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Measurements and Significant Digits


 

One of the most important requirements of a good scientist is the ability to properly record measurements to the correct number of significant digits and with the correct units. Examples of the types of volumetric glassware and instruments encountered in the laboratory are given below with directions on how to correctly record measurements.



Beakers
Burettes
Digital Balance
Erlenmeyer Flasks
Graduated Cylinders


Hydrometers
Micropipettes
Mohr Pipettes
Thermometers
Volumetric Flasks
                                 

Beakers

Beakers are designed to hold a particular amount of liquid and are not designed to accurately measure volumes of liquids. If an approximate volume is needed with little accuracy, a beaker may be used to contain the liquid. Beakers come in various sizes with different calibrations. An examination of the 100-mL beaker shown reveals calibration lines every 10 mL between 20 and 80 mL. These measurements have a ±5% error and are therefore approximations.

The 250-mL beaker has calibration lines every 25 mL between 25 and 200 mL. The error again is ±5%. The amount of liquid in the beaker is therefore 125 mL ±5%.

To determine the size of the beaker needed during an experiment, determine the maximum amount of liquid the beaker needs to contain. Multiply that number by two and select a beaker size that comes closest to that number.

EXAMPLE: An experiment calls for 25 mL of one solution to be added to 30 mL of another solution. The beaker must safely hold 25 mL + 30 mL or 55 mL of liquid total. Multiplying 55 x 2 gives 110 mL. A 150-mL beaker would be a good choice.

Erlenmeyer Flasks

Erlenmeyer flasks are also designed to hold a particular amount of liquid and and are not designed to accurately measure volumes of liquids. Since the sides taper to a narrow opening, these flasks are useful for stirring solutions during titrations and synthetic work.

Erlenmeyer flasks come in various sizes. An examination of the 250-mL flask shown gives calibration lines every 25 mL between 50 and 200 mL. These measurements have a ±5% error and are therefore approximations.

The 500-mL Erlenmeyer flask has calibration lines every 50 mL between 200 and 500 mL. The error again is ±5%. The amount of liquid in the flask is therefore 500 mL ±5%.

To determine the size of the flask needed during an experiment, determine the maximum amount of liquid the flask needs to contain. Multiply that number by two and select a flask size that comes closest to that number.

EXAMPLE: An experiment calls for 25 mL of a solution to be titrated to its endpoint with 25 mL of another solution. The flask must safely hold 25 mL + 25 mL or 50 mL of liquid total. Multiplying 50 x 2 gives 100 mL. A 125-mL Erlenmeyer flask would be a good choice.










Graduated Cylinders

Graduated cylinders are calibrated to contain (TC) or to deliver (TD) a precise amount of liquid at a given temperature, usually 20 °C. Tolerances vary with the size of the graduated cylinder. Tolerances may or may not be given at the top of the cylinder neck. If a tolerance is not shown, a good rule of thumb is to use one half the interval division. The 50-mL graduated cylinder shown has 1-mL divisions and a tolerance of ±0.50 mL and is calibrated to contain the measured volume. 

A graduated cylinder marked TC will hold the volume measured but will not deliver that volume to the container when transferred. Some of the liquid will remain behind in the graduated cylinder. If an exact amount is to be transferred, the graduated cylinder should be marked. TD.




                  

 

 

MEASUREMENTS: Water and aqueous solutions will form a concave meniscus when placed in a graduated cylinder as the water molecules are more strongly attracted to the glass than each other.  The bottom of the curved surface is read at eye level and the volume measurement is read to the proper number of significant digits.  Follow these steps to make the volume measurement with the proper number of significant digits:

1.  Determine the smallest division marked on the graduated cylinder:  (1) find two adjacent markings that have a numeric label, (2) subtract and divide by the number of divisions between the numeric labels. 

2.  Numbers are scaled to increase from bottom to top of the graduated cylinder.  Using the scale markings, determine the value of the certain digits.

3.  Estimate the distance the meniscus lies between markings as a decimal fraction.  Multiply the fraction times the division increment from Step 1.  Add this to the certain digits to provide the uncertainty in the measurement.  Record this number.






                    
                        Read scale at eye level.
EXAMPLE 1 

Step 1.  The labeled scale markings are 8 mL and 6 mL.  There are 10 divisions between the numeric labels.  [(8-6)/10] mL = 0.2 ml is the increment value.
Step 2.  The first certain digit is 6 mL since the meniscus is below 8 mL.  There are three smaller scale divisions below the meniscus: 3 x 0.2 mL/division = 0.6 mL  The known digits are (6 + 0.6 ) mL = 6.6 mL
Step 3.  The meniscus lies 0.1 of the distance between the markings: 0.1 x 0.2 mL = 0.02 mL  The volume should be recorded as (6.6  + 0.02) mL = 6.62 mL





Example 1.  10-mL graduated cylinder

EXAMPLE

Step 1.  The labeled scale markings are 20 mL and 10 mL.  There are 10 divisions between the numeric labels.    [(20-10)/10] mL = 1 ml is the increment value.
Step 2.  The first certain digit is 10 mL since the meniscus is below 20 mL.  There are seven smaller scale divisions below the meniscus: 7 x 1 mL/division = 7 mL  The known digits are (10 + 7) mL = 17 mL
Step 3.  The meniscus lies on the scale division:  0.0 x 1 mL = 0.0 mL  The volume should be recorded as (17 + 0.0) mL = 17.0 mL






Example 2.  50-mL graduated cylinder



Burettes

Like graduated cylinders, burettes come in various volume sizes. Burettes are calibrated to deliver a very precise volume of liquid, often to the hundredths of a mL.  Care must be taken when filling the buret that the tip contains no air bubbles.

 

 

MEASUREMENTS: Water and aqueous solutions will form a concave meniscus when placed in a burette similar to a graduated cylinder. The bottom of the curved surface is read at eye level and the volume measurement is read to the proper number of significant digits.  Use the steps given for graduated cylinders to make the volume measurement with the proper number of significant digits. Note that, unlike the graduated cylinder, the numbers are scaled to increase from top to bottom.

                 

           Burette with holder and stand                              Tip with no air bubbles                        

EXAMPLE 1

Step 1.  The labeled scale markings are 14 mL and 15 mL.  There are 10 divisions between the numeric labels.    [(15-14)/10] mL = 0.1 ml is the increment value.
Step 2.  The first certain digit is 14 mL since the meniscus is below 14 mL.  There are zero smaller scale division above the meniscus: 0 x 0.1 mL/division = 0.0 mL  The known digits are (14 + 0.0 ) mL = 14.0 mL
Step 3.  The meniscus lies 0.5 of the distance between the markings: 0.5 x 0.1 mL = 0.05 mL  The volume should be recorded as (14.0  + 0.05) mL = 14.05 mL






Example 1

EXAMPLE 2

Step 1.  The labeled scale markings are 20 mL and 21 mL.  There are 10 divisions between the numeric labels.    [(21-20)/10] mL = 0.1 ml is the increment value.
Step 2.  The first certain digit is 20 mL since the meniscus is below 20 mL.  There are eight scale divisions above the meniscus: 8 x 0.1 mL/division = 0.8 mL  The known digits are (20 + 0.8 ) mL = 20.8 mL
Step 3.  The meniscus lies 0.9 of the distance between the markings: 0.9 x 0.1 mL = 0.09 mL  The volume should be recorded as (20.8  + 0.09) mL = 20.89 mL






Example 2

 

 

 

Mohr Pipettes

A Mohr pipette differs from a volumetric pipette in that the barrel has a series of marked lines, much like a buret, to measure the amount of volume dispensed.  A pipette pump is used to draw liquid into the barrel and dispense into a container.  To insure the correct volume of liquid is dispensed, the tip of the pipette is touched to side of the container to release the final drop.  Be careful to check the scale markings on Mohr pipettes.  The last calibration mark on the 10-mL pipette shown is 9.5 mL.  The entire volume is drained from the barrel if 10.00 mL is required.  The last calibration mark on the 1-mL pipette is 1.00 mL.  The liquid is drained until the meniscus reaches 1.00 mL if 1.000 mL is required.  











10-mL Mohr pipette





1-mL Mohr pipette

 

 

MEASUREMENTS: Water and aqueous solutions will form a concave meniscus when placed in a pipette similar to the burette. The bottom of the curved surface is read at eye level and the volume measurement is read to the proper number of significant digits.  Use the steps given for graduated cylinders to make the volume measurement with the proper number of significant digits. Note that, like the burette, the numbers are scaled to increase from top to bottom.

EXAMPLES

Step 1.  The labeled scale markings are 0.4 mL and 0.5 mL.  There are 10 divisions between the numeric labels.    [(0.5-0.4)/10] mL = 0.01 ml is the increment value.
Step 2.  The first certain digit is 0.4 mL since the meniscus is below 0.4 mL.  There are four smaller scale division above the meniscus: 4 x 0.01 mL/division = 0.04 mL  The known digits are (0.4 + 0.04 ) mL = 0.44 mL
Step 3.  The meniscus lies 0.3 of the distance between the markings: 0.3 x 0.01 mL = 0.003 mL  The volume should be recorded as (0.44  + 0.003) mL = 0.443 mL

 

 

                          

 

Micropipettes

Micropipettes are used to accurately measure and dispense small volumes of liquid, usually in the 1 µL - 1000 µL range.  Micropipettes work on an air displacement principle and use detachable tips to hold the liquid sample.  Adjustable volume micropipettes, like the one, shown have a window to display the volume required for pipetting.  

 


MEASUREMENTS: Note the volume units given on the micropipette.  The micropipette shown measures volumes in the 100-1000 µL range.  The window indicates that the liquid volume in the tip is 400.0 µL (0.4000 mL).






Volumetric Flasks

Volumetric flasks are calibrated to contain a  precise volume of solution and are, therefore, often used to prepare solutions needed for quantitative analysis.  The neck of the flask has a calibration mark to indicate the fill level for the volume of solution needed.  The bottom of the meniscus is lined up with this mark to insure accuracy. 

 



MEASUREMENTS: Note the volume given on the volumetric flask.  Tolerances are usually within a few hundredths of a mL.  When filled to the calibration mark, the flask shown would contain 100.00 mL (0.10000 L) of solution.




 

 

Digital Balance

Accurate mass measurements are made using a digital balance.  The maximum capacity of the balance is typically found on the front label.  Typical laboratory balances measure in grams (some may be set to measure in mg or kg) and feature a tare function. Tare allows the user to zero the instrument to cancel the mass of a container (such as a weighing dish) from the reading of the instrument.  This allows the actual mass of the product being weighed to be read directly from the digital display.  A draft shield is often placed over the balance pan to minimize air currents and stabilize readings.

MEASUREMENTS: The mass is read directly from the digital display.  The balance shown is set to display mass in grams to three decimal places.  All digits must be recorded for each weighing performed during the laboratory .  Note that the last digit in the display is an estimated digit.





Hydrometers

Hydrometers are used to measure the specific gravity of a liquid. Hydrometers work on Archimedes principle that a submerged or floating body experiences a buoyant force equal to the mass of the liquid volume it displaces. Since this force depends on the mass and volume of liquid displaced, it is directly proportional to the specific gravity of the liquid.  The more dense the liquid, the higher the hydrometer floats.  A specialized hydrometer known as a urinometer can be used to measure the specific gravity of a urine sample.






               

          urinometer                                    urinometer in a urine sample

MEASUREMENTS: If a hydrometer is floated in a liquid of known temperature, the specific gravity of the liquid can be read from the scale on the stem of the hydrometer. The density of the liquid can then be determined from the equation for specific gravity. Figure 1 shows the use of a hydrometer to measure specific gravity. If attractions exist between the hydrometer surface and the liquid being measured, a meniscus occurs.  This is a common occurrence when the solution contains water and the hydrometer is made of glass.  The reading is taken at the lowest level of the meniscus. Note that the scale markings increase from top to bottom.

EXAMPLE:

Step 1.   The labeled scale markings are .95 and 1.000.  There are five divisions between the numeric markings.  (1.000-0.95)/5 = 0.01 is the increment value.
Step 2.  The first significant digit is 0.95 since the meniscus lies below 0.95.  There are three smaller scale divisions above the meniscus: (3 x 0.01) = 0.03  The known significant digits are (0.95 + 0.03) = 0.98
Step 3.  The meniscus lies 0.2 of the distance between the markings: (0.2 x 0.01) = 0.002  The specific gravity should be reported as (0.98 + 0.002) = 0.982  







                                 

Thermometers

Thermometers are used to measure the temperature of a system.  A thermometer has two important elements: the temperature sensor in the bulb of the thermometer in which some physical change occurs with temperature, plus some means of converting this physical change into a numerical value on the scale.   To protect against the hazards associated with mercury, many lab thermometers are filled with alcohol.  In the laboratory the Celsius temperature scale is used.  Temperatures ranges vary from one thermometer to another.  It is important to select a thermometer in which the expected temperature is within the range of the scale markings.







                                           

-20 -110 °C thermometer                  -10 - 260 °C thermometer

 

 

MEASUREMENTS: Use the steps given for graduated cylinders to make the temperature measurement with the proper number of significant digits.

  

EXAMPLE

Step 1.  The labeled scale markings are 80 °C and 90 °C.  There are 10 divisions between the numeric labels.[(90-80)/10] °C = 1 °C is the increment value.
Step 2.The first certain digit is 80 °C since the meniscus is below 90 °C.  There are seven smaller scale divisions below the liquid: 7 x 1 °C/division=7 °C.  The known digits are (80 + 7) °C = 87°C
Step 3.The meniscus lies 0.5 of the distance between the markings: 0.5 x 1 °C = 0.5 °C. The temperature should be recorded as (87  + 0.5) °C = 87.5 °C