Definitions

# formula

[fawr-myuh-luh]
formula, in chemistry, an expression showing the chemical composition of a compound. Formulas of compounds are used in writing the equations (see chemical equations) that represent chemical reactions. Compounds are combinations in fixed proportions of the chemical elements. The smallest unit of an element is the atom.

## Formulas for Compounds

The formula of a well-known compound, water, is H2O. Water is made up of molecules, and the formula shows that each molecule consists of two atoms of hydrogen, H, bonded to an atom of oxygen, O. The subscript 2 indicates that there are two atoms of hydrogen in the molecule; where no subscript appears, as after the O, the subscript 1 is implied. It should be kept in mind that not all compounds are molecular. For example, sodium chloride, NaCl, is an ionic rather than a molecular compound. Solid sodium chloride consists of a collection of sodium ions and chloride ions arranged in a regular, three-dimensional pattern called a crystalline structure. One cannot say that a certain sodium ion and a certain chloride ion are grouped together into a unit, since each sodium ion is equally associated with all its neighboring chloride ions and each chloride ion is equally associated with all its neighboring sodium ions. The formula NaCl, therefore, cannot be taken as showing the composition of some particular unit, such as a molecule. Rather, it shows the proportion of the atoms of each element making up the compound—in this case, one atom of sodium to every atom of chlorine; such a formula is called an empirical formula.

## Molecular and Empirical Formulas

If a compound is molecular, the molecular formula is preferred to the empirical formula since it gives more information. A molecule of glucose, for example, consists of 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms. Its molecular formula, C6H12O6, displays this information explicitly; the empirical formula is CH2O. From the formula one can also deduce the proportion of the atoms of each element making up the compound: one atom of carbon to every two atoms of hydrogen to every one atom of oxygen (6 : 12 : 6=1 : 2 : 1). The empirical formula of glucose, CH2O, shows only the proportion, not the actual number of atoms.

Many compounds may have the same empirical formula. For example, formaldehyde, each molecule of which consists of one carbon atom, two hydrogen atoms, and one oxygen atom, has the molecular formula CH2O, which is identical to the empirical formula of glucose. Another example is furnished by ethyne (acetylene), whose molecular formula is C2H2, and benzene, whose molecular formula is C6H6. Both have the same empirical formula, CH.

In addition to showing the actual number of atoms, molecular formulas are also more useful than empirical formulas in that they explicitly show radicals. For example, the molecular formula for the compound aluminum sulfate, Al2(SO4)3, shows that it contains three sulfate radicals (SO4). The empirical formula, Al2S3O12, does not show this. When only one radical is present in the molecule, the parentheses and subscript are omitted, e.g., CuSO4 for cupric sulfate. Other groups are also shown in molecular formulas, e.g., the water molecules in the mineral chalcanthite (blue vitriol), which consists of cupric sulfate atoms to each of which are attached five water molecules. Its molecular formula is CuSO4·5H2O, its empirical formula CuSO9H10.

## Structural Formulas

In many cases, especially with organic compounds, even the molecular formula does not provide enough information to identify a compound, so that structural formulas are needed. For example, both ethanol (ethyl alcohol) and dimethyl ether have the molecular formula C2H6O (see isomer). Their structural formulas are:In these formulas each line represents a single covalent chemical bond. A double bond is represented by a double line and a triple bond by a triple line. In ethene (ethylene), C2H4, the carbon atoms are joined by a double bond. The structural formula of ethene is:(In many representations of structural formulas, the angles of the lines indicating bonds do not necessarily have meaning.) In ethyne (acetylene), C2H2, the carbon atoms are joined by a triple bond. The structural formula of ethyne is:

Semistructural Formulas

Structural formulas are often simplified so that they can be written on a single line; the simplified formulas are often called semistructural formulas. The semistructural formula for ethanol is CH3CH2OH, or more simply C2H5OH. In such a semistructural formula the OH is written explicitly to indicate that the oxygen has a hydrogen bonded to it. The C2H5 indicates that the two carbon atoms are bonded to one another. The semistructural formula for dimethyl ether may be written CH3OCH3. Here the O is placed between the two carbon atoms to show that the carbons are bonded to the oxygen. A carbon often has three hydrogens bonded to it, and the H3 is written after the C. In some cases the H3 is written before the C for clarity; thus the formula for dimethyl ether might be written H3COCH3.

Electron Dot Diagrams

Dots are used in a type of formula called the electron dot diagram, where each pair of dots represents a pair of shared electrons in a covalent bond. The diagrams for ethane (CH3CH3), ethene, and ethyne are:

formula, in mathematics and physics, equation expressing a definite fixed relationship between certain quantities. The quantities are usually expressed by letters, and their relationship is indicated by algebraic symbols. For example, Ar2 is the formula for the area A of a circle of radius r, and s=1/2at2 is the formula for the distance s traveled by a body experiencing an acceleration a during a time interval t.

Sum of the atomic weights of all atoms in a chemical formula. The term is generally applied to a substance that consists of ions (see ionic bond) rather than individual molecules (and thus does not have a molecular weight). An example of such a substance is sodium chloride (table salt). Such a substance's chemical formula describes the simplest ratio of the number of atoms of the constituent elements. Seealso stoichiometry.

Expression of the composition or structure of a chemical compound. Formulas for molecules use chemical symbols with subscript numbers to show the number of atoms of each element: O2 for molecular oxygen, O3 for ozone, CH4 for methane, C6H6 for benzene. Parentheses may enclose atoms that act as a group. General formulas show the proportions of atoms in members of a class (e.g., Cmath.nH2math.n+ 2 for alkanes). If the substance does not exist as molecules (see ionic bond), empirical formulas show the relative proportions of the constituents (e.g., NaCl for sodium chloride). Structural formulas show bonds (see bonding) between atoms in a molecule as short lines between symbols; they are particularly useful for showing how isomers differ. A projection formula also indicates the three-dimensional arrangement of the atoms (see Fischer projection; stereochemistry).

In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. One of many famous formulae is Albert Einstein's E = mc² (see special relativity).

## In mathematics

In mathematics, a formula is a key to solve an equation with variables. For example, the problem of determining the volume of a sphere is one that requires a significant amount of integral calculus to solve. However, having done this once, mathematicians can produce a formula to describe the volume in terms of some other parameter (the radius for example). This particular formula is:

$V =frac\left\{4\right\}\left\{3\right\} pi r^3.$

Having determined this result, and having a sphere of which we know the radius we can quickly and easily determine the volume. Note that the quantities $V$, the volume, and $r$ the radius are expressed as single letters. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate larger and more complex formulae.

In general mathematical use there is no essential difference in meaning with the term "expression", although the word "formula" tends to be reserved for an expression that "can stand on its own", that has a meaning outside of the immediate context in which it appears and a significance that can be grasped intuitively.

The majority of all mathematical study revolves around formulae in many different forms from quadratic equations to the equations of motion (mainly used in mechanical mathematics and physics). In a general context, formulae are applied to provide a mathematical solution for real world problems. Some may be general formulae designed to explain a phenomenon experienced everywhere - an example is force = mass × acceleration. It is a formula which applies anywhere in the universe. Other formulae may be specially created to solve a particular problem - for example using the equation of a sine curve to model the movement of the tides in a bay. In all cases however, formulae form the basis for all calculations.

## In computing

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

Total fruit = number of Apples + number of Oranges.

In computer spreadsheet terminology, a formula is usually a text string containing cell references, e.g.

=A1+A2

where both A1 and A2 describe "cells" (column A, row 1 or 2) within the spreadsheet. The result appears within the cell containing the formula itself (possibly A3, at end of values in column A). The = sign precedes the right hand side of the formula indicating the cell contains a formula rather than data. The left hand side of the formula is, by convention, omitted because the result is always stored in the cell itself and would be redundant.

## Formula with prescribed units

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

In the example above, for the volume of a sphere, we may wish to compute with r =2.0 cm, which yields

$V = 33.51~bold\left\{cm\right\}^\left\{3\right\}.$

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

However, the vast majority of computations with measurements is done in computer programs with no facility for retaining a symbolic computation of the units. Only the numerical quantity is used in the computation. This requires that the universal formula be converted to a formula that is intended to be used only with prescribed units, meaning 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 the formula is to require that $V equiv mathrm\left\{VOL\right\}~bold\left\{tbsp\right\}$, where tbsp is the U.S. tablespoon (as seen in conversion of units) and VOL is the name for the number used by the computer. Similarly, the formula is to require $r equiv mathrm\left\{RAD\right\}~bold\left\{cm\right\}$. The derivation of the formula proceeds as:

$mathrm\left\{VOL\right\}~bold\left\{tbsp\right\} = frac\left\{4\right\}\left\{3\right\} pi mathrm\left\{RAD\right\}^3~ bold\left\{cm\right\}^3.$

Given that $1~bold\left\{tbsp\right\} = 14.787~bold\left\{cm\right\}^3$, the formula with prescribed units is

$mathrm\left\{VOL\right\} = 0.2833~mathrm\left\{RAD\right\}^3.$

The formula is not complete without words such as: "VOL is volume in tbsp and RAD is radius in cm". Other possible words are "VOL is the ratio of $V$ to tbsp and RAD is the ratio of $r$ to cm."

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

$V = 0.2833~r^3.$
and the accompanying words could be: "where V is volume (tbsp) and r is radius (cm)".

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