pascal

SI coherent derived unit with special name and symbol

Name

Symbol

Derived quantity

Expressed in terms of SI base units

pascal

pressure,
stress

kg m⁻¹ s⁻²

Definition

The pascal, symbol Pa, is the SI coherent derived unit of pressure. It is the special name for the newton per square metre, symbol N/m².

One pascal is defined as the pressure exerted by a perpendicular force of one newton on an area of one square metre.

The pascal is named after the French mathematician and physicist, Blaise Pascal (1623 – 1662).

Pressure

The pressure, p, exerted by a perpendicular force, F, over an area, A, is proportional to the product of the force and the area:

$p \propto \dfrac{F}{A}$

Using SI coherent units, the proportionality constant is 1, Thus:

$p = \dfrac{F}{A}$

where:

p is the pressure in pascals, symbol Pa,
F is the force applied in newtons, symbol N,
A is the area in square metres, symbol m².

$1 \ \text{Pa} = 1 \ \text{N} \ \text{m}^{-2}$

Meteorology

In weather forecasts, atmospheric pressure is measured in hectopascals, symbol hPa. The hectopascal is equivalent to the non-SI unit millibar, symbol mbar. Mean sea-level atmospheric pressure is about 101 325 Pa, or about 1013 hPa.

Energy density

A system under pressure has the potential to perform work on its surroundings. As such, pressure is a measure of potential energy per unit volume, or energy density.

Using SI coherent units,

$p = \dfrac{F \times distance}{A \times distance} = \dfrac{E}{V}$

where:

p is the pressure in pascals, symbol Pa,
F is the force applied in newtons, symbol N,
A is the area in square metres, symbol m²,
E is the potential energy in joules, symbol J,
V is the area in cubic metres, symbol m³.

$1 \ \text{Pa} = 1 \ \text{J} \ \text{m}^{-3}$

The SI coherent unit of energy density, E/V, is the joule per cubic metre, symbol J/m³.

Boyle’s law

Boyle’s law, or the Boyle–Mariotte law, states that the absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies, if the temperature and amount of gas remain unchanged within a closed system:

$p \propto \dfrac{1}{V}$

The law can be expressed as an equation, where k is the proportionality constant:

$pV = k$

For comparing the same substance under two different sets of conditions, the law can be usefully expressed as:

$p_1 V_1 = p_2 V_2$

The equation shows that, as volume increases, the pressure of the gas decreases in proportion. Similarly, as volume decreases, the pressure of the gas increases.

Ideal gas law

Boyle’s law, Charles’s law, and Gay-Lussac’s law form the combined gas law. The three gas laws in combination with Avogadro’s law can be generalised by the ideal gas law.

The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It approximates the behaviour of gases under many conditions.

Using SI coherent units,

$p V = n R T$

where:

p is the pressure in pascals, symbol Pa,
V is the volume in cubic metres, symbol m³,
T is the absolute temperature in kelvins, symbol K,
n is the amount of gas in moles, symbol mol,
R is the ideal gas constant, in J K⁻¹ mol⁻¹.

The ideal gas constant, R, is equal to the product of two of the SI defining constants – the Boltzmann constant, k, and the Avogadro constant, N_A.

$R = k N_A\\ \\ R = 1.380\ 649 \times 10^{-23} \mspace{6mu} \text{J K}^{-1} \ \times \ 6.022\ 140\ 76 \times 10^{23} \mspace{6mu} \text{mol}^{-1}\\ \\ R=8.314\ 462\ 618\ 153\ 24 \mspace{6mu} \text{J K}^{-1} \text{mol}^{-1}$

Sound

In measurements of sound pressure or loudness of sound, one pascal is equal to 94 decibels SPL. The quietest sound a human can hear, known as the threshold of hearing, is 0 dB SPL, or 20 µPa.

Materials science

In materials science and engineering, the pascal is used to quantify internal pressure, stress, Young’s modulus and ultimate tensile strength of materials.

Shear stress

In contrast to conventional stress, which arises from a force perpendicular to the material cross section on which it acts, a shear stress arises from a force parallel to the material cross section. The force that produces the shear stress is called a shear force.

Shear stress is proportional to the shear force, and inversely proportional to the cross-sectional area of material to which the shear force is applied:

$\tau \propto \dfrac{F}{A}$

Using SI coherent units,

$\tau = \dfrac{F}{A}$

where:

τ is the shear stress in pascals, symbol Pa,
F is the shear force in newtons, symbol N,
A is the cross-sectional area of material (parallel to the applied force) in square metres, symbol m².