# Pressure

Created | Updated May 27, 2005

The definition of pressure as taught in school is *force per area*. Written as a formula:

*p* = * F*/

*A*

It's measured in Newtons per square metre(N/m^{2}), more usually called 'pascals' (Pa). A more detailed description of pressure, its uses and its units is found after this first illustrating example.

65kg on a surface of 2cm^{2}(eg, high heel shoes) will result in a pressure of 3 250 000 Pa (beneath the high heels, if the person is standing on the surface of planet Earth). A four ton elephant, on the other hand, standing on one foot will cause a pressure of only 250 000 Pa under that foot. As an exercise try to calculate the area of the aforementioned foot. The solution can be found in this footnote^{1}.

### Details

The easy definition above can be seen as a descriptive mechanistic definition of pressure. There are, however, more ways to interpret the phenomenon of pressure, especially when the objects exerting the pressure cannot be weighed, or have their area measured, in the conventional way. Gases and liquids, for example, are not as easy to weigh. Furthermore, they can be compressed and they can flow, facts which change some aspects of the interpretation of pressure.

#### Pressure and Thermodynamics

In thermodynamics, pressure can be seen as a potential, especially when The Gas State Equation is used to calculate work^{2} and heat. Thus, if one has two compartments containing gases with different pressures, such as a full scuba-diving compressed-air cylinder and an empty one, then one can propel a turbine and generate energy by letting the gas stream out of the full cylinder into the empty one. When both compartments have the same pressure, then no more energy can be gained as no gas would flow. This is because there is no potential difference between the compartments.

Pressure can also be defined as the transmission of the momentum of every particle stuff is made of. In other words, the force of every single atom that bumps into a surface: the overall pressure is given by the sum over all pressures of all individual particles. This individual pressure is again the quotient of force and area. The force is given by the particle's velocity and mass, while the area is dictated by the particle's size. From this relationship, people can calculate the effective size of gas molecules^{3}.

#### Dynamic and Other Forms of Pressure

Forces can also increase and decrease with time. A phenomenon which varies with time is known as a 'dynamic'. There is, therefore, also a dynamic dimension to pressure: 'dynamic pressure'. This is crucial for the mechanisms involved in flying, but dynamic pressure is also involved in calculating football banana-kicks, among many other things. There are more abstract forms of pressure, such as the pressure of a solvent upon a semi-permeable membrane: a phenomenon called osmosis.

Sometimes one may find a pressure measurement given as 'head pressure', in metres. Head pressure is the pressure exerted by a column of a fluid with a certain height. For example, a 30m head pressure for water would be equivalent to the pressure on the bottom of a 30m high water column^{4}. To calculate the pressure, one must know the density (*p*) of the fluid^{5} and the gravitational acceleration (**g**^{6}. The pressure (*P*) is then directly proportional to the height (*h*).

*P = p*

**g**·

*h*

The pressure is then given in Pascals (Pa). For water, 1 metre of head pressure is roughly^{7} equivalent to 9800 Pa. Humans can suck up to 7m high water column with a straw, though this figure is usually a lot less. The formula above is also useful to calculate pressures at certain depths, for instance when scuba diving: a 10m depth corresponds roughly to 1 atm.

It is also worth noting that there is no such thing as an absolute negative pressure, just as with absolute temperatures. A 'negative' pressure can only represent a pressure *difference* (most commonly against atmospheric pressure). Those pressures are sometimes termed *gauge pressure*, ie, the pressure indicated. This is in contrast to 'absolute pressure', which refers to all pressures together - *atmospheric pressure* plus *gauge pressure*.

#### Measuring Pressure

Since pressure is dictated by two parameters, namely force and area, all one must do to measure the pressure is to keep one of them constant, and proceed to measure the other one. In 99% of cases the area is kept constant and the force is measured. Normally, this is done by measuring the deformation of a membrane or of a coil. In some cases this is done by measuring the height of a fluid column, like mercury (hence the mmHg unit below) or water.

The gadgets used to measure pressure are called barometers. Some are called manometers, which are used to measure gas pressure. They work according to the following principle: the pressure of a gas pushes a piston, a membrane or a liquid against a coil or another gas with a known pressure. The mechanic deformation of those materials is visualized electronically, with a needle or on a scale behind the coil and converted into the appropriate units of pressure. As with most measuring apparatus, most of the barometers must be recalibrated every now and then, to make sure what they are indicating is correct.

#### Units

For those who want some proof that physicists are human, the proof is in the idiocy of all the different units which they use...^{8}- RP Feynman

Units are one of the biggest problems in science. There are many allegedly clever ways to relate pressure with a unit that any average Joe can understand. However, a 'Newton per square metre' can get Americans confused. In America the unit psi, which is 'pounds per square inch', is used instead. This is why car tyre pressures are often measured in psi. One psi corresponds approximately to 6894.757 Pa. Many people also use the unit atm, which stands for 'atmosphere', because it's something they apparently relate to. One atm is 101,325 Pa. There are even more units used for inexplicable reasons. In any case here goes a smart conversion table:

Smart Unit Conversion Table for PressurePa | bar | atm | mmHg
(= Torr) | psi | |

Pa | 1 | 0.00001 | 9.8692·10^{-6} | 0.0075 | 1.4504·10^{-4} |

bar | 100,000 | 1 | 0.98692 | 0.75006 | 14.504 |

atm | 101,325 | 1.01325 | 1 | 760 | 14.69594 |

mmHg | 133.322 | 0.00133 | 0.00132 | 1 | 0.01934 |

psi | 6894.757 | 0.068948 | 0.068046 | 51.7151 | 1 |

There are even more obscure units like 'foot of H_{2}O' (0.00034 Pa) or 'Pound-force per square foot' (0.02088 Pa), but these are used only by very special freaks in very special contexts.

### Pressure in Everyday Life (Some Figures)

- 10
^{-20}atm - The pressure in space-vacuum - 10
^{-16}atm - The lowest pressure ever achieved by a man made gizmo - 10
^{-6}atm - Ordinary vacuum pumps - 10
^{-2}atm - The pressure in a common light bulb - 0.5 - 1.5 atm - Atmospheric pressure
- 1.5 - 2.4 atm - Car tyres
- 3 - 7 atm - Flatus
- 4 - 12 atm - Bicycle tyres
- 10 atm - The pressure inside the cylinder cavity in a car's engine
- 100 - 500 atm - Compressed gas cylinders
- 500 atm - The impact pressure of a karate fist punch
- 1000 atm - The pressure at the bottom of the Mariana Trench
- 7000 atm - Water compressors
^{9} - 10
^{6}atm - The pressure at the centre of the earth, and also the highest pressure ever achieved by a man-made machine (diamond anvil) - 10
^{11}atm - The pressure at the centre of the sun -enough to ignite fusion reactions - Approx. 10
^{29}atm - The pressure at the centre of a neutron star.

^{1}0.04 m

^{2}, or roughly 40 by 40cm.

^{2}If the volume remains constant work is dw = V·dp.

^{3}Note that the molecules are actually a lot smaller, and they can also harbour some extra energy in the form of rotation and vibration. So, at that scale of length they act as if they were a little bit larger. In serious calculations one must compensate for that, and even take the shape of the molecules into consideration.

^{4}This would occur when the water in a water tower is 30m higher than the tap that is its eventual destination.

^{5}Water has a density close to 1000kg/m

^{3}.

^{6}Usually around 9.8 kg·m/s

^{2}.

^{7}There is no exact conversion factor as the head pressure depends on the density, which in its turn depends on a lot of other factors such as salt concentration and temperature.

^{8}The complete quote goes on: '...for measuring energy', but the sentiment is applicable to pressure, too.

^{9}Such high pressure water jets can be used to cut 60 mm steel plates - an impressive technique called abrasive waterjet cutting.