Applied Example of Random and systematic errors in titrimetric analysis:

The example of the students’ titrimetric experiments showed clearly that randomand systematic errors can occur independently of one another, and thus presumablyarise at different stages of the experiment. A complete titrimetric analysis inaqueous solution with a colorimetric indicator can be regarded as having the followingsteps.

Making up a standard solution of one of the reactants. This involves (a) weighinga weighing bottle or similar vessel containing some solid material,

(b) transferringthe solid material to a standard flask and weighing the bottle again to obtain by subtraction the weight of solid transferred (weighing by difference), and

(c) fillingthe flask up to the mark with water.

d-measuring aliquot.an aliquot of the standard material to a titration flask with the aid of a pipette. This involves both filling and draining the pipette properly.

e-Titrating the liquid in the flask with a solution of the other reactant, added from a burette. This involves

(i) filling the burette and allowing the liquid in it to drainuntil the meniscus is at a constant level,

(ii) adding a few drops of indicator solutionto the titration flask,

(iii) reading the initial burette volume,

(iv) adding liquidto the titration flask from the burette a little at a time until, using a colour change, the end-point is judged to have been reached, and

(v) measuring the final level of liquid in the burette (e.p.).

Weighing Errors:

A.Random Errrors

Weighing procedures are normally associated with very small random errors. Inroutine laboratory work a ‘four-place’ balance is commonly used, and the randomerror involved should not be greater than ca. 0.0002 g. If the quantity being weighed is normally ca. 1 g or more, it is evident that the random error, expressed as a percentage of the weight involved, is not more than 0.02%.

A good standard material for volumetric analysis should (amongst other characteristics) have as high a formula weight as possible, in order to minimize these random weighing errorswhen a solution of a given molarity is being made up. In some analyses ‘microbalances’are used to weigh quantities of a few milligrams – but the errors involved arelikely to be only a few micrograms.

B- Systematic Error. Systematic errors in weighings can be appreciable, and have a number of wellEstablished sources. (i)These include adsorption of moisture on the surface of the weighing vessel; (ii)failure to allow heated vessels to cool to the same temperature asthe balance before weighing; (iii)corroded or dust-contaminated weights; and For the most accurate work, weights must be calibrated against standards furnished by standards authorities .

Some simple experimental precautions can be taken to minimize thesesystematic weighing errors. Weighing by difference cancels systematicerrors arising from (for example) the moisture and other contaminants on thesurface of the bottle . If such precautions are taken, the errorsin the weighing steps will be small, and it is probable that in most volumetric experimentsweighing errors will be negligible compared with the errors arising from theuse of volumetric equipment. Indeed, gravimetric methods are generally used forthe calibration of an item of volumetric glassware, by weighing (in standard conditions)the water that it contains or delivers, and standards for top-quality calibrationexperiments are made up by using weighings rather than volumeMeasurements.

Volume measuring Errors:A- Random error:

In volumetric steps random errors arise in the use of volumetric glassware.

The error in reading a burette graduated in 0.1 ml divisionsis ca. 0.01–0.02 ml. Each titration involves two such readings . If the titration volume is ca. 25 ml,the percentage error is again very small. The experimental conditions should bearranged so that the volume of titrant is not too small (say not less than 10 ml),otherwise the errors will become appreciable. (This precaution is analogous to choosing a standard compound of high formula weight to minimize the weighing error.) Even though a volumetric analysis involves several steps, in each of which a piece of volumetric glassware is used, it is apparent that the random errors shouldbe small if the experiments are performed with care. In practice a good volumetricanalysis should have a relative standard deviation of not more thanabout 0.1%. Until recently such precision was not normally attainable in instrumentalanalysis methods, and it is still not common.

B- Systemetic ErrorsVolumetric procedures incorporate several important sources of systematic error.Chief amongst these are:-the drainage errors in the use of volumetric glassware, -Calibration errors in the glassware, and- ‘indicator errors’. -Perhaps the commonest error in routine volumetric analysis is to fail to allow enough time for a pipette to drain properly, or a meniscus level in a burette to stabilize. -Pipette drainage errors have a systematic as well as a random effect: the volume delivered is invariably less than it should be. -The temperature at which an experiment is performed has two effects.Volumetric equipment is conventionally calibrated at 20°C, but the temperature inan analytical laboratory may easily be several degrees different from this, and many experiments, for example in biochemical analysis, are carried out in ‘cold rooms’ at ca. 4°C. The temperature affects both the volume of the glassware and the density of liquids.

Indicator errors can be quite substantial – perhaps larger than the random errors in a typical titrimetric analysis. For example, in the titration of 0.1 M hydrochloric acid with 0.1 M sodium hydroxide, we expect the end-point to correspond to a pH of 7. In practice, however, we estimate this end-point by the use of an indicator suchas methyl orange. Separate experiments show that this substance changes colourover the pH range ca. 3–4. If, therefore, the titration is performed by adding alkalito acid, the indicator will yield an apparent end-point when the pH is ca. 3.5, i.e.just before the true end-point. The systematic error involved here is likely to be asmuch as 0.2%. Conversely, if the titration is performed by adding acid to alkali, theend-point indicated by the methyl orange will actually be a little beyond the trueend-point. In either case the error can be evaluated and corrected by performing ablank experiment, i.e. by determining how much alkali or acid is required to producethe indicator colour change in the absence of the acid (alkali).

In any analytical procedure, classical or instrumental, it should be possible to consider and estimate the sources of random and systematic error arising at each separate stage of the experiment, as outlined above for titrimetric methods. It is very desirable for the analyst to do this, in order to avoid major sources of error by careful experimental design .

Various Terms used in the Treatment of Experimental Data

Pooled Standard deviation

For median calculation: the data are arranged in ascending or descending orderThus in ascending : 0.1019 , 0.1021, 0.1023, 0.1025 In descending: 0.1025. 0.1023, 0.1021, 0.1019Since data are even we take the mean of the two middle value

n1 0.1019 – 0.1022= 0.00032

0.1021-0.1022= 0.0001

30.1023-0.1022= 0.0001

4 0.1025-0.1022= 0.0003

∑ = 0.0008

= 0.00000009

Standard error = 0.0003/ 2 = 0.00015

First set of data