Scientific Measurement And Dimensional Analysis Chapter 3 & 4

Measurement Qualitative Measurements: Give results in descriptive, nonnumeric form Example: The baby feels hot Quantitative Measurements: Give results in definite form as units and numbers Example: 273 K

Scientific Notation A number written as the product of two numbers A coeffficient and 10 raised to a power Numbers larger than 1 have positive exponents Numbers less than 1 have negative exponents Move your decimal between the first and second digits. Example: 36,000 = 3.6 x 104 Example: .00036 = 3.6 x 10-4

Break it down 3.6 x 104 = 3.6 x 10 x 10 x 10 x 10 = 36, 000! 3.6 x 10-4 = 3.6 10x10x10x10 = .00036!

Try some! 1,100 = 250,000 = 68,000,000 = 250 = .000,000,56 = .003,4 = .000,000,001,1 =

Answers 1,100 = 1.1 x 103 250,000 = 2.5 x 105 68,000,000 = 6.8 x 107 250 = 2.5 x 102 .000,000,56 = 5.6 x 10-7 .003,4 = 3.4 x 10-3 .000,000,001,1 =1.1 x 10-9

ROUNDING >5  round previous digit up one What if the digit to be dropped is: >5  round previous digit up one <5  do not change previous digit

Multiplication Division Multiply numbers then add the exponents Example: (2.0 x 102) (2.0 x 105) 2.0 x 2.0 = 4.0 102 + 105 = 107  Answer = 4.0 x 107

Division Divide numbers then subtract the exponents. Example: 5.0 x 10-10 2.5 x 10-2 5.0/2.5 = 2.0 (-10 – (-2) ) = -10 + 2 = -8  Answer = 2.0 x 10-8

Try some! (3.0 x 103)(2.0 x 10-5) = (2.5 x 10-2)(3.0 x 10-8) =

Answers (3.0 x 103)(2.0 x 10-5) = 6.0 x 10-2 (2.5 x 10-2)(3.0 x 10-8) = 7.5 x 10-10 (5.0 x 10-10)(2.0 x 100) = 10. x 10-10 (2.0 x 10-2)/(1.0 x 10-4) = 2.0 x 102 (8.0 x 105)/(4.0 x 10-10) = 2.0 x 1015 (5.0 x 10-1)/(1.0 x 10-1) = 5.0 x 100

Addition & Subtraction Exponents must be the same Fix them if different Example: 5.40 x 103 + 6.00 x 102 or .600 x 101)102 = 103 Move your decimal + or - and increase or decrease your exponent. Then add!

Addition & Subtraction 5.40 x 103 + .600 x 103 6.00 x 103 8.00 x 104 - 2.00 x 105 20.0 x 10-1)105 = 104 = 12.0 x 104 (not in scientific notation) = 1.20 x 105

Try Some! (4.0 x 10-10) + (2.0 x 10-9) = (5.0 x 10-2) + ( 6.0 x 10-2) = (1.0 x 102) – (5.0 x 101) = (6.0 x 10-3) – (6.0 x 10-2) = (10. x 101) + (9.0 x 101) =

Answers! (4.0 x 10-10) + (2.0 x 10-9) = 2.4 x 10-9 (5.0 x 10-2) + ( 6.0 x 10-2) = 1.1 x 10-1 (1.0 x 102) – (5.0 x 101) = 5 x 101 (6.0 x 10-3) – (6.0 x 10-2) = 5.4 x 10-2 (10. x 101) + (9.0 x 101) = 1.9 x 102

Accuracy vs. Precision Even the most carefully taken measurements are always inexact. This can be a consequence of inaccurately calibrated instruments, human error, or any number of other factors. Two terms are used to describe the quality of measurements: precision and accuracy.

What is Precision? Precision is a measure of how closely individual measurements agree with one another.

What is Accuracy? Accuracy refers to how closely individually measured numbers agree with the correct or "true" value.

CALCULATING % ERROR actual value  experimental value x 100

SAMPLE CALCULATION In a mass/volume experiment to determine the density of gold, a student calculated the density to be 18.75 g/mL. The actual value for the density of gold is 19.32 g/mL. What is the percent error?

% ERROR CALCULATION | 19.32 – 18.75| x 100 = 3.0 % 19.32

WHAT IS AN ACCEPTABLE % ERROR? Is 3.0% error a good or a bad result? That depends upon The precision of the instruments used And ultimately the expectation of the teacher. In this case, 3.0% is very good because of the instruments available in a typical school lab.

BEFORE YOU CAN CRUNCH You must know which digits are significant Because they are going to control the number of digits in a calculated figure

WHAT IS A SIGNIFICANT FIGURE? Significant figures are all the digits in a measurement that are known with certainty plus a last digit that must be estimated.

WHICH NUMBERS ARE SIGNIFICANT? For the purposes of significant figures there are two major categories Nonzero digits: 1,2,3,4,5,6,7,8,9 Zero digits:

NONZERO DIGITS All nonzero digits are significant 3269 cm – 4 significant figures 257 L – 3 significant figures 1.234567 mm – 7 significant figures

ZEROS Zeros take three forms Leading zeros Trapped zeros Trailing zeros

LEADING ZEROS Leading zeros are zeros that come before the nonzero digits in a number. They are place holders only and are never considered significant. 0.123 L – 3 significant digits 0.000012 m – 2 significant digits 0.003578 mL – 4 significant digits

TRAPPED ZEROS Trapped zeros are zeros between two nonzero digits. Trapped zeros are always significant. 101 s – 3 significant figures 20013 m – 5 significant figures 0.3006 cm – 4 significant figures (the leading zero is not significant)

TRAILING ZEROS Trailing zeros are zeros that follow nonzero digits. They are only significant if there is a decimal point in the number. 0.1200 mm – 4 significant figures 3000 s – 1 significant figure 250. mL – 3 significant figures 50.000 g – 5 significant figures

Exact Numbers Numbers that were not obtained using measuring devices but by counting. They can also arise from definitions. They can be assumed to have an infinite number of significant figures.

Examples of Exact Numbers 2 in 2r 3 and 4 in ¾ r3 Avogadro’s number is exactly 6.02 x 1023 One inch is exactly 2.54 cm

Now you try How many significant digits in each of the following: 12 apples 3000 m 69 people

Answers 1.034 s - 4 significant figures 0.0067 g - 2 significant figures 12 apples - exact number 3000 m - 1 significant figure 69 people - exact number