Basic Math Skills

 

In introductory chemistry basic math skills are required to complete many of the chemistry problems.This instructional sheet will cover these basic math skills.

 

Using known relationships or equations:

 

Density(d) is defined as the mass(m) divided by the volume(v).

Ex: If an object has a mass of 195g and occupies a volume of 15 mL this would correspond to a density of 13. g/mL.

 

The expression for density contains 3 terms. If you know any 2 of the terms, you can calculate the third. The density expression can be rearranged for any term to be calculated.

 

Since the density expression involves dividing(¸) the mass by the volume you use the opposite operation, multiplication(´) to rearrange the expression.

m = d´v

 

This may also be written as m = dv or m = d·v where it is understood that d is multiplied by v.

 

Ex: If a liquid has a density of 0.985 g/mL and a volume of 60.0 mL this would correspond to a mass of 59.1 g.

 

To solve for volume you simply solve the above expression for v by using the opposite operation and dividing both sides.

For more complex equations that use ¸, ´, +, and -, all operations in parenthesis() are done first, followed by ¸ and ´ and finally  – and +.

 

Consider the relationship y = mx + b. If m = 2, b = 5, and x = 3, then y = 11.

 

If you rearrange to solve for x the first thing that must be done is to get the mx term by itself. This will invlove subtraction.

y – b = mx

Finally to get x by itself you use division.

Consider the following equation:  y = ax2 + c. Rearrange the following equstion and solve for x.

Must first get the ax2 term by itself and then divide by a to get x2 by itself.

The opposite operation of x2 is the square root so by taking the square root of both sides you solve for x.

 

Scientific Notation:

 

Scientific notation is the way that scientists easily handle very large or very small numbers. For example, instead of writing 0.000000005, we write 5´10-9. Think of 5´10-9 as the product of two numbers: 5 (the digit term) and 10-9 (the exponential term).

 

Here are some examples of scientific notation.

 

10000 = 1´104

24327 = 2.4327´104

1000 = 1´103

7354 = 7.354´103

100 = 1´102

482 = 4.82´102

10 = 1´101

89 = 8.9´101 (not usually done)

1 = 100

 

1/10 = 0.1 = 1´10-1

0.32 = 3.2´10-1 (not usually done)

1/100 = 0.01 = 1´10-2

0.053 = 5.3´10-2

1/1000 = 0.001 = 1´10-3

0.0078 = 7.8´10-3

1/10000 = 0.0001 = 1´10-4

0.00044 = 4.4´10-4

 

As you can see, the exponent of 10 is the number of places the decimal point must be shifted to give the number in long form. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left.

 

On your scientific calculator:

Make sure that the number in scientific notation is put into your calculator correctly.
Read the directions for your particular calculator. For inexpensive scientific calculators:

  1. Punch the number (the digit number) into your calculator.
  2. Push the EE or EXP button. Do NOT use the ´ (times) button!!
  3. Enter the exponent number. Use the +/- button to change its sign.
  4. Treat any number in scientific notation as a normal number in all subsequent calculations.

To check yourself, multiply 6.0´105 times 4.0´103 on your calculator. Your answer should be 2.4´109.

 

Dimensional Analysis:

 

Dimensional Analysis (also called Factor-Label Method or the Unit Factor Method) is a problem-solving method used to convert from one set of units to another. It uses the fact that any number or expression can be multiplied by one without changing its value. Consider the direct proportion relating inches(in) to centimeters(cm). It is known that there are exactly2.54 cm in one inch. In otherwords, 1 in = 2.54 cm. Since 1 in is the same quantity as 2.54 cm if you use them in a fraction it should equal one because anything divided by itself is equal to one.

If you were to convert 6.00 in into cm you simply multiply by this fraction. Since you are multiplying by one you are not changing the quantity, just the units. The problem is solved by multiplying the given data and its units by the appropriate unit factors so that only the desired units are present at the end.

If you set the fraction up wrong the units will not cancel properly.

 

You can also string many unit factors together.

Ex: Calculate how many seconds are in 2.0 years.

 

Logs:

 

Two kinds of logarithms are often used in chemistry: common (or Briggian) logarithms and natural (or Napierian) logarithms. The power to which a base of 10 must be raised to obtain a number is called the common logarithm (log) of the number. The power to which the base e (e = 2.718281828.......) must be raised to obtain a number is called the natural logarithm (In) of the number.

In simpler terms, LOGS ARE EXPONENTS!!

  1. Using log10 ("log to the base 10"):
    log10100 = 2 is equivalent to 102 = 100
    where 10 is the base, 2 is the logarithm (i.e., the exponent or power) and 100 is the number.
  2. Using natural logs (loge or ln):
    Carrying all numbers to 5 significant figures,
    ln 30 = 3.4012 is equivalent to e3.4012 = 30 or 2.71833.4012 = 30

 

To find the logarithm of a number you need to use your scientific calculator. On most calculators, you obtain the log (or ln) of a number by entering the number, then pressing the log or ln button. On some calculators this is reversed so you must press the log or In button, type in the number and press the equal key.

 

FINDING ANTILOGARITHMS (also called Inverse Logarithm)

Sometimes we know the logarithm (or ln) of a number and must work backwards to find the number itself. This is called finding the antilogarithm or inverse logarithm of the number. To do this using most simple scientific calculators, enter the number, press the inverse (inv) or shift button, then press the log (or ln) button. It might also be labeled the 10x (or ex) button. On some calculators this is reversed so you must press the inverse (inv) or shift button, then press the log (or ln) button, enter the number and press the equal key.

Natural logarithms work in the same way: