I have read the following definition of enthalpy in my textbook:
A substance has to occupy some space in its surroundings depending upon its volume ($V$). It does against the compressing influence of the atmospheric pressure ($p$). Due to this,the substance possesses an additional energy called $pV$ energy which is given by the product of pressure ($p$) and volume ($V$) of the system.
The sum of internal energy and $pV$ energy of any system, under given set of conditions, is called enthalpy. It is denoted by $H$ and is also called heat content of the system. Mathematically, it may be put as $$H = U + pV$$
Let's suppose we have some water in a test tube at room temperature open to atmosphere. It will have internal energy which consist of kinetic energy of molecules, chemical energy, rotational energy and some other forms of energy. In order to occupy some space in its surroundings, the molecules of the water should vibrate in such a way that they cancel out the compressing influence of the air molecules present in the atmosphere.
So if we try to measure the internal energy of the system, isn't the energy required to make space for the surroundings already included in the internal energy of the system. Why do we add an extra term $pV$ along with internal energy $U$ to measure the heat content of the system?
Answer
The energy U of a system consisting of a single component and phase is
$ U = -PV + TS + \mu N$
As you rightly point out, U includes the energy required to create the volume by expanding against the surrounding (constant) pressure, namely the -PV term. Removing this contribution gives H.
Therefore
$ H = U + PV = TS + \mu N$
Equating TS with the "heat content" of the system leads to the statement in the textbook (this is my interpretation of this textbook's author's meaning). This kind of algebraic manipulation can lead to all sorts of confusion.
EDIT
Some people recoil (and downvote) when they see an expression such as
$ U = -PV + TS + \mu N$
as if this were somehow unphysical. Well, it does suggest that there is such a thing as an absolute value of U rather than only relative values, and it is hammered into our heads in school that there are only relative, not absolute, energies. But the point here is that there is an absolute value, relative to an empty system, and you can use N (the number of particles) as your simple scaling constant. The expression above captures the extensive property of U. You could equally write
$ \Delta U = -P \Delta V + T \Delta S + \mu \Delta N$
and no one will blink. Well, if you define
$\Delta U = \Delta N \times U_m$
$\Delta S = \Delta N \times S_m$
$\Delta V = \Delta N \times V_m$
with
$\Delta N = N-0$
you obtain an expression equal in spirit to the sacriligeous one above.
Usage of terminology such as "heat content" is potentially confusing, but we can use similar arguments to those used in explaining the PV term's meaning as work to create the system from scratch by increasing the particle number, while keeping intensive properties such as T and P constant, to suggest that the term TS is the heat that must be exchanged with the surroundings during creation of the system (from 0 to N particles). This is more of theoretical than experimental interest, in the same way we can hypothetically charge an object from 0 to a finite charge to compute its energy (relative to no charge).
Note that saying that H represents the heat content is confusing, since the application of the equation is not usually in the spirit I have just outlined, and if you look at
$ H = U + PV = TS + \mu N$
there is an additional term $\mu N$ term which cannot be easily associated with the concept of "heat content".
So, being generous, I would think the author of the book meant by the statement was, very loosely, that changes in H, that is $\Delta H$, are usually associated with the heat exchanged by a system at constant P (subject also to other restraints but more on that in other questions and answers referring specifically to the meaning of H).
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