James Richard Fromm
The experimental observations of physical science are summarized in theories or laws. Many of these are expressed in mathematical language in the form of equations. For example, Newton's first law of motion is usually written as F = ma where F symbolizes force, a acceleration, and m mass. The equation can be used equally well in general discussion, employing the general symbols F, m, and a, or in a particular calculation when these symbols are each replaced by a numeric value and a unit to which that numeric value applies. Since many physical quantities exist, the list of symbols is quite long. To describe a more specific example of a general quantity, additional qualifications are attached to the general symbol as superscripts, as subscripts, or simply in parentheses following the symbol.
The symbols used in these sections, which follow the usage prescribed by the International Union of Pure and Applied Chemistry, are carefully chosen to provide maximum clarity of use with one-character symbols wherever possible. The student should realize that symbols which represent real physical quantities are generally given as italic Roman letters or as Greek letters, while symbols which represent units or prefixes are given as normal (upright) Roman letters. In some cases the same symbol is used to represent both a physical quantity and a physical constant. For example, c represents both the real physical quantity of molar concentration and a constant, the speed of light in vacuum. Likewise, the symbol F represents both the real physical quantity of force and the Faraday constant. Then the meaning intended should be clear from the context in which the symbol is used.
Mathematical expression of physical theories or laws permits use of the mathematical apparatus of arithmetic, algebra, geometry, calculus, and other areas to explain or describe the physical world. At the level of introductory university chemistry, arithmetic, algebra, geometry, and elementary calculus will suffice. In the area of arithmetic, it is assumed that basic numeric operations such as addition, subtraction, multiplication, division, powers, and roots are already familiar. Logarithms have already been discussed in an earlier section. We now take up elementary algebra. Geometric considerations are introduced as needed in the different sections, and elementary calculus is the subject of the following section.
Algebra is that area of mathematics which deals with the manipulation of the relation of equality. An equation is a statement that two quantities separated by an equals sign are equal to each other. Any mathematical operation can be performed upon the quantity on one side of the equation if the same operation is also performed upon the quantity on the other side because the equality of the two sides is preserved. Thus the acceleration of a mass due to an applied force can explicitly be obtained from Newton's first law of motion simply by dividing both sides by mass, a = F/m. Algebraic manipulation of this type, together with use of additional physical laws in mathematical form, permits calculation of many useful quantities.
Traditionally, algebra has reserved the symbols x, y, and z for quantities whose value is usually unknown or variable and the symbols a, b, and c for quantities which are constants. When algebra is applied to the physical world, these reservations are sometimes followed - and quite often not followed, since x and y are often used for distance and the other symbols above have other meanings.
Among the algebraic relationships we find useful are the equation of a straight line, giving a linear relationship or direct relationship of the dependent variable y upon the independent variable x. This equation is y = ax + b, which may equally well be written in terms of x, x = (y - b)/a.
Another useful algebraic relationship is the quadratic equation. The general form of any quadratic equation is ax2 + bx + c = 0, which can be rearranged in terms of x to x = (-b +/- the square root of (b2 - 4ac))/2a. A quadratic equation has two roots, or solutions for x, or values of x that will satisfy the equation. One of these is obtained when the positive sign of the +/- is used and the other is obtained when the negative sign of the +/- is used. Algebraically, both are equally valid, but in the physical world some quantities can have only positive (or only negative) values. Concentrations, masses, and absolute temperatures, for example, cannot be physically negative. When x represents one of these quantities, any root having the sign which cannot be physically real must be discarded. This criterion is often sufficient to determine which of the two possible roots should be used.