Measurements and Calculations Using Scientific Measurements

Scientific Notation M x 10n M is a number ≥ 1, but less than 10 n is a whole number Can only add or subtract if n is the same Multiplication – multiply the M’s, add the n’s Division – divide the M’s, subtract the n’s OR, use your calculator!

Direct Proportions Two quantities are directly proportional to each other if dividing one by the other gives a constant value. y/x = k (k is a constant) Can be rewritten as y = kx When the value of y increases, the value of x also increases, and vice versa

Direct Proportions Straight line Passes through origin

Inverse Proportions Two quantities are inversely proportional to each other if their product is a constant xy = k If x increases, y decreases and vice versa. Graph produces a curve called a hyperbola.

Accuracy and Precision Accuracy refers to the closeness of measurements to the correct or accepted value of the quantity measured.

Accuracy and Precision Precision refers to the closeness of a set of measurements of the same quantity made in the same way. Multiple measurements are precise if they are close to each other.

Precision of a Measuring Tool The precision of a measuring tool is based on how fine the markings are on the tool. The finer the markings, the more precise the tool.

Percent Error A measure of the accuracy of a measurement Subtract the experimental value from the accepted value, divide by the accepted value and multiply by 100

Percent Error What is the percent error for a mass measurement of 17.7g given that the correct value is 21.2 g?

Percent Error A volume is measured experimentally as 4.26 mL. What is the percent error, given that the correct value is 4.15 mL?

Error in Measurement There’s always some uncertainty in measurement. Results can be affected by the skill of the measurer and the precision of the instrument.

When you use a measuring device, you can be certain of a particular number of digits based on the markings on the device. The last digit can be estimated by where the value is between the finest markings. To repeat: a measurement includes all digits based on the markings on the measuring device PLUS one more figure which is estimated

The number of digits you include for your measurement is an indication of how precise the measuring tool was. Therefore, for clear communication with other scientists, it is important that EVERYTIME you measure anything in the lab, you MUST record the measurement with all the digits you are sure of PLUS an estimate for the last digit.

Significant Figures Significant figures are all the digits in a measurement that are known for sure, plus one final digit that is somewhat uncertain, or estimated. Significant ≠ certain

Significant Figures Significant figures are only used in measurements Counting numbers or conversion factors are considered to have an infinite number of significant figures.

Significant Figures All nonzero numbers are significant. 36.4 595 72395

Significant Figures All zeroes that come between non-zero numbers are significant 10.5 145.09 900.4 40.006

Significant Figures Preceding, or leading, zeroes are not significant 0.005 0.6 0.0000009 0.00405

Significant Figures Zeros at the end of a number are significant only if there is a decimal point somewhere in the number 30.0 20. 509.00 600 75000 75000. 75000.00

Significant Figures In scientific notation, do not consider the exponent (or factor of 10) in the number of significant figures. 1.45 x 105 3.0 x 102 9.006 x 10-3

Rounding When you perform calculations with sig figs, you may have to round your answer to get the correct number of sig figs. Rules for rounding: See table 2-6 on p. 48 Pay special attention to the rule for rounding 5’s

Addition or Subtraction The answer must have the same precision as the least precise measurement being added or subtracted. The key is to look at place value.

Multiplication and Division Can’t have more sig figs than the measurement with the fewest number of sig figs.

Multistep Calculations Some problems involve several steps. Don’t round to correct number of sig figs in the middle of your calculations. Keep a few extra digits. Round to the correct number of sig figs for the final answer.