Given below are two statements:
$R=8.314$ J K$^{-1}$ mol$^{-1}$ and $1$ cal $=4.2$ J
Statement I: When $E_a=12.6$ kcal/mol, the room temperature rate constant is doubled by a $10\,^\circ$C increase in temperature ($298$ K to $308$ K)
Statement II: For a first order reactions A $\rightarrow$ B,

Here $[A]_o$ is the initial concentration of A and $t_{1/2}$ is half life of reaction.
In the light of the above statements, choose the correct answer from the options given below:

For Statement I:
According to the Arrhenius equation:
$\log \left( \dfrac{k_2}{k_1} \right) = \dfrac{E_a}{2.303 R} \left( \dfrac{T_2 - T_1}{T_1 T_2} \right)$

Given that the rate constant doubles, $k_2 = 2k_1$. The temperatures are $T_1 = 298$ K and $T_2 = 308$ K.
The gas constant $R = 8.314$ J K$^{-1}$ mol$^{-1} = \dfrac{8.314}{4.2}$ cal K$^{-1}$ mol$^{-1} \approx 1.98$ cal K$^{-1}$ mol$^{-1}$.

Substituting the values:
$\log(2) = \dfrac{E_a}{2.303 \times 1.98} \left( \dfrac{308 - 298}{298 \times 308} \right)$

$0.301 = \dfrac{E_a}{4.56} \left( \dfrac{10}{91784} \right)$

$E_a = \dfrac{0.301 \times 4.56 \times 91784}{10} \approx 12598$ cal/mol $\approx 12.6$ kcal/mol.
Thus, Statement I is true.

For Statement II:
For a first order reaction, the half-life is given by:
$t_{1/2} = \dfrac{0.693}{k}$
This indicates that the half-life of a first order reaction is independent of the initial concentration $[A]_o$. The graph of $t_{1/2}$ versus $[A]_o$ should be a horizontal line parallel to the concentration axis.
The given graph shows $t_{1/2}$ directly proportional to $[A]_o$, which corresponds to a zero order reaction ($t_{1/2} = \dfrac{[A]_o}{2k}$).
Thus, Statement II is false.

Therefore, Statement I is true but Statement II is false.

Answer: Statement I is true but Statement II is false