The following study guides are linked to Anne McKenna's website (http://employees.csbsju.edu/amckenna/CH123), with permission by the author.

Units, Scientific Notation, Density

Topics:     Units, Scientific Notation, Calculators 
                  Algebra and Equations
                  Density 

Units

In the last class, you learned about the metric system of measurement.  Any measurement made must contain two pieces of information:  the measured number and the unit of the measurement.  A measurement of 1.5 is meaningless unless the unit of measurement (meters, kilograms, liters, pounds) is included.  It is important to include the unit with the number as you do calculations. 

Scientific Notation

Measurements made in science range from incredibly tiny to exceptionally large.  For example, about 80 million iron atoms can be placed side by side on a line 1 cm in length.  About 500 billion of these atoms could be placed in a box with a measurement of 1 cm per side.  This means that the size of an atom must be very small. 

To get an idea of how large 1 billion is, let's calculate the number of years it would take to spend $1 billion if you spent $1000 per day (a Bill Gates problem).  To simplify this problem, we will assume that you have the $1 billion under your mattress, so it is not invested or earning interest.

Although the process of solving this problem (dimensional analysis) will be discussed in more detail later, let's talk about how to solve the problem.  If we know the total amount of money ($1 billion) and the rate of spending ($1000 per day) we can calculate the number of days needed to spend the money:

Since we know that there are 365 days in one year, we can also calculate how many years it would take to spend the money.

So, we should either plan to leave a substantial inheritance to someone or increase our rate of spending.

In chemistry, we are frequently dealing with numbers larger than 1 billion.  Scientists handle large and small numbers using scientific notation.  A number written in this format has several parts:

                  1.23x10⁴

1.23 part is referred to as the "coefficient"
10 part is referred to as the "base"
4 part is referred to as the "exponent"

The coefficient is a number between one and ten.  The coefficient is multiplied by the exponential part, consisting of a base and an exponent.  In scientific notation, the base is 10 and the exponent is a positive or negative whole number.  The coefficient is multiplied by the base the number of times given by the exponent.  So, for example, the number expressed in scientific notation above could be rewritten as:

                        1.23 x 10 x 10 x 10 x 10 = 12,300

If the exponent is negative, we divide by 10 instead of multiplying. 

Important exponential parts used in scientific notation and their values and meanings are given in Table 1.

Table 1:  Values and meanings of exponential numbers

             1.  Write the following in scientific notation:

                         a.  Five million = __________________________________

                         b.  Four thousandths = _____________________________

                         c.  3 tenths = ____________________________________

                         d.  400 = _______________________________________

                         e.  0.000009 = ___________________________________

Of course, numbers aren't usually this simple.  Usually the coefficient contains more than one number and we frequently encounter exponential parts not listed in the table above.  Therefore, we need a system to allow us to write any number in scientific notation.

Writing Numbers in Scientific Notation

    1.  Express the coefficient as a number between 1 and 10 by moving the decimal point.

2.  Indicate in the exponential part the number of places the decimal was moved.  Each place is one power of ten.

                        a.  if the decimal place was moved to the left, the exponent will be positive

                        b.  if the decimal place was moved to the right, the exponent will be negative.

2.  Express the following numbers in scientific notation

            a.  356 = ________________________________

            b.  0.00474 = _____________________________

                        c.  1,200,000 = ____________________________

3.  Express the following numbers in decimal notation:

            a.  1.492 x 10⁴ = ___________________________

            b.  5.703 x 10^-6 = __________________________

Scientific Notation and Calculators

 Many calculators will express numbers in scientific notation and will allow you to enter numbers in scientific notation form.  To enter a number in scientific notation on a calculator, key in the coefficient, press the EXP or EE key, then enter the exponent.  The EXP or EE key contains the base (x10) so you do not press the multiplication key.  If the exponent is negative, use the +/- key on your calculator.

Multiplication and Division of Numbers in Scientific Notation

To multiply numbers in scientific notation, we multiply the two parts of the number separately.  First, we multiply the coefficients.  To multiply the exponential parts, we simply add the exponents.  Finally, we check to see if the answer is in scientific notation.  If not, we move the decimal point so that the coefficient is between 1 and 10 and adjust the exponent to indicate how the decimal point was moved.

The example below illustrates this process:

             4.  Perform the multiplication problems below using the process above, then check your answers using scientific notation on your calculator.

             a.  (6.39 x 10²) x (5.92 x 10⁸)

             b.   (1.2 x 10^-3) x (3.5 x 10⁵)

             c.   (6.39 x 10²) x (592 x 10⁸)

 Addition and Subtraction with Scientific Notation

 When you learned to add and subtract decimal numbers in elementary school, you discovered that you must line up the decimal points before beginning to add the numbers.  For numbers written in scientific notation, the position of the decimal point is determined by the exponential part of the number.  If you are adding these numbers manually, all of the numbers must have identical exponential parts (i.e. be raised to the same power of 10) before lining up the decimal points and adding.  Because of this extra step, it is easier to let our calculators work for us.  So, if you key in the numbers correctly, your calculator will handle the movement of the decimal point.  For some calculators, the answer will automatically be expressed in scientific notation; for others you must express the number in scientific notation yourself.

            5.  Perform the addition and subtraction problems below and express the answer in scientific notation.

            a.  (4.2 x 10 ^-3) - (3.4 x 10^-4)

            b.  (7.3 x10⁵) + (5.8 x 10⁴)

Density

Let's apply what we have learned about manipulations of numbers using density problems.  Density is defined as mass per unit volume:

                        Density = mass/volume

In symbols,

                        D=m/V

Units commonly used for mass are grams and volume is commonly expressed in milliliters, cubic centimeters and liters.  Remember that one milliliter is equal to one cubic centimeter and that there are 1000 milliliters in one liter. 

Density is commonly used to identify a substance because it is an intensive property.  Properties of substances generally fall into two types

¨      Extensive, in which the value depends on the amount of substance present

¨      Intensive, in which the value is constant regardless of the amount of substance present.

Mass and volume are both extensive properties.  Obviously, the more of a substance you have, the greater its mass and volume.  However, by dividing the mass by the volume, the dependence on the amount of substance "cancels out" and we have a property which is intensive.  The density values of some common substances are given in the table below;

Table 2.  Densities of some common substances

             6.  If you had samples of aluminum, copper and gold, each with a volume of 1.00 cm³, which substance would have the highest mass?  ______________________

            7.  If you had 1.00 gram samples of the lead, magnesium and mercury, which substance would have the largest volume?  __________________

You can use mass and volume measurements to calculate densities of substances as illustrated in the problems below:

            8.  A piece of wood has a mass of 30.0 g and a volume of 35.0 cm³.  Calculate the density.

            9.  Calculate the density of a metal bar with a mass of 113 grams and a volume of 12.7 cm³.  Identify the metal using the density values in Table 2.

Densities of common substances also serve uses other than identification.  If you are dealing with liquids in the lab, it is much easier to measure a volume than a mass.  Density allows one to interconvert between mass and volume.  To do this, it is necessary to manipulate the density formula to get the variable needed. 

            10.  If D=m/V,

a.  solve the formula for mass.

b.  solve the formula for volume.

The only tricky part is to remember that the mass and volume units must match those in the density value.

            11.  Calculate the mass of 2.5L of mercury.

            12. In the lab, you need 200 grams of octane.  How many milliliters should you use?

Other algebraic equations

Frequently it is necessary to solve algebraic equations in chemistry.  If you have an equation such as

                        3x + 23 = 50

you must first isolate the term containing the unknown (x in this case) on one side of the equation.  In this case, you would subtract 23 from each side:

            3x+23-23 = 50-23

            3x = 27

Now that the term containing the variable is isolated, you can now solve for x by dividing both sides of the equation by 3.

            3x/3 = 27/3

            x = 9

Looking at a slightly more complicated case, let's solve the equation below for F

First, we need to begin to isolate F on one side of the equation.  Because F is in the denominator of the expression, we must first multiply both sides of the equation by the expression containing F.

Again, let's try to isolate F by multiplying both sides of the equation by 9/5.

Now, to isolate F, we would add 32 to both sides:

This is the equation for the conversion from the Fahrenheit to the Celcius temperature scale.

13.  Using the equation above, calculate the Celcius temperature if the Fahrenheit temperature is 55.

14.  Solve the following equation for t