# The heat content of a system is equal to the enthalpy only for a system that is at constant what?

##### 1 Answer

Only for a system at constant pressure... By definition,

We can begin from the **first law of thermodynamics**

#DeltaU = q + w# ,where

#q# and#w# are the heat flow and work, respectively,

and the relation of the **internal energy** **enthalpy**

#DeltaH = DeltaU + Delta(PV)# where

#P# and#V# are pressure and volume.

When one plugs in the expression for

#DeltaH = q + w + Delta(PV)#

The common convention is that the **work** is defined by *system* (i.e. negatively-signed when the system does work on the surroundings by expanding, with

Furthermore, we apply the product rule from calculus (plus a bit extra) to see that

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

As a result, the **change in enthalpy** is related to the **heat flow** as:

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

Clearly, when the pressure is constant, we can see that

#DeltaH = q_P# ,

if both quantities are in

#color(blue)(nDeltabarH = q_P)#