SCIENTIFIC ALGEBRA Designation of Variables

A.1. SCIENTIFIC ALGEBRA Designation of Variables

To solve an algebraic equation such as

5x + 25 = 165

we first isolate the term containing the unknown (5x) by addition or subtraction on each side of the equation of any terms not containing the unknown. In this case, we subtract 25 from each side:

5x + 25 − 25 = 165 − 25 5x = 140

We then isolate the variable by multiplication or division. In this case, we divide by 5:

If values are given for some variables, for example, for the equation

y x= z

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if y = 14 and z = 5.0, then we simply substitute the given values and solve:

In chemistry and other sciences, such equations are used continually. Since density is defined as mass divided by volume, we could use the equation

y x= z

with y representing the mass and z representing the volume, to solve for x, the density. However, we find it easier to use letters (or combinations of letters) that remind us of the quantities they represent. Thus, we write

m d=

with m representing the mass, V representing the volume, and d representing the density. In this way, we do not have to keep looking at the statement of the problem to see what each variable represents. However, we solve this equation in exactly the same way as the equation in x, y, and z.

Chemists need to represent so many different kinds of quantities that the same letter may have to represent more than one quantity. The necessity for duplication is lessened in the following ways:

Method Example

Using italic letters for variables and roman (regular)

m for mass and m for meter

letters for units Using capital and lowercase (small) letters to mean

V for volume and v for velocity different things

Using subscripts to differentiate values of the same

V 1 for the first volume, V 2 for the second, and so on type

Using Greek letters

π (pi) for osmotic pressure

Using combinations of letters

MM for molar mass

Each such symbol is handled just as an ordinary algebraic variable, such as x or y. EXAMPLE A.1. Solve each of the following equations for the first variable, assuming that the second and third variables

are equal to 15 and 3.0, respectively. For example, in part a, M is the first variable, n is the second, and V is the third. So n is set equal to 15, and V is set equal to 3.0, allowing you to solve for M.

(a) M=n /V (b) n=m /MM

(where MM is a single variable) (c) v = λν

(λ and ν are the Greek letters lambda and nu.) (d ) P 1 V 1 = P 2 ( 2.0)

( P 1 and P 2 represent different pressures.)

Ans. (a) M = 15/3.0 = 5.0

(c) v = (15)(3.0) = 45

(b) n = 5.0 (d ) P 1 = ( 3.0)(2.0)/15 = 0.40 EXAMPLE A.2. Solve the following equation, in which each letter stands for a different quantity, for R:

P V = n RT

Ans. Dividing both sides of the equation by n and T yields

PV R= nT

EXAMPLE A.3. Solve the following equation for F in terms of t:

= F− 32.0 9

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Ans. Inverting each side of the equation yields

F− 32.0 9

Simplifying gives

F= 9 5 t+ 32.0

EXAMPLE A.4. Using the equation in Example A.3, find the value of F if t = 25.0. Ans.

Perhaps the biggest difference between ordinary algebra and scientific algebra is that scientific measurements (and most other measurements) are always expressed with units. Like variables, units have standard symbols. The units are part of the measurements and very often help you determine what operation to perform.

Units are often multiplied or divided, but never added or subtracted. (The associated quantities may be added or subtracted, but the units are not.) For example, if we add the lengths of two ropes, each of which measures

4.00 yards (Fig. A-1a), the final answer includes just the unit yards (abbreviated yd). Two units of distance are multiplied to get area, and three units of distance are multiplied to get the volume of a rectangular solid (such as

a box). For example, to get the area of a carpet, we multiply its length in yards by its width in yards. The result has the unit square yards (Fig. A-1 b):

Yard × yard = yard 2

Fig. A-1. Addition and multiplication of lengths (a) When two (or more) lengths are added, the result is a length, and the unit is a unit of length, such as yard. (b) When two lengths are multiplied, the result is an area, and the unit is the square of the unit of length, such as square yards.

Be careful to distinguish between similarly worded phrases, such as 2.00 yards, squared and 2.00 square yards (Fig. A-2).

1 yd 2 1 yd 2 2 yd

2 square yards

2 yd 2 yards, squared

Fig. A-2. An important difference in wording Knowing the difference between such phrases as 2 yards, squared and 2 square yards is important. Multiplying 2 yards by 2 yards gives 2 yards, squared, which is equivalent to 4 square yards, or four blocks with sides 1 yard

long, as you can see. In contrast, 2 square yards is two blocks, each having sides measuring 1 yard.

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EXAMPLE A.5. What is the unit of the volume of a cubic box whose edge measures 2.00 ft? Ans.

A cube has the same length along each of its edges, so the volume is

The unit is: ft × ft × ft = ft 3

The unit of a quantity may be treated as an algebraic variable. For example, how many liters of soda are purchased if someone buys a 1.00-L bottle of soda plus three 2.00-L bottles of soda?

1.00 L + 3(2.00 L) = 7.00 L

The same result would have been obtained if L were an algebraic variable instead of a unit. In dollars, how much will the 7.00 L of soda cost if the average price is 89 cents per liter?

EXAMPLE A.6. What is the unit of the price of ground meat at the supermarket? Ans.

The price is given in dollars per pound. If two or more quantities representing the same type of measurement—for example, a distance—are multi-

plied, they are usually expressed in the same unit. For example, to calculate the area of a rug that is 9.0 ft wide and 4.0 yd long, we express the length and the width in the same unit before they are multiplied. The width in yards is

The area is

4.0 yd)(3.0 yd) = 12 yd 2

If we had multiplied the original measurements without first converting one to the unit of the other, we would have obtained an incomprehensible set of units:

9.0 ft)(4.0 yd) = 36 ft·yd

EXAMPLE A.7. What is the cost of 8.00 ounces (oz) of hamburger if the store charges $2.98 per pound? Ans.

Do not simply multiply:

23.84 oz·dollar

Instead, first convert one of the quantities to a unit that matches that of the other quantity:

The same principles apply to the metric units used in science.

EXAMPLE A.8.

A car accelerates from 10.0 mi/h to 40.0 mi/h in 10.0 s. What is the acceleration of the car? (Acceleration is the change in velocity per unit of time.)

Ans. The change in velocity is

40.0 mi/h − 10.0 mi/h = 30.0 mi/h

30.0 mi/h

3.00 mi/h

10.0 s

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The acceleration is 3.00 mi/h per second. This is an example of one of the few times when two different units are used for the same quantity (time) in one value (the acceleration).

QUADRATIC EQUATIONS

A quadratic equation is an equation of the form

ax 2 + bx + c = 0

Two solutions are given by the equation

x= b± b 2 − 4ac

2a

This equation giving the values of x is known as the quadratic formula. Two answers are given by this equa- tion (depending on whether the plus or minus sign is used), but often, only one of them has any physical significance.

EXAMPLE A.9. Determine the values of a, b, and c in each of the following equations (after it is put in the form ax 2 + bx + c = 0). Then calculate two values for x in each case.

(a) x 2 − x − 12 = 0

(b) x 2 + 3x = 10

Ans. (a) Here a = 1, b = −1, and c = −12.

Using the plus sign before the square root yields

Using the minus sign before the square root yields

The two values for x are 4 and −3. Check:

(b) First, rearrange the equation into the form

ax 2 + bx + c = 0

In this case, subtracting 10 from each side yields

x 2 + 3x − 10 = 0

Thus, a = 1, b = 3, and c = −10. The two values of x are

Conversion to Integral Ratios

It is sometimes necessary to convert a ratio of decimal fraction numbers to integral ratios (Chaps. 3 and 8.) (Note that you cannot round a number more than about 1%.) The steps necessary to perform this operation follow, with an example given at the right.

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Table A-1 Certain Decimal Fractions and Their Common Fraction Equivalents

Decimal Part

Common Fraction

of Number

Equivalent

Multiply by:

1. Divide the larger of the numbers by the smaller to get a new

1.333 ratio of equal value with a denominator equal to 1.

2. Recognize the fractional part of the numerator of the new frac- 0.333 is the decimal equivalent of 1 3 tion as a common fraction. (The whole-number part does not matter.) Table A-1, lists decimal fractions and their equivalents.

3. Multiply both numerator and denominator of the calculated

4 ratio in step 1 by the denominator of the common fraction of