Chapter 2 Data Analysis 2.1 Units of Measurement 2.2 Scientific Notation and Dimensional Analysis 2.3 How reliable are measurements

2.1 Units of Measurement Objectives Define SI base units for time, length, mass, and temperature. Explain how adding a prefix changes a unit. Compare the derived units for volume and density.

SI Units International System of Units (SI Units)has seven base units of measure: Length – meter (m) Time – second (s) Amount of substance – mole (mol) Electric current – Ampere (A) Temperature – Kelvin (K) Luminous Intensity – Candela (cd) Mass – kilogram (kg)

Time The SI base unit for time is the second (s). The frequency of microwave radiation given off by a cesium-133 (Cs- 133) atom is the physical standard used to establish the length of a second. Cesium clocks are more reliable than the ones we use every day. Many chemical reaction take place in less than a second.

Length The SI base unit for length is the meter (m). A meter is the distance that light travels through a vacuum in 1/299 792 458 of a second.

Mass Kilogram (kg) is the SI base unit for mass. Kilogram is defined by the platinum-iridium metal sphere kept in Sevres, France. It is the only base unit whose standard is currently a physical object.

Prefixes Used with SI Units Symbol Factor Scientific Notation Example giga G 1,000,000,000 109 gigameter(Gm) mega M 1,000,000 106 megameter(Mm) kilo k 1,000 103 kilometer(km) deci d 1/10 10-1 decimeter(dm) centi c 1/100 10-2 centimeter(cm) milli m 1/1000 10-3 millimeter(mm) micro µ 1/1,000,000 10-6 micrometer(µm) nano n 1/1,000,000,000 10-9 nanometer(nm) pico p 1/1,000,000,000,000 10-12 picometer(pm)

Derived Units Derived units are units defined by a combination of base units. Speed m/s (distance and time) Volume is the space occupied by an object – cm3 Volume of liquid is measured in mL (1 mL is equal to 1cm3) Density is the ratio that compares the mass of an object to its volume. g/ cm3 D=m/V 1000 mL Volume of cotton balls and 1000 mL Volume of marbles….Do they have the same density? Why?

Density of Aluminum Mass – 13.5 g Volume – 5.0 cm3 Density = mass/Volume D= 13.5g/5.0 cm3 D=2.7g/cm3

Temperature Hot or cold are qualitative descriptions of temperature. In order to get a quantitative description you need a thermometer. A thermometer is a narrow tube with a liquid that expands when heated and contracts when cooled.

Celsius Celsius Devised by Anders Celsius, a Swedish astronomer. Uses the temperature at which water freezes and boils to establish his scale. 0ºC 100ºC

Kelvin Kelvin scale was devised by William Thomson, a Scottish physicist and mathematician, who was known as Lord Kelvin. Water freezes at 273K and boils at 373K Kelvin is the SI unit for temperature.

Kelvin  Celsius or Celsius  Kelvin 10ºC + 273 = 283 K 293 K – 273 = 20ºC

2.2 Scientific Notation and Dimensional Analysis Express numbers in scientific notation Use dimensional analysis to convert between units.

Scientific Notation Scientific notation expresses numbers as a multiple of two factors: a number between 1 and 10; and ten raised to a power, or exponent. The exponent tells you how many times the first factor must be multiplied by ten. Greater than 1 – positive exponent Less than 1 – negative exponent

Scientific Notation 1,392,000 0.000000028

Adding and subtracting using scientific notation When adding or subtracting using scientific notation the exponents must be the same. Ex. 7.35 x 102 m + 2.43 x 102 m = 9.78 x 102 m

Let’s try 15.6 x 106 + 0.165 x 108 + 2.70 x 107 2.4 x 103 – 0.23 x 104

Multiplying and dividing using scientific notation When multiplying or dividing the exponents do not have to be the same. For multiplication First multiply the first factors Then add the exponents For division First divide the first factors Then subtract the exponents

Example Multiplication (2 x 103) x (3x102) Division (9x108) ÷ (3x10-4)

Dimensional Analysis Dimensional Analysis is a method of problem-solving that focuses on the units used to describe matter. A conversion factor is a ratio of equivalent values used to express the same quantity in different units. Often used in dimensional analysis

Example Convert 48 km to m Convert 550m/s to km/min

2.3 How reliable are measurements Objectives Define and compare accuracy and precision. Use significant figures and rounding to reflect the certainty of data. Use percent error to describe the accuracy of experimental data.

ACCURACY/PRECISION Accuracy Precision How close a measurement is to the actual value Precision How close a set of measurements are to each other

Accuracy and Precision The density of a white solid was determined by students. The same was sucrose, which has a density of 1.59 g/cm3 Who collected the most accurate data? Who collected the most precise data? Student A Student B Student C Trial 1 1.54g/cm3 1.40g/cm3 1.70g/cm3 Trial 2 1.60g/cm3 1.68g/cm3 1.69g/cm3 Trial 3 1.57g/cm3 1.45g/cm3 1.71g/cm3 Average 1.51g/cm3

Percent Error Percent error is the ratio of an error to an accepted value. Percent error = error/accepted value x 100 Student A Student B Student C Trial 1 -0.05g/cm3 -0.19g/cm3 +0.11g/cm3 Trial 2 +0.01g/cm3 +0.09g/cm3 +0.10g/cm3 Trial 3 -0.02g/cm3 -0.14g/cm3 +0.12g/cm3

Percent Error – Student A Trial Density (g/cm3) Error (g/cm3) 1 1.54 -0.05 2 1.60 +0.01 3 1.57 -0.02 % error = 0.05 g/cm3/1.59 g/cm3 x 100 = 3.14% % error = 0.01 g/cm3/1.59 g/cm3 x 100 = 0.63% % error = 0.02 g/cm3/1.59 g/cm3 x 100 = 1.26%

SIGNIFICANT DIGITS or Figures This is a process used to determine the number of digits to round to when measuring an object. Use this process when Measuring mass (on the scale) – g, kg, etc. Measuring volume (in a graduated cylinder) – ml L, etc. Measuring length (with a ruler) – cm, m Is used to communicate to other scientists how accurate your measurement is: Does your scale measure to the hundredths place, tenths place or whole number? Referred to as “Sig Figs”

How to determine the number of Sig Figs in a measured value Atlantic-Pacific Method A = decimal Absent, begin counting from right P = decimal Present, begin counting from left Try these: 1,000 1 sig fig 0.001 0.0010 2 sig fig 1000.0 5 sig fig

Rules for Using Sig Figs Multiplication/Division Do all calculations, then round to the same number of digits as the number with the smallest number of sig figs 4.56 x 1.4 = 6.384 Round to 2 sig figs: 6.4 8.315/298 = 0.0279027 Round to 3 sig figs: 0.0279 Addition/Subtraction Do the calculations, then round to the place of the number with the smallest number of decimal places 12.11 + 18.0 + 1.013 = 31.123 Round to 31.1 88.88 – 2.2 = 86.68 Round to 86.7 (note: if the number after 6 is > 5, round up)

Rules for Using Sig Figs Multiple step calculations Use an overbar to keep track of the significant figures from step to step. Round only when reporting the final answer Example: 88.88 – 86.66 2.22 .024977 (calculator 88.88 88.88 answer) Based on 2.22, round to 3 sig figs .024977 If the number after the place you want to round to is > 5, round up (in this case 7). Ignore the other 7. Answer = .0250 The zero after the 5 is significant. You must show it! = =