Question

Identify the correct graph for a first-order reaction (A → P) Given:
- \( x \)-axis = time (\( t \))
- \( a \) = Initial concentration of A
- \( (a-x) \) = Concentration of A at time \( t \)

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

- First-order reactions follow logarithmic decay behavior.
- A plot of \(\log(a-x)\) vs. \( t \) gives a straight line with a negative slope.
- This property is useful in determining the rate constant from experimental data.

Answer 1

The Correct Option is D

Step 1: First-Order Reaction Integrated Rate Law The rate equation for a first-order reaction is: \[ \frac{d[A]}{dt} = - k[A] \] Upon integration, we obtain: \[ \ln [A] = \ln [A_0] - kt \] Or in terms of concentrations, \[ \log (a-x) = \log a - \frac{k}{2.303} t \]

 Step 2: Interpretation of Graphs 
- The equation shows that \(\log(a-x)\) vs. \( t \) gives a straight line with a negative slope. 
- Option (4) correctly represents this linear decrease in the concentration of reactant \( A \) over time. 
- Other options do not follow this logarithmic behavior. 

Step 3: Conclusion 
- The correct graphical representation for a first-order reaction is Option (4). 
- The slope of the straight-line graph gives \(- k/2.303\), which can be used to determine the rate constant \( k \).