The unit of rate constant for a first order reaction

Options

A. `L^2"mol"^(-2)s^(-1)`
B. `s^(-1)`
C. `"mol"L^(-1)s^(-1)`
D. `L "mol"^(-1) s^(-1)`

Answer

To determine the unit of the rate constant (k) for a first-order reaction, we can follow these steps:

### Step 1: Write the Rate Law for a First-Order Reaction
For a first-order reaction, the rate law can be expressed as:
\[ \text{Rate} = k[A]^1 \]
where:
- \( \text{Rate} \) is the rate of the reaction,
- \( k \) is the rate constant,
- \( [A] \) is the concentration of the reactant A.

### Step 2: Express the Rate in Terms of Concentration and Time
The rate of the reaction can also be expressed as the change in concentration of A over time:
\[ \text{Rate} = -\frac{d[A]}{dt} \]
This means that the rate has units of concentration per time.

### Step 3: Determine the Units of Rate
If we consider concentration in terms of moles per liter (mol/L), the units of the rate can be expressed as:
\[ \text{Units of Rate} = \frac{\text{mol}}{\text{L} \cdot \text{s}} \]

### Step 4: Substitute the Units into the Rate Law
From the rate law, we can equate the units:
\[ \frac{\text{mol}}{\text{L} \cdot \text{s}} = k \cdot \left(\frac{\text{mol}}{\text{L}}\right) \]
Here, we have the concentration of A in the units of mol/L.

### Step 5: Solve for the Units of k
Rearranging the equation to solve for the units of \( k \):
\[ k = \frac{\frac{\text{mol}}{\text{L} \cdot \text{s}}}{\frac{\text{mol}}{\text{L}}} \]
When we simplify this, the units of concentration (mol/L) cancel out:
\[ k = \frac{1}{\text{s}} \]

### Conclusion
Thus, the unit of the rate constant \( k \) for a first-order reaction is:
\[ k = \text{s}^{-1} \]

### Final Answer
The unit of the rate constant for a first-order reaction is \( \text{s}^{-1} \).

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