### Why do this problem?

This
task is an exercise in understanding rates of change, graphs
and (possibly) calculus in a way which does not use the usual route
of speed-time graphs.

### Possible approach

The question will need to be read carefully, as it involves
the rates of change as used in chemistry. Can learners explain what
is happening clearly? Having a good mental image of the process
will aid in the solution of this problem. Getting a sense of the
'limits' of the problem will help: what happens with a very, very
fast rate of heating? What happens with a very, very slow rate of
heating? By understanding these processes, it should become clear
that there will be a rate which will minimise the time taken for
the reaction to produce 100 mol.

### Key questions

The key questions should be used to help learners to understand
that the amount catalysed equals the area under the graph of Rate
against Time. For the straight line graphs involved here, simple
calculation of areas will be enough. Learners should be given
plenty of thinking or discussion time before being prompted with
these key questions:

- What does the graph of rate of reaction against temperature
look like?
- Will the reaction always lead to at least 100 mol being
catalysed?
- Why would a very fast rate of heating give rise to a very small
amount of chemical being catalysed?
- If the oven heated the compound to exactly 21 degrees, how long
would it take to catalyse 100 moles?
- If the oven heated the compound at 1 degree per minute, how
much would be catalysed in total?
- If the oven heated the compound at 2 degrees per minute, how
much would be catalysed in total?

### Possible extension

The extension in the question suggests using non-constant
rates of heating. There are several possible levels of
engagement with this: How can you phrase the question in this case?
How would the charts change qualitatively (draw a sketch)? Can you
write down the equations? Can you solve them to find the
answer?

### Possible support

First draw charts of temperature against time and rate against
temperature for a fixed rate of 1 degree per minute. From these we
need to work out a graph of rate against time. Try to get learners
to solve this part first and then repeat for a 2 degree per minute
rate.