Scientific Measurement Chapter 2 Scientific Measurement

Chapter 2 Goals – Scientific Measurement Calculate values from measurements using the correct number of significant figures. List common SI units of measurement and common prefixes used in the SI system. Distinguish mass, volume, density, and specific gravity from one another. Evaluate the accuracy of measurements using appropriate methods.

Introduction Everyone uses measurements in some form Deciding how to dress based on the temperature; measuring ingredients for a recipes; construction. Measurement is also fundamental in the sciences and for understanding scientific concepts It is important to be able to take good measurements and to decide whether a measurement is good or bad

Introduction In this class we will make measurements and express their values using the International System of Units or the SI system. All measurements have two parts: a number and a unit.

2.1 The Importance of Measurement Qualitative versus Quantitative Measurements Qualitative measurements give results in a descriptive, nonnumeric form; can be influenced by individual perception Example: This room feels cold.

2.1 The Importance of Measurement Qualitative versus Quantitative Measurements Quantitative measurements give results in definite form usually, using numbers; these types of measurements eliminate personal bias by using measuring instruments. Example: Using a thermometer, I determined that this room is 24°C (~75°F) Measurements can be no more reliable than the measuring instrument.

2.1 Concept Practice 1. You measure 1 liter of water by filling an empty 2-liter soda bottle half way. How can you improve the accuracy of this measurement? 2. Classify each measurement as qualitative or quantitative. a. The basketball is brown b. the diameter of the basketball is 31 centimeters c. The air pressure in the basketball is 12 lbs/in2 d. The surface of the basketball has indented seams

2.2 Accuracy and Precision Good measurements in the lab are both correct (accurate) and reproducible (precise) accuracy – how close a single measurement comes to the actual dimension or true value of whatever is measured precision – how close several measurements are to the same value Example: Figure 2.2, page 29 – Dart boards….

2.2 Accuracy and Precision All measurements made with instruments are really approximations that depend on the quality of the instruments (accuracy) and the skill of the person doing the measurement (precision) The precision of the instrument depends on the how small the scale is on the device. The finer the scale the more precise the instrument. 2.2 Demo, page 28

2.2 Concept Practice 3. Which of these synonyms or characteristics apply to the concept of accuracy? Which apply to the concept of precision? a. multiple measurements b. correct c. repeatable d. reproducible e. single measurement f. true value

2.2 Concept Practice 4. Under what circumstances could a series of measurements of the same quantity be precise but inaccurate?

2.3 Scientific Notation In chemistry, you will often encounter numbers that are very large or very small One atom of gold = 0.000000000000000000000327g One gram of H = 301,000,000,000,000,000,000,000 H molecules Writing and using numbers this large or small is calculations can be difficult It is easier to work with these numbers by writing them in exponential or scientific notation

2.3 Scientific Notation scientific notation – a number is written as the product of two numbers: a coefficient and a power of ten Example: 36,000 is written in scientific notation as 3.6 x 104 or 3.6e4 Coefficient = 3.6 → a number greater than or equal to 1 and less than 10. Power of ten / exponent = 4 3.6 x 104 = 3.6 x 10 x 10 x 10 x 10 = 36,000

2.3 Scientific Notation When writing numbers greater than ten in scientific notation the exponent is positive and equal to the number of places that the exponent has been moved to the left.

2.3 Scientific Notation Numbers less than one have a negative exponent when written in scientific notation. Example: 0.0081 is written in scientific notation as 8.1 x 10-3 8.1 x 10-3 = 8.1/(10 x 10 x 10) = 0.0081 When writing a number less than one in scientific notation, the value of the exponent equals the number of places you move the decimal to the right.

2.3 Scientific Notation To multiply numbers written in scientific notation, multiply the coefficients and add the exponents. (3 x 104) x (2 x 102) = (3 x 2) x 104+2 = 6 x 106 To divide numbers written in scientific notation, divide the coefficients and subtract the exponent in the denominator (bottom) from the exponent in the numerator (top). (6 x 103)/(2 x 102) = (6/2) x 103-2 = 3 x 101

2.3 Scientific Notation Before numbers written in scientific notation are added or subtracted, the exponents must be made the same (as a part of aligning the decimal points). (5.4 x 103)+(6 x 102) = (5.4 x 103)+(0.6 x 103) = (5.4 + 0.60) x 103 = 6.0 x 103

2.3 Concept Practice 5. Write the two measurements given in the first paragraph of this section in scientific notation. a. mass of a gold atom = 0.000000000000000000000327g b. molecules of hydrogen = 301,000,000,000,000,000,000,000 H molecules

2.3 Concept Practice 6. Write these measurements in scientific notation. The abbreviation m stands for meter, a unit of length. a. The length of a football field, 91.4 m b. The diameter of a carbon atom, 0.000000000154 m c. The radius of the Earth, 6,378,000 m d. The diameter of a human hair, 0.000008 m e. The average distance between the centers of the sun and the Earth, 149,600,000,000 m

2.4 Significant Figures in Measurement The significant figures in a measurement include all the digits that are known precisely plus one last digit that is estimated. Example: With a thermometer that has 1° intervals, you may determine that the temperature is between 24°C and 25°C and estimate it to be 24.3°C. You know the first two digits (2 and 4) with certainty, and the third digit (3) is a “best guest” By estimating the last digit, you get additional information

Rules 1. Every nonzero digit in a recorded measurement is significant. - Example: 24.7 m, 0.743 m, and 714 m all have three sig. figs. 2. Zeros appearing between nonzero digits are significant. - Example: 7003 m, 40.79 m, and 1.503 m all have 4 sig. figs.

Rules 3. Zeros appearing in front of all nonzero digits are not significant; they act as placeholders and cannot arbitrarily be dropped (you can get rid of them by writing the number in scientific notation). - Example: 0.0071 m has two sig. figs. And can be written as 7.1 x 10-3 4. Zeros at the end of the number and to the right of a decimal point are always significant. - Example: 43.00 m, 1.010 m, and 9.000 all have 4 sig. figs.

Rules 5. Zeros at the end of a measurement and to the left of the decimal point are not significant unless they are measured values (then they are significant). Numbers can be written in scientific notation to remove ambiguity. - Example: 7000 m has 1 sig. fig.; if those zeros were measured it could be written as 7.000 x 103

Rules 6. Measurements have an unlimited number of significant figures when they are counted or if they are exactly defined quantities. - Example: 23 people or 60 minutes = 1 hour * You must recognize exact values to round of answers correctly in calculations involving measurements.

Significant Figures – Example 1 How many significant figures are in each of the following measurements? a. 123 m b. 0.123 cm c. 40506 mm d. 9.8000 x 104 m e. 4.5600 m f. 22 meter sticks g. 0.07080 m h. 98000 m

2.4 Concept Practice 7. Write each measurement in scientific notation and determine the number of significant figures in each. a. 0.05730 m b. 8765 dm c. 0.00073 mm d. 12 basketball players e. 0.010 km f. 507 thumbtacks

Significant Figures in Calculations The number of significant figures in a measurement refers to the precision of a measurement; an answer cannot be more precise than the least precise measurement from which it was calculated. Example: The area of a room that measures 7.7 m (2 sig. figs.) by 5.4 m (2 sig. figs.) is calculated to be 41.58 m2 (4 sig. figs.) – you must round the answer to 42 m2

Rounding – The Rule of 5 If the digit to the right of the last sig. fig is less than 5, all the digits after the last sig. fig. are dropped. Example: 56.212 m rounds to 56.21 m (for 4 sig. figs.) If the digit to the right is 5 or greater, the value of the last sig. fig. is increased by 1. Example: 56.216 m rounds to 56.22 m (for 4 sig. figs.)

Rounding – Example 2

Addition and Subtraction The answer to an addition or subtraction problem should be rounded to have the same number of decimal places as the measurement with the least number of decimal places.

Multiplication and Division In calculations involving multiplication and division, the answer is rounded off to the number of significant figures in the least precise term (least number of sig. figs.) in the calculations

2.6 SI Units The International System of Units, SI, is a revised version of the metric system Correct units along with numerical values are critical when communicating measurements. The are seven base SI units (Table 2.1) of which other SI units are derived. Sometimes non-SI units are preferred for convenience or practical reasons

2.6 SI Units – Table 2.2 Quantity SI Base or Derived Unit Non-SI Unit Length meter (m) Volume cubic meter (m3) liter Mass kilogram (kg) Density grams per cubic centimeter (g/cm3); grams per mililiter (g/mL) Temperature kelvin (K) degree Celcius (°C) Time second (s) Pressure Pascal (Pa) atmosphere (atm); milimeter of mercury (mm Hg) Energy joule (J) calorie (cal)

Common SI Prefixes Units larger than the base unit Tera T terameter (Tm) Giga G e-9 = 0.000000001 gigameter (Gm) Mega M e-6 = 0.000001 megameter (Mm) Kilo k e-3 = 0.001 kilometer (km) Hecto h e-2 = 0.01 hectometer (hm) Deka da e-1 = 0.1 decameter (dam) Base Unit e0 = 1 meter (m)

Common SI Prefixes Units smaller than the base unit e0 = 1 meter (m) Deci d e1 = 10 decimeter (dm) Centi c e2 = 100 centimeter (cm) Milli m e3 = 1000 millimeter (mm) Micro μ e6 = 1,000,000 micrometer (μm) Nano n e9 = 1,000,000,000 Nanometer (nm) Pico p e12 = 1,000,000,000,000 picometer (pm)

Common SI Prefixes A mnemonic device can be used to memorize these common prefixes in the correct order: The Great Monarch King Henry Died By Drinking Chocolate Mocha Milk Not Pilsner

2.7 Units of Length The basic unit of length is the meter Prefixes can be used with the base unit to more easily represent small or large measurements Example: A hyphen (12 point font) measures about 0.001 m or 1 mm. Example: A marathon race is approximately 42,000 m or 42 km.

2.7 Concept Practice 15. Use the tables in the text to order these lengths from smallest to largest. a. centimeter b. micrometer c. kilometer d. millimeter e. meter f. decimeter

2.8 Units of Volume The space occupied by any sample of matter is called its volume The volume of rectangular solids can be calculated by multiplying the length by width by height Units are cubed because you are measuring in 3 dimensions Volume of liquids can be measured with a graduated cylinder, a pipet, a buret, or a volumetric flask

2.8 Units of Volume A convenient unit of measurement for volume in everyday use is the liter (L) Milliliters (mL) are commonly used for smaller volume measurements and liters (L) for larger measurements 1 mL = 1 cm3 10 cm x 10 cm x 10 cm = 1000 cm3 = 1 L

2.8 Units of Volume

2.8 Concept Practice 17. From what unit is a measure of volume derived?

2.8 Practice 18. What is the volume of a paperback book 21 cm tall, 12 cm wide, and 3.5 cm thick? 19. What is the volume of a glass cylinder with an inside diameter of 6.0 cm and a height of 28 cm? V=πr2h

2.9 Units of Mass A person on the moon would weigh 1/6 of his/her weight on Earth. This is because the force of gravity on the moon is approximately 1/6 of its force of Earth. Weight is a force – it is a measure of the pull on a given mass by gravity; it can change by location. Mass is the quantity of matter an object contains Mass remains constant regardless of location. Mass v. Weight

2.9 Units of Mass The kilogram is the basic SI unit of mass It is defined as the mass of 1 L of water at 4°C. A gram, which is a more commonly used unit of mass, is 1/1000 of a kilogram 1 gram = the mass of 1 cm3 of water at 4°C.

2.9 Concept Practice 20. As you climbed a mountain and the force of gravity decreased, would your weight increase, decrease, or remain constant? How would your mass change? Explain. 21. How many grams are in each of these quantities? a. 1 cg b. 1 μg c. 1 kg d. 1mg

2.10 Density Density is the ratio of the mass of an object to its volume. Equation → D = mass/volume Common units: g/cm3 or g/mL Example: 10.0 cm3 of lead has a mass 114 g Density (of lead) = 114 g / 10.0 cm3 = 11.4 g/cm3 See Table 2.7, page 46

2.10 Density Density determines if an object will float in a fluid substance. Examples: Ice in water; hot air rises Density can be used to identify substances See Table 2.8, page 46

2.10 Concept Practice 22. The density of silver is 10.5 g/cm3 at 20°C. What happens to the density of a 68-g bar of silver that is cut in half?

2.10 Concept Practice 23. A student finds a shiny piece of metal that she thinks is aluminum. In the lab, she determines that the metal has a volume of 245 cm3 and a mass of 612 g. Is the metal aluminum? 24. A plastic ball with a volume of 19.7 cm3 has a mass of 15.8 g. Would this ball sink or float in a container of gasoline?

2.10 Specific Gravity (Relative Density) Specific gravity is a comparison of the density of a substance to the density of a reference substance, usually at the same temperature. Water at 4°C, which has a density of 1 g/cm3, is commonly used as a reference substance. Specific gravity = density of substance (g/cm3) density of water (g/cm3) Because units cancel, a measurement of specific gravity has no units A hydrometer can be used to measure the specific gravity of a liquid.

2.11 Concept Practice 25. Why doesn’t a measurement of specific gravity have a unit? 26. Use the values in Table 2.8 to calculate the specific gravity of the following substances. a. Aluminum b. Mercury c. ice

2.12 Measuring Temperature Temperature determines the direction of heat transfer between two objects in contact with each other. Heat moves from the object at the higher temperature to the object at a lower temperature. Temperature is a measure of the degree of hotness or coldness of an object. Almost all substances expand with an increase in temperature and contract with a decrease in temperature An important exception is water

2.12 Measuring Temperature There are various temperature scales On the Celsius temperature scale the freezing point of water is taken as 0°C and the boiling point of water at 100°C

2.12 Measuring Temperature The Kelvin scale (or absolute scale) is another temperature scale that is used On the Kelvin scale the freezing point of water is 273 K and the boiling point is 373 K (degrees are not used). 1°C = 1 Kelvin The zero point (0 K) on the Kelvin scale is called absolute zero and is equal to -273°C Absolute zero is where all molecular motion stops

2.12 Measuring Temperature Converting Temperatures: K = °C + 273 °C = K - 273

2.12 Concept Practice 27. Surgical Instruments may be sterilized by heating at 170°C for 1.5 hours. Convert 170°C to kelvins. 28. The boiling point of the element argon is 87 K. What is the boiling point of argon in °C?

2.13 Evaluating Measurements Accuracy in measurement depends on the quality of the measuring instrument and the skill of the person using the instrument. Errors in measurement could have various causes In order to evaluate the accuracy of a measurement, you must be able to compare it to the true or accepted value.

2.13 Evaluating Measurements accepted value – the true or correct value based or reliable references experimental value – the measured value determined in the experiment The difference between the accepted value and the experimental value is the error. error = accepted value – experimental value

2.13 Evaluating Measurements The percent error is the error divided by the accepted value, expressed as a percentage of the accepted value. Percent Error = x 100 An error can be positive or negative, but an absolute value of error is used so that the percentage is positive |error| AV

2.13 Concept Practice 32. A student estimated the volume of a liquid in a beaker as 208 mL. When she poured the liquid into a graduated cylinder she measured the value as 200 mL. What is the percent error of the estimated volume from the beaker, taking the graduated cylinder measurement as the accepted value?