Formula

In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the term formula in science refers to the general construct of a relationship between given quantities.

On the left is a sphere, whose volume is given by the mathematical formula

V = 4 / 3 π r 3

. On the right is the compound isobutane, which has chemical formula (CH₃)₃CH.

The plural of formula can be spelled either as formulas (from the most common English plural noun form) or, under the influence of scientific Latin, formulae (from the original Latin).[1]

In mathematics, a formula generally refers to an identity which equates one mathematical expression to another. with the most important ones being mathematical theorems.[2] Syntactically, a formula is an entity which is constructed using the symbols and formation rules of a given logical language.[3] For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion.[4] However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius:

V={\frac {4}{3}}\pi r^{3}

.

Having obtained this result, the volume of any sphere can be computed as long as its radius is known. Here, notice that the volume V and the radius r are expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex.[5] Mathematical formulas are often algebraic, analytical or in closed form.[6]

In modern chemistry, a chemical formula is a way of expressing information about the proportions of atoms that constitute a particular chemical compound, using a single line of chemical element symbols, numbers, and sometimes other symbols, such as parentheses, brackets, and plus (+) and minus (−) signs.[7] For example, H₂O is the chemical formula for water, specifying that each molecule consists of two hydrogen (H) atoms and one oxygen (O) atom. Similarly, O⁻
₃ denotes an ozone molecule consisting of three oxygen atoms[8] and a net negative charge.

In a general context, formulas are a manifestation of mathematical model to real world phenomena, and as such can be used to provide solution (or approximated solution) to real world problems, with some being more general than others. For example, the formula

F = m a

is an expression of Newton's second law, and is applicable to a wide range of physical situations. Other formulas, such as the use of the equation of a sine curve to model the movement of the tides in a bay, may be created to solve a particular problem. In all cases, however, formulas form the basis for calculations.

Expressions are distinct from formulas in that they cannot contain an equals sign (=).[9] Expressions can be liken to phrases the same way formulas can be liken to grammatical sentences.

Chemical formulas

{\ce {H-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-H}}

The structural formula for butane. There are three common non-pictorial types of chemical formulas for this molecule:

the empirical formula C₂H₅
the molecular formula C₄H₁₀ and
the condensed formula (or semi-structural formula) CH₃CH₂CH₂CH₃.

A chemical formula identifies each constituent element by its chemical symbol, and indicates the proportionate number of atoms of each element.

In empirical formulas, these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound—as ratios to the key element. For molecular compounds, these ratio numbers can always be expressed as whole numbers. For example, the empirical formula of ethanol may be written C₂H₆O,[10] because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom. Some types of ionic compounds, however, cannot be written as empirical formulas which contains only the whole numbers. An example is boron carbide, whose formula of CB_n is a variable non-whole number ratio, with n ranging from over 4 to more than 6.5.

When the chemical compound of the formula consists of simple molecules, chemical formulas often employ ways to suggest the structure of the molecule. There are several types of these formulas, including molecular formulas and condensed formulas. A molecular formula enumerates the number of atoms to reflect those in the molecule, so that the molecular formula for glucose is C₆H₁₂O₆ rather than the glucose empirical formula, which is CH₂O. Except for the very simple substances, molecular chemical formulas generally lack needed structural information, and might even be ambiguous in occasions.

A structural formula is a drawing that shows the location of each atom, and which atoms it binds to.

In computing

In computing, a formula typically describes a calculation, such as addition, to be performed on one or more variables. A formula is often implicitly provided in the form of a computer instruction such as.

Degrees Celsius = (5/9)*(Degrees Fahrenheit - 32)

In computer spreadsheet software, a formula indicating how to compute the value of a cell, say A3, could be written as

=A1+A2

where A1 and A2 refer to other cells (column A, row 1 or 2) within the spreadsheet. This is a shortcut for the "paper" form A3 = A1+A2, where A3 is, by convention, omitted because the result is always stored in the cell itself, making the stating of the name redundant.

Formulas with prescribed units

A physical quantity can be expressed as the product of a number and a physical unit, while a formula expresses a relationship between physical quantities. A necessary condition for a formula to be valid is the requirement that all terms have the same dimension, meaning that every term in the formula could be potentially converted to contain the identical unit (or product of identical units).[11]

For example, in the case of the volume of a sphere ( $\textstyle {V={\frac {4}{3}}\pi r^{3}}$ ), one may wish to compute the volume when $r=2.0{\text{ cm}}$ , yielding that:

V={\frac {4}{3}}\pi (2.0{\mbox{ cm}})^{3}\approx 33.51{\mbox{ cm}}^{3}.

[12]

There is a vast amount of educational training about retaining units in computations, and converting units to a desirable form (such as the case of units conversion by factor-label).

In most likelihood, the vast majority of computations with measurements are done in computer programs, with no facility for retaining a symbolic computation of the units. Only the numerical quantity is used in the computation, which requires the universal formula to be converted to a formula intended to be used with prescribed units only (i.e., the numerical quantity is implicitly assumed to be multiplying a particular unit). The requirements about the prescribed units must be given to users of the input and the output of the formula.

For example, suppose that the aforementioned formula of the sphere's volume is to require that $V\equiv \mathrm {VOL} ~\mathbf {tbsp}$ (where $\mathbf {tbsp}$ is the US tablespoon and $\mathrm {VOL}$ is the name for the number used by the computer) and that $r\equiv \mathrm {RAD} ~\mathbf {cm}$ , then the derivation of the formula would become:

\mathrm {VOL} ~\mathbf {tbsp} ={\frac {4}{3}}\pi \mathrm {RAD} ^{3}~\mathbf {cm} ^{3}.

In particular, given that $1~\mathbf {tbsp} =14.787~\mathbf {cm} ^{3}$ , the formula with prescribed units would become

\mathrm {VOL} \approx 0.2833~\mathrm {RAD} ^{3}.

[13]

Here, the formula is not complete without words such as: " $\mathrm {VOL}$ is volume in $\mathbf {tbsp}$ and $\mathrm {RAD}$ is radius in $\mathrm {cm}$ ". Other possible words are " $\mathrm {VOL}$ is the ratio of $V$ to $\mathbf {tbsp}$ and $\mathrm {RAD}$ is the ratio of $r$ to $\mathrm {cm}$ ."

The formula with prescribed units could also appear with simple symbols, perhaps even with identical symbols as in the original dimensional formula:

V=0.2833~r^{3}.

and the accompanying words could be: "where $V$ is volume ( $\mathbf {tbsp}$ ) and $r$ is radius ( $\mathrm {cm}$ )".

If the physical formula is not dimensionally homogeneous, it would be erroneous. In fact, the falsehood becomes apparent in the impossibility to derive a formula with prescribed units, as it would not be possible to derive a formula consisting only of numbers and dimensionless ratios.

In science

Formulas used in science almost always require a choice of units.[14] Formulas are used to express relationships between various quantities, such as temperature, mass, or charge in physics; supply, profit, or demand in economics; or a wide range of other quantities in other disciplines.

An example of a formula used in science is Boltzmann's entropy formula. In statistical thermodynamics, it is a probability equation relating the entropy S of an ideal gas to the quantity W, which is the number of microstates corresponding to a given macrostate:

S=k\cdot \log W

(1) S= k ln W

where k is Boltzmann's constant equal to 1.38062 x 10⁻²³ joule/kelvin, and W is the number of microstates consistent with the given macrostate.

References

"formula". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or UK public library membership required.)
"The Definitive Glossary of Higher Mathematical Jargon — Theorem". Math Vault. 2019-08-01. Retrieved 2019-11-26.
Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York, NY: Springer Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6
Smith, David E. (1958). History of Mathematics. New York: Dover Publications. ISBN 0-486-20430-8.
"Why do mathematicians use single letter variables?". math.stackexchange.com. 28 February 2011. Retrieved 31 December 2013.
"List of Mathematical formulas". andlearning.org. 24 August 2018.
Atkins, P.W., Overton, T., Rourke, J., Weller, M. and Armstrong, F. Shriver and Atkins inorganic chemistry (4th edition) 2006 (Oxford University Press) ISBN 0-19-926463-5
"Ozone Chemistry". www.chm.bris.ac.uk. Retrieved 2019-11-26.
Hamilton, A.G. (1988), Logic for Mathematicians (2nd ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-36865-0
PubChem. "Ethanol". pubchem.ncbi.nlm.nih.gov. Retrieved 2019-11-26.
Lindeburg, Michael R. (1998). Engineering Unit Conversions, Fourth Edition. Professional Publications. ISBN 159126099X.
To derive V ~= 33.51 cm³ (2.045 cu in), then calculate the formula for volume: 4/3 × 3.1415926535897 × 2.0³ or ~= 33.51032163829 and round to 2 decimal digits.
To derive VOL ~= 0.2833 RAD³, the tbsp is divided out as: 4/3 × 3.1415926535897 / 14.787 ~= 0.2832751879885 and rounded to 4 decimal digits.
Haynes, William M., ed. (2013) [1914]. CRC Handbook of Chemistry and Physics, 94th Edition. Boca Raton: CRC Press. ISBN 1466571144.

External links

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[oxford-1] "formula". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or UK public library membership required.)

[2] "The Definitive Glossary of Higher Mathematical Jargon — Theorem". Math Vault. 2019-08-01. Retrieved 2019-11-26.

[3] Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York, NY: Springer Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6

[4] Smith, David E. (1958). History of Mathematics. New York: Dover Publications. ISBN 0-486-20430-8.

[5] "Why do mathematicians use single letter variables?". math.stackexchange.com. 28 February 2011. Retrieved 31 December 2013.

[6] "List of Mathematical formulas". andlearning.org. 24 August 2018.

[7] Atkins, P.W., Overton, T., Rourke, J., Weller, M. and Armstrong, F. Shriver and Atkins inorganic chemistry (4th edition) 2006 (Oxford University Press) ISBN 0-19-926463-5

[8] "Ozone Chemistry". www.chm.bris.ac.uk. Retrieved 2019-11-26.

[9] Hamilton, A.G. (1988), Logic for Mathematicians (2nd ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-36865-0

[10] PubChem. "Ethanol". pubchem.ncbi.nlm.nih.gov. Retrieved 2019-11-26.

[11] Lindeburg, Michael R. (1998). Engineering Unit Conversions, Fourth Edition. Professional Publications. ISBN 159126099X.

[12] To derive V ~= 33.51 cm³ (2.045 cu in), then calculate the formula for volume: 4/3 × 3.1415926535897 × 2.0³ or ~= 33.51032163829 and round to 2 decimal digits.

[13] To derive VOL ~= 0.2833 RAD³, the tbsp is divided out as: 4/3 × 3.1415926535897 / 14.787 ~= 0.2832751879885 and rounded to 4 decimal digits.

[14] Haynes, William M., ed. (2013) [1914]. CRC Handbook of Chemistry and Physics, 94th Edition. Boca Raton: CRC Press. ISBN 1466571144.