Dimensionless quantity
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Dimensionless quantities, also known as quantities of dimension one
The number one is recognized as a dimensionless base quantity.[4] Radians serve as dimensionless units for angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference.[5]
Dimensionless quantities play a crucial role serving as parameters in differential equations in various technical disciplines. In calculus, concepts like the unitless ratios in limits or derivatives often involve dimensionless quantities. In differential geometry, the use of dimensionless parameters is evident in geometric relationships and transformations. Physics relies on dimensionless numbers like the Reynolds number in fluid dynamics,[6] the fine-structure constant in quantum mechanics,[7] and the Lorentz factor in relativity.[8] In chemistry, state properties and ratios such as mole fractions concentration ratios are dimensionless.[9]
History
Quantities having dimension one, dimensionless quantities, regularly occur in sciences, and are formally treated within the field of
Numerous dimensionless numbers, mostly ratios, were coined in the early 1900s, particularly in the areas of
There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017
Buckingham π theorem
The Buckingham π theorem
Another consequence of the theorem is that the
Integers
Number of entities | |
---|---|
Common symbols | N |
1 |
Ratios, proportions, and angles
Dimensionless quantities can be obtained as ratios of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.[19][20] Examples of quotients of dimension one include calculating slopes or some unit conversion factors. Another set of examples is mass fractions or mole fractions, often written using parts-per notation such as ppm (= 10−6), ppb (= 10−9), and ppt (= 10−12), or perhaps confusingly as ratios of two identical units (kg/kg or mol/mol). For example, alcohol by volume, which characterizes the concentration of ethanol in an alcoholic beverage, could be written as mL / 100 mL.
Other common proportions are percentages
It has been argued that quantities defined as ratios Q = A/B having equal dimensions in numerator and denominator are actually only unitless quantities and still have physical dimension defined as dim Q = dim A × dim B−1.[21] For example,
Dimensionless physical constants
Certain universal dimensioned physical constants, such as the
- engineering strain, a measure of physical deformation defined as a change in length divided by the initial length.
- electromagnetic interactionbetween electrons.
- β (or μ) ≈ 1836, the rest mass of the proton divided by that of the electron. An analogous ratio can be defined for any elementary particle;
- Strong forcecoupling strength αs ≈ 1;
- Planck massratio of the mass of any given elementary particle, .
List
Physics and engineering
- Lorentz Factor[23] – parameter used in the context of special relativity for time dilation, length contraction, and relativistic effects between observers moving at different velocities
- Fresnel number – wavenumber(spatial frequency) over distance
- Mach number – ratio of the speed of an object or flow relative to the speed of sound in the fluid.
- Beta (plasma physics)– ratio of plasma pressure to magnetic pressure, used in magnetospheric physics as well as fusion plasma physics.
- Damköhler numbers (Da) – used in chemical engineering to relate the chemical reaction timescale (reaction rate) to the transport phenomena rate occurring in a system.
- Thiele modulus – describes the relationship between diffusion and reaction rate in porous catalyst pellets with no mass transfer limitations.
- Numerical aperture – characterizes the range of angles over which the system can accept or emit light.
- Sherwood number – (also called the mass transfer Nusselt number) is a dimensionless number used in mass-transfer operation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport.
- Schmidt number – defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes.
- Reynolds number is commonly used in fluid mechanics to characterize flow, incorporating both properties of the fluid and the flow. It is interpreted as the ratio of inertial forces to viscous forces and can indicate flow regime as well as correlate to frictional heating in application to flow in pipes.[24]
- Zukoski number, usually noted , is the ratio of the heat release rate of a fire to the enthalpy of the gas flow rate circulating through the fire. Accidental and natural fires usually have a . Flat spread fires such as forest fires have . Fires originating from pressured vessels or pipes, with additional momentum caused by pressure, have .[25]
- Eckert number
- Biot number
- Grashof number
Chemistry
- Relative density – density relative to water
- Relative atomic mass, Standard atomic weight
- Equilibrium constant (which is sometimes dimensionless)
Other fields
- Cost of transport is the efficiency in moving from one place to another
- Elasticity is the measurement of the proportional change of an economic variable in response to a change in another
- Basic reproduction number is a dimensionless ratio used in epidemiology to quantify the transmissibility of an infection.
See also
- List of dimensionless quantities
- Arbitrary unit
- Dimensional analysis
- Normalization (statistics) and standardized moment, the analogous concepts in statistics
- Orders of magnitude (numbers)
- Similitude (model)
References
- ^ "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. Retrieved 2011-03-22.
- ^ "SI Brochure: The International System of Units, 9th Edition". BIPM. ISBN 978-92-822-2272-0.
- ^ Mohr, Peter J.; Phillips, William Daniel (2015-06-01). "Dimensionless units in the SI". Metrologia. 52.
- ISSN 0026-1394.
- ISBN 978-0-8135-2898-4.
- ISBN 978-0-07-717359-3.
- .
- .
- ISSN 0743-7463.
- .
- (PDF) from the original on 2022-12-21. Retrieved 2022-12-21. (1 page)
- S2CID 52806576.
- ^ "BIPM Consultative Committee for Units (CCU), 15th Meeting" (PDF). 17–18 April 2003. Archived from the original (PDF) on 2006-11-30. Retrieved 2010-01-22.
- ^ "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). Archived from the original (PDF) on 2006-11-30. Retrieved 2010-01-22.
- PMID 15588029.
- .
- ISBN 978-1-107-00127-5. Retrieved 2021-11-30.
- ISBN 978-0-12-801909-2. Retrieved 2021-11-30.
- ^ a b "ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23.
- ^ "7.3 Dimensionless groups" (PDF). Massachusetts Institute of Technology. Retrieved 2023-11-03.
- S2CID 122242959.
- Baez, John Carlos (2011-04-22). "How Many Fundamental Constants Are There?". Retrieved 2015-10-07.
- .
- Naval Research Laboratory. pp. 23–25. Archived from the originalon 2021-04-27. Retrieved 2015-10-07.
- ^ Zukoski, Edward E. (1986). "Fluid Dynamic Aspects of Room Fires" (PDF). Fire Safety Science. Retrieved 2022-06-13.
Further reading
- Flater, David (October 2017) [2017-05-20, 2017-03-23, 2016-11-22]. Written at (15 pages)
External links
- Media related to Dimensionless numbers at Wikimedia Commons