# Is energy density and pressure fundamentally the same thing?

I've been trying to fully understand energy density in terms of the equations that explain it. Unfortunately, the internet hasn't been very helpful in clarifying my misunderstanding. One website defines energy density(u) as energy(E) over volume(V):

u=E/V

I also noticed that pressure(P) can be defined in the same way:

P=E/V

This leads me to believe that they must be describing the same thing. However, I doubt my sources and my own basic knowledge of physics.

From a statistical mechanics point of view, the energy density is really defined as:

$$u=\frac{E}{v}$$

The pressure however is the conjugate variable of the volume, thus:

$$P=\frac{\partial E}{\partial v}$$

The two are the same only when the energy is linear in the volume. This indeed may depend on the momentum as written above for some systems, But may also depend on other things. Think for example what happens when a piston is compressing gas.

• Would this suggest that pressure is equal to the 'local energy density' and not the 'overall energy density'? Aug 30 at 10:31
• I guess this is a valid interpretation. Generally speaking though, in statistical mechanics, free energy (Helmholtz or Gibbs) is a quantity that minimizes the path and satisfies Hamilton’s equation (or Euler Lagrange equation in Lagrangian mechanics). Its components are a sum multiplications of what is known as conjugate pairs (spin and magnetic field for instance is another example, as well as electric voltage and and charge). For each pair taking the derivative of energy with respect to one variable can be thought of as a distribution of the other. Sep 1 at 9:35

Yeah, where is the definition $P=\frac{E}{V}$ from? It most definitely does not hold for all systems.

There are systems for which $P=\frac{2E}{3V}$ (example: ideal classical gas) and systems for which $P=\frac{E}{3V}$ (example: photon gas) and, generalizing these cases, systems for which $P=\frac{sE}{3V}$ where the relation between energy and momentum is $E\propto p^{s}$ (independent of whether boson or fermions are in discussion).

So yes, they are closely related but they most definitely aren't one and the same thing.

There s also another aspect to energy density, that was not mentioned in previous answers. From "chemical" point of view you can say that for example gasoline has larger energy density that water (energy is "stored" in bonds between atoms and can be released by breaking some of these bonds).

I came to this same question when thinking about the ideal gas equation, $$PV= NkT$$. If temperature is a kind of average kinetic energy per gas molecule, then the right side of the ideal gas equation is essentially number of molecules times average kinetic energy per molecule, which is the total kinetic energy in your system of gas. If one side of the equation represents total kinetic energy, it would make sense for the other side also to represent total kinetic energy, and the units of $$PV$$ bear that out. If you divide both sides by volume $$V$$, your left with $$P = NkT/V$$. So, not getting too concerned about $$k$$, which is a conversion factor between degree K energy units and Joule energy units, for an ideal gas you really can view pressure as kinetic energy density - average kinetic energy per unit volume, instead of average kinetic energy per molecule. The ideal gas equation seems to relate these two basies for considering how energy is distributed.

Based on your comment, I think what you are getting at is that if there is displacement $\delta x$ under pressure $p$ exerted over area $A$, such that reduction in systems's volume $A~\delta x$ takes place, then \begin{align} P=\frac{F}{A}=\frac{F~\delta x}{A~\delta x}=\frac{\textrm{Displacement work done}}{\textrm{Change in volume}} \end{align} However energy density is \begin{align} E=\frac{\textrm{Energy content}}{\textrm{Current total volume}} \end{align} There is no resemblance between the two.

I see why you are confused, especially based on your comment where you shared this link.

Just because the units are the same, it does not mean that they describe the same thing. Pressure can be rearranged to have the same units as energy density; but it doesn't represent the whole energy density of the system. It contributes, but there are many other factors which can add internal energy density without changing pressure.

The link you showed also says some other things that are complete oversimplifications to the point of being untrue.

If you are peeling an apple, then pressure is the key variable

That's definitely not true, for example. They are using pressure in an extremely generic way whenever the units work, when often it would be considered more as a distributed force (or in their really bad example, internal energy).