STEM Chemistry Basic Quantitative Skills Measurement precision & accuracy significant digits scientific notation

Observations & Measurements We make QUALITATIVE observations of reactions Example: changes in color and physical state. 2. We also make QUANTITATIVE MEASUREMENTS, which involve numbers.

Observation & Measurement Units A measurement should include a numeric quantity and a base scale (unit). To claim a measurement of 72 is not communicating any worthwhile information. 72 seconds? 72 feet? 72 gallons? The reported measurement should always be written to reflect the limitation of the measuring device.

Observation & Measurement The units of a measurement depend on the tool used. There is an international system of agreed upon usage of units so that when scientists are speaking to one another and use the term “mass” then they understand they are all using the same units. This system is called Le Systéme International d’unités and is commonly referred to SI Units (system international) To measure mass To measure volume To measure length Grams Cups Inches Kilograms Liters Miles Pounds Gallons meters

Observation & Measurement Accidents happens when SI units are not used… Click on the picture to see a video

Precision & Accuracy When taking measurements, precision refers to the repeatability of the measurements. If you are consistently getting the same measurement over and over again, your measurements are considered precise. How close your measurement is to the true value is a reflection of its accuracy.

Precision & Accuracy Precise, but not accurate Neither accurate nor precise Precise AND accurate

Measurement Error Measurement error can be divided into two categories: Random error - Random error is always present in a measurement. It is caused by inherently unpredictable fluctuations in the readings of a measurement apparatus or in the experimenter's interpretation of the instrumental reading. Random errors can be reduced by averaging multiple measurements. Systematic error - Systematic error is predictable and typically constant or proportional to the true value. If the cause of the systematic error can be identified, then it usually can be eliminated. Example: a thermometer is consistently low by 5 oC in every reading; all measurements are low by 5 oC.

Observation & Measurement What is the length of the red line (assuming it begins at the zero mark and increments are centimeters)? First digit 2 Second digit 2.3 Third digit (estimated) 2.34 Final measurement 2.34 cm All measurements must include the known digits and one estimated digit

Observation & Measurement Your turn to practice. Indicate the measurements at each location marked on the ruler. A = B = C = 1.49 mm D = E = F = 5.29 mm G = 11.98 mm 2.89 mm 6.60 mm 4.21 mm 8.07 mm

Which measurement is the best? Significant Figures Which measurement is the best? What is the measured value? What is the measured value? What is the measured value?

Significant Digits When we record a piece of data we are communicating something about the tool used to take the measurement. the communicated data is then used to determine an answer to a question. The data communicated has an impact on the Quality of the Final answer. There are rules in place for how to communicate precise data for accurate outcomes.

Significant Digits Significant digits are numbers that are reported in a measurement and are limited by the measuring tool. Significant digits in a measurement include the known digits (there are marks on the tool for them) AND one digit that is estimated.

Significant Digits Rule 1: Rule 2: All non-zero numbers in a measurement are significant. Example: 437 has three significant digits (none are zeros) Rule 2: Zeros are sometimes significant depending on the placement of the zero in the number Leading zeros are NEVER significant Captive zeros are ALWAYS significant Trailing zeros are SOMETIMES significant – only if a decimal is present.

Significant Digit Examples 0.0486 This number has leading zeros (placeholders) which are never significant. Therefore, this number has 3 significant digits – the 4, 8 and 6. Estimated digit is 6. 16.07 This number has a captive zero (between two non-zero numbers) and IS significant. There are 4 significant digits. Estimated digit is 7. 9.300 This number has trailing zeros AND a decimal is present. There are 4 significant digits. The estimated digit is the last zero. 9300 This number has trailing zeros and NO decimal point. There are only 2 significant digits here. The estimated digit is the 3.

Two Special Cases Counted items Example: 23 people, 14 pencils Two special situations have an unlimited number of significant digits: Counted items Example: 23 people, 14 pencils Exactly defined quantities Example: 60 seconds = 1 minute (most conversion factors)

Significant Digits in Calculated Answers In general, a calculated answer cannot be more precise than it’s least precise measurement used to calculate it. “The team is only as strong as its weakest link” Multiplying/Dividing When calculating an answer from a multiplication and division problem, you need to pay attention to how many significant digits the data measurements have. Example: 2434 cm / 12.2 seconds = 199.180328 My calculator reports this answer with many digits. Looking at the two numbers I divided, I see that 2434 has 4 significant digits and 12.2 has three significant digits. The measurement with the fewer significant digits will dictate how many digits can be reported in my final answer. Remember to check for rounding. Final answer: 199 cm/s Here, with only 3 digits allowed, I look to the 4th digit to see if it is 5 or greater. Since it is a 1, it does not make the 9 round up.

Significant Digits in Calculated Answers Adding/Subtracting Like multiplying and dividing, I must look to the original measurements to determine how many digits can be reported in my final answer. This time, I only look at the number of digits to the right of the decimal. Example: 12.479 cm + 14.2 cm + 15.0022 cm = 41.682 My calculator reports all of these digits. Looking at the three numbers I added, I look at how precisely the measurements were taken, and the least precise measurement dictates how many digits can be written in my answer. 12.479 has three digits to the right of the decimal, 14.2 has one and 15.0022 has 4. The least precise is 14.2. 185.96 m – 16.005 m = 169.955 m Since my answer can have only two digits to the right of the decimal, I look to the thousandths place to see if it will cause the hundredths place to round up. The 5 makes the 5 round up to a 6. Answer: 169.96 m Significant Digits in Calculated Answers

Practice Problem Calculator Says Answer 3.24 cm x 7.20 cm 142 g / 14.31 cm3 0.03 mi / 140.2 s 1934.20 cm x 1200 cm 0.0200 m x 18002 m (14.338 + 120.24) x 0.3 23.328 23.3 cm2 9.92313068 9.92 g/cm3 0.0002 mi/s 0.00021398 2321040 2300000 cm2 360.04 360. m2 40.3734 40

Scientific Notation Scientific notation is a way of expressing really big numbers or really small numbers. For very large and very small numbers, scientific notation is more concise and the numbers are easier to work with. Scientific Notation consists of two parts: A number between 1 and 10 (N) A power of 10 N x 10x

Scientific Notation To change a number from standard form to scientific notation: Place the decimal point so that there is one non-zero digit to the left of the decimal point. Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10. If the original number was less than 1, then the exponent is negative. If the original number was greater than 1, then the exponent is positive. Registered Nurses Love Patients! If you move the decimal to the right (R) then the exponent is negative (N) If you move the decimal to the left (L) then the exponent is positive (P)

Scientific Notation Examples Given: 289,800,000 Use: 2.898 (moved 8 places) Answer: 2.898 x 108 Given: 0.000567 Use: 5.67 (moved 4 places) Answer: 5.67 x 10-4 For any problem, the number of significant digits must be conserved! If the original number in standard form has 4 significant digits, then the number in scientific notation form must also have 4 significant digits. They are the same number!

(Use zeros to fill in places.) Scientific Notation To change a number from scientific notation to standard form: Simply move the decimal point to the right for positive exponent 10. Move the decimal point to the left for negative exponent 10. (Use zeros to fill in places.)

Scientific Notation Examples Given: 5.093 x 106 Answer: 5,093,000 (moved 6 places to the right) Given: 1.976 x 10-4 Answer: 0.0001976 (moved 4 places to the left)

Practice Express these numbers in Scientific Notation: Answer 405789 0.003872 3000000000 2 0.478260 4.05789 x 105 3.872 x 10-3 3 x 109 2 x 100 4.78260 x 10-1