For the homogeneous elementary reaction, `A+B rarrC`, the unit of rate constant is

Options

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

Answer

To determine the unit of the rate constant for the homogeneous elementary reaction \( A + B \rightarrow C \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Reaction**: The reaction given is \( A + B \rightarrow C \). This is a homogeneous elementary reaction, meaning all reactants are in the same phase (solid, liquid, or gas).

2. **Define Rate of Reaction**: The rate of a reaction is defined as the change in concentration of reactants or products per unit time. The unit of concentration is moles per liter (mol/L), and the unit of time is seconds (s). Therefore, the unit of rate is:
\[ \text{Rate} = \frac{\text{mol}}{\text{L} \cdot \text{s}} \]

3. **Write the Rate Law**: For an elementary reaction, the rate law can be expressed as:
\[ \text{Rate} = k [A]^1 [B]^1 \]
Here, \( k \) is the rate constant, and the exponents correspond to the stoichiometric coefficients of the reactants in the balanced equation.

4. **Substitute Units into the Rate Law**: Rearranging the rate law to solve for the rate constant \( k \):
\[ k = \frac{\text{Rate}}{[A][B]} \]

5. **Determine the Units of Concentration**: The concentration of \( A \) and \( B \) is also in moles per liter (mol/L). Thus, the unit of \( [A] \) and \( [B] \) is:
\[ [A] = \text{mol/L} \quad \text{and} \quad [B] = \text{mol/L} \]

6. **Substitute Units into the Equation for \( k \)**:
\[ k = \frac{\text{mol/L/s}}{(\text{mol/L})(\text{mol/L})} \]

7. **Simplify the Units**:
- The numerator is \( \text{mol/L/s} \).
- The denominator is \( \text{(mol/L)}^2 = \text{mol}^2/\text{L}^2 \).

Therefore, we can simplify:
\[ k = \frac{\text{mol/L/s}}{\text{mol}^2/\text{L}^2} = \frac{\text{mol/L} \cdot \text{L}^2}{\text{mol}^2 \cdot \text{s}} = \frac{\text{L}}{\text{mol} \cdot \text{s}} \]

8. **Final Unit of Rate Constant**: Thus, the unit of the rate constant \( k \) for the reaction \( A + B \rightarrow C \) is:
\[ \text{Unit of } k = \text{L} \cdot \text{mol}^{-1} \cdot \text{s}^{-1} \]

### Conclusion:
The unit of the rate constant for the homogeneous elementary reaction \( A + B \rightarrow C \) is \( \text{L} \cdot \text{mol}^{-1} \cdot \text{s}^{-1} \).