Question 5

For a first-order reaction R $\rightarrow$ P:

1. Evaluating Graph (A):

The concentration of the product P at any time $t$ is given by the integrated rate law: $[\text{P}] = [\text{R}]_0(1 - e^{-kt})$

As time $t$ increases, $[\text{P}]$ increases but at a gradually decreasing rate until it reaches a maximum constant value ($[\text{R}]_0$). The curve should be concave down.

Graph (A) displays an exponential growth curve (concave up, increasing slope), which is incorrect.

2. Evaluating Graph (B):

The rate of disappearance of the reactant R is given by the differential rate law:

$\text{Rate} = -\frac{d[\text{R}]}{dt} = k[\text{R}]$

Rearranging this for the y-axis variable shown in the graph:

$\frac{d[\text{R}]}{dt} = -k[\text{R}]$

This is an equation of a straight line ($y = mx$) passing through the origin, but with a negative slope ($-k$). Since $[\text{R}]$ is always positive, $\frac{d[\text{R}]}{dt}$ must be negative.

Graph (B) shows a straight line with a positive slope in the first quadrant, which is mathematically and chemically incorrect.

3. Evaluating Graph (C): The rate of formation of the product P is:

$\frac{d[\text{P}]}{dt} = k[\text{R}]$

Since $[\text{R}] = [\text{R}]_0 e^{-kt}$ for a first-order reaction, substituting this gives:

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

This equation shows that the rate of product formation exponentially decays with time.

Graph (C) perfectly represents this exponential decay curve. This is correct.

4. Evaluating Graph (D):

The rate constant $k$ is a proportionality constant that depends only on the temperature of the reaction. It does not change with the passage of time $t$.

Graph (D) shows $k$ as a constant horizontal line parallel to the time axis. This is correct.

Conclusion:

The correct graphical representations are (C) and (D).