Question

Fresh catalyst is loaded into a reactor for the reaction \(A \rightarrow \text{products}\). The catalyst deactivates with time. The instantaneous activity \(a(t)\) is defined as the ratio of the rate \(-r_A'(t)\) (with deactivated catalyst) to the rate with fresh catalyst. Experimental correlation: \[ -r_A'(t) = -0.5\,t + 10 \quad \text{(mol\,(g cat)\(^{-1}\)\,hr\(^{-1}\))}, \] with \(t\) in hours. The activity of the catalyst at \(t=10\ \text{hr}\) is __________ (rounded off to one decimal place).

Hint:

Catalyst activity is commonly normalized to the initial (fresh) rate: \(a(t)=r(t)/r(0)\).
With linear deactivation, read rates directly from the given line and take the ratio.

Answer 1

Correct Answer: 0.5

Step 1: Activity is the rate relative to the fresh rate: \(a(t)=\dfrac{-r_A'(t)}{-r_A'(0)}\). Step 2: Evaluate the rates from the correlation: \[ -r_A'(0)=10,\qquad -r_A'(10)=-0.5(10)+10=5. \] Step 3: Compute activity: \[ a(10)=\frac{5}{10}=0.5 \;\Rightarrow\; \boxed{0.5}. \]