For the following reaction scheme (homogeneous), the rate constant has units : `A + B oversetKto C`

Options

A. `sec^(-1)"mole"`
B. `sec^(-1)`
C. `sec^(-1).L per mole
D. sec

Answer

To determine the units of the rate constant \( K \) for the reaction \( A + B \overset{K}{\rightarrow} C \), we can follow these steps:

### Step 1: Write the Rate Expression
For a reaction where two reactants \( A \) and \( B \) combine to form product \( C \), the rate of the reaction can be expressed as:
\[ \text{Rate} = K [A][B] \]
where \( [A] \) and \( [B] \) are the concentrations of reactants \( A \) and \( B \).

### Step 2: Identify the Units of Rate
The rate of a reaction is defined as the change in concentration over time. The units of concentration are typically expressed in moles per liter (mol/L), and time is measured in seconds (s). Therefore, the units of rate are:
\[ \text{Units of Rate} = \frac{\text{moles}}{\text{liters} \cdot \text{seconds}} = \text{mol L}^{-1} \text{s}^{-1} \]

### Step 3: Identify the Units of Concentration
The concentrations of \( A \) and \( B \) are both expressed in moles per liter:
\[ \text{Units of } [A] = \text{Units of } [B] = \text{mol L}^{-1} \]

### Step 4: Substitute Units into the Rate Expression
Now, substituting the units back into the rate expression:
\[ \text{Rate} = K [A][B] \]
This can be rearranged to solve for \( K \):
\[ K = \frac{\text{Rate}}{[A][B]} \]

### Step 5: Substitute the Units
Substituting the units into the equation gives:
\[ K = \frac{\text{mol L}^{-1} \text{s}^{-1}}{(\text{mol L}^{-1})(\text{mol L}^{-1})} \]

### Step 6: Simplify the Units
Now simplifying this expression:
\[ K = \frac{\text{mol L}^{-1} \text{s}^{-1}}{\text{mol}^2 \text{L}^{-2}} = \frac{\text{mol L}^{-1} \text{s}^{-1}}{\text{mol}^2} \cdot \text{L}^2 = \frac{\text{L}^2}{\text{mol} \cdot \text{s}} = \text{mol}^{-1} \text{L} \text{s}^{-1} \]

### Conclusion
Thus, the units of the rate constant \( K \) for the reaction \( A + B \rightarrow C \) are:
\[ \text{Units of } K = \text{mol}^{-1} \text{L} \text{s}^{-1} \]