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FCS Physics and Chemistry

Intro to Chemistry : Measurements and Calculations
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This topic is an introduction to the world of measurements in chemistry.  Here we explore scientific notation, units, significant figures and dimensional analysis. 
 
Scientific Notation
 
Scientific notation is an easy way to represent large numbers.  There are only a few basic rules that you need to learn.  See below..
 

Scientific Notation


Convert from Scientific Notation to Real Number:
5.14 x 105 = 514000.0

    Scientific notation consists of a coefficient (here 5.14) multiplied by 10 raised to an exponent (here 5). To convert to a real number, start with the base and multiply by 5 tens like this: 5.14 x 10 x 10 x 10 x 10 x 10 = 514000.0. Multiplying by tens is easy: one simply moves the decimal point in the base (5.14) 5 places to the right, adding extra zeroes as needed.

  • Convert from Real Number to Scientific Notation:
    0.000345 = 3.45 x 10-4
    • Here we wish to write the number 0.000345 as a coefficient times 10 raised to an exponent. To convert to scientific notation, start by moving the decimal place in the number until you have a number between 1 and 10; here it is 3.45. The number of places to the left that you had to move the decimal point is the exponent. Here, we had to move the decimal 4 places to the right, so the exponent is -4.
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    Practice Questions!

    Significant Figures

    It is important to be honest when reporting a measurement, so that it does not appear to be more accurate than the equipment used to make the measurement allows. We can achieve this by controlling the number of digits, or significant figures, used to report the measurement.

    Determining the Number of Significant Figures

    The number of significant figures in a measurement, such as 2.531, is equal to the number of digits that are known with some degree of confidence (2, 5, and 3) plus the last digit (1), which is an estimate or approximation. As we improve the sensitivity of the equipment used to make a measurement, the number of significant figures increases.

    Postage Scale 3 1 g 1 significant figure
    Two-pan balance 2.53 0.01 g 3 significant figures
    Analytical balance 2.531 0.001 g 4 significant figures

    Rules for counting significant figures are summarized below.

    Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures.

    Zeros that do nothing but set the decimal point are not significant. Thus, 470,000 has two significant figures.

    Trailing zeros that aren't needed to hold the decimal point are significant. For example, 4.00 has three significant figures.

    If you are not sure whether a digit is significant, assume that it isn't. For example, if the directions for an experiment read: "Add the sample to 400 mL of water," assume the volume of water is known to one significant figure.


    Addition and Subtraction with Significant Figures

    When combining measurements with different degrees of accuracy and precision, the accuracy of the final answer can be no greater than the least accurate measurement. This principle can be translated into a simple rule for addition and subtraction: When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.

    150.0 g H2O (using significant figures)
    + 0.507 g salt
    150.5 g solution


    Multiplication and Division With Significant Figures

    The same principle governs the use of significant figures in multiplication and division: the final result can be no more accurate than the least accurate measurement. In this case, however, we count the significant figures in each measurement, not the number of decimal places: When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement.

    Example: To illustrate this rule, let's calculate the cost of the copper in an old penny that is pure copper. Let's assume that the penny has a mass of 2.531 grams, that it is essentially pure copper, and that the price of copper is 67 cents per pound. We can start by from grams to pounds.

    equation

    We then use the price of a pound of copper to calculate the cost of the copper metal.

    equation

    There are four significant figures in both the mass of the penny (2.531) and the number of grams in a pound (453.6). But there are only two significant figures in the price of copper, so the final answer can only have two significant figures.

    Practice Problem 7

    Calculate the length in inches of a piece of wood 1.245 feet long. Determine the correct number of significant figures.

    Answer:

    14.94 in


    Rounding Off

    When the answer to a calculation contains too many significant figures, it must be rounded off.

    There are 10 digits that can occur in the last decimal place in a calculation. One way of rounding off involves underestimating the answer for five of these digits (0, 1, 2, 3, and 4) and overestimating the answer for the other five (5, 6, 7, 8, and 9). This approach to rounding off is summarized as follows.

    If the digit is smaller than 5, drop this digit and leave the remaining number unchanged. Thus, 1.684 becomes 1.68.

    If the digit is 5 or larger, drop this digit and add 1 to the preceding digit. Thus, 1.247 becomes 1.25.

    What are units and why are units important?

    Units give numbers meaning!   

    3 means what?  3 lemons..3 mph...we have no idea without units

    WE CAN USE UNITS TO MAKE SOLVING EQUATIONS EASY!

    WE CALL THIS DIMENSIONAL ANALYSIS

    For Example:

    distance can be measured in m

    so, given the following information..we can solve for distance and show by cancelling units that we have the correct answer (as long as we input the numbers correctly into the calculator)

    EXAMPLE

    A car travels at 25 m/s for 10 seconds, how far did it go?

    Step 1.  Solve the equation

    V  =  d / t   , we can multiply each side by t to solve for d

    d = Vt  =  (25 m/s) (10 s) = 250 m

    Step 2.  Check units

         

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