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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

G Tony Jacobs
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Adam_thunder
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1 Answers1

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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.

G Tony Jacobs
  • 31,218