Question

Identify the unit of rate constant for a first-order reaction.

• A. \(\text{mol dm}^{-3}\text{s}^{-1}\)
• B. \(\text{s}^{-1}\)
• C. \(\text{mol}^{-1}\text{dm}^3\text{s}^{-1}\)
• D. \(\text{mol dm}^{-3}\)

Hint:

Remember the common units of rate constants:
• Zero order: \[ \text{mol dm}^{-3}\text{s}^{-1} \]
• First order: \[ \text{s}^{-1} \]
• Second order: \[ \text{mol}^{-1}\text{dm}^3\text{s}^{-1} \] The unit changes with reaction order.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The rate constant (\(k\)) is a fundamental parameter in chemical kinetics that quantifies the speed of a reaction. Unlike the reaction rate itself, which always has units of change in concentration per unit time (e.g., \(mol \cdot L^{-1} \cdot s^{-1}\)), the dimensions of the rate constant are variable. The units of \(k\) must adjust so that the overall Rate Law equation remains dimensionally consistent. For a reaction of order \(n\), the rate is proportional to the concentration raised to the power of \(n\). Therefore, as the order changes, the units of the concentration term change, necessitating a corresponding change in the units of \(k\) to ensure the product always results in units of "Rate."
Step 2: Key Formula or Approach:
The general formula to determine the unit of the rate constant for any order \(n\) is:
\[ \text{Unit of } k = (\text{Concentration})^{1-n} \cdot (\text{Time})^{-1} \]
Substituting the standard units for concentration (\(mol \cdot dm^{-3}\)) and time (\(s\)):
\[ \text{Unit of } k = (mol \cdot dm^{-3})^{1-n} \cdot s^{-1} \]
Step 3: Detailed Explanation:
Let us apply this general formula to a first-order reaction (\(n = 1\)):
1. Plug \(n = 1\) into the formula:
\[ \text{Unit of } k = (mol \cdot dm^{-3})^{1-1} \cdot s^{-1} \]
2. Since any non-zero number raised to the power of zero is 1:
\[ \text{Unit of } k = (mol \cdot dm^{-3})^{0} \cdot s^{-1} \]
\[ \text{Unit of } k = 1 \cdot s^{-1} = s^{-1} \]
3. This can be intuitively understood through the Rate Law for first order:
\[ \text{Rate} = k[A] \]
Rearranging for \(k\):
\[ k = \frac{\text{Rate}}{[A]} \]
Substituting the units:
\[ \text{Unit of } k = \frac{mol \cdot dm^{-3} \cdot s^{-1}}{mol \cdot dm^{-3}} \]
The concentration units in the numerator and denominator cancel out perfectly. This leaves only the reciprocal of time. This is a unique characteristic of first-order reactions; the rate constant is purely a frequency term (per second, per minute, etc.) and is independent of the initial concentration of the reactants. This property is why radioactive decay (which is always first-order) is characterized by a "half-life" that does not depend on the amount of material present.
Step 4: Final Answer:
The unit for the rate constant of a first-order reaction is \(s^{-1}\) (per second).
Therefore, option (2) is correct.