CH141 Algebra Review

Algebra is a prerequisite for this Chemistry course. Because a decent familiarity with algebra is essential to success for this course, I have summarized the basic math skills you'll need to achieve success. Weak math skills make understanding the chemical concepts much more difficult. If this material seems foreign to you, there are a number of alternatives available to you. One choice is to get a tutor. Another choice is to enroll in a refresher algebra course, either concurrent with this course or in place of this course.

Scientific Notation (see also Chapter 1 and Appendix A of Silberberg)

Another way to express the number 1000 is 10³. Other common ways of saying this are: 1 x 10³ or 1e3 or 10^3 or 1 x 10^3

On most calculators (yours may be different, check your manual), you "punch in" the following sequence: 1 EE 3 or sometimes 1 EXP 3

Other examples:

Number	Scientific Notation	Calculator Sequence
10,000	1 x 10^4	1 EE 4
642	6.42 x 10^2	6.42 EE 2
0.1	1 x 10^-1	1 EE -1 (sometimes 1 EE 1 ±)
0.023	2.3 x 10^-2	2.3 EE -2 (sometimes 2.3 EE 2 ±)

Using Scientific notation.

Examples:

(6.42 x 10²) * (1 x 10^-1) = 6.42 x 10¹ = 64.2 (2.3 x 10^-2) / (1 x 10⁴) = 2.3 x 10^-6 = 0.0000023

The metric system is another way to shorten long numbers by using prefixes. Common examples:

X x 10⁶ Y = X MY (megaY) X x 10³ Y = X kY (kiloY) X x 10^-2 Y = X cY (centiY) X x 10^-3 Y = X mY (milliY) X x 10^-6 Y = X mY (microY) X x 10^-9 Y = X nY (nanoY)

Therefore, 1.2 x 10^-3 g is the same thing as 1.2 mg and 12 mg is the same thing as 1.2 x 10^-2 g.

Cross multiplying and dimensional analysis:

Let's start with something you could probably do intuitively. If you know the rate of movement (i.e., speed in distance per time) is 60 miles per hour and you know that you want to go 90 miles, you can find how long it will take. Set it up as follows:

60 miles = 90 miles 1 hour x hours

You then multiply diagonally to get: (60 miles) ( x hours) = (90 miles) (1 hour) (You can check the setup at this point, because the units are the same on both sides of the equation.) Solving for x, gives you: x = (90 miles) (1 hour) = 1.5 hours (60 miles)

Another example:

50 miles = x miles (x = 680 miles) 1 gallon 13.6 gallons

A more common way to phrase these types of questions follows. Given 13.6 gallons of gas, and your car gets 25 miles per gallon. How many miles can you drive with that amount of gas? You set it up like the previous example, or more conveniently:

x miles = 13.6 gallons * (25 miles / 1 gallon) = 340 miles

This is because the gallons units cancel (dimensional analysis). If your units cancel, you can have some assurance that you've set up the problem properly. If you look at your units at the end and have gallons²/mile, you put in the miles per gallon upside-down.

In general, these problems are given as a known quantity (e.g., gallons) and a conversion factor (e.g., miles per gallon).

More examples: Given a speed of 60 miles per hour, how many hours will it take to travel 42 miles?

x hours = 42 miles * (1 hour / 60 miles) = 0.7 hours

How many minutes for the previous example?

x minutes = 0.7 hours * (60 minutes / 1 hour) = 42 minutes

Logarithms

Base-10 logarithms (LOGs) are simply 10 to some power which equals some number.

Examples: 100 = 10² Therefore, 2 is the logarithm of 100. 0.01 = 10^-2 Therefore, -2 is the logarithm of 0.01

The above LOGs can be done in your head, but most numbers are not "perfect LOGs." For those, you would require a calculator.

Examples: 367 = 10^2.56 Therefore, 2.56 is the logarithm of 367

0.03 = 10^-1.52 Therefore, -1.52 is the logarithm of 0.03

The opposite process of taking LOGs is the ANTILOG. This the number that you get by putting some number as a power of 10. Example: The ANTILOG of 2 is 100, because 100 = 10². In general, you get the ANTILOG on a calculator by entering a number (e.g., 2) then pressing ANTILOG or 10^x or FNC LOG or 2nd LOG.

Also, ln is not the same as LOG. ln is a "natural" where the base is not 10.

Synthesis Questions

Given a bicyclist moving at 28.0 km/hour (kilometers/hour), how long will it take (in minutes) to travel 1.50 x 10³ m?

Multi-step problem. Try writing a flow chart. First, you need to convert the 1.50 x 10³ m to km. Then you can solve for hours. Then you can convert hours to minutes.

x km = 1.50 x 10³ m * (1 km / 1000 m) = 1.50 km

x hours = 1.50 km * (1 hour / 28 km) = 0.05357 hours

x minutes = 0.05357 hours * ( 60 minutes / 1 hour) = 3.21 minutes

Alternatively, you could solve this in one long problem:

x minutes = 1.50 x 10³ m * (1 km / 1000 m) * ( 1 hour / 28.0 km) * ( 60 minutes / 1 hour) = 3.21 minutes