$\text{A}\to \text{P}$ is a first order reaction. At $300 \text{ K}$ this reaction was started with $[\text{A}]=0.5 \text{ mol}\text{L}^{-1}$. The rate constant of reaction was $0.125 \text{ min}^{-1}$. The same reaction was started separately with $[\text{A}]=1 \text{ mol}\text{L}^{-1}$ at $300 \text{ K}$. The rate constant (in $\text{min}^{-1}$) now is
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The rate constant ($k$) for a chemical reaction is independent of the reactant concentrations. It is affected only by temperature (Arrhenius equation) and the presence of a catalyst. If the temperature and catalyst remain the same, the rate constant must remain the same.
Step 1: State the definition and properties of the rate constant.
The rate constant ($k$) is the proportionality constant in the rate law expression (Rate $= k[\text{A}]^n$).
The rate constant is specific to a given reaction and depends only on the temperature and the presence of a catalyst.
The rate constant is independent of the initial concentration of the reactants.
Step 2: Identify the given information.
The reaction is a first-order reaction, $\text{A}\to \text{P}$.
Initial concentration in the first experiment: $[\text{A}]_1 = 0.5 \text{ mol}\text{L}^{-1}$.
Rate constant in the first experiment: $k_1 = 0.125 \text{ min}^{-1}$.
Temperature in the first experiment: $T_1 = 300 \text{ K}$.
Initial concentration in the second experiment: $[\text{A}]_2 = 1 \text{ mol}\text{L}^{-1}$.
Temperature in the second experiment: $T_2 = 300 \text{ K}$.
Step 3: Determine the rate constant for the second experiment.
Since the reaction (A$\to$P) and the temperature ($300 \text{ K}$) are the same in both experiments, the rate constant must be the same, regardless of the change in the initial concentration of $\text{A}$.
\[
k_2 = k_1.
\]
\[
\boxed{k_2 = 0.125 \text{ min}^{-1}}.
\]
