Making Measurements and Using Numbers

The guide to lab calculations

Not just numbersScientists express values that are obtained

in the lab. In the lab we use scales, 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 ThermometersThermometers 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 LinesWe 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 thermometer, you may estimate one additional digit (with a few exceptions)

Estimation TipsWhen markings go up or down by ones,

estimate your measurement to the tenths placeWhen 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 VolumeBecause 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 figuresIn 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

equipmentWe cannot make estimates without being

preciseEstimating multiple measurements can

add up to a lot of error

Insignificant digits are not reported, ever. Scientist don’t write down all the numbers the calculator displays.

To determine if a value is significant the following rules are applied:

Rule 1All non-zero numbers are significant. Examples:

123 L has 3 significant figures (sigfigs)7.896 m3 has 4 sig figs8 meters has 1 sig figs

Rule 2Zeros appearing between non zero digits

are significant.Examples:

40.7 L has 3 sig figs87,009 km has 5 sig figs

Rule 3Zeros appearing in front of all non zero

digits are not significant.Examples

0.095897 m has 5 sig figs0.0009 kg has 1 sig figThese zeros are place holders

Rule 4Zeros at the end of a number AND to the

right of a decimal point are significant.Examples

85.00 g has 4 sig figs9.000000000 mm has 10 sig figs

Rule 5Zeros 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

Examples2000 m has only 1 sig fig2000. M has 4 sig fig (decimal at the end)

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

Calculations with Sig FigsWhen multiplying and dividing, limit and

round to the the number with the fewest sig figs.

5.4 x 17.2 x 0.0005467 =?

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 =?

Working with numbers continued

How many sig figs are in the number 23000000000?

Do we need to write all of the zeros?

Scientific notationScientist 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.

150000 becomes 1.5 x 105

43500000 becomes 4.35 x 107

0.0034 becomes 3.4 x 10-3

0.000000000005687 becomes 5.687 x 10-12

More examples…

Multiplying & Dividing Using Scientific Notation

Ex: (4.58 x 105) (6.8 x 10-3)

Multiply the basesAdd the exponentsAdjust value to correct scientific notation

formatDetermine sig figs from quantities listed in

the original problem

Ex: (2.8 x 10-5) / (1.673 x 10-2)

Divide the basesSubtract the exponentsAdjust value to correct scientific notation

formatDetermine sig figs from quantities listed in

the original problem

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 exponentAdd or subtract the base numbersAdjust value to correct scientific notation formatDetermine sig figs from quantities listed when all

exponents have been adjusted.

Accuracy vs. PrecisionAccuracy 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 ProblemA 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.

DensityDensity is a mass to volume ratioD = m/v m = Dv v = m/DDensity 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 = 1mLWater 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

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)

theoreticalX 100

Ideally, you would shoot for <5% error in any lab experimentTheoretical values are given by teacher or text

Calculations With Conversions

78.6 mm + 68.350 cm =

55 L + 25 cm3 =

Calculations With Conversions

The density of cork is .193 g/cm3. What is the mass in pounds of 7.0 x 103 mL of cork? (1 lb = 16 oz) , (1 g = .0353 oz)