Question

Show that, time required for 99.9 percent completion of a first order reaction is three times the time required for 90 percent 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:

For first-order reactions, use \( \ln \frac{[A]_0}{[A]} = kt \); nanomaterials like carbon black improve mechanical properties in applications.

Answer 1

i. Time for First Order Reaction: For a first-order reaction, the rate law is \( \ln \frac{[A]_0}{[A]} = kt \).
For 90 percent completion, 10% of reactant remains: \( \frac{[A]}{[A]_0} = 0.1 \).
\[ \ln \frac{1}{0.1} = kt_{90%} \quad \Rightarrow \quad kt_{90%} = \ln 10 \approx 2.3026. \] For 99.9% completion, 0.1% remains: \( \frac{[A]}{[A]_0} = 0.001 \).
\[ \ln \frac{1}{0.001} = kt_{99.9%} \quad \Rightarrow \quad kt_{99.9%} = \ln 1000 \approx 6.9078. \] Ratio: \[ \frac{t_{99.9%}}{t_{90%}} = \frac{\ln 1000}{\ln 10} = \frac{3 \ln 10}{\ln 10} = 3. \] Thus, \( t_{99.9%} = 3 t_{90%} \).
ii. Electronic Configuration of Gd (Z = 64): Gadolinium: [Xe] 4f\(^7\) 5d\(^1\) 6s\(^2\).
iii. Nanostructured Material in Car Tyres: Carbon black nanoparticles.
Solution: Carbon black nanoparticles are used in car tyres to enhance durability, improve traction, and increase resistance to wear.