Towards a Procedural-Instrumental Discourse on

Quantitative Chemical Analysis:

Chemistry Insights into the Validity of

Objective Investigations and Student Evaluations

Rituraj Kalita, Research Journal of Contemporary Concerns (Cotton College Research Council, Cotton College, Guwahati–781001, Assam, India), Vol. 3, page 46 (2005)

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Abstract:

The extent and common examples of procedural errors (errors of method) and instrumental errors in quantitative chemical analysis, particularly in South Asian educational context, has been outlined. Similar procedural and instrumental errors arise also in non-chemical quantitative experiments, say, in physics. An outline for creation of a formal branch of science that would find the least-cumbersome yet rigorous method of correction for such procedural-instrumental errors has been delineated, the name of this branch being suggested as 'Procedural-Instrumental Discourse' (PID). Even though knowledge of the subject matter of this branch definitely pre-exists in many scientific discussions, there is a need for formalization and rigorous treatments. As a corollary to such discourses, there arises the related issue of countering the detrimental effects of procedural-instrumental errors during any proper evaluation of students’ skills in performing quantitative experiments. A concept of ‘Evaluator-Evaluated Harmony’ (EEH) has been developed, intending that the EEH must be preserved in any such proper evaluation. As a concluding remark, it may be safely wondered that similar sort of procedural-instrumental drawbacks hinder (from time to time) our knowledge of non-quantitative science topics as well, and even that of social sciences and humanities.

Quantitative chemical analysis, the branch of science dealing with measurements of the amounts and concentrations of chemical substances, is prone to diverse kinds of errors [1,2]: determinant errors such as reagent errors, instrumental errors, errors of method (procedure), personal and operational errors, and also the indeterminate (random) errors. The magnitudes of some of these determinate errors are so significant that they are a matter of practical concern, not only a subject matter of philosophical discussions (random errors are generally insignificant). For example, the instrumental error arising out of the faults in weights used for weighing with common chemical balances in India is frequently observed to become several tens of milligrams, which may easily introduce an error of one in twenty during chemical analysis! This is a disastrous extent of error, which has the potential to cause colossal loss and/or disaster when applied to most chemistry-related industries! In volumetric apparatus such as pipette also, an error of the range of half a millilitre in a 25-mL pipette is quite common!

The search for the true value of the quantity to be measured so, naturally, involves correcting those errors. While personal and operational errors, such as due to overheating or overcooling, bumping of solutions etc., can be eliminated by care in manipulations and through experience, the instrumental, reagent-origin and procedural errors in the analysis must be corrected by specific techniques that go beyond the care and expertise of the individual analyst. However, in a given analysis one needn't go for a complete, thorough correction of all the instruments, reagents and procedures involved. (Such an exercise is even an impractical one, as in South Asia, say, it might even be quite difficult to find a reference standard burette for which the millilitre reading conforms to the internationally recognised millilitre, by a required accuracy of say one in a thousand!). For example, if the aim of the experiment is to find just the volume of an acid solution that would exactly neutralise a given volume of a given alkali solution, the impurity corrections for the reagents is uncalled for, while calibration of the burette & pipette (the volumetric instruments used) in terms of absolute (actual) millilitres is unnecessary. Assuming the homogeneity of burette markings and of burette bore, what suffices is simply the determination of the burette-measured volume of the pipette used, and this process of correction saves a great deal of labour in correction, even though it is not an iota less efficient than any other process of correction. To mention another example, if the aim is to find the ratio of volumes of an unknown alkali solution and of a known alkali solution that would neutralise equal volumes of a given acid, it can be shown that simply no corrective measures are necessary provided the said homogeneity assumption about the burette is valid (and the same burette-pipette pair is used)!

Thus there is a necessity to have a branch of science that would aim to find the minimum requirement of corrective measures in any given quantitative determination experiment. Generalizing to include non-chemical quantitative experiments in our discussion, we note that reagents, volumetric glassware and electronic instruments are all tools in our hand to measure something within the universe, and so can be denoted by the general term 'instruments'. Similarly, analytical methods and procedures of measurements etc. can be denoted by the generalized term 'procedure'. Thus, this branch of science may be named as 'Procedural-Instrumental Discourse' (PID), and it would analyse the generalized procedural and generalized instrumental background of the given quantitative determination (whether chemical or non-chemical) experiment, and suggest the set of the necessary corrective measures that would require just the minimum effort and/or trouble.

This is not to say that the knowledge coming within this proposed branch of science didn't exist in earlier scientific discussions within physical and chemical science, but its resulting decisions were mostly assumed or considered as understood, and hardly were formally and rigorously derived. However, there is possible a rigorous, mathematical derivation of the minimum procedural and instrumental corrective requirement for any quantitative determination, though the space constraint forbids such derivations here (detailed in an annexure kept towards the end). Thus, there has been a lack of formalisation in the area of knowledge pertaining to this discussion, and so recognition of this area as a branch of science would certainly help.

Such derivations would start with finding the appropriate expression of the quantity to be determined (say, an acid solution’s molarity) as a function of the measured and known values (say, acid volume, base volume & base molarity) as well as of the factors associated with the instrumental and procedural errors (say, proportional factors such as ratio of the burette mL & actual mL, that of pipette mL & actual mL, and minute additive factors such as correction for indicator’s intrinsic acidity/ basicity). In proceeding to do so, the derivations would generally employ some simple assumptions (practically valid in most situations using some sort of calibrated weights) such as the Burette Uniformity Assumption (BUA) and the Weight-Balance Proportionality Assumption (WBPA). BUA would state that the values of volumes (say, 20.4 mL & 10.2 mL) measured by the single burette being used are proportional to their actual volumes (say, 20.2 mL & 10.1 mL). Similarly, WBPA would state that the values of weights (say, 2.446 g & 4.892 g) measured by the single weight-balance system being used are proportional to their actual weights (say, 2.442 g & 4.884 g).

Other than the help offered for easily searching for the exact truth in case of objective investigations about nature, another immediate and more down to earth application of the concepts of Procedural-Instrumental Discourse (PID) is in the case of student evaluations, such as in (chemistry or other science) practical examinations. Applying PID to the teacher's (evaluator's) determination experiment, we may know what instrumental and procedural shortcomings the evaluator is facing. Then, again applying PID to the student's experiment, we may similarly know about the procedural and instrumental background of the evaluated. So, the evaluator must plan his/her own procedural-instrumental (PI) corrections and also must instruct the evaluated (s) to perform their PI corrections in such a way so that in absence of personal, operational and indeterminate errors, the evaluator and evaluated gets the same result. Such sort of a planning may be said to contain Evaluator-Evaluated Harmony (EEH). Thus, application of PID to the field of student evaluations give rise to the concept of EEH, and EEH is a must in an examination process because in a practical examination the evaluators should be looking for practical skills reflected by minimum operational and personal errors, and not any propensity of being favoured by the blind forces of chance (about procedural & instrumental factors). Measurement of the extent of operational and personal errors made by the evaluated (the student) is not at all possible if instrumental and procedural factors introduce large errors in a non-EEH examination setting.

As an example, in the glucose determination experiment involving titration with the Fehling solution, there won't be EEH if the students determine unknown glucose concentrations using known given Fehling solution, while the teacher determines directly from weighing and then compares. From the procedural side, EEH is always protected if the teacher and the students determine the unknown by the same procedure. However, there may be other simpler ways, to be known by a PID discourse, to protect EEH. Thus, EEH is not violated from the procedural side even if the teacher finds glucose amount from weighing (only weighing), provided the students prepare a standard glucose solution by weighing, and then standardize the Fehling solution with that, and finds the unknown glucose by titration with the Fehling solution. (EEH must also need to be protected from the instrumental side – this in this case means that the weight-balance system should have at most uniformly proportional error equal for both the evaluator and the evaluated, the lone student burette be homogeneous, and the pipettes be burette-calibrated.) Most teachers in chemistry have known and applied such principles of EEH intuitively without giving the concept a name, but formalisation of this idea would surely be better.

EEH can't be termed an unnecessary figment of idle contemplations. Till around one or two decades earlier, the undergraduate students of an Eastern Indian university (that this author personally knows) with chemistry as a general subject had to appear for a practical examination that involved a large fraction of the marks at stake on the result of a quantitative determination - with no provision of, and with no knowledge ever imparted to the students about, instrumental and procedural corrections. The non-EEH instrumental errors from the weights used naturally amounted to a few percent, while no marks in that experiment was given if overall error was more than three percent. All this resulted in a large proportion of the meritorious students failing at the chemistry practical examination for no fault or incompetence of theirs!

It seems we can possibly generalise the concept of EEH to discuss qualitative determinations in exams, and to discuss theory examinations as well. For example, in qualitative analysis in chemistry, the students need to find what constituents (inorganic ions or organic functional groups etc.) are present in a given chemical sample. Here, however, the teacher generally knows about the constituents from the manufacturer, while the students find them by analysis. In case of significant impurities introduced due to manufacturer’s fault, and in the face of certain unrecognised problems in some rare combinations of ions or groups (e.g., lead nitrate solution unexpectedly giving a precipitate with barium chloride reagent solution) can we always be sure that there is EEH in such an exam? Similarly, in a theory exam for some school students who have a fixed set of prescribed textbooks, can we expect them to go beyond the mistakes in their textbooks and answer correctly the questions that involve the wrongly written portions in the textbooks? Something akin to EEH must be missing in such situations. Thus, the concept of EEH takes us beyond the domain of chemistry & natural sciences to that of philosophy and education!

Similar philosophical generalisation seems to be possible also for the concept of PID dealing with search for the objective truth. Recognition of and discourse on the procedural-instrumental drawbacks of quantitative determinations naturally lead us to doubt our qualitative findings as well, and there may be situations where such doubt is not unfounded. Similarly, this concept also enlightens us about some sort of procedural-instrumental drawbacks (within our minds and within the social institutions) that surely hinder and deform even our knowledge on the social, socio-political and socio-economic fronts, and so may wrongly affect our everyday decisions in those vital fields as well!

References:

1. “A Text-Book of Quantitative Inorganic Analysis”, 3rd ed.; Vogel, A. I.; Longman Group Limited, London, 1972

2. “Errors, Measurements and Results in Chemical Analysis”; Eckschlager, K.; Van Nostrand Reinhold, London, 1961

Annexure:

Deductions in Procedural-Instrumental Discourse about some Quantitative Chemistry Experiments

Deduction 1: In determining the ratio of solution concentrations of a strong acid and a strong base by volumetric titration, using the calibrated pipette volume measured (calibrated) in terms of burette volume is enough as the lone corrective measure, provided the additive indicator error (procedural error) is negligible, and that the burette is a uniformly graduated one (i.e., one that obeys BUA - Burette Uniformity Assumption).

Let V_a, V_b are the solution volumes of the acid and base respectively (having basicity & acidity unity for each), and S_a, S_b are their molarities. So V_a S_a = V_b S_b + d, where d is the additive indicator (procedural) error. As d is considered negligible, so we have V_a S_a = V_b S_{b .}  Hence the concentration ratio  S_a / S_b = V_b / V_a (where V_b & V_a are actual volumes, not necessarily the instrument-measured volumes).

Let acid volume be measured with a definite burette, and base volume with a definite pipette. Assuming BUA, let b (let's call it the burette-factor) is the ratio of the burette-marked mL and the actual, true mL value. So V_b = b V'_b , where V'_b is the burette-measured volume-value. Similarly let p (let's call it the pipette-factor) is the ratio of the pipette-marked mL and the actual mL value. So V_b = p V'_b , where V'_b is the pipette-measured volume-value (For single-mark pipettes, there's no need of something like a pipette uniformity assumption). Thus we get, 

S_a / S_b = (b V'_b) / (p V'_a) = (b / p).(V'_b / V'_a) 

Now, if the pipette volume is calibrated in terms of the burette volume, i.e., if the pipette filled up to its mark is discharged into the burette so as to measure the pipette volume (e.g., 24.6 mL instead of the manufacturer-defined 25.0 mL) in terms of burette readings, then the corresponding newly defined pipette-marked mL will become exactly same as the burette-marked mL, which means that the factors b & p will now have identical values. Thus (b / p) will now equal unity, and S_a / S_b equals simply (V'_b / V'_a). Thus, by this simple corrective procedure the instrumental error has been completely eliminated (if BUA is valid).  

N. B. Any corrective procedure that makes p equal b equally works here. In the college where the author       works, at the 12th class (Grade XII) science-stream level, for every BUA-obeying burette we look for a pair-forming pipette having practically the same p, and thus form 25-30 number of burette-pipette pairs. Alternatively, the manufacturer's pipette-marking circular-line could also be changed to make the pipette volume-unit conform to that of the burette.

Deduction 2: In student's determination of the amount of glucose supplied (in the form of its solution) to the student (evaluated person) by the teacher (evaluator), EEH is preserved (a) from the procedural side if the teacher finds the glucose amount from weighing and the student prepare an standard glucose solution of comparable concentration by weighing using the same glucose source, then standardize the Fehling solution with that, and then finds the unknown glucose by a similar titration with the Fehling solution. Following this procedure, EEH is preserved (b) from the instrumental side if the weight-balance system has an uniformly proportional error (i.e., obeys WBPA), that proportionality of error being equal for both the teacher & the student, the student's lone burette obeys BUA (even if the lone pipette-volume is not burette-measured), and the student's volumetric-flask volume must be volume-measured (or newly volume-marked) using the student's burette, provided the Fehling solution is measured with the burette only (and glucose solutions by the lone pipette only), and that the solid glucose source has an uniform purity in all its samples.

(a) As in the final analysis the student and the teacher both have determined glucose by weighing only, any procedural difference error can't appear that way. As the Fehling solution is titrated with both the standard & unknown glucose solutions, any proportional procedural error related with the titration is ruled out (cancellation effect). Vaguely speaking, this is why EEH is procedurally preserved. To talk in a more rigorous way, let's define the following symbols: Let the molarities of Fehling solution (i.e., Cu⁺⁺ ion's molarity), standard glucose solution and supplied glucose solution be m_f, m_s & m_g respectively; and their reacting volumes (to be found by the evaluated) be V_f1, V_f2 , V_s & V_g respectively (such that V_s = V_g , and m_s & m_g are comparable). Assuming no instrumental error (for procedural analysis alone), we have (from stoichiometry of the Fehling-glucose reaction)   m_s V_s  = g m_f V_f1    and   m_g V_g  = g m_f V_f2 , where we assume that the glucose-Fehling titrimetric procedure involves an essentially proportional procedural error denoted by a near-unity factor g (safe assumption for V_s = V_g &  m_s ~ m_g).

Thus m_g  = g m_f V_f2  / V_g  = m_s (V_s / V_f1) ( V_f2 / V_g) = m_s (V_s / V_g) ( V_f2 / V_f1)  = m_s (V_f2 / V_f1). So, mass of glucose supplied is this m_g into solution volume into molar mass. As weighing has no procedural error, and no volumetric (or any other) instrumental error is assumed, the standard solution concentration m_s has no procedural barrier to accuracy. So knowledge of m_g has no procedural barrier. As here we assume no instrumental error, the solution volume is correctly known, and so the mass is correctly known (in spite of the already cancelled-out procedural error-factor g within the above expression for m_g). As the evaluator has directly measured the glucose mass without any procedural error, (a weighing operation has no intrinsic associated procedural error), we see that there is EEH protected from the procedural side.

(b) Let the above-defined burette-factor & pipette-factor be b & p respectively. Let's consider the evaluated first. For V'_s = V'_g &  m_s ~ m_g we can use relation m_g  = m_s (V_s / V_g) ( V_f2 / V_f1) derived above. 

As V_s = p V'_{s ,}  V_g = p V'_{g ,}  V_f2 = b V'_f2   and   V_f1 = b V'_f1 where V'_f2 and V'_f1 are the burette-measured (Fehling-solution) volume values as measured by the evaluated, and as b is the same (lone burette) in both relations we have m_g  = m_s (V'_s / V'_g) ( V'_f2 / V'_f1) i.e., m_g  = m_s ( V'_f2 / V'_f1) where we have used the experimental simplification V'_s = V'_g . Let the evaluated weigh out W_s mass (in grams, say) of impure glucose (of fraction of purity r, i.e., 100r % w/w pure) to prepare the standard solution in a volumetric flask of volume V_f. Considering WBPA, we see that the actual mass W_s is proportional to the measured mass W'_s, i.e., W_s = w W'_s  where w is our so-called weighing-factor. The mass of pure glucose is  W⁰_s  = r W_s = r w W'_s . If molar mass of glucose is M, we get that molarity m_s  = W⁰_s / (M V_f)  =  r w W'_s / (M V_f). The volume V_f is however equal to b V'_f   as the 'flask-factor' has been made to equal the burette-factor b because of its volume already getting burette-measured, V'_f being the burette-observed flask-volume (say 251.3 mL). 

Thus we have m_s  =  (r w W'_s) / (M b V'_f). Now let the supplied glucose solution was supplied within some other volumetric flask having actual volume V_v (burette-measured volume V'_v so that V_v  = b V'_v), the volume of which was to be made up by the student. So the mass of glucose present W_p is (m_gV_v).M =

m_g b M V'_v = b V'_v M ( V'_f2 / V'_f1). (r w W'_s) / (M b V'_f) =

(r w W'_s) ( V'_f2 / V'_f1).(V'_v / V'_f)

However, as the student is ignorant (supposed to be at dark) about the weighing-factor w & the purity-factor r, she'll report (using r = 1 & w = 1) the glucose-mass as W_r = W'_s ( V'_f2 / V'_f1).(V'_v / V'_f).

Next let's consider the evaluator (teacher). Let the mass of glucose present is W_p the value of which is already stated from evaluated-considerations (assuming no personal or operational errors). However, W_p = r w W'_t where W'_t is the teacher-measured mass of glucose supplied to the student. As we assume the same glucose source of uniform purity r, and the same value of weighing-factor r encountered by the teacher, so W'_t equals W_p / (r w), which equals W_r . The teacher is here equally ignorant about the weighing-factor w & the purity-factor r, so he’ll take the mass supplied as simply W'_t . Or in other words the evaluator-found mass W'_t equals the evaluated-reported W_r in the absence of any personal or operational error(s). So there is EEH protected in this stated corrective-analytical method, as is found from the above PID.

N. B. If the teacher makes a high-concentration stock solution of glucose and distributes the students, via a lone, teacher's BUA-burette, measured volumes v'_t  of glucose solution, provided the teacher's stock volumetric-flask volume has been burette-measured (as V'_t) by that teacher's burette, and takes the mass of glucose supplied to the student as (W'_t v'_t / V'_t) , it can be shown that EEH will still remain protected.