50 MEASUREMENTS 



acid labeled 0.100406281 n. I was told that this was prepared by weigh- 

 ing potassium acid phthalate to five significant figures. The solution 

 prepared from this substance was used to standardize an alkaline solution, 

 the akali concentration being expressed to seven figures. The alkali was 

 then used as a standard for the final acid solution, which became even 

 two figures better. Possibly this acid was 0.10041 n, but certainly the 

 extra numbers are meaningless. In fact, the weight of the potassium acid 

 phthalate could be no better than the set of balance weights, and the 

 several titrations can be no better than the volumetric glassware used. 



Certain kinds of measurement require all the precision available. 

 Biological materials rarely demand such precision, however, so the extra 

 effort is wasted. It is quite possible to spend too much time in careful 

 measurement if, for example, the living material changes during the 

 measurement. Beginners sometimes handle solutions so slowly and with 

 such care that evaporation significantly changes the concentration. A 

 reaction rate might be found by measuring the amount of one of the 

 reactants, say every five minutes, but the estimate of the rate is not very 

 good if it takes three minutes to obtain each value. Nearly always the 

 largest error in any biological measurement is in the living material. 

 Measurements should not introduce error larger than the biological 

 error, but there is no point in using measurements a good deal more 

 precise than needed. 



Dimensions: Measurements of physical quantities are given in terms 

 of a unit, and the label defining the unit is as important as the number. 

 We commonly use the term dimension when referring to these labels 

 Admittedly, dimension usually refers to length, but it takes only a little 

 imagination to think of seconds, degrees, and grams as dimensions also. 



When calculations are performed using dimensional values, that is, 

 numbers together with their dimensions, the values will change but 

 frequently the dimensions change also. The dimensions of velocity are 

 in terms of length units per unit of time, or cm/sec. This relationship 

 could also be expressed exponentially as cm^ X sec~^ If an object travels 

 X cm in y sec, its velocity is x/y cm/sec. Here "cm/sec" is a new kind 

 of dimension. 



Calculations can be shown by equations, as A = B, where the "equals" 

 sign means that A and B are identical. If A and B have dimensions, the 

 dimensions must be identical as well as the accompanying numbers. 

 Many measurements made in the laboratory lead to expressions of pro- 

 portionality. For example, we find that a given material obeys the rela- 

 tionship "mass is proportional to volume." The relationship is true 



