Consider the first order reaction $\mathrm{R} \rightarrow \mathrm{P}$.

The fraction of molecules decomposed in the given first order reaction can be expressed as

Consider the reaction aX → bY, for which the rate constant at 30°C is $1 \times 10^{-3}\ \text{mol}^{-1}\ \text{L}\ \text{s}^{-1}$. Which of the following statements are true?

A. When concentration of ‘X’ is increased to four times, the rate of reaction becomes 16 times.

B. The reaction is a second order reaction.

C. The half-life period is independent of the concentration of X.

D. Decomposition of N$_2$O$_5$ is an example of the above reaction.

E. vs time is valid for the above reaction.

Choose the correct answer from the options given below:

t_100% is the time required for the 100% completion of the reaction while t_1/2 is the time required for 50% of the reaction to be completed. Which of the following option correctly represents the relation between t_100% and t_1/2 for zero and first order reactions respectively?

An organic compound undergoes first order decomposition. The time taken for decomposition to $\left(\frac{1}{8}\right)^{\text {th }}$ and $\left(\frac{1}{10}\right)^{\text {th }}$ of its initial concentration are $\mathrm{t}_{1 / 8}$ and $\mathrm{t}_{1 / 10}$ respectively.

What is the value of $\frac{\mathrm{t}_{1 / 8}}{\mathrm{t}_{1 / 10}} \times 10$ ?

$$ (\log 2=0.3) $$