Question

Show that, time required for 99.9% completion of a first order reaction is three times the time required for 90% completion. Give electronic configuration of Gd (Z = 64). Write the name of nano structured material used in car tyres to increase the life of tyres. 
 

Hint:

In first-order reactions, the time for completion is logarithmic, and the time for higher completion (such as 99.9\%) increases significantly compared to lower completion (like 90\%).

Answer 1

Step 1: First Order Integrated Rate Law.
For a first-order reaction, the integrated rate law is: \[ \ln \left( \frac{[A]_0}{[A]} \right) = kt \] Where: - \([A]_0\) is the initial concentration of the reactant, - \([A]\) is the concentration of the reactant at time \(t\), - \(k\) is the rate constant, - \(t\) is the time. Step 2: Time for 90\% Completion.
For 90\% completion, the concentration of the reactant is 10\% of its initial concentration: \[ \ln \left( \frac{[A]_0}{0.1[A]_0} \right) = kt_{90} \] \[ \ln(10) = k t_{90} \] \[ t_{90} = \frac{\ln(10)}{k} \] Step 3: Time for 99.9\% Completion.
For 99.9\% completion, the concentration of the reactant is 0.1\% of its initial concentration: \[ \ln \left( \frac{[A]_0}{0.001[A]_0} \right) = kt_{99.9} \] \[ \ln(1000) = k t_{99.9} \] \[ t_{99.9} = \frac{\ln(1000)}{k} = 3 \times t_{90} \] Thus, the time required for 99.9\% completion is three times the time required for 90\% completion. Step 4: Electronic Configuration of Gd (Z = 64).
The electronic configuration of Gd (Z = 64) is: \[ \text{Gd}: [Xe] 4f^7 5d^1 6s^2 \] Step 5: Nano Structured Material in Tyres.
The nano-structured material used in car tyres to increase their life is silica.