Pressure

Pressure is a physical quantity that describes the normal force exerted per unit area of a surface. Usually represented by $p$ or $P$, it is fundamental to physics, particularly the study of fluids and thermodynamics. Pressure is a scalar: it has a magnitude but no direction, although the force it produces on a surface acts perpendicular to that surface. Its SI unit is the pascal, equivalent to one newton per square metre. (openstax.org)

Definition and units

For a normal force distributed uniformly over an area,

where $F_\perp$ is the perpendicular component of force and $A$ is the area. If the distribution is nonuniform, local pressure is defined through the derivative

Thus, the same force produces greater pressure when concentrated on a smaller area. This explains why a sharp point can produce effects that a broad surface cannot under the same applied force. In a fluid at rest, pressure at a given point acts equally in every direction, and pressure forces on boundaries are normal to their surfaces. (openstax.org)

The International System of Units expresses pressure as

The pascal is named after Blaise Pascal. Common multiples include the kilopascal and megapascal. Other units include the bar, exactly $100{,}000$ Pa; the standard atmosphere, exactly $101{,}325$ Pa; and the pound-force per square inch, approximately $6{,}894.757$ Pa. A hectopascal equals 100 Pa and is numerically equivalent to a millibar. (openstax.org)

Absolute, gauge, and differential pressure

Pressure measurements require a reference. Absolute pressure is measured relative to a perfect vacuum. Gauge pressure is measured relative to local atmospheric pressure:

A gauge reading of zero therefore does not mean that no pressure exists: it means that the measured pressure equals the surrounding atmospheric pressure. Negative gauge pressure indicates pressure below that reference, as in an evacuated chamber. (openstax.org)

Differential pressure is the difference between pressures at two specified locations. Liquid-column instruments can measure such differences by balancing them against the weight of a column. These distinctions matter because a numerical pressure value without its reference can be ambiguous. A tire gauge, for example, normally reports gauge rather than absolute pressure. (openstax.org)

Pressure in stationary fluids

A stationary fluid in a gravitational field develops a pressure gradient because lower layers support the weight of material above them. For constant density and gravitational acceleration,

where $p_0$ is pressure at the reference surface, $\rho$ is density, $g$ is acceleration due to gravity, and $h$ is depth below that surface. For approximately incompressible water, pressure therefore increases nearly linearly with depth. More generally, taking height $z$ upward gives $dp/dz=-\rho g$, allowing density to vary with height. (openstax.org)

Pascal’s principle states that an applied pressure change is transmitted undiminished through an enclosed fluid under static conditions. It concerns the change in pressure, not identical pressure at every elevation. In an ideal hydraulic device, equal pressure increments on pistons imply

A larger piston can consequently deliver greater force, but moves through a smaller distance for the same displaced fluid volume; force amplification does not create energy. (openstax.org)

Pressure differences across an immersed object also produce buoyancy. The resulting upward force equals the weight of the fluid displaced, as expressed by Archimedes’ principle. (openstax.org)

Microscopic origin and gas behavior

In kinetic theory, gas pressure arises from particles transferring momentum to container walls through collisions. For an ideal gas with isotropic molecular motion,

where $N$ is particle number, $V$ is volume, $m$ is particle mass, and $\langle v^2\rangle$ is mean squared speed. This relates a macroscopic pressure to microscopic motion. (openstax.org)

The corresponding equation of state is

where $k_{\mathrm B}$ is the Boltzmann constant and $T$ is absolute temperature. At fixed particle number and temperature, pressure varies inversely with volume; at fixed volume, it increases with temperature. This model is an approximation, generally most effective when gas particles are widely separated and intermolecular interactions are relatively unimportant. (openstax.org)

Pressure, work, and flowing fluids

Pressure enables a gas to perform work by moving a piston. For a quasistatic expansion, the work done by the gas is

The integral is the area under the process curve on a pressure–volume diagram. Work depends on the path between states, not merely their initial and final pressures and volumes. Compression reverses the direction of energy transfer. (openstax.org)

For steady, incompressible flow with negligible viscous losses, Bernoulli’s equation relates pressure, speed, and elevation along a streamline:

The terms represent pressure, kinetic energy per unit volume, and gravitational potential energy per unit volume. At equal elevation, greater flow speed corresponds to lower static pressure under these assumptions. This relationship is not a universal rule for all flows: viscosity, compressibility, and energy supplied by machinery can alter the balance. (openstax.org)

Measurement

A barometer measures atmospheric pressure, while a manometer commonly compares pressures through liquid-column heights. Mechanical and electronic instruments instead use pressure-dependent deformation, capacitance, piezoelectric response, or other calibrated properties. Instrument choice depends on pressure range and measurement conditions; some gauges are designed specifically for highly evacuated chambers. (openstax.org)