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 (k) for the homogeneous elementary reaction \( A + B \rightarrow C \), we can follow these steps:

### Step 1: Write the Rate Law Expression
For the elementary reaction \( A + B \rightarrow C \), the rate law can be expressed as:
\[ \text{Rate} = k [A]^1 [B]^1 \]
Here, the stoichiometric coefficients (1 for both A and B) indicate that the reaction is first-order with respect to each reactant.

### Step 2: Understand the Rate of Reaction
The rate of reaction is defined as the change in concentration of a reactant or product per unit time. The unit of rate can be expressed as:
\[ \text{Rate} = \frac{\Delta [C]}{\Delta t} \quad \text{(change in concentration over time)} \]
The unit of concentration is typically moles per liter (mol/L), and the unit of time is seconds (s). Thus, the unit of rate is:
\[ \text{Unit of Rate} = \frac{\text{mol/L}}{\text{s}} = \text{mol L}^{-1} \text{s}^{-1} \]

### Step 3: Substitute into the Rate Law Expression
Now, substituting the unit of rate into the rate law expression:
\[ \text{Rate} = k [A]^1 [B]^1 \]
We can rearrange this to solve for the unit of the rate constant (k):
\[ k = \frac{\text{Rate}}{[A][B]} \]

### Step 4: Determine the Units of Concentration
Since both A and B are concentrations, their units are also in mol/L. Therefore, we can express the units of the concentrations as:
\[ [A] = \text{mol/L}, \quad [B] = \text{mol/L} \]

### Step 5: Substitute the Units into the Expression for k
Substituting the units into the expression for k:
\[ k = \frac{\text{mol L}^{-1} \text{s}^{-1}}{(\text{mol L}^{-1})(\text{mol L}^{-1})} \]
This simplifies to:
\[ k = \frac{\text{mol L}^{-1} \text{s}^{-1}}{\text{mol}^2 \text{L}^{-2}} = \text{L mol}^{-1} \text{s}^{-1} \]

### Final Answer
Thus, the unit of the rate constant \( k \) for the reaction \( A + B \rightarrow C \) is:
\[ \text{L mol}^{-1} \text{s}^{-1} \]

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