# Why enthalpy change at constant volume is being stated as change in internal energy?

My textbook, NCERT Chemistry page-167 (PDF), states that change in enthalpy at a constant volume is give by:

$$\Delta H = \Delta U =Q_{_V}$$

Whereas I think that it should be:

$$\Delta H = \Delta U+V\Delta P$$

So:

• Which equation is correct one?

• if my equation isn't correct, where might I be going wrong?

I ask this because some people whom I have asked about this have said that the one given by the book is correct (though they didn't justified why so.)

The statement made in the book is that if $$P$$ is constant, then (equation 6.8) $$\Delta H = \Delta\big(U + PV) = \Delta U + P\Delta V$$

From there, if the volume is also constant, equation 6.8 becomes $$\Delta H = \Delta U = q_V$$

The point being made in the passage is that if both pressure and volume are constant, then there is not an appreciable difference between thinking about $$U$$ and thinking about $$H$$. This would be the case in a solid or liquid exposed to the atmosphere (or some other source of constant pressure).

• I think it would also be worthwhile mentioning that, even for an incompressible solid or liquid, at constant V, $\Delta H\neq \Delta U$ if $\Delta P\neq 0$ Commented Feb 26, 2020 at 3:26
• Thanks for the answer. Can you give an example in which pressure and volume both are constant? In such a situation how might the internal energy change?
– user249968
Commented Feb 26, 2020 at 4:16
• @JohanLiebert Sure - consider adding heat to a solid exposed to the atmosphere. The change in the internal energy is purely due to heat, because the solid does not expand (or at least, does so by a negligible amount for the purposes of this discussion), and is manifested in an increase in temperature. The change in the enthalpy during such a process would be precisely the same. For an example of nontrivial thermodynamic interest, you could imagine that the solid in question is ferromagnetic and magnetized, and the heating takes the solid beyond its Curie temperature. Commented Feb 26, 2020 at 4:24

Enthalpy is defined by

$$H=U+PV$$

A differential change in enthalpy is

$$dH=dU + d(PV)$$

$$dH=dU+PdV+VdP$$

For a constant volume process $$dV=0$$, therefore

$$dH=dU+VdP$$

or in $$\Delta$$ form

$$\Delta H=\Delta U + V\Delta P$$

Hope this helps.