Calibration
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Consider a typical mercury thermometer. What we measure with this piece of analytical equipment is of the length of a glass capillary that happens to be filled by mercury. At a first encounter with such a device, a person might wonder why anyone would care to have that particular bit of information.
Of course, we are interested instead in knowing the temperature, a property several steps removed from our perception of that length. For all physical measurements, in fact, multiple factors influence the path between the physical property of our interest and the qualia of our perceptions. In the case of our thermometer, we have mercury's coefficient of thermal expansion, the diameter of the capillary, our parallax error, and so on.
Such factors behave as a chain of translators, and any one of them may or may not speak the right language, so to speak. If those translators are not up to the task, and the ticks on our thermometer's length have no relationship to units of temperature, then we simply do not have a thermometer. All we have then is a tube of marked-up glass and a bit of poisonous liquid.
In short, any single quantitative physical measurement is, at most, as accurate as our model of the relationship between observations and reality, as accurate as our calibration. To find this translation between the physical world and our sense of it, to calibrate, we have several options.
If we did, for example, have a defective thermometer, we could always attempt to calibrate and bring the device back to use. The most significant barrier to such an attempt is our ironic need to know temperatures before our device for measuring temperatures can be made functional. Nevertheless, with several basic assumptions about the physical world, this problem is surmountable. For example, we could place the thermometer in boiling DI water at atmospheric pressure and mark the resulting mercury height as 100 °C. We then assume that each time we experience the sense of seeing the mercury reach that mark on the capillary (at 1 atm), the thermometer is at the boiling temperature of water. The same could be done with ice water and 0 °C, or with other well-characterized solutions undergoing a phase change.
If the thermometer is constructed in such a way that the height of mercury can be reasonably assumed to be linearly proportional to the temperature, we could then mark ticks along the length of the glass to to indicate temperatures for which we have no standards. Such a procedure is, of course, known as calibrating your equipment and doing so is a vital component of laboratory experience.
Interactive pH Meter Example:
Suppose we have a PH meter that hasn't been used for months, and we wish to begin using it again. Buried somewhere within the electronics of the device are calibration constants as old as its last use. After such a stretch in storage, you should reasonable assume the chemical makeup of the pH electrode has changed and the calibration constants are now inaccurate.
In this instance, we will also assume there is a linear relationship between the pH and the voltage across the electrode, and therefore we have two values in need of determination: the slope, m, and the intercept, b. Presently, m=0.01 pH/mV and b=5.8 pH. The black line in Figure A shows this relationship between voltage and PH. However, this is not the correct relationship. Figure B shows the estimated pH plotted as a function of the true pH (a property which we could not know with certainty in reality).
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We, of course, wish to correct, or at least test the calibration before we use the pH meter in experimentation. While we do not have 100% certain knowledge of true pH values, any lab with a pH meter will have access to at least three standards. These standards are buffered solutions, typically formulated by their manufacture to have a pH of 4, 7, and 10. To calibrate our pH meter we would immerse our electrode in aliquots of these standard solutions and measure the voltages (we would never directly use the bulk standard solutions, because it is vitally important for any good calibration to avoid contaminating our standards). You may simulate these voltage measurements by clicking the Measure Voltage buttons in the example above.
Once we have these three points on the line relating mV to pH (Figure A), we can then perform a linear least-squares fit to find the calibration constants, m and b. Click the Fit Calibration Line button to perform this fit and see the new, improved calibration line (such a fit is typically done within the software of the PH meter). Notice, with the new calibration, the line in Figure B is a 45 degree diagonal, as it should be for an accurate meter.
Notice also that, in this example, if you were to only test the pH 4 standard, you might conclude that the old calibration was sufficient--the meter gives the same voltage for both the accurate and poor calibration at a pH of 4. Depending upon the device being calibrated, it may be, as it is in this example, necessary to test multiple standards. For instance, some thermocouple measurements require a high-order polynomial to model the calibration curve, even if, within a certain range, a linear fit will do.
Simply, you need at least as many standards as you have unknowns in your calibration curve, and you may have several conditions at which you obtain deceptively accurate readings and yet have a poor calibration. Care should be taken to know how, exactly, your analytical equipment is calibrated, and measurements outside the range in which you calibrate (lower than a pH of 4 and above a pH of 10 in our example) should be regarded with added suspicion.
