To determine the time required for 50% of molecule A to change into its isomeric form B, follow these steps:

Half-life Formula for First Order Kinetics:

The half-life ($ t_{1/2} $) for a first-order reaction is given by:

$ t_{1/2} = \frac{0.693}{K} $

Calculate the Rate Constant (K):

The rate constant $ K $ can be calculated using the Arrhenius equation:

$ K = A \cdot e^{-\frac{E_a}{RT}} $

Given:

$ A $ (frequency factor) = $ 10^{20} $

$ E_a $ (activation energy) = $ 191.48 \, \text{kJ/mol} = 191.48 \times 10^3 \, \text{J/mol} $

$ R $ (universal gas constant) = $ 8.314 \, \text{J/mol} \cdot \text{K} $

$ T = 1000 \, \text{K} $

Substitute the values into the Arrhenius equation:

$ K = 10^{20} \times e^{-\frac{191.48 \times 10^3}{8.314 \times 1000}} $

Simplify this calculation:

$ K = 10^{20} \times e^{-23.031} $

Simplifying further by recognizing that $ e^{-23.031} $ is a very small number, gives:

$ K \approx \frac{10^{20}}{10^{10}} = 10^{10} \, \text{sec}^{-1} $

Calculate the Half-life:

Using the calculated value of $ K $:

$ t_{1/2} = \frac{0.693}{10^{10}} = 6.93 \times 10^{-11} \, \text{seconds} $

Convert to Picoseconds:

Since $ 1 \, \text{second} = 10^{12} \, \text{picoseconds} $:

$ t_{1/2} = 6.93 \times 10^{-11} \times 10^{12} \, \text{picoseconds} = 69.3 \, \text{picoseconds} $

Therefore, the time required for 50% of the molecules of A to become B is approximately 69 picoseconds (nearest integer).