Help me to understand why we have $pV$ in the enthalpy equation

Question

I'm learning (slowly) about enthalpy. My understanding is that enthalpy is the sum of internal energy and the work done on the environment, in order to occupy a volume $V$ at pressure $p$?

$$H = U + pV$$

where:

• $H$ is enthalpy
• $U$ is internal energy
• $p$ is pressure
• $V$ is volume

$U$ is basically a function of temperature: at a constant temperature, $U$ doesn't change?

$H$ is a "state function", meaning that it does not depend on the path one takes, only on the final state.

Then, thinking about the "work done on the environment", starting from zero volume, represented by the $pV$ term, we can consider two possible paths, which should therefore give the same enthalpy? And therefore the work done should be identical. Let's consider the case of a perfectly incompressible liquid or solid. Two paths are:

1. Expand the volume to $V$, in a vacuum, zero pressure. Then add pressure
   • The work done to get to volume $V$ should be zero, since there's no pushing against any pressure.
   • The work done to increase the pressure should be zero, since work is (force $\times$ distance), and here, $V$ doesn't change, so the distance is zero.
   • So the total work done on environment = $0$?
2. Expand the volume to $V$, at pressure $p$
   • Presumably the work done is $pV$, as per the enthalpy equation earlier

Something clearly seems to be wrong with my understanding of enthalpy somewhere, since these two paths result in different amounts of work. But since enthalpy is a state variable, shouldn't different paths give the same enthalpy?

Please help me to understand where my understanding of enthalpy is flawed in various ways.

Note:

• when I first drafted this, I forgot to mention I'm considering the case of a perfectly incompressible liquid or solid. Updated to specify this.

Answer 1

Enthalpy, (H), can best be described by starting from simple terminology and then going into the more scientific explanation.

Basic Explanation:

The enthalpy equation is expressed as: [ H = U + pV ] Where:

• (H) = Enthalpy
• (U) = Internal energy of the system
• (p) = Pressure of the system
• (V) = Volume of the system

Why is there (pV) in the equation?

The (pV) term represents the work a system must exert to 'push' its way out against the surrounding pressure to make room for itself. That is, if a system undergoes a change in volume, it must expand (pushing the surrounding air out of the way) or compress (letting the surroundings push in). This (pV) term represents this type of 'pressure-volume work.'

Enthalpy is convenient because it includes, in one quantity, all the energy that must be brought into the system to form it as it is, both the energy constituent in the internal energy of the molecules and the energy which must be expended to clear space for the system in the surroundings.

Scientific Explanation:

Now, we will discuss in detail why the (pV) term enters the equation of enthalpy.

1. Internal Energy ((U)):

The internal energy (U) is the total energy of the system that is in the motions of its molecules, their vibrations, and the interactions between molecules. This excludes the energy associated with the interaction of the system with surrounding pressure.

2. (pV) Work (Pressure-Volume)

To help illustrate what this (pV) term means, consider a process in which you create or expand a system from the surroundings:

Imagine inflating a balloon. As this occurs, the balloon must push outward on the surrounding air to make room for itself. The work done will depend on one quantity, the pressure (p), applied in creating a change in another quantity, the volume (V).

The work done by or on the system is:

$$\text{Work} = p \Delta V$$ This work forms part of the energy that should be considered when one looks at the total energy content of a system.

3. Combination of Internal Energy and (pV)-Work:

Enthalpy includes the internal energy (U) and the (pV) term; hence, it is a more complete measure of the energy of a system, especially in problems involving constant pressure. The expression (H = U + pV) can be understood as:

• (U): Internal energy of the system's molecules.
• (pV): The work that must be done to push the surrounding pressure such that it occupies its real volume.

Why is Enthalpy Useful?

Enthalpy simplifies the calculations for many thermodynamic processes because, in these processes, which occur at constant pressure (usual for real processes such as chemical reactions carried out in open vessels), ($$\Delta H$$) is equal to the heat absorbed or lost by the system:

$$\Delta H = \Delta U + p \Delta V$$ At constant pressure, the energy exchanged as heat equals the change in enthalpy.

In summary, the (pV) in the enthalpy equation accounts for the work done to establish the volume of the system against an external pressure. Enthalpy is an important measure of total energy change when heat transfer and expansion or compression are under consideration in processes. I hope that helps!

Answer 2

The definition of an enthalpy value itself, $U+PV$, doesn't have much meaning. The reason it is useful is when discussing a change in enthalpy:

$$\Delta H = \Delta U + \Delta (PV)$$

Internal energy, to a good approximation in general and particularly for gases, is solely a function of temperature $U(T)$ and essentially measures how hot the item is.

The $H$ function accounts for both the temperature changes and pressure-volume changes that account for the energy input or output from the gas or the item.