Empirical formulas are not derived theoretically and, as a rule, do not make much sense in scientific understanding. The form of such a dependence is chosen by the researcher. A characteristic feature of such formulas expressing empirical laws is the presence of empirical coefficients - parameters of the empirical formula, the numerical values of which are selected by the researcher in order to most closely match the calculation results to empirical data.
In chemistry [edit |
Themes 8, 9. Choosing empirical formulas for nonlinear dependencies
Goal: To familiarize with the methodology for choosing empirical formulas for nonlinear dependencies
8.1. The need for experimental data processing
8.2. Experimental data processing sequence
If the desired function on the graph does not lie directly, it is difficult to say what analytical form it has. Therefore, you can use the following recommendations:
Let be yfunction of one variable with two parametersaandb. As a set of functions from which we will choose an empirical dependence, we can consider:
1) a linear function y = a + bx,
2) exponential function y = a * b x,
3) rational function y = 1 / (a + bx),
4) the logarithmic function y = a + b * ln (x),
5) the power function y = a * x b (it determines the parabolic dependence if the parameter b> 0, and the hyperbolic dependence if b x.
3) In the case where the smallest of the absolute errors is e3, the desired empirical dependence determines the fractional rational function y = 1 / (a + bx).
4) If the smallest of the absolute errors is e4, then the logarithmic function y = a + b * ln (x) is a good approximation.
5) For the case where the smallest absolute error is e5, the power function y = a * x b is selected as an empirical dependence.
6) If the smallest of the absolute errors is e6, then for the desired dependence one should choose a hyperbolic function of the form y = a + b / x.
7) In the case where the smallest of the absolute errors is e7, as an analytical dependence, a fractional rational function of the form y = x / (a + bx) is selected.
Example. Find the empirical dependence for the function given in the table