van der Waals equation: see gas laws.

equation, chemical: see chemical equation.

equation of time: see solar time.

equation, in mathematics, a statement, usually written in symbols, that states the equality of two quantities or algebraic expressions, e.g., x+3=5. The quantity x+3, to the left of the equals sign (=), is called the left-hand, or first, member of the equation, that to the right (5) the right-hand, or second, member. A numerical equation is one containing only numbers, e.g., 2+3=5. A literal equation is one that, like the first example, contains some letters (representing unknowns or variables). An identical equation is a literal equation that is true for every value of the variable, e.g., the equation (x+1)²=x²+2x+1. A conditional equation (usually referred to simply as an equation) is a literal equation that is not true for all values of the variable, e.g., only the value 2 for x makes true the equation x+3=5. To solve an equation is to find the value or values of the variable that satisfy it. Polynomial equations, containing more than one term, are classified according to the highest degree of the variable they contain. Thus the first example is a first degree (also called linear) equation. The equation ax²+bx+c=0 is a second degree, or quadratic, equation in the unknown x if the letters a, b, and c are assumed to represent constants. In algebra, methods are evolved for solving various types of equations. To be valid the solution must satisfy the equation. Whether it does can be ascertained by substituting the supposed solution for the variable in the equation. The simultaneous solution of two or more equations is a set of values of the variables that satisfies each of the equations. In order that a solution may exist, the number of equations (i.e., conditions) must generally be no greater than the number of variables. In chemistry an equation (see chemical equation) is used to represent a reaction.

chemical equation, group of symbols representing a chemical reaction.

Basic Notation Used in Equations

The chemical equation 2H₂+O₂→2H₂O represents the reaction of hydrogen and oxygen to form water. The arrow points in the direction of the reaction—from the reactants (substances that react) toward the product or products. In this case the reactants are hydrogen (written H₂ because each molecule consists of two atoms of hydrogen) and oxygen (written O₂ because each molecule consists of two atoms of oxygen) and the product is water. The coefficient 2 before the H₂ indicates that two molecules of hydrogen take part in the reaction, and the 2 before the H₂O indicates that two molecules of water are produced. When no number is written, as in front of the O₂, a one is assumed; one molecule of oxygen takes part in the reaction. The equation shows that two molecules of hydrogen react with one molecule of oxygen to form two molecules of water. Because of the relationship between molecules and the mole, the equation also shows that two moles of hydrogen react with one mole of oxygen to form two moles of water. The same sort of relationship holds with the gram-formula weight.

Methodology for Writing an Equation

There are three steps involved in writing a chemical equation. The first step is to decide which substances are the reactants and which are the products. For example, natural gas (cooking gas) burns in air, providing heat and producing no visible products. The natural gas is principally methane, and the portion of the air that reacts (supports combustion) is oxygen. These are the reactants. Products of the reaction are heat and two invisible gases, carbon dioxide and water vapor. We can now write the word equation methane + oxygen → carbon dioxide + water vapor + heat. The next step is to determine the correct formula for each substance and substitute it for the name. The equation now becomes CH₄+O₂→CO₂+H₂O. (A notation for heat is often omitted.)

The final step is to balance this equation. As the equation is now written, three oxygen atoms are produced from two, and four hydrogen atoms become only two. This cannot occur, since atoms are not created or destroyed in chemical reactions. The equation is already balanced for carbon, since there is one carbon atom on the reactant side and one carbon atom on the product side. There are four hydrogen atoms in the methane molecule on the reactant side, so there must be four hydrogen atoms in water molecules on the product side (since water is the only product containing hydrogen); thus there must be two water molecules, each containing two hydrogen atoms. The equation can now be written CH₄+O₂→CO₂+2H₂O. It is not yet balanced, since there are only two oxygen atoms shown as reactants and four as products. The equation is completely balanced by showing two oxygen molecules (four atoms) as reactants: CH₄+2O₂→CO₂+2H₂O.

Additional Symbols Used in Chemical Equations

There are a number of other symbols used in chemical equations. A symbol written above or below the reaction arrow indicates special reaction conditions. For example, when mercuric oxide is heated it decomposes into mercury metal and oxygen gas; this reaction is shown by the equation 2HgO Δ⃗ 2Hg + O₂↑. The Greek letter delta under the arrow represents the heating. The upward-pointing arrow after the O₂ indicates that this product is gaseous and escapes. When a precipitate is formed by a reaction, the substance that precipitates is often followed by a downward-pointing arrow, e.g., AgNO₃ + NaCl H₂O͢; AgCl↓ + NaNO₃. The H₂O above the arrow shows that the reaction takes place in the presence of water—in this case, in water solution. The formulas AgNO₃, NaCl, and NaNO₃ do not represent molecules, since these substances are almost completely ionized in water solution (see ion).

When chemical equilibrium occurs in a reaction, the double arrow is used instead of the single arrow. For example, liquid water dissociates to form hydronium ions (H₃O⁺) and hydroxide ions (OH^-). These ions exist in equilibrium with water molecules. The equation is 2H₂O &rlhar2H2O; H₃O⁺ + OH^-. The sign = is sometimes used in place of the double arrow.

Any of a class of equations that relate the pressure math.P, volume math.V, and temperature math.T of a given substance in thermodynamic equilibrium. For example, the equation math.Pmath.V = math.nmath.Rmath.T, where math.n is the number of moles of gas and math.R is the universal gas constant, relates the pressure, volume, and temperature of a perfect gas. Real gases, solids, and liquids have more complicated equations of state. Seealso thermodynamics.

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Algebraic equation of particular importance in optimization. A more descriptive name is second-degree polynomial equation. Its standard form is math.amath.x² + math.bmath.x + math.c = 0, and its solution is given by the quadratic formula which guarantees two real-number solutions, one real-number solution, or two complex-number solutions, depending on whether the discriminate, math.b² − 4math.amath.c, is greater than, equal to, or less than 0.

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In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable. It can be read as a statement about how a process evolves without specifying the formula defining the process. Given the initial state of the process (such as its size at time zero) and a description of how it is changing (i.e., the partial differential equation), its defining formula can be found by various methods, most based on integration. Important partial differential equations include the heat equation, the wave equation, and Laplace's equation, which are central to mathematical physics.

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Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). Because the derivative is a rate of change, such an equation states how a function changes but does not specify the function itself. Given sufficient initial conditions, however, such as a specific function value, the function can be found by various methods, most based on integration.

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Mathematical formula that describes the motion of a body relative to a given frame of reference, in terms of the position, velocity, or acceleration of the body. In classical mechanics, the basic equation of motion is Newton's second law (see Newton's laws of motion), which relates the force on a body to its mass and acceleration. When the force is described in terms of the time interval over which it is applied, the velocity and position of the body can be derived. Other equations of motion include the position-time equation, the velocity-time equation, and the acceleration-time equation of a moving body.

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Relationship between mass (math.m) and energy (math.E) in Albert Einstein's special theory of relativity, expressed math.E = math.mmath.c², where math.c equals 186,000 mi/second (300,000 km/second), the speed of light. Whereas mass and energy were viewed as distinct in earlier physical theories, in special relativity a body's mass can be converted into energy in accordance with Einstein's formula. Such a release of energy decreases the body's mass (see conservation law).

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In mathematics, an equation with an unknown function within an integral. An example is where math.f(math.x) is known and phiv(math.t) is to be found, given certain conditions on math.f. Such equations are useful in solving differential equations.

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Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. Differential equations are very common in physics, engineering, and all fields involving quantitative study of change. They are used whenever a rate of change is known but the process giving rise to it is not. The solution of a differential equation is generally a function whose derivatives satisfy the equation. Differential equations are classified into several broad categories. The most important are ordinary differential equations (ODEs), in which change depends on a single variable, and partial differential equations (PDEs), in which change depends on several variables. Seealso differentiation.

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Equation involving differences between successive values of a function of a discrete variable (i.e., one defined for a sequence of values that differ by the same amount, usually 1). A function of such a variable is a rule for assigning values in sequence to it. For example, math.f(math.x + 1) = math.xmath.f(math.x) is a difference equation. Methods developed for solving such equations have much in common with methods for solving linear differential equations, which difference equations are often used to approximate.

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Mathematical statement of equality between algebraic expressions. An expression is algebraic if it involves a finite combination of numbers and variables and algebraic operations (addition, subtraction, multiplication, division, raising to a power, and extracting a root). Two important types of such equations are linear equations, in the form math.y = math.amath.x + math.b, and quadratic equations, in the form math.y = math.amath.x² + math.bmath.x + math.c. A solution is a numerical value that makes the equation a true statement when substituted for a variable. In some cases it may be found using a formula; in others the equation may be rewritten in simpler form. Algebraic equations are particularly useful for modeling real-life phenomena.

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