CHEMISTRY CHAPTER 2, SECTION 3. USING SCIENTIFIC MEASUREMENTS Accuracy and Precision Accuracy refers to the closeness of measurements to the correct or accepted value of the quantity measured.

Precision refers to the closeness of a set of measurements of the same quantity made in the same way. (Another definition: how exact a measurement is.)

Accuracy and Precision

Example: 5 measurements of the density of water made by two groups. density (g/cm 3 ) Group 1Group Average Which is more accurate? More precise?

Percentage error is calculated by subtracting the accepted value from the experimental value, dividing the difference by the accepted value, and then multiplying by 100.

Sample Problem: A student measures the mass and volume of a substance and calculates its density as 1.40 g/mL. The correct, or accepted, value of the density is 1.30 g/mL. What is the percentage error of the student’s measurement? Note that units will cancel out.

Note that if the measured value is lower than the accepted value, the percentage error will be a negative number.

Error in Measurement Some error or uncertainty always exists in any measurement. skill of the measurer conditions of measurement measuring instruments

Significant Figures Significant figures in a measurement consist of all the digits known with certainty plus one final digit, which is somewhat uncertain or is estimated.

Reporting Measurements Using Significant Figure s

Rules for Significant Figures Any digit that is not zero counts. Any zero between two digits that are not zero counts. Numbers greater than 1: don’t count zeroes at the right, unless they are after the decimal point or are followed by a decimal point. examples: 7, significant figures 7, significant figures 7, significant figures

Numbers less than 1: don’t count zeroes at the left (before the first digit that is not zero). Zeroes to the right are counted. examples:.0038 – 2 significant figures – 2 significant figures – 3 significant figures

How many significant figures?

How many significant figures?

All digits (not including the 10 x ) are significant when a number is written in scientific notation. Example: we want to express 300 with two significant figures. If we write 300, this has 1 significant figure, but 300. has 3.

Solution: write the number as 3.0 x This has 2 significant figures. How many significant figures in these? 6 x x x 10 -3

Solution: write the number as 3.0 x This has two significant figures. How many significant figures in these? 6 x x x

Rounding We use rounding to express calculated numbers to the correct number of significant figures, and not suggest greater precision than actually occurred.

Rules for Rounding (see Tab. 6, p. 48) If the number following the last digit that you want to keep is 6, 7, 8, or 9, round up. Examples: 16 → → to 2 sig. fig. → → 0.10

If the number following the last digit that you want to keep is 0, 1, 2, 3, or 4, keep the digit the same. Examples: 183 → to 2 s.f. → If the number following the last digit that you want to keep is 5 followed by a nonzero digit, round up. Examples: 1851 to 2 s.f. → to 3 s.f. →

If the number following the last digit that you want to keep is 5 exactly: some people say always round up; others say round up or down to give an even number (to reduce slight errors that can occur if you have lots of numbers that are rounded up). Examples: 2.5 rounds to 3 or to rounds to to 1 significant figures rounds to rounds to 870 or to 860

Addition or Subtraction with Significant Figures When adding or subtracting decimals, the answer must have the same number of digits to the right of the decimal point as there are in the measurement having the fewest digits to the right of the decimal point.

Example: g g = g Since the second number has only 1 digit to the right of the decimal point, the sum must be rounded to 1 decimal place =

Example: g g = g Since the second number has only1 digit to the right of the decimal point, the sum must be rounded to 1 decimal place = 57.6 g

For numbers with no decimal points, round so that the last significant digit is in the same place as the leftmost uncertain digit. Example: – = Since the leftmost uncertain digit is in the 10 thousands place (the “3” of ), the difference must be rounded to the same place =

For numbers with no decimal points, round so that the last significant digit is in the same place as the leftmost uncertain digit. Example: – = Since the leftmost uncertain digit is in the 10 thousands place (the “3” of ), the difference must be rounded to the same place =

For multiplication or division, the answer can have no more significant figures than are in the measurement with the fewest number of significant figures. Multiplication or Division with Significant Figures

Examples: 6.50 m x 3 m = m 2 The first number has 3 s.f., the second only 1, so the answer rounds to 1 s.f. =

Examples: 6.50 m x 3 m = m 2 The first number has 3 s.f., the second only 1, so the answer rounds to 1 s.f. = 20 m 2

Of the two numbers being divided, one has 4 s.f. and the other 2, so the answer must be rounded to 2 s.f. =

Of the two numbers being divided, one has 4 s.f. and the other 2, so the answer must be rounded to 2 s.f. = 1.4 g/cm 3

Exact Numbers and Significant Figures Exact numbers are not considered when determining significant figures. 1. Conversion factors: example: You don’t need to round this because 1000 is an exact number.

2. Integers. Example: you determine the mass of 3 pennies as g, g, and g. What is the average mass? The “3” in the denominator is an integer. Round to 4 significant figures, not 1. Answer =

2. Integers. Example: you determine the mass of 3 pennies as g, g, and g. What is the average mass? The “3” in the denominator is an integer. Round to 4 significant figures, not 1. Answer = 2.526

In scientific notation, numbers are written in the form M × 10 n, where the factor M is a number greater than or equal to 1 but less than 10 and n is a whole number (positive or negative). 1. Determine M by moving the decimal point in the original number to the left or the right so that only one nonzero digit remains to the left of the decimal point. Scientific Notation

2. Determine n by counting the number of places that you moved the decimal point. If you moved it to the left, n is positive (the number is greater than 1). If you moved it to the right, n is negative (the number is less than 1).

example: mm = 1.2 × 10 −4 mm Move the decimal point 4 places to the right and multiply the number by 10 −4.

Examples = x 10 8 move decimal pt. 8 places 2.36 x 10 4 = = x x =

Mathematical Operations Using Scientific Notation 1. Addition and subtraction —These operations can be performed only if the values have the same exponent (n factor). ex.: 4.2 × 10 4 kg × 10 3 kg (Why?)

or: 42 x 10 3 kg 7.9 x 10 3 kg 49.9 x 10 3 kg rounded to 50. x 10 3 kg Note that 50. x 10 3 is not in scientific notation, so it must be changed to: 5.0 x 10 4 kg

2. Multiplication —The M factors (number parts) are multiplied, and the exponents (powers of 10) are added algebraically. ex.: (5.23 × 10 6 µm) (7.1 × 10 −2 µm) = (5.23 × 7.1) (10 6 × 10 −2 ) = × 10 4 µm 2 = 3.7 × 10 5 µm 2 (convert to scientific notation and round to 2 significant figures)

3. Division — The M factors are divided, and the exponent of the denominator is subtracted from that of the numerator. example: = × 10 3 = 6.7  10 2 g/mol

Direct Proportions Two quantities are directly proportional to each other if dividing one by the other gives a constant value. We can also write: read as “y is proportional to x.”

Slope = k Graph is a straight line passing through (0,0)

Inverse Proportions Two quantities are inversely proportional to each other if their product is constant: x y = k If one increases, the other decreases. read as “y is proportional to 1 divided by x.”

The graph is a hyperbola. P V = k