Making Measurements

Precision vs Accuracy  Accuracy : A measure of how close a measurement comes to the actual, accepted or true value of whatever is measured.  Example: Over two trials, a student measures the boiling point of ethanol and then calculates the average. Trial No. °C Average79.0  She then checks a chemistry handbook (CRC) to see how close her measurements are to the actual value. Accepted Value = 78.4 °C

Precision vs Accuracy  Precision : A measure of how close a series of measurements are to one another. This is best determined by the deviation of the data points.  Example: Two students independently determine the average boiling point of ethanol. Trial No. Student A Student B Average Deviation  Student A’s data is less accurate but more precise than student B’s. Accepted Value = 78.4 °C

Determining Error  Error : The difference between the accepted value and the experimental value. Formula: Error = experimental value – accepted value  Calculate the % error for the data given below. Accepted Value = 78.4 °C  Percent error : The absolute value of the error divided by the accepted value, multiplied by 100. Formula: % Error = ׀ error ׀ Accepted value X 100 Student A = 79.0 °C %E = 0.8 % ׀ ׀ 78.4 X 100 %E =

Scientific Notation

 Scientific Notation : An expression of numbers in the form m x 10 n where m is ≥ 1 and < 10 and n is an integer..  Example #1: A single gram of hydrogen contains approximately atoms which can be rewritten in scientific notation as 6.02 x  Example #2: The mass of an atom of gold is  Using scientific notation makes it easier to work with numbers that are very large and very small gram which can be rewritten in scientific notation as 3.27 x

Scientific Notation  In scientific notation, there is a coefficient and an exponent, or power x Coefficient Exponent 6 02  For example: When changing a large number to scientific notation, the decimal is moved left until one non-zero digit remains  The exponent value is determined by moving a decimal point. atoms

Scientific Notation  Remember to count the number of place values as you move the decimal. 23  Now drop the “trailing” zeros and add an “x 10”. atoms places

Scientific Notation  Now drop the “trailing” zeros and add an “x 10”.  Place the 23 as an exponent of the “10”. x 10  Remember to count the number of place values as you move the decimal.  You have just converted standard notation to scientific notation x atoms places

Scientific Notation 3 27 gram  When changing a very small number to scientific notation, the decimal is moved to the right until it passes a non-zero number places  Again, count the number of place values the decimal moved.

Scientific Notation 3 27 gram  When changing a very small number to scientific notation, the decimal is moved to the right until it passes a non-zero number places  Again, count the number of place values the decimal moved.  Now drop the “leading” zeros and add an “x 10”.

x 10 Scientific Notation  When changing a very small number to scientific notation, the decimal is moved to the right until it passes a non-zero number places  Again, count the number of place values the decimal moved.  Now drop the “leading” zeros and add an “x 10”. gram

x 10 Scientific Notation  When changing a very small number to scientific notation, the decimal is moved to the right until it passes a non-zero number  Again, count the number of place values the decimal moved  Now drop the “leading” zeros and add an “x 10”.  Finally, place the -22 as an exponent of the “10”.  Again you have converted standard notation to scientific notation x gram

Scientific Notation  Change the following number to proper scientific notation  Change the following number to proper standard notation. 4.3 x 10 5 = 1.2 x =  Normally, scientific notation is not used for measurements that produce an exponent of +1 or -1. Examples : 0.32 = 3.2 x (not usually done) 89.1 = 8.91 x 10 1 (not usually done) Practice problems

Recording Measurements  Assume you are using a thermometer and want to record a temperature. To do this, you must first determine the instrument’s precision, or I.P. The precision of an instrument is equal to the smallest division on the instrument’s scale. The I.P. is = 0.1 °C  Pictured to the right is a thermometer scale. What is the scale’s precision? °C Smallest division

Recording Measurements  The next thing to do is record the temperature and unit. How would you record the temperature shown?  If you recorded the temperature as 24.3 °C then you were not being as precise as you could be with the scale given. °C  If you look carefully at the top of the thermometer’s fluid you will see it rises a bit higher than 24.3 °C. You might now record the temperature as °C or °C

Recording Measurements °C °C  The first three digits in our measurements are known with certainty. However, the last digit was estimated and involves some uncertainty.  All measurements consist of known digits and one estimated digit. Together, they are called significant digits. Estimated digit Certain or known digits Significant digits

Recording Measurements  We will now express our measurement as °C  Error in measurement may be represented by a tolerance interval.  Machines used in manufacturing often set tolerance intervals, or ranges in which product measurements will be tolerated before they are considered flawed.  To determine the tolerance interval in a measurement, add and subtract (±) one-half of the precision of the measuring instrument to the measurement °C ± 0.05 °C (±T.I.) Unit Significant figures Tolerance interval

Recording Measurements  For example: How would you read the temperature shown to the right?  Always round the experimental measurement or result to the same decimal place as the uncertainty °C ± 0.1 °C (±T.I.) °C It would be confusing (and perhaps dishonest) to suggest that you knew the digit in the hundredths (or thousandths) place when you admit that you’re unsure of the tenth’s place. It would be confusing (and perhaps dishonest) to suggest that you knew the digit in the hundredths (or thousandths) place when you admit that you’re unsure of the tenth’s place. It should be read as… This 3 rd digit was rounded down to match the place value of the T.I.

Recording Measurements Time(min)°C (T.I. = ± 0.05°C) Average =  Note: It is not necessary to write the tolerance interval for each measurement of a series of measurements made using the same instrument.  For example: If you record a series of temperature measurements in a data table, then you need only state the tolerance interval once.  Keep in mind there are many ways of showing uncertainty in measurement. This is why you must indicate the source of the uncertainty, such as… Tolerance Interval (T.I.) Standard deviation (± SD) Standard error (± SE) ± SD = ± 5.67 ± SE = ± 2.53 % E = 5.4 % Percent error (%E) Accepted value = °C

Statistic What it is Statistical interpretation Symbol Average An estimate of the "true" value of the measurement The central value X Standard deviation A measure of the "spread" in the data You can be reasonably sure (about 70% sure) that if you repeat the same measurement one more time, that next measurement will be less than one standard deviation away from the average. SD Standard error An estimate in the uncertainty in the average of the measurements You can be reasonably sure (about 70% sure) that if you do the entire experiment again with the same number of repetitions, the average value from the new experiment will be less than one standard error away from the average value from this experiment. SE Recording Measurements

Significant Figures

 The rules for recognizing significant figures are as follows:  Significant figures are all the digits that can be known precisely in a measurement, plus a last estimated digit. Determining Significant Figures √ Zeros within a number are always significant. √ Zeros that do nothing but set the decimal point are not significant. √ Trailing zeros that aren’t needed to hold a decimal point are significant. Both 4308 and contain four sig. figs and and 310 contain two sig. figs. Both 4.00 and contain three sig. figs.

In our example of , a decimal is present.  Here is a “trick” that can help you with significant figures. Determining Significant Figures  Question: How many sig. figs. are in ? Step 1: Check to see if the number has a decimal. If yes, think “present.” If no, think “absent.” “present.” If no, think “absent.” Step 2: Note that Present starts with a “P” and so does Pacific. Pacific Ocean (decimal present)

Determining Significant Figures Step 3: Now place the number inside the U.S.A. pictured below. Step 4:Draw an arrow from the Pacific Ocean through the number until you encounter a non-zero digit. Pacific Ocean (decimal present) √ Rule: All digits to the right of the arrow tip are significant. In our example, has three significant figures.

Determining Significant Figures Pacific Ocean (decimal present) Atlantic Ocean (decimal absent) In our example of , a decimal is absent.  New Question: How many sig. figs. are in ? Step 1: Check to see if the number has a decimal. If yes, think “present.” If no, think “absent.” “present.” If no, think “absent.” Step 2: Note that Absent starts with an “A” and so does Atlantic.

Determining Significant Figures Pacific Ocean (decimal present) Atlantic Ocean (decimal absent) Step 3: Now place the number inside the U.S.A. pictured below. Step 4:Draw an arrow from the Atlantic Ocean through the number until you encounter a non-zero digit. √ Rule: All digits to the left of the arrow tip are significant. In our example, has four significant figures.

 How many sig. figs are in each of the following measurements? Determining Significant Figures x x 10 2 five five two x two one three seven two

 Now change this measurement into scientific notation. How would you record the following measurement? Determining Significant Figures It should have been recorded as mm ± 0.05 mm mm It should have been written as x 10 2 ± 0.05 mm  Note that the three zeros after the numeral 5 must be retained in order to uphold the precision of the measurement.  How many significant figures does x 10 2 contain? It contains five.  Let’s apply what you have learned.

Calculating with Significant Figures

This is analogous to saying that a chain cannot be stronger than its weakest link. This is analogous to saying that a chain cannot be stronger than its weakest link.  In general, a calculated answer cannot be more precise than the least precise measurement from which it is calculated. Calculating with Significant Figures Round the answer to the same number of decimal places (not digits) as the measurement with the least number of decimal places.  Here is the rule for Adding or Subtracting significant figures  Here is the rule for Multiplying or Dividing significant figures Round the answer to the same number of significant figures as the measurement with the least number of significant figures.

 Example addition problem: Calculating with Significant Figures Step 1: Stack the numbers and align them by decimal location Step 2: Add the numbers Step 3: Locate the measurement with the least number of digits to the right of the decimal point The first measurement (349.0 meters) has the least number of digits (one) to the right of the decimal point. Step 4:The answer must be rounded to one digit after the decimal point or x 10 2

 Example multiplication problem: Calculating with Significant Figures 2.10 m 0.70 m x 1.47 m 2 The first measurement (0.70) has smallest number of significant figures (two). Step 3: The answer must be rounded to the same number of significant figures as the measurement with the least number of significant figures 1.5 m 2 Step 1: Multiply the numbers Step 2: Locate the measurement with the least number of significant figures m 2

 Example addition problem in scientific notation Calculating with Scientific Notation Evaluate 5.2 x x x x x x x x x  Example subtraction problem in scientific notation Evaluate 7.0 x x x x x x x x 10 5 Change to same exponents 6.5 x x x x th ’s place

 Example multiplication problem in scientific notation Calculating with Scientific Notation  Example division problem in scientific notation x 10 1 Evaluate 2 x x x 3.6 x 10 4 x 3.6 x 10 4 Evaluate 4.7 x x 10 7 Add the exponents Multiply the coefficients Subtract the exponents Divide the coefficients x x x 10 1 Round to 1 sig. fig. 7.2 = Round to 2 sig. figs.

Problem Calculating with Significant Figures Scientific notation Rounded answer Calculator answer x x x x x x x x * x x x 1.2 x 10 2 (2 sf) (1 sf) (1 p) (0.1 p) (1 p) (2 sf) * Change to x 10 0 – x 10 0