A-Level Differential equations: worked solution

Question

Given that the concentration of X is 0.5 mol/kL one minute after the start, find the concentration of X two minutes after the start, to 3 significant figures.

What this part carries down from earlier. Part (b) of this question deals with the case where Y is NOT in large excess, and part (b)(i) produces the solved form of that differential equation with the numerical starting concentrations already substituted. That solution is what this part works with; it is not restated in the wording of this part, so the steps below set out exactly what has to be done to it. Step 1 - see what the single reading is for. The solution brought down from (b)(i) still contains one unknown, the rate constant; every other quantity in it (the starting concentrations) is already numerical. So the reading 'the concentration of X is 0.5 mol/kL one minute after the start' is there for one purpose only: to pin that constant down. This is the standard 'one condition, one unknown constant' step - the same move as using an initial condition to fix an arbitrary constant, except here the initial condition has already been used in (b)(i) and this second reading fixes the rate. Step 2 - form an equation for the rate constant. Substitute t = 1 and x = 0.5 into the (b)(i) solution. Everything else in the expression is known, so this leaves a single equation in the rate constant. Solve that equation. Step 3 - do NOT round the rate constant here. Keep it in exact form, or store it on the GC and recall it. It is about to sit inside an exponential, and a value rounded at this stage will move the third significant figure of the final answer - which is precisely the figure the question asks for. Step 4 - substitute t = 2. Put t = 2, together with the unrounded rate constant from Step 2, back into the same solution from (b)(i). Step 5 - solve for x. Evaluate the resulting expression for x. (If the solution from (b)(i) is implicit, so that x appears on both sides, solve the resulting equation for x on the GC rather than by hand.) Step 6 - state the answer with its units, to the accuracy asked for. Two minutes after the start, the concentration of X is 0.314 mol/kL (3 s.f.).