedly gives the impression that it could be a variable, making the expression unnecessarily general). If P is a constant, expression one reduces to expression two.

As for expression three, W (the 'work of expansion') is merely another designation for the second term in expression two. Expression three is in fact the first law of thermodynamics; the statement that an overall energy change (for a closed system) comprises two terms, one for heat and the other for work. If no work is done (there is no volume change), then all the change in energy takes the form of heat, which gives you your final expression. This expression is valid if the product of P and V is constant (i.e. no work is done). I don't think it's necessary that both pressure and volume be constant here, just that their product be constant (i.e. the two variables are codependent), though I stand corrected by anyone who knows better.


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