For a first-order reaction R → P at a given temperature, k is the rate constant. For this reaction, at the given temperature, the concentrations of R and P at a time t are [R] and [P], respectively. The correct graphical representation(s) for this reaction is(are)

The correct graphical representations for the given first-order reaction are d[P]/dt vs t (decreasing curve) and k vs t (horizontal line).

Step-by-step Analysis and Derivations:

For a first-order reaction $\text{R}\to \text{P}$ with rate constant $k$:

1. Analysis of Rate Constant vs Time (k vs t):
The rate constant $k$ of a chemical reaction depends only on temperature and catalyst. It is independent of time $t$ or the initial concentration of reactants.
Therefore, a plot of $k$ versus $t$ is a horizontal straight line parallel to the time axis.
This corresponds to graph (D) shown in the image.

2. Analysis of Rate of Formation vs Time (d[P]/dt vs t):
For a first-order reaction, the rate of disappearance of reactant R or the rate of formation of product P is directly proportional to the concentration of reactant R at that instant:

$\frac{\text{d}\left[\text{P}\right]}{\text{d}t}=-\frac{\text{d}\left[\text{R}\right]}{\text{d}t}=k\left[\text{R}\right]$

The concentration of reactant R at any time $t$ for a first-order reaction is given by the integrated rate law:

$\left[\text{R}\right]={\left[\text{R}\right]}_0{\text{e}}^{-kt}$

Substituting $\left[\text{R}\right]$ into the rate expression gives:

$\frac{\text{d}\left[\text{P}\right]}{\text{d}t}=k{\left[\text{R}\right]}_0{\text{e}}^{-kt}$

This equation represents an exponentially decreasing curve with respect to time $t$.
This corresponds to graph (C) shown in the image.

3. Analysis of Other Graphs for Comparison:
- [P] vs t: The concentration of product P is given by $\left[\text{P}\right]={\left[\text{R}\right]}_0(1-{\text{e}}^{-kt})$. This is an exponentially increasing curve that approaches ${\left[\text{R}\right]}_0$ asymptotically with a decreasing slope (concave downward), whereas graph (A) shows an upward exponential growth curve, which is incorrect.
- d[R]/dt vs [R]: Graph (B) shows a straight line starting from the origin, but rate of consumption is technically $-\text{d}\left[\text{R}\right]/\text{d}t=k\left[\text{R}\right]$.

Thus, graphs (C) and (D) correctly represent the behavior of the given first-order reaction.