Can Boyle's law be used to approximate energy storage? We know from Boyle's law that $PV=k$, where pressure $P$ has units of Pascals, volume $V$ has units of cubic meters, and $k$ is a constant.  Multiplying and simplifying the units from the left side of this equation, we have
$$\mbox{Pa}\cdot\mbox{m}^3$$
$$=\frac{\mbox{N}}{\mbox{m}^2}\mbox{m}^3$$
$$=\mbox{N}\cdot\mbox{m}$$
$$=\mbox{J}$$
so $k$ has units of Joules.
Given the pressure and volume of a compressed air system, can $k$ be seen as an approximation of the total energy in the system (my theoretical question)?
Additionally, could we write an available stored energy approximation as
$$E\approx(P-p_0)V$$
where $E$ is the available stored energy and $p_0$ is the atmospheric pressure outside the compressed air system (my pragmatic question)?
 A: It has the right units, but doesn't really work. If you want the internal energy available for doing useful work then that will depend on the technique used for extracting that work. For example, if the gas is expanded isothermally then $P = P_i V_i / V$ and the work the gas does in expanding is: $$\begin{align} W & = \int P \operatorname{d}V \\
& = P_i V_i \int_{V_i}^{V_f} V^{-1} \operatorname{d}V \\
& = P_i V_i \ln\left(\frac{V_f}{V_i}\right) = P_i V_i \ln\left(\frac{P_i}{p_0}\right).\end{align}$$ With isothermal expansion part of that energy came from whatever was keeping the gas warm during the expansion. Adiabatic expansion, where the system isn't allowed to absorb heat, will get a different answer. 
The total amount of energy stored in the gas will depend on the type of gas it is and the temperature of the gas. For a simple ideal gas the heat energy stored in it is:
$$U = \frac{3}{2} NkT = \frac{3}{2} PV.$$ That fraction, $3/2$, goes up when the gas has more ways to store energy than just movement of its molecules (like when the molecules can rotate or vibrate).
A better understanding of pressure comes from the first law of Thermodynamics: $$\operatorname{d}U = - P \operatorname{d}V.$$ That is, pressure is the generalized force which drives changes in volume for a system. Even better is the following relation:$$\mathbf{f} = -\nabla P,$$ which says that spatial gradients in pressure (differences in pressure with location) exert a force density (force per unit volume). This last relation leads directly to the more common definition of pressure as force per unit area when there is a sudden drop in pressure across some surface.
Edit: fix an incorrect subscript.
