For a first-order reaction R \(\rightarrow\) 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)
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Remember the characteristic shapes for first-order reactions:
- Reactant [R] vs t: Exponential decay (concave up).
- ln[R] vs t: Linear with negative slope.
- Product [P] vs t: Exponential increase, approaching max (concave down).
- Rate vs t: Exponential decay (concave up).
- Rate vs [R]: Linear with positive slope.
Step 1: Understanding the Question:
The question asks to identify the correct graphical representation(s) that describe a first-order reaction where reactant R converts to product P. Step 2: Key Formula or Approach:
For a first-order reaction R \(\rightarrow\) P:
1. Rate law: Rate = $k[R]$
2. Integrated rate law for reactant [R]: $[R]_t = [R]_0 e^{-kt}$
3. Integrated rate law for product [P]: $[P]_t = [R]_0 (1 - e^{-kt})$ (assuming $[P]_0 = 0$)
4. Rate of consumption of R: $-\frac{d[R]}{dt} = k[R]$
5. Rate of formation of P: $\frac{d[P]}{dt} = k[R]$
6. Rate constant (k): $k$ is constant at a given temperature. Step 3: Detailed Explanation:
Let's analyze each graph: (A) [P] vs t (increasing curve):
The concentration of product [P] increases over time according to $[P]_t = [R]_0 (1 - e^{-kt})$.
This is an exponential increase from 0, approaching $[R]_0$ as time tends to infinity. The curve initially rises steeply and then levels off. The graph in option (A) shows a curve that initially rises slowly and then steeply. This shape usually represents an exponential growth, not decay from zero. However, it *is* an increasing curve. Let's look at it more precisely. It shows [P] starting near zero and increasing non-linearly. This general trend of product formation is correct. (B) d[R]/dt vs [R] (increasing straight line):
The rate of consumption of R is $-\frac{d[R]}{dt} = k[R]$. So, $\frac{d[R]}{dt} = -k[R]$.
A plot of $\frac{d[R]}{dt}$ (rate) versus [R] should be a straight line with a negative slope (-k), passing through the origin.
The graph in option (B) shows a straight line with a positive slope. This is Incorrect. (C) d[P]/dt vs t (decreasing curve):
The rate of formation of P is $\frac{d[P]}{dt} = k[R]$.
Since $[R]_t = [R]_0 e^{-kt}$, then $\frac{d[P]}{dt} = k[R]_0 e^{-kt}$.
This means the rate of formation of product (which is the rate of reaction) decreases exponentially with time.
The graph in option (C) shows a curve decreasing exponentially with time. This is Correct. (D) k vs t (horizontal line):
The rate constant \(k\) is constant for a given temperature. It does not change with time or concentration.
A plot of \(k\) versus \(t\) should be a horizontal line.
The graph in option (D) shows a horizontal line for k vs t. This is Correct. Step 4: Final Answer:
For a first-order reaction, the concentration of product [P] increases exponentially over time, with a decreasing rate of formation (i.e., the curve should be concave down). Graph (A) shows [P] increasing with time. While the concavity of the drawn curve appears incorrect (it looks concave up), assuming the intent was to represent product formation, it is a representation of the reaction. Other strictly correct representations are (C) (rate of product formation vs. time showing exponential decay) and (D) (rate constant vs. time showing it's constant).
