I derived the equation below for a problem/project i am currently working on. The purpose of the equation is to determine the time required to reach a given % concentration of oxygen. The upper end limit of the equation will is 21% but what I am trying to confirm is the lower or in this case the equilibrium concentration of oxygen. The derived equation is as follows:
$$\ t = -m \ln\left({0.21mn-x\over 0.21mn-0.21}\right)$$
where t = time, x = oxygen concentration, m & n are constants.
What i deduced is that with the natural log function in the above equation, the equilibrium or lower limit will occur when concentration leads to;
$$\ ln(0) $$
Which is undefined. Therefore equating the above limit gives
$$\ {0.21mn-x\over 0.21mn-0.21} = 0 $$
simplified gives
$$\ x = 0.21mn$$
So i can use this result to determine what my equilibrium oxygen concentration is.
What i wanted to confirm from the maths community is if my logic above is sound/corect? Secondly it there a more formal way to derive my limit equation such as using l'hopital's rule?
Many thanks in advance
If I'm understanding you correctly, you want to know what happens with $x$ as $t$ gets large. A standard approach would be to solve the equation for $x$ that you currently have solved for $t$, giving: $$x=0.21mn-(0.21mn-0.21)e^{-t/m}.$$
Now, we can ask what the limit of $x$ is as $t\to\infty$. Assuming that $m$ is a positive constant, the growth of $t$ causes the term on the right to decay to $0$, yielding $x\to 0.21mn$, as you already found.