Identify True $(T)$ and False $(F)$ statements for the following expressions regarding a first-order reaction $R \rightarrow P$:
Statement $I$: $\text{Rate} = -\frac{d[R]}{dt} = k[R]$
Statement $II$: $\text{Rate} = -\frac{d[R]}{dt} = -k[R]$

For a first-order reaction involving gaseous reactants and gaseous products,what will be the unit of its rate constant?

The decomposition of $O_{3(g)}$ follows first order kinetics and is given by $O_{3(g)} \longrightarrow O_{2(g)} + O_{(g)}$. The rate constant for this reaction is $1.0 \times 10^{-3} \ s^{-1}$. The initial pressure of $O_{3(g)}$ is $100 \ atm$. What will be the partial pressure (in $atm$) of $O_3, O_2, O$ respectively after $38.38 \ minutes$?

An organic compound undergoes first-order decomposition. The time taken for its decomposition to $1/8$ and $1/10$ of its initial concentration are $t_{1/8}$ and $t_{1/10}$ respectively. What is the value of $\frac{t_{1/8}}{t_{1/10}} \times 10$? (Given: $\log_{10} 2 = 0.3$)

In a first order reaction,the concentration of the reactant is reduced to $1/8$ of the initial concentration in $75 \ minutes$. The $t_{1/2}$ of the reaction (in minutes) is $(\log 2 = 0.30, \log 3 = 0.47, \log 4 = 0.60)$.

The reaction $A + B \xrightarrow{k}$ product is first order with respect to $A$ and zero order with respect to $B$. If $a_0$ and $a_t$ are the concentrations of $A$ at $t = 0$ and after time $t \, sec$ respectively,then select the correct relationship -