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# Reaction Types

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Challenge Level

In chemistry, rates of reaction for complicated reactions are often approximated using simple polynomials. It is very useful to understand the qualitative nature of such algebraic representations. These concepts are explored in this task.

Consider the following algebraic forms for an approximate rate of a reaction $R$ in terms of $t$. When $R(t)$ is negative, it can be assumed that the reaction has either not started or has stopped.

$A: R_1(t) = -t +0.1t^3$

$B: R_2(t) = 2+ 2t - 2t^2$

$C: R_3(t) = 2 +t+0.1 t^2$

$D: R_4(t) = 5t - t^2$

$E: R_5(t) = t + t^2-0.1t^3$

$F: R_6(t) = -t + t^2$

Which of these start reacting immediately? Start slowly? Keep speeding up? Speed up to a peak and then slow down? Eventually stop? How would you best describe the reactions in words? Can you think of reactions which might be modelled by these sorts of equations?

Without doing any detailed calculations, can you work out which reaction will take the longest to get started? Which reaction will be the fastest after a long time?

Each reaction is simultaneously started and left to run. Assuming that the algebraic forms continue to hold, which of the reactions will, at some time, be the fastest reaction?

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x