Density Measurement, Calibration
of a Thermometer and a Pipette
Accuracy and Precision in
Measurements
Objectives
•
•
•
•
To measure the density of an unknown solid
To calibrate your alcohol thermometer
To calibrate your volumetric pipette
To gain an appreciation for precision and
accuracy in temperature, volume and mass
measurements in this lab
Reporting Figures in Science
• Scientists agree to a standard way of reporting
measured quantities in which the number of
reported digits reflect the precision in the
measurement
• More digits more precision, fewer digits less
precision
• Numbers are usually written so that the uncertainty
is indicated by the last reported digit.
Counting Significant Numbers in the
Lab, Precision
• The rule is that every digit in the number
reported except the last one is certain. So if
the mass is reported as
• 45.872 g
• we are certain about 45.87 but the 2 is
estimated.
Mass in the Lab
• Instruments generally have a precision in their
measurement. For example the scales in the
lab measure the mass to 0.0001g they are
digital so that is that. So the scale cannot
measure something that has a mass of
0.00000001g say.
Temperature
• When we use a regular thermometer to measure temperature
– how precisely can we measure the temperature?
Temperature
• With a typical thermometer, the best we can do is to
estimate the temperature to within maybe a tenth of
a degree Celcius or Farenheit, so we can specify the
degree with certainty but the tenth we are not
certain about.
• 60.6oC or 21.3F
Alcohol Thermometers
• In your locker you most likely have an alcohol
thermometer
• Unlike a mercury-in-glass thermometer, the contents of
an alcohol thermometer are less toxic and will evaporate
away fairly quickly
• The liquid used can be pure ethanol, toluene, kerosene
or isoamyl acetate, depending on the manufacturer and
the working temperature range. The liquids are all
transparent, so a red or blue dye is added yours is
suppose to measure to 110oC
• Less Costly
• Less Accurate at high temperatures!
Resistance Temperature Detector
• Temperature sensors are
• An RTD is often made of a thin film of Pt – the thin-film’s
resistance depends on temperature. Knowing the
resistance as a function of temperature means it can be
used as a thermometer.
• Needs software that knows the resistance vs. temperature
characteristic for your temperature sensor
• Pay attention to the type of sensor you use and make sure
you use the correct program with it
• Precision of 0.01oC
• Accuracy of 0.15oC at 0oC
Volume
• The most accurate way to measure volume in the lab is using
either a pipette or a burette
• burettes read the volume to 0.1mL, so the best we can do is
report the volume to a hundredth of a mL.
Significant Figures in Calculations
• When we use measured quantities in calculations, the results of the
calculations must reflect the precision of the measure quantities.
• We should not lose or gain precision
Significant Figures
Multiplication/Division
In multiplication we keep the precision of the lowest precision number
5.02
(3 sig figs)
x
89.663
(5 sig figs)
x
0.10 = 45.0118 =
45
(2 sig figs)
(2 sig figs)
In Division we follow the same rule
5.892
/ 6.10
(4 sig figs) (3 sig figs)
= 0.96590
= 0.966
(3 sig figs)
When reducing the number of significant figures how do we round?
Rounding
When we round to the correct number of figures we
• round down if the last digit is 4 or less
• round up is 5 or more
– 1.01 x 0.12 x 53.5 / 96 = 0.067556
– = 0.068 (since 0.12 and 96 are two sig figs)
– 9.4 x 10 = 94 = 90 (10 1 sig. fig)
– 0.096 x 1000 = 100 (100 1 sig. fig)
Addition and Subtraction
• In addition and subtraction carry the fewest decimal
places
5.74 (2 decimal places)
0.823 (3 decimal places)
+ 22.651 (3 decimal places)
29.214 = 29.21 (2 decimal places)
Calculations Involving
Multiplication/Division and
Addition/Subtraction
• Same as division/multiplication (lowest sig figs of any
number in the equation)
3.489 x (5.67 – 2.3)
3.489 x (3.37)
11.758 = 12 (2 sig figs same as 2.3)
• Round to the appropriate sig. figs at the end!
Sampling (random) Errors
• whenever we attempt to measure a certain quantity we will find that the
observed result is not always the same
• If we histogram the measured answer versus how many times we get the
answer we get a distribution like this one
• This occurs either because the measured quantity varies in nature (like
the height of adult males)
• Or because of random errors that affect the measurement of the result
(marble to roll down an incline)
Sampling (Random) Errors
• When reporting a value, we need to report the mean
value along with information about the distribution
of measured values
σ
x ± s
where
x is the mean value
Sampling (Random) Errors
• The standard deviation σ is a measure of random
error
• The smaller σ is the more precise the result is
• For a given set of results σ can be calculated using
the formula where the sum is over your data values
1 N
2
s=
xi - x )
(
å
N -1 i=1
Accuracy vs. Precision
Measuring the density of an ‘unknown’
• This method works only for solids, insoluble in water and
more dense than water
• Take an unknown and record its number
• Weigh a Volumetric flask m1=mflask
• Add your unknown to the flask
• Reweigh the flask and its contents m2=mflask+munk
• Add water up to neck of flask – remove any trapped air
by gently tapping the flask or using a stirring rod
• Use a dropper to fill the volumetric up to the meniscus
• Remove any water adhered to neck of the flask above
the meniscus
• Reweigh the flask m3=mflask+munk+mwater
Calculation
To calculate the density of the unknown we need the mass
of the unknown munk and volume of the unknown Vunk.
Assuming that dwater = 0.998203 g/mL, (true at 20.0oC), and
Vtot = volume of volumetric flask (50.00mL)
 50 mL = Vwater+Vunk
 Vwater = mwater/dwater
 Mwater = m3-m2
 Vwater= (m3-m2)/(0.998203 g/mL)
 Vunk = 50.00 mL – Vwater
 dunk = (m2-m1)/Vunk
Note we know V to 3-4 sig figs and mass to 6 sig. figs so we know d to
3-4 sig figs.
Thermometer Calibration
• By varying the temperate of water over the range from freezing to boiling,
create a table of thermometer readings Tr and sensor reading Ts
• Assuming that the sensor reading may have a small systematic error over
the range of the experiment – correct using the formula
é( 0 o C - Ts (ice)) + (Tactual (boiling) - Ts (boiling)) ù
û
Tc = ë
2
Tactual = Ts + Tc
• Plot TR vs Tactual and make a linear fit, displaying the equation of the line –
this is your calibration curve