The concentration C C\,C of a catalyst in a chemical reaction vessel, measured in mg/L, is modelled by the equation
C=1005(3t−k),t>k3 C = \frac{100}{5(3t - k)}, \quad t > \frac{k}{3} C=5(3t−k)100,t>3kwhere t t\,t is the time in seconds since the start of the reaction, k k\,k is a positive constant, and k≠3k \neq 3k=3.
Find dCdt\displaystyle \frac{dC}{dt}dtdC, giving your answer in simplest form in terms of kkk.
The rate of change of the concentration at time t=1t = 1t=1 is -15 mg/L/s.
Find the two possible values of kkk.
Given also that k<3k < 3k<3,
find the equation of the normal to the curve of C C\,C against t t\,t at the point where t=1t = 1t=1, writing your answer in the form at+bC+c=0at + bC + c = 0at+bC+c=0, where a,b a, b\,a,b and c c\,c are integers to be found.
Show, using algebraic integration, that
∫131005(3t−k) dt=λln2 \int_{1}^{3} \frac{100}{5(3t - k)} \, dt = \lambda \ln 2 ∫135(3t−k)100dt=λln2where λ \lambda\,λ is a constant to be found.
