pressure

Definition

Pressure is the amount of force applied perpendicular to the surface of an object per unit area. The symbol for it is p or P.

The IUPAC recommendation for pressure is a lower-case p.

However, upper-case P is widely used. The usage of P versus p depends upon the field in which one is working, on the nearby presence of other symbols for quantities such as power and momentum, and on writing style.

Formula

Mathematically:

p

=

F

A

,

or

P

=

F

A

,

{\displaystyle p={\frac {F}{A}},{\text{ or }}P={\frac {F}{A}},}

where:

p

{\displaystyle p}

is the pressure,

F

{\displaystyle F}

is the magnitude of the normal force,

A

{\displaystyle A}

is the area of the surface on contact.

Pressure is a scalar quantity. It relates the vector area element (a vector normal to the surface) with the normal force acting on it. The pressure is the scalar proportionality constant that relates these two normal vectors:

d

F

n

=

−

p

Units

The SI unit for pressure is the pascal (Pa), equal to one newton per square metre (N/m2, or kg·m−1·s−2). This name for the unit was added in 1971; before that, pressure in SI was expressed in newtons per square metre.

Other units of pressure, such as pounds per square inch (lbf/in2) and bar, are also in common use. The CGS unit of pressure is the barye (Ba), equal to 1 dyn·cm−2, or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre ("g/cm2" or "kg/cm2") and the like without properly identifying the force units. But using the names kilogram, gram, kilogram-force, or gram-force (or their symbols) as units of force is deprecated in SI. The technical atmosphere (symbol: at) is 1 kgf/cm2 (98.0665 kPa, or 14.223 psi).

Pressure is related to energy density and may be expressed in units such as joules per cubic metre (J/m3, which is equal to Pa).

Mathematically:

p

=

F

⋅

distance

A

⋅

distance

=

work

volume

=

energy (J)

volume

(

m

3

)

.

{\displaystyle p={\frac {F\cdot {\text{distance}}}{A\cdot {\text{distance}}}}={\frac {\text{work}}{\text{volume}}}={\frac {\text{energy (J)}}{{\text{volume }}({\text{m}}^{3})}}.}

Examples

As an example of varying pressures, a finger can be pressed against a wall without making any lasting impression; however, the same finger pushing a thumbtack can easily damage the wall. Although the force applied to the surface is the same, the thumbtack applies more pressure because the point concentrates that force into a smaller area. Pressure is transmitted to solid boundaries or across arbitrary sections of fluid normal to these boundaries or sections at every point. Unlike stress, pressure is defined as a scalar quantity. The negative gradient of pressure is called the force density.

Another example is a knife. If the flat edge is used, force is distributed over a larger surface area resulting in less pressure, and it will not cut. Whereas using the sharp edge, which has less surface area, results in greater pressure, and so the knife cuts smoothly. This is one example of a practical application of pressure.

For gases, pressure is sometimes measured not as an absolute pressure, but relative to atmospheric pressure; such measurements are called gauge pressure. An example of this is the air pressure in an automobile tire, which might be said to be "220 kPa (32 psi)", but is actually 220 kPa (32 psi) above atmospheric pressure. Since atmospheric pressure at sea level is about 100 kPa (14.7 psi), the absolute pressure in the tire is therefore about 320 kPa (46 psi). In technical work, this is written "a gauge pressure of 220 kPa (32 psi)".

Where space is limited, such as on pressure gauges, name plates, graph labels, and table headings, the use of a modifier in parentheses, such as "kPa (gauge)" or "kPa (absolute)", is permitted. In non-SI technical work, a gauge pressure of 32 psi (220 kPa) is sometimes written as "32 psig", and an absolute pressure as "32 psia", though the other methods explained above that avoid attaching characters to the unit of pressure are preferred.

Gauge pressure is the relevant measure of pressure wherever one is interested in the stress on storage vessels and the plumbing components of fluidics systems. However, whenever equation-of-state properties, such as densities or changes in densities, must be calculated, pressures must be expressed in terms of their absolute values. For instance, if the atmospheric pressure is 100 kPa (15 psi), a gas (such as helium) at 200 kPa (29 psi) (gauge) (300 kPa or 44 psi [absolute]) is 50% denser than the same gas at 100 kPa (15 psi) (gauge) (200 kPa or 29 psi [absolute]). Focusing on gauge values, one might erroneously conclude the first sample had twice the density of the second one.

Scalar nature

In a static gas, the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant random motion. Because there are an extremely large number of molecules and because the motion of the individual molecules is random in every direction, no motion is detected. When the gas is at least partially confined (that is, not free to expand rapidly), the gas will exhibit a hydrostatic pressure. This confinement can be achieved with either a physical container, or in the gravitational well of a large mass, such as a planet, otherwise known as atmospheric pressure.

In the case of planetary atmospheres, the pressure-gradient force of the gas pushing outwards from higher pressure, lower altitudes to lower pressure, higher altitudes is balanced by the gravitational force, preventing the gas from diffusing into outer space and maintaining hydrostatic equilibrium.

In a physical container, the pressure of the gas originates from the molecules colliding with the walls of the container. The walls of the container can be anywhere inside the gas, and the force per unit area (the pressure) is the same. If the "container" is shrunk down to a very small point (becoming less true as the atomic scale is approached), the pressure will still have a single value at that point. Therefore, pressure is a scalar quantity, not a vector quantity. It has magnitude but no direction sense associated with it. Pressure force acts in all directions at a point inside a gas. At the surface of a gas, the pressure force acts perpendicular (at right angle) to the surface.

A closely related quantity is the stress tensor σ, which relates the vector force

F

{\displaystyle \mathbf {F} }

to the

vector area

A

{\displaystyle \mathbf {A} }

via the linear relation

F

Types

Fluid pressure

Fluid pressure is most often the compressive stress at some point within a fluid. (The term fluid refers to both liquids and gases – see below for more information specifically about liquid pressure or gas pressure.)

Fluid pressure occurs in one of two situations:

An open condition, called "open channel flow", e.g. the ocean, a swimming pool, or the atmosphere.

A closed condition, called "closed conduit", e.g. a water line or gas line.

Pressure in open conditions usually can be approximated as the pressure in "static" or non-moving conditions (even in the ocean where there are waves and currents), because the motions create only negligible changes in the pressure. Such conditions conform with principles of fluid statics. The pressure at any given point of a non-moving (static) fluid is called the hydrostatic pressure.

Closed bodies of fluid are either "static", when the fluid is not moving, or "dynamic", when the fluid can move as in either a pipe or by compressing an air gap in a closed container. The pressure in closed conditions conforms with the principles of fluid dynamics.

The concepts of fluid pressure are predominantly attributed to the discoveries of Blaise Pascal and Daniel Bernoulli. Bernoulli's equation can be used in almost any situation to determine the pressure at any point in a fluid. The equation makes some assumptions about the fluid, such as the fluid being ideal and incompressible. An ideal fluid is a fluid in which there is no friction, it is inviscid (zero viscosity). The equation for all points of a system filled with a constant-density fluid is

p

γ

+

v

2

2

Explosion or deflagration pressures

Explosion or deflagration pressures are the result of the ignition of explosive gases, mists, dust/air suspensions, in unconfined and confined spaces.

Negative pressures

While pressures are, in general, positive, there are several situations in which negative pressures may be encountered:

When dealing in relative (gauge) pressures. For instance, an absolute pressure of 80 kPa may be described as a gauge pressure of −21 kPa (i.e., 21 kPa below an atmospheric pressure of 101 kPa). For example, abdominal decompression is an obstetric procedure during which negative gauge pressure is applied intermittently to a pregnant woman's abdomen.

Negative absolute pressures are possible. They are effectively tension, and both bulk solids and bulk liquids can be put under negative absolute pressure by pulling on them. Microscopically, the molecules in solids and liquids have attractive interactions that overpower the thermal kinetic energy, so some tension can be sustained. Thermodynamically, however, a bulk material under negative pressure is in a metastable state, and it is especially fragile in the case of liquids where the negative pressure state is similar to superheating and is easily susceptible to cavitation. In certain situations, the cavitation can be avoided and negative pressures sustained indefinitely, for example, liquid mercury has been observed to sustain up to −425 atm in clean glass containers. Negative liquid pressures are thought to be involved in the ascent of sap in plants taller than 10 m (the atmospheric pressure head of water).

The Casimir effect can create a small attractive force due to interactions with vacuum energy; this force is sometimes termed "vacuum pressure" (not to be confused with the negative gauge pressure of a vacuum).

For non-isotropic stresses in rigid bodies, depending on how the orientation of a surface is chosen, the same distribution of forces may have a component of positive stress along one surface normal, with a component of negative stress acting along another surface normal. The pressure is then defined as the average of the three principal stresses.

The stresses in an electromagnetic field are generally non-isotropic, with the stress normal to one surface element (the normal stress) being negative, and positive for surface elements perpendicular to this.

Stagnation pressure

Stagnation pressure is the pressure a fluid exerts when it is forced to stop moving. Consequently, although a fluid moving at higher speed will have a lower static pressure, it may have a higher stagnation pressure when forced to a standstill. Static pressure and stagnation pressure are related by:

p

0

=

1

2

ρ

v

2

+

p

{\displaystyle p_{0}={\frac {1}{2}}\rho v^{2}+p}

where

p

0

{\displaystyle p_{0}}

is the stagnation pressure,

ρ

{\displaystyle \rho }

is the density,

v

{\displaystyle v}

is the flow velocity,

p

{\displaystyle p}

is the static pressure.

The pressure of a moving fluid can be measured using a Pitot tube, or one of its variations such as a Kiel probe or Cobra probe, connected to a manometer. Depending on where the inlet holes are located on the probe, it can measure static pressures or stagnation pressures.

Surface pressure and surface tension

There is a two-dimensional analog of pressure – the lateral force per unit length applied on a line perpendicular to the force.

Surface pressure is denoted by π:

π

=

F

l

{\displaystyle \pi ={\frac {F}{l}}}

clarification: the expression does not evaluate to the constant pi (π).

and shares many similar properties with three-dimensional pressure. Properties of surface chemicals can be investigated by measuring pressure/area isotherms, as the two-dimensional analog of Boyle's law, πA = k, at constant temperature.

Surface tension is another example of surface pressure, but with a reversed sign, because "tension" is the opposite to "pressure".

Gas pressure

In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume:

p

=

n

R

T

V

,

{\displaystyle p={\frac {nRT}{V}},}

where:

p is the absolute pressure of the gas,

n is the amount of substance,

T is the absolute temperature,

V is the volume,

R is the ideal gas constant.

Real gases exhibit a more complex dependence on the variables of state.

Vapour pressure

Vapour pressure is the pressure of a vapour in thermodynamic equilibrium with its condensed phases in a closed system. All liquids and solids have a tendency to evaporate into a gaseous form, and all gases have a tendency to condense back to their liquid or solid form.

The atmospheric pressure boiling point of a liquid (also known as the normal boiling point) is the temperature at which the vapor pressure equals the ambient atmospheric pressure. With any incremental increase in that temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and lift the liquid to form vapour bubbles inside the bulk of the substance. Bubble formation deeper in the liquid requires a higher pressure, and therefore higher temperature, because the fluid pressure increases above the atmospheric pressure as the depth increases.

The vapor pressure that a single component in a mixture contributes to the total pressure in the system is called partial vapor pressure.

Liquid pressure

When a person swims under the water, water pressure is felt acting on the person's eardrums. The deeper that person swims, the greater the pressure. The pressure felt is due to the weight of the water above the person. As someone swims deeper, there is more water above the person and therefore greater pressure. The pressure a liquid exerts depends on its depth.

Liquid pressure also depends on the density of the liquid. If someone was submerged in a liquid more dense than water, the pressure would be correspondingly greater. Thus, we can say that the depth, density and liquid pressure are directly proportionate. The pressure in a liquid of uniform density is represented by the following formula:

p

=

ρ

g

h

,

{\displaystyle p=\rho gh,}

where:

p is liquid pressure,

ρ is density of the liquid,

g is the acceleration due to gravity,

h is depth within the liquid.

Another way of saying the same formula is the following:

p

=

density

×

gravitational acceleration

×

depth

.

Kinematic pressure

P

=

p

ρ

0

{\displaystyle P={\frac {p}{\rho _{0}}}}

is the kinematic pressure, where

p

{\displaystyle p}

is the pressure and

ρ

0

{\displaystyle \rho _{0}}

constant mass density. The SI unit of P is m2/s2. Kinematic pressure is used in the same manner as kinematic viscosity

ν

{\displaystyle \nu }

in order to compute the Navier–Stokes equation without explicitly showing the density

ρ

0

{\displaystyle \rho _{0}}

.

∂

u

∂

t

+

(

u

∇

)

Generalized pressure

Generalized pressure is a physical concept that describes a thermodynamic ensemble of all factors that produce the same volumetric strain in a material. The concept was proposed in a 2022 paper published in Physical Review B by the Lei Li group at Sichuan University.

The framework is inspired by the equivalence principle in Einstein's theory of general relativity and aims to unify different sources of volumetric strain. Within this approach, different driving forces—such as chemical pressure and mechanical pressure—can be considered equivalent if they generate the same magnitude of volume strain.

Based on a stress–strain model, an equation of state under generalized pressure conditions was derived. The theory was further examined using the ternary semiconductor alloy AlxGa1−xN as a case study. It was reported that the alloy system and its corresponding end-member materials exhibit identical phase-transition points when expressed in terms of generalized pressure.