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

### Step 2: Define the Units of Rate
The rate of a reaction is defined as the change in concentration per unit time. The units for concentration are typically expressed in moles per liter (mol/L), and time is usually expressed in seconds (s). Therefore, the unit of rate can be expressed as:
\[ \text{Unit of Rate} = \frac{\text{moles}}{\text{liters} \cdot \text{seconds}} = \frac{\text{mol}}{\text{L} \cdot \text{s}} \]

### Step 3: Rearrange the Rate Law to Solve for the Rate Constant
From the rate law, we can rearrange the equation to solve for \( k \):
\[ k = \frac{\text{Rate}}{[A]} \]

### Step 4: Substitute the Units into the Equation
Substituting the units of rate and concentration into the equation gives us:
\[ k = \frac{\frac{\text{mol}}{\text{L} \cdot \text{s}}}{\frac{\text{mol}}{\text{L}}} \]

### Step 5: Simplify the Units
When we simplify this expression:
- The moles (mol) in the numerator and denominator cancel out.
- The liters (L) in the numerator and denominator also cancel out.

Thus, we are left with:
\[ k = \frac{1}{\text{s}} \]

### Step 6: Express the Final Unit
The unit of the rate constant \( k \) for a first-order reaction is therefore:
\[ k = \text{s}^{-1} \]
This can also be expressed as "per second" or "second inverse".

### Conclusion
The unit of the rate constant for a first-order reaction is \( \text{s}^{-1} \) (or "per second").

---