tool 3 mathssolikeibmathsishardrightthisisresourcesforyou.pptx

tool 3 math ppt

1. Tool 3: MATHEMATICS

2. UNITS SI units should be used and these conventions followed: A space between a numerical value and its unit Decimal marker should be preceded by a number, even if this number is zero Different units should be separated by a space

3. Types and Sources of Error Systematic errors: are errors due to identifiable causes in the experimental design give results that are consistently higher or lower than the true value are not reduced by repetition of the experiment can, in principle, be reduced by modifications to the experiment. Examples of causes of systematic error include: error caused by poor insulation during thermochemical experiments error caused when measuring gas volume by collection over water, assuming the gas is insoluble .

4. Types and Sources of Error

5. Types and Sources of Error Random errors: arise from the limit of the precision of the experimental apparatus lead to measurements that are equally likely to be higher or lower than the true value can be minimized by measurement repetition and averaging the measurements, leading to cancelling out of the variation can be reduced with the use of more precise measuring equipment can be quantified in all measurements and are expressed as a ± range of values

6. Accuracy and Precision Accuracy is how close a measured value is to the correct value. Experiments with smaller systematic errors are more accurate . Precision indicates how many significant figures there are in a measurement. Data with smaller random errors are more precise .

7. For example, the normal boiling point of water is 100°C. Measurements from two experiments are provided in the table on boiling point versus uncertainty. Describe the accuracy and precision of the data.

8. Estimating and R ecording U ncertainty in Raw D ata Example 1 34.0 cm 3  ± 0.5 cm 3 34.10 cm 3  ± 0.05 cm 3 Fluctuations in the balance Changes in the surrounding environment such as temperature variations and air currents Misinterpreting the reading and Insufficient data reaction time of the experimenter when measuring time* judging the colour change at the end point of an indicator during a titration* *These may not be quantifiable but should be noted as additional sources of error.

9. In general, random error can be estimated as follows. The uncertainty for a specified temperature is often stated by the manufacturer of the instrument or glassware. Where not stated, for digital equipment , the uncertainty is the smallest scale division (sometimes known as “the least count”). Where not stated, for analogue equipment , the uncertainty is half the smallest division. Estimating and Recording Uncertainty in Raw Data

10. Estimating and R ecording U ncertainty in Raw D ata Example 2 As the figure of the glassware shows, the smallest division (or least count) is 0.1 cm 3 . In this case, the uncertainty is taken as ± 0.05 cm 3 . Therefore, the volume poured is expressed as 48.80 cm 3  ± 0.05 cm 3 .

11. Error propagation for addition or subtraction of measurements When data values are added or subtracted, the uncertainties associated with each value must be added together. This is because the total error must include the range from the possible maximum to the possible minimum of each reading . Temperature (°C ± 0.1°C) Final temperature 29.9 Initial temperature 27.9 Temp . change = (29.9°C ± 0.1°C) − (27.9 ± 0.1°C) = 2.0°C ± 0.2°C Final temperature is in the range 29.8°C to 30.0°C Initial temperature is in the range 27.8°C to 28.0°C Therefore, the temperature difference could be as high as 2.2°C (30.0°C − 27.8°C) or as low as 1.8°C (29.8°C − 28.0°C), namely 2.0°C ± 0.2°C.

12. Error propagation for multiplication or division of measurements Convert each absolute uncertainty into a percentage uncertainty : Add the percentage uncertainties for each data value. Express overall uncertainty as percentage uncertainty, or convert it back into a final absolute uncertainty. Data value Absolute uncertainty Percentage uncertainty Concentration 1.00 mol dm −3 ± 0.05 mol dm −3 0.05 / 1.00 × 100 = 5% Volume 10.0 cm 3 ± 0.1 cm 3 0.1 / 10.0 × 100 = 1 %

13. Error propagation for multiplication or division of measurements F inal percentage uncertainty is greater than or equal to 2 %, 1 s.f. Final percentage uncertainty is less than 2%, not more than 2 s.f. % error < final % uncertainty, the difference is due to random errors. % error > final % uncertainty, then random errors alone do not explain the discrepancy and systematic errors must be involved.    % error = |Theo – Exp|   x  100                           Theo

14. Error consideration when taking averages of data values

15. ERROR PROPAGATION Example for specific heat: Mass is 130.093 +/- .001g      Temperature is 37.5˚C +/- .2 ˚C Find the heat if q = mcΔT   and  c = 4.186 J/ g˚C         q = (130.093)(4.186)(37.5) = 20.4 kJ

16. ERROR PROPAGATION First find % uncertainties for mass and temp. 0.001/130.093 x 100 = 7.67x10 -4 % for mass 0.2/37.5 x 100 = .533 % for temp (Note – look how much more the temperature reading affects the error.  This is what you have to mention in your conclusion.) Add the %’s: 7.67x10 -4 % + 0.533 % = .534% To convert back into an absolute uncertainty .534%/100 x 20.4kJ = 0.109kJ Answer is now 20.4 +/- 0.1kJ

17. Mean Titre Volume/ cm −3  ± 0.10 cm −3 Ethanoic Acid Concentration/ mol dm −3   Uncertainty/ mol dm −3 14.18 14.18 14.25 14.44 14.64 ERROR PROPAGATION