### Basic functions: Rational functions

### Applications

Rational functions are often used in mathematical models of biological processes of change. But there are also simple applications; we give three examples.

Use of the thin lens formula The thin lens formula for a thin lens with a focal length \(f\) is \[\frac{1}{b}+\frac{1}{v}=\frac{1}{f},\] where \(v\) is the distance from the object to the lens and \(b\) is the distance of the image to the lens. Given an object distance \(v\) and focal length \(f\) you can compute the image distance \(b\).

Verify that the image distance as a function of the object distance \(v\) is equivalent to the following rational function: \[b(v)=\frac{f\cdot v}{v-f}\]

Reaction kinetics In an enzyme-catalysed biochemical reaction, the reaction rate \(v\) depends on the substrate concentration \(s.\)

The Michaelis-Menten equation for this relation is \[v=\frac{V\cdot s}{K+s}\] where \(V\) is the maximum reaction rate and \(K\) the Michaelis-Menten constant.

Verify that \(v\approx V\) for large \(s\) and \(\displaystyle v\approx \frac{V}{K}\cdot s\) for small \(s\).

Degree of protolysis of a weak acid In the calculation of the **degree of protolysis of a weak acid,** that is, the fraction of the original molecules (or ions) that donate a proton, rational functions also play a role.

Suppose that we dissolve \(n\) moles of an acid \(A\) in pure water up to \(V\) liter, then the reaction \[A+\mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{H}_3\mathrm{O}^{+} +\mathrm{B^-}\] where \(A\) and \(B\) is an acid-base pair (for example acetic acid \(\mathrm{HAc}\) with \(\mathrm{Ac}^{-}\) as short notation for the acetate ion \((\mathrm{CH}_3)\!\mathrm{COO}^{-}\). Under equilibrium conditions, the following holds: \[K_a=\frac{\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]\cdot\bigl[B\bigr]}{\bigl[A\bigr]},\] where \(K_a\) is the acidity dissociation constant (for example, \(K_{\mathrm{HAc}}=1.7\times 10^{-5}\) at room temperature).

Suppose that the degree of protolysis is equal to \(\alpha\), the volume of the solution is equal to \(V\), and the original number of acid molecules is equal to \(n\), then: \[\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]=\bigl[B^-\bigr]=\frac{n\alpha}{V}, \quad \bigl[A\bigr]=\frac{n(1-\alpha)}{V}\] It follows that: \[K_a=\frac{n}{V}\cdot \frac{\alpha^2}{1-\alpha}\] The acidity dissociation constant \(K_a\) is a rational function in the degree of protolysis \(\alpha\). When \(K_a\) is known, one can calculate the degree of protolysis \(\alpha\) from the data.

If the degree of protolysis \(\alpha\) is small, then you may use in the denominator of the above formula instead of \(1-\alpha\) also the number 1. It follows then (check yourself): \[\alpha\approx \sqrt{\frac{V}{n}\cdot K_{a}}\;,\quad \text{for small }\alpha\text.\]