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 will follow these steps:

### Step 1: Write the Rate Law
The rate law for the reaction can be expressed as:
\[ \text{Rate} = k [A]^m [B]^n \]
For the reaction \( A + B \rightarrow C \), we assume it is a simple elementary reaction, which means the order of the reaction is the sum of the stoichiometric coefficients. Here, both \( A \) and \( B \) have a coefficient of 1, so:
- \( m = 1 \) (for \( A \))
- \( n = 1 \) (for \( B \))

Thus, the rate law can be written as:
\[ \text{Rate} = k [A]^1 [B]^1 = k [A][B] \]

### Step 2: Express the Rate in Terms of Concentration
The rate of a reaction is defined as the change in concentration of a reactant or product per unit time. Therefore, we can express the rate as:
\[ \text{Rate} = \frac{\Delta [C]}{\Delta t} \]
In terms of concentration, the unit of rate is:
\[ \text{Rate} = \text{mol/L/s} \quad \text{(moles per liter per second)} \]

### Step 3: Substitute the Rate into the Rate Law
Now, substituting the expression for the rate into the rate law:
\[ \frac{\Delta [C]}{\Delta t} = k [A][B] \]

### Step 4: Rearrange to Solve for \( k \)
Rearranging the equation to solve for \( k \):
\[ k = \frac{\text{Rate}}{[A][B]} \]

### Step 5: Substitute Units into the Equation
Substituting the units into the equation:
- The unit of rate is \( \text{mol/L/s} \)
- The unit of concentration \( [A] \) and \( [B] \) is \( \text{mol/L} \)

Thus, the units for \( k \) become:
\[ k = \frac{\text{mol/L/s}}{(\text{mol/L})(\text{mol/L})} \]
This simplifies to:
\[ k = \frac{\text{mol/L/s}}{\text{mol}^2/\text{L}^2} = \frac{\text{mol/L/s}}{\text{mol}^2/\text{L}^2} = \frac{\text{L}^2}{\text{mol}^2 \cdot \text{s}} \cdot \text{mol} = \frac{1}{\text{mol} \cdot \text{s}} \cdot \text{L} = \text{L/s/mol} \]

### Final Answer
The units of the rate constant \( k \) for the reaction \( A + B \rightarrow C \) are:
\[ \text{L} \cdot \text{s}^{-1} \cdot \text{mol}^{-1} \]