Pressure

Pressure is a fundamental physical quantity defined as the force applied perpendicular to the surface of an object divided by the area over which that force is distributed. Represented by the symbol p (or P in some fields), it plays a central role in mechanics, fluid dynamics and thermodynamics. Because pressure reflects how force is concentrated over an area, it can vary widely even when the applied force is constant, depending on the size and shape of the contact surface.

Definition and Mathematical Formulation

In its simplest form, pressure is defined by the relationship  pressure = force ÷ area.Mathematically, this can be written as  p = F/A,where F is the magnitude of the normal force and A is the surface area over which it acts.
Pressure is a scalar quantity: although the force resulting from pressure has a direction, the pressure itself does not. In fluid systems, pressure acts equally in all directions at a point and is always normal to the surface under consideration.
A more complete formulation uses vector notation. Writing dFₙ for the normal force acting on a surface element and dA for the outward surface vector, the relationship becomes:  dFₙ = −p dA,with the minus sign indicating that the force exerted by the fluid onto the surface is directed inward relative to the outward-pointing normal. Integrating this expression over any surface in contact with a fluid yields the total normal force acting upon it.
Pressure also has a thermodynamic definition. It is the conjugate variable to volume and may be expressed as  p = (∂U/∂V)ₛ,ₙ,the rate of change of internal energy U with respect to volume V while entropy S and particle number N remain constant.

Units of Pressure

The International System of Units defines the pascal (Pa) as the standard unit of pressure. One pascal equals one newton per square metre (N m⁻²). The term “pascal” was adopted in 1971; previously, SI pressure was expressed directly in newtons per square metre.
A wide range of alternative units remains in everyday and specialist use:

• bar: 1 bar = 100 kPa.
• pound-force per square inch (psi): used in the imperial and US customary systems.
• barye (Ba): 1 dyn cm⁻², used in the centimetre–gram–second (CGS) system.
• technical atmosphere (at): 1 kgf cm⁻² ≈ 98.0665 kPa.
• atmosphere (atm): a defined constant approximating average sea-level air pressure.
• torr: equal to 1/760 of a standard atmosphere.
• joules per cubic metre (J m⁻³): used when relating pressure to energy density, since Pa = J m⁻³ exactly.

Specialist disciplines often prefer certain units. Meteorologists use hectopascals (hPa), identical to millibars (mbar). Oceanographers favour the decibar (dbar), because underwater pressure increases by roughly one decibar per metre depth. Aviation continues to use inches of mercury (inHg) for altimetry.
Some industries append suffixes to differentiate between absolute pressure (a), gauge pressure (g) and differential pressure (d), producing terms such as kPaa, barg or psid. However, metrology guidance recommends attaching such qualifiers to the measured quantity rather than the unit to avoid ambiguity.

Gauge Pressure and Absolute Pressure

Gauge pressure, traditionally spelled gauge in British English, refers to pressure measured relative to ambient atmospheric pressure. A gauge reading of zero therefore corresponds to local atmospheric conditions rather than a vacuum.
Absolute pressure measures pressure relative to a perfect vacuum. The relationship is:  absolute pressure = gauge pressure + atmospheric pressure.
This distinction is crucial in engineering systems such as boilers, vacuum pumps and diving equipment, where atmospheric effects must be accounted for precisely.

Manometric Units and Fluid Columns

Because pressure can displace a column of liquid, many units historically derived from the height of fluid supported in a manometer. Common units include:

• millimetres or inches of mercury (mmHg, inHg),
• centimetres or metres of water (cmH₂O, mH₂O),
• feet of sea water (fsw) and metres of sea water (msw) in diving contexts.

In a static fluid, the hydrostatic pressure generated by a column of height h and density ρ is given by  p = ρgh,where g is gravitational acceleration.
Mercury is frequently used due to its high density, allowing shorter columns for the same pressure. Water-based units remain common in medicine, for example in blood pressure or respiratory pressure measurement.
Modern definitions no longer rely on physical columns of liquid but instead assign exact conversions into SI units. One millimetre of mercury is approximately equivalent to one torr.

Pressure in Thermodynamics and Fluids

Pressure plays a central role in fluid mechanics, governing buoyancy, fluid flow, and the transmission of forces within liquids and gases. It also determines equilibrium conditions across interfaces, influences weather systems and controls the behaviour of gases through relationships such as the ideal gas law.
In thermodynamics, pressure interacts with volume and temperature to define the state of a system. Changes in pressure can perform work, as illustrated by engines, compressors and expanding gases.

Applications and Examples

Pressure differences govern many familiar phenomena. A fingertip pressed against a wall exerts only a small pressure because the force is spread over a wide area. The same force applied to a thumbtack concentrates on a much smaller tip, producing a significantly higher pressure and easily penetrating the surface.
Similarly, high heel shoes exert greater pressure on the ground than flat shoes, despite the wearer’s weight being unchanged, and deep-sea environments subject equipment and organisms to extreme pressures due to the increasing weight of overlying water.

Overview of Common Units

Widely used pressure units include:

• SI units: Pa, kPa, MPa.
• Manometric units: mmHg, cmH₂O, mH₂O, inHg.
• Atmospheric-related units: atm, bar, millibar, hectopascal.
• Diving-related units: msw, fsw.
• Imperial units: psi, psf, tonforce per square inch.
• CGS units: barye, dyne cm⁻².