Making Measurements and Using Numbers The guide to lab calculations

Not just numbers Scientists express values that are obtained in the lab. In the lab we use balances, thermometers and graduated cylinders to record mass, temperature and volume. It may seem simple to read the instruments but it is actually more difficult than you think.

Reading Thermometers Thermometers measure temperature. Key points: Temperature is read from the bottom to top. Lines on a thermometer are only so accurate. We as scientist are allowed to estimate between the lines. The unit on thermometers is Celsius

Estimating Lines We are allowed to estimate one additional digit to make the reading more significant. No matter what the last line of reading may be on the measuring device being used, you may estimate one additional digit (with a few exceptions)

Estimation Tips When markings go up or down by ones, estimate your measurement to the tenths place When markings go up or down by tenths, estimate your measurement to the hundreths place When markings go up or down by 2 ones or 2 tenths, estimate your measurement to that place

Why do we estimate lines? Some errors or uncertainty always exists in measurements. The measuring instruments place limitations on precision. When using a device we can be almost certain of a particular number or digit. Simply leaving the estimated digit out would be misleading because we do have some indication of the value’s likely range.

Reading Liquid Volume Because of certain physical properties, liquids are attracted or repelled from glass surfaces. Water is especially attracted to glass. Due to this attraction a meniscus forms when water is in glass tubing. Meniscus is the upside down bubble that forms when water is in glass

When reading glass volumes, the volume is of liquid is read at the bottom of the meniscus. Not only is the liquid read at the bottom of the meniscus but the last digit of the reading is estimated. The estimation tips are the same for all measuring devices No matter the guess you are right. As long as you include the guess in your answer

Water Meniscus

Significant figures In science, measured values are reported in terms of significant figures. Significant Figures in a measurement consist of all the digits known with certainty plus one final digit, which is somewhat uncertain or is estimated.

Why Use Sig Figs? We can only measure as well as our equipment We cannot make estimates without being precise Estimating multiple measurements can add up to a lot of error

Insignificant digits may still be recorded if they act as placeholders. Scientist don’t write down all the numbers the calculator displays. To determine if a value is significant the following rules are applied:

Rule 1 All non-zero numbers are significant. Examples: 123 L has 3 significant figures (sigfigs) 7.896 m3 has 4 sig figs 8 meters has 1 sig figs

Practice #1 Determine how many sig figs are in the following numbers: 885 mL 4 589 452 Km 96 g

Rule 2 Zeros appearing between non zero digits are significant. Examples: 40.7 L has 3 sig figs 87,009 km has 5 sig figs

Practice #2 Determine how many sig figs are in the following numbers: 305 sec 200 015 mol 908 066 molecules

Rule 3 Zeros appearing in front of all non zero digits are not significant. These zeros are place holders Examples 0.095897 m has 5 sig figs 0.0009 kg has 1 sig fig

Practice #3 Determine how many sig figs are in the following numbers: .000568 atm .02965 mm .0000591 psi

Rule 4 Zeros at the end of a number AND to the right of a decimal point are significant. Examples 85.00 g has 4 sig figs 9.000000000 mm has 10 sig figs .006700 L has 4 sig figs

Practice #4 Determine how many sig figs are in the following numbers: 98.0000 cm .0042800 kg 4.016050 mol

Rule 5 Zeros at the end of a number but to the left of a decimal point may or may not be significant. If a zero has not been measured or estimated but is just a placeholder, it is not significant. A decimal point placed after zeros indicates they are significant Examples 2000 m has only 1 sig fig 2000. M has 4 sig fig (decimal at the end)

Practice #5 Determine how many sig figs are in the following numbers: 450.000 Hz

Why Do Insignificant Figures Need To Sometimes Be Recorded? Insignificant figures may still be recorded if they act as placeholders to keep the value of how large or small the number is credible. 5000 m only has 1 sig fig but we still need to keep the 3 zeros as placeholders

Cumulative Practice #1 Determine how many sig figs are in the following numbers: 385 sec 2 000 000 J 980 510 ft 80019 cal 6.0 mm 90.02500 in .00580010 L 100 000. yd 458900 mL

When To Apply Sig Fig Rules? Sig fig rules only apply to situations where a measurement was made by an instrument. For all other situations, all measurements are exact, and therefore contain an unlimited amount of significant figures. 300 mL = 1 sig fig 300 people = 3 sig figs 300 pennies = 3 sig figs

Cumulative Practice #2 Determine how many sig figs are in the following numbers: 400 sheep 400 mg of C12H22O11 105.80 lbs 80 bananas

Rounding With Sig Figs Round each of the following calculations to 3 significant figures: 1.0030570 ~ 1.00 1008 ~ 1010 .005051879 ~ .00505 54556 ~ 54600 Look at the number following the 3rd sig fig to determine rounding If this number is _ 5, round 3rd sig fig up If this number is < 5, keep 3rd sig fig the same >

Calculations with Sig Figs When multiplying and dividing, limit and round to the the number with the fewest sig figs. 5.4 x 17.2 x 0.0005467 = 1.500 / 2.00 x .00500 =

When adding and subtracting, limit and round your answer to the least number of decimal places in any of the numbers that make up your answer 142.3 + 12 - 0.61 = 60.59 + 489.000 + 2.7795 =

Units For calculations involving multiplication, add exponents of units if the same Ex: cm1 x cm1 x cm1 = cm3 For calculations involving division, subtract exponents of units if the same Ex: cm3 / cm2 = cm1 or cm For calculations involving adding or subtracting, exponents must be the same Ex: cm + cm + cm = cm

Practice Working With Units m2 - m2 = m3 / m2 = mm + mm + mm = mm x mm x mm =

Scientific notation Scientist often deal with very small and very large numbers, which can lead to a lot of confusion about counting zeros. Scientist notation takes the from of M x 10n where 1 <M<10 and “n” represents the number of decimal places moved.

How Are Sig Figs Applied To Numbers In Scientific Notation? M x 10n -All numbers included in the “M” factor are ALWAYS significant -All numbers included in “n” are NEVER significant

150000 becomes 1.5 x 105 43500000 becomes 4.35 x 107 0.0034 becomes 3.4 x 10-3 .000005687 becomes 5.687 x 10-12

Scientific Notation Practice Write the following in scientific notation or in standard form, given the following: 6.350 x 102 .003671 4.7 x 10-2 4.00000

Multiplying & Dividing Using Scientific Notation Ex: (4.58 x 105) (6.8 x 10-3) Multiply the bases 4.58 x 6.8 = 31.144 Add the exponents 105 + 10-3 = 102 Adjust value to correct scientific notation format 31.144 x 102  3.1144 x 103 Determine sig figs from quantities listed in the original problem 3.1 x 103

Ex: (2.8 x 10-5) / (3.673 x 10-2) Divide the bases 2.8 / 3.673 = .76231963 Subtract the exponents 10-5 - 10-2 = 10-3 Adjust value to correct scientific notation format .76231963 x10-3  7.6231963 x 10-4 Determine sig figs from quantities listed in the original problem 7.6 x 10-4

Adding & Subtracting Using Scientific Notation Ex: (3.52 x 106) + (5.9 x 105) – (6.447 x 104) Convert all quantities so that they all have the same largest exponent (3.52 x 106) + (.59 x 106) - (.06447 x 106) Add or subtract the base numbers 3.52 + .59 - .06447 = 4.04553 x 106 Adjust value to correct scientific notation format 4.04553 x 106 Determine sig figs from quantities listed when all exponents have been adjusted. (red) 4.05 x 106

Practice Calculations With Scientific Notation (7.36 x 102) + (2.9 x 10-2) = (3 x 102) x (2.9 x 10-1) = (1.20 x 102) / (5.000 x 105) =

Accuracy vs. Precision Accuracy is the ability of a tool or technique to measure close to the accepted value of the quantity being measured (how close it is to being right) Precision is the ability of a tool or technique to measure in a consistent way (how close the measurements are to each other)

Example Problem A student measured a magnesium strip 3 times and recorded the following measurements: 5.49cm, 5.48cm, 5.50cm The actual length of the strip is 5.98cm. Describe the results in terms of accuracy and precision.

Density Density is a mass to volume ratio D = m/v m = Dv v = m/D Density is an intensive property and will not change regardless of the amount of matter present. Each substance has its own defined density value ex. H2O = 1g/cm3

How Can Density Be Determined In The Lab? You must know the mass and volume if you want to experimentally determine the density of a sample of matter Mass can be found using a scale (g) Volume can be found by one of two ways: For regular shaped objects, use a ruler to find l x w x h (measurement will be in cm3) For irregular shaped objects, use water displacement (measurement will be in mL)

Remember… 1cm3 = 1mL Water displacement is a process in which an object is submerged in water. The difference between the water level before and after the object is submerged in the water will be the volume of the object

Example Problems A sample of metal has a mass of 12.90 g and a volume of 45.0 cm3. What is the density of this metal? What is the volume of of a piece of zinc with a mass of 25.69 g? DZn= 7.14 g/mL

The water level in a graduated cylinder stands at 13 The water level in a graduated cylinder stands at 13.5 mL before a copper sample is lowered into the cylinder. The water level then rises to 19.8 mL after the sample is submerged. What is the mass of this sample? DCu = 8.92 g/cm3

Percent Error… How Wrong Are You? Once your densities are determined experimentally, you can then compare your lab results to the theoretical value by using the following equation: % error = (theoretical – experimental) theoretical X 100 Ideally, you would shoot for <5% error in any lab experiment Theoretical values are given by teacher or text

Remember… Theoretical values are definitive. These values will be given by the teacher or can be found in a published source (textbook). Experimental results are always found in the lab.

Example Problem After calculating the density of ethanol as .801 g/mL in the lab, you want to see how well you performed. Would Mrs. Rustad be happy with these results? Why or why not? Dethanol = .791 g/mL