= Pressure
{title2=$P=F_\perp/A$}
{wiki}
Pressure measures the compressive <normal force> per unit <surface area>. For a uniformly loaded surface of <surface area> $A$, the magnitude of the perpendicular <force> is $F_\perp=PA$; locally, $P=dF_\perp/dA$. Its <SI> unit is the pascal, $1\,\mathrm{Pa}=1\,\mathrm{N\,m^{-2}}$. A <force> parallel to the surface contributes shear <stress>, rather than <pressure>. In a <gas>, the steady macroscopic <force> comes from many microscopic transfers of <momentum>; <ideal gas pressure from wall collisions> makes this connection explicit. https://openstax.org/books/university-physics-volume-2/pages/2-2-pressure-temperature-and-rms-speed[OpenStax University Physics Volume 2, section 2.2: microscopic origin of pressure]{ref}
For a computed example, a uniform <pressure> of $100\,\mathrm{kPa}$ on a piston of <surface area> $10\,\mathrm{cm^2}=10^{-3}\,\mathrm{m^2}$ produces a perpendicular <force> of $100\,\mathrm N$. If the opposite face is exposed to $80\,\mathrm{kPa}$, the net <force> due to the two <pressures> is $(100-80)\times10^3\times10^{-3}=20\,\mathrm N$, directed toward the lower-<pressure> side. The <pressure> on either face and their difference answer different questions; <gauge pressure> records a difference from a specified reference, often <atmospheric pressure>.
A scalar <pressure> describes a <gas> in local <thermodynamic equilibrium> with <isotropic> random particle <velocities>: every wall orientation has the same normal <force> per unit <surface area>. Without this <isotropy>, normal <stresses> can depend on direction and a <pressure tensor> is needed. Thus random motion alone is insufficient; a preferred direction matters.
In <thermodynamics>, <pressure> and <volume (thermodynamics)> are <thermodynamic conjugate variables>. For a <simple compressible system>, reversible expansion delivers <work> $\delta W=P\,dV$, and at fixed <entropy> $S$ and particle number $N$,
$$
P=-\left(\frac{\partial U}{\partial V}\right)_{S,N},
$$
where $U$ is <internal energy>. A large <pressure> therefore means both a large wall <force> per unit <surface area> and a large amount of expansion <work> per small increase in <volume (thermodynamics)>.
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