Step 1: Understanding the Concept:
Graphical analysis of reaction kinetics relies on plotting the integrated rate equation in various linear and exponential formats.
Step 2: Key Formula or Approach:
Evaluate the standard integrated rate equation for a first-order reaction: $[R] = [R]_{0} e^{-kt}$. Take natural and base-10 logarithms to check the straight-line dependencies $y = mx + c$.
Step 3: Detailed Explanation:
1. Option (A): From $[R] = [R]_{0} e^{-kt}$, concentration shows an exponential decay over time. A plot of $[R]$ vs $t$ is indeed a downward sloping curve. (Applicable)
2. Option (B): Taking the natural log gives $ln[R] = ln[R]_{0} - kt$. This is a straight line equation ($y = c + mx$) with a negative slope ($m = -k$). It forms a downward sloping straight line. (Applicable)
3. Option (C): Converting to base-10 log gives $log_{10}[R] = log_{10}[R]_{0} - \frac{kt}{2.303}$. This is a straight line with a negative slope ($-\frac{k}{2.303}$). An "upward sloping" line implies that the concentration of the reactant increases over time, which is physically impossible. (Not Applicable)
4. Option (D): Rearranging the log equation gives $log_{10}(\frac{[R]_{0}}{[R]}) = \frac{k}{2.303}t$. This fits $y = mx$, which is a straight line passing through the origin with a positive (upward) slope. (Applicable)
Step 4: Final Answer:
Graph (C) is the one that is not applicable.