Question

The rate of reaction \(A \rightarrow P\) is \(1.25 \times 10^{-2}\ \text{mol dm}^{-3}\text{s}^{-1}\) when \([A] = 0.5\ \text{M}\). Calculate the rate constant if the reaction is second order in \(A\).

• A. \(0.05\)
• B. \(0.04\)
• C. \(0.03\)
• D. \(0.01\)

Hint:

For reaction orders:
• First order: \[ \text{Rate} = k[A] \]
• Second order: \[ \text{Rate} = k[A]^2 \]
• Third order: \[ \text{Rate} = k[A]^3 \] Always substitute concentration carefully before solving for \(k\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Chemical kinetics deals with the speed or rate at which a chemical reaction occurs. The "Rate Law" is a mathematical expression that relates the rate of a reaction to the molar concentration of its reactants. For any reaction, the dependency on concentration is defined by the "order" of the reaction. In this problem, the reaction is explicitly stated to be "second order" with respect to reactant A. This means that if the concentration of A is doubled, the rate of the reaction will increase by a factor of four (\(2^{2}\)). The proportionality constant in this relationship is the rate constant (\(k\)), which is specific to a particular reaction at a given temperature. The units of the rate constant change depending on the overall order to ensure the rate always has units of \(mol \cdot dm^{-3} \cdot s^{-1}\).
Step 2: Key Formula or Approach:
For a second-order reaction involving a single reactant A, the rate law is written as:
\[ \text{Rate} = k[A]^{2} \]
Where:
- Rate = Velocity of the reaction (\(1.25 \times 10^{-2} \text{ mol dm}^{-3} \text{s}^{-1}\))
- \(k\) = Rate constant (to be determined)
- \([A]\) = Concentration of reactant A (0.5 M or \(0.5 \text{ mol dm}^{-3}\))
To find the rate constant, we rearrange the formula:
\[ k = \frac{\text{Rate}}{[A]^{2}} \]
Step 3: Detailed Explanation:
Let's plug in the numerical values provided in the question:
1. The given rate is \(1.25 \times 10^{-2}\). This can also be written as \(0.0125\) for simpler decimal division if preferred.
2. The concentration \([A]\) is 0.5.
3. First, calculate the square of the concentration:
\[ [A]^{2} = (0.5)^{2} = 0.5 \times 0.5 = 0.25 \]
4. Now, substitute these into the rearranged rate constant equation:
\[ k = \frac{1.25 \times 10^{-2}}{0.25} \]
5. To simplify the calculation, express both numbers in scientific notation or as fractions:
\[ k = \frac{0.0125}{0.25} \]
If we divide 1.25 by 0.25, we get 5 (since \(0.25 \times 4 = 1.00\) and one more 0.25 makes 1.25).
Thus, \[ k = 5 \times 10^{-2} \]
6. Converting this back to a standard decimal:
\[ k = 0.05 \]
The units for this rate constant would be \(M^{-1}s^{-1}\) or \(dm^{3} \cdot mol^{-1} \cdot s^{-1}\). The magnitude 0.05 corresponds to option (1). This value tells us how quickly the molecules of A are successfully colliding and reacting under the given conditions.
Step 4: Final Answer:
The calculated value for the rate constant \(k\) is 0.05.
Therefore, the correct option is (1).