1. Density Measurement, Calibration of a Thermometer and a Pipette
Accuracy and Precision in Measurements

2. 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

3. Precision and Accuracy
pre·ci·sion (prəˈsiZHən)
noun: precision
Technical:
refinement in a measurement or calculation, especially as represented by the number of digits
ac·cu·ra·cy (akyərəsē)
noun: accuracy
Technical: the degree to which the result of a measurement, calculation, or specification conforms to the correct value or a standard.

4. 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.

5. 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   g
we are certain about but the 2 is estimated
The precision is g for this measurement

6. Mass in the Lab
Instruments generally have a precision in their measurement. For example the scales in the lab measure the mass to g
Our scales have a precision of g
The scales cannot measure something that has a mass of g

7. Temperature
With a typical thermometer which has a scale marks every 1oC, the best we can do is to estimate the temperature to within maybe a tenth of a degree Celcius (oC)
We can specify the degree with certainty but the tenth we are not certain about.
41.5oC (since the reading is midway between 41oC and 42oC)
For our thermometers we have a precision of 0.1oC

8. Alcohol Thermometers We use alcohol thermometers
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 iso-amyl 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!

9. Resistance Temperature Detector
A temperature sensor is an RTD
These have 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

10. Volume
The most accurate way to measure volume in the lab is using either a pipette or a burette
Scale marks on burettes read the volume every 0.1mL, so the best we can do is report the volume to a hundredth of a mL
The precision of a burette or a pipette is 0.01mL

11. 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

12. Significant Figures Multiplication/Division
 In multiplication we keep the precision of the lowest precision number
x x = =
(3 sig figs) (5 sig figs) (2 sig figs) (2 sig figs)
In division we follow the same rule
/ = =
(4 sig figs) (3 sig figs) (3 sig figs)
When reducing the number of significant figures how do we round?

13. 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 =
= (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)

14. Addition and Subtraction
In addition and subtraction carry the fewest decimal places
5.74 (2 decimal places)
0.823 (3 decimal places)
(3 decimal places)
= (2 decimal places)

15. 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)
= 12 (2 sig figs same as 2.3)
Round to the appropriate sig. figs at the end!

16. Errors in Science
Unfortunately there are always errors in any measurement
There may be more that one source of error
Errors fall into two categories
Random Error
Systematic Error

17. Sampling (random) Errors
whenever measure a certain quantity multiple times 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 (height of people in the room)
random errors affect the measurement of the result (marble to roll down an incline)

18. Sampling (Random) Errors
When reporting a value, we need to report the mean value along with information about the distribution of measured values σ
s is the standard deviation (a measure of the random error)

19. 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

20. Accuracy vs. Precision
Precision is related to the random error and is measured with the standard deviation
Accuracy is a measure of systematic error, a bias in the experiment that always either overestimates or underestimates the true value
Accuracy is calculated as the % error
% 𝑒𝑟𝑟𝑜𝑟= 𝑦𝑜𝑢𝑟 𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒−𝑟𝑒𝑓 𝑣𝑎𝑙𝑢𝑒 𝑟𝑒𝑓 𝑣𝑎𝑙𝑢𝑒 ×100%

21. Part I: Measuring the density of an ‘unknown’ solid
This method works only for solids, insoluble in water and more dense than water, or liquids immiscible in 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

22. 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 = 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)/( g/mL)
Vunk = 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.

23. Part II: Temperature Calibration
to channel A
Open either BLUE/bluevc0002.ds, or RTD/RTDvc0002.ds or BLACK/blackvc0002.ds depending on your sensor
record the temperature according to the sensor Ts and the temperature according to the thermometer Tr at roughly 20oC, 40oC, 60oC, 80oC, and boiling
To record 0oC, fill beaker 1/3 tap water and then fill up with ice, and stir till the temperature goes as low as possible

24. Part II: Thermometer Calibration
Create a table of thermometer readings Tr and sensor reading Ts
The sensor will likely have a small systematic error over the range of the experiment. To assess it we need to know some temperatures exactly then we can assess this error. Here we use the freezing point of water (0.00oC) and the boiling point of water. We then calculate the average systematic error for these 2 points
Using Excel, plot TR vs Tactual (TR is the x-axis and Tactual the y-axis) and make a linear fit, displaying the equation of the line – this is your calibration curve.
Fit the curve to a linear trendline and print the fitted equation for a straight line on the graph (print one for each member of the group)

25. Part III: Using the Calibrated Thermometer to correct the density
Now we are going to correct our answer from part A for the density duncorr
The calculation assumed that we were at 20.0oC for which the density of water = g/mL
Use your measured temperature Tr of the water from part A to obtain the true temperature (using your calibration curve)
Use the CRC Handbook to find the true value for the density of water at room temperature and recalculate the density of your unknown let’s call the dcorr
Calculate the error

26. Part IV: Volumetric Pipette Calibration