Christopher G. Hamaker, Illinois State University, Normal IL © 2005, Prentice Hall Scientific Measurements

Scientific Investigations Science is the methodical exploration of nature followed by a logical explanation of the observations. Scientific investigation entails: –planning an investigation –carefully recording observations –gathering data –analyzing the results

The Scientific Method The scientific method is a systematic investigation of nature and requires proposing an explanation for the results of an experiment in the form of a general principle. The initial, tentative proposal of a scientific principle is called a hypothesis. After further investigation, the original hypothesis my rejected, revised, or elevated to the status of a scientific principle.

It is a series of steps (not always in this order) Making observations => question Formulating hypotheses => answering question inferring, predicting. Testing hypotheses. => experimenting, communicating, collecting data, and measuring. Formulating theories => Confirming hypotheses that are supported by data. It includes constructing models, and predicting. The Scientific Method

Uncertainty in Measurements A measurement is a number with a unit attached. It is not possible to make exact measurements, and all measurements have uncertainty. We will generally use metric system units, these include. –the meter, m, for length measurements –the gram, g, for mass measurements –the liter, L, for volume measurements

Length Measurements Lets measure the length of a candy cane. Ruler A has 1 cm divisions, so we can estimate the length to ± 0.1 cm. The length is 4.2 ± 0.1 cm. Ruler B has 0.1 cm divisions, so we can estimate the length to ± 0.05 cm. The length is 4.25 ± 0.05 cm.

Uncertainty in Length Ruler A: 4.2 ± 0.1 cm; Ruler B: 4.25 ± 0.05 cm. Ruler A has more uncertainty than Ruler B. Ruler B gives a more precise measurement.

Mass Measurements The mass of an object is a measure of the amount of matter it posses. Mass is measured with a balance and is not affected by gravity. Mass and weight are not interchangeable.

Volume Measurements Volume is the amount of space occupied by a solid, liquid, or gas. There are several instruments for measuring volume, including: –graduated cylinder –syringe –buret –pipet –volumetric flask

Significant Digits Each number in a properly recorded measurement is a significant digit (or significant figure). The significant digits express the uncertainty in the measurement. When you count significant digits, start counting with the first non-zero number. Lets look at a reaction measured by three stopwatches.

Significant Digits Cont. Stopwatch A is calibrated to seconds (±1 s), Stopwatch B to tenths of a second (±0.1 s), and Stopwatch C to hundredths of a second (±0.01 s).

Significant Digits and Placeholders If a number is less than one, a placeholder zero is never significant. Therefore, 0.5 cm, 0.05 cm, and cm all have one significant digit. If a number is greater than one, a placeholder zero is usually not significant. Therefore, 50 cm, 500 cm, and 5000 cm all have one significant digit.

Exact Numbers When we count something, it is an exact number. Significant digit rules do not apply to exact numbers. An example of an exact number: there are 3 coins on this slide.

Rounding Numbers All numbers from a measurement are significant. However, we often generate nonsignificant digits when performing calculations. We get rid of nonsignificant digits by rounding off numbers. There are four rules for rounding off numbers.

Rules for Rounding Numbers 1.If the first nonsignificant digit is less than 5, drop all nonsignificant digits. Example: A calculator displays and 3 significant digits are justified. The first nonsignificant digit is a 4, so we drop all nonsignificant digits and get 12.8 as the answer.

2.If the first nonsignificant digit is greater than or equal to 5, increase the last significant digit by 1 and drop all nonsignificant digits. A calculator display and 3 significant digits are justified. The first nonsignificant digit is a 5, so the last significant digit is increased by one to 9, all the nonsignificant digits are dropped, and we get 12.9 as the answer. Rules for Rounding Numbers

Rounding Numbers 3. a) If the last digit is 5 and is preceded by an odd number, then the last digit should be increased by. Example: is rounded to 4.64 b) If the last digit is 5 but is preceded by an even number, then it stays the same or is rounded down by 1. Example: is rounded to If a calculation has two or more operations, retain all nonsignificant digits until the final operation and then round off the answer.

Adding & Subtracting Measurements When adding or subtracting measurements, the answer is limited by the value with the most uncertainty. 5g 5.0g +5.00g 15.00g Lets add three mass measurements. The measurement 5 g has the greatest uncertainty (± 1 g). The correct answer is 15 g.

Multiplying & Dividing Measurements When multiplying or dividing measurements, the answer is limited by the value with the fewest significant figures. Lets multiply two length measurements.  5.15 cm × 2.3 cm = cm 2 The measurement 2.3 cm has the fewest significant digits, two. The correct answer is 12 cm 2.

Exponential Numbers Exponents are used to indicate that a number has been multiplied by itself. Exponents are written using a superscript; thus, 2×2×2×2 = 2 4. The number 4 is an exponent and indicates that the number 2 is multiplied by itself 4 times. It is read “2 to the fourth power”.

Powers of Ten A power of 10 is a number that results when 10 is raised to an exponential power. The power can be positive (number greater than 1) or negative (number less than 1).

Scientific Notation Numbers in science are often very large or very small. To avoid confusion, we use scientific notation. Scientific notation utilizes the significant digits in a measurement followed by a power of ten. The significant digits are expressed as a number between 1 and 10.

Applying Scientific Notation To use scientific notation, first place a decimal after the first nonzero digit in the number followed by the remaining significant digits. Indicate how many places the decimal is moved by the power of 10. –A positive power of 10 indicates that the decimal moves to the left. –A negative power of 10 indicates that the decimal moves to the right.

Scientific Notation Continued There are 26,800,000,000,000,000,000,000 helium atoms in 1.00 L of helium gas. Express the number in scientific notation. Place the decimal after the 2, followed by the other significant digits. Count the number of places the decimal has moved to the left (22). Add the power of 10 to complete the scientific notation × atoms

Another Example The typical length between two carbon atoms in a molecule of benzene is m. What is the length expressed in scientific notation? Place the decimal after the 1, followed by the other significant digits. Count the number of places the decimal has moved to the right (7). Add the power of 10 to complete the scientific notation × m

Accuracy = proximity of a measurement to the true value of a quantity. Precision = proximity of several measurements to each other. Accuracy versus Precision

Summary A measurement is a number with an attached unit. All measurements have uncertainty. The uncertainty in a measurement is dictated by the calibration of the instrument used to make the measurement. Every number in a recorded measurement is a significant digit.

Summary Continued Place holding zeros are not significant digits. If a number does not have a decimal point, all nonzero numbers and all zeros between nonzero numbers are significant If a number has a decimal place, significant digits start with the first nonzero number and all digits to the right are also significant.

Summary Continued When adding and subtracting numbers, the answer is limited by the value with the most uncertainty. When multiplying and dividing numbers, the answer is limited by the number with the fewest significant figures. When rounding numbers, if the first nonsignificant digit is less than 5, drop the nonsignificant figures…If the number is 5 or more, raise the first significant number by one and drop all of the nonsignificant digits.

Summary Continued Exponents are used to indicate that a number is multiplied by itself n times. Scientific notation is used to express very large or very small numbers in a more convenient fashion. Scientific notation has the form D.DD × 10 n, where D.DD are the significant figures (and is between 1 and 10) and n is the power of ten. Accuracy refers to the proximity of a measurement to the true value of a quantity. Precision refers to the proximity of several measurements to each other.