`2NO+2H_(2)rarr N_(2)+2H_(2)O`. The experimental rate law for above reaction is , Rate `=k[NO]^(2)[H_(2)]`. When time is in minutesand the concentration is in moles `//L`, the units for `k` are

Options

A. `("moles"^(3))/(L^(3)-"min")`
B. `("moles")/(L-"min")`
C. `("moles"^(2))/(L^(2)-"min")`
D. `(L^(2))/("moles"^(2)-"min")`

Explanation

To determine the units of the rate constant \( k \) for the reaction \( 2NO + 2H_2 \rightarrow N_2 + 2H_2O \) with the rate law given as \( \text{Rate} = k[NO]^2[H_2] \), we can follow these steps:

### Step 1: Identify the Rate Expression
The rate law for the reaction is given as:
\[ \text{Rate} = k[NO]^2[H_2] \]

### Step 2: Determine the Units of Rate
The rate of a reaction is typically expressed in terms of concentration change over time. In this case, the units of rate (R) can be expressed as:
\[ \text{Rate} = \frac{\text{moles}}{\text{liter} \cdot \text{second}} \quad \text{(or moles per liter per second)} \]
Since we are considering time in minutes, we can express the rate as:
\[ \text{Rate} = \frac{\text{moles}}{\text{liter} \cdot \text{minute}} \]

### Step 3: Determine the Units of Concentration
The concentrations of \( [NO] \) and \( [H_2] \) are given in moles per liter (mol/L). Therefore, we can express their units as:
\[ [NO] \text{ and } [H_2] = \frac{\text{moles}}{\text{liter}} \]

### Step 4: Substitute Concentration Units into the Rate Law
Substituting the units of concentration into the rate law:
\[ \text{Rate} = k \left( \frac{\text{moles}}{\text{liter}} \right)^2 \left( \frac{\text{moles}}{\text{liter}} \right) \]
This simplifies to:
\[ \text{Rate} = k \left( \frac{\text{moles}^2}{\text{liter}^2} \cdot \frac{\text{moles}}{\text{liter}} \right) = k \left( \frac{\text{moles}^3}{\text{liter}^3} \right) \]

### Step 5: Rearrange to Solve for \( k \)
Now, we can rearrange the equation to solve for \( k \):
\[ k = \frac{\text{Rate}}{\left( \frac{\text{moles}^3}{\text{liter}^3} \right)} \]
Substituting the units of rate:
\[ k = \frac{\frac{\text{moles}}{\text{liter} \cdot \text{minute}}}{\frac{\text{moles}^3}{\text{liter}^3}} = \frac{\text{moles} \cdot \text{liter}^3}{\text{liter} \cdot \text{minute} \cdot \text{moles}^3} \]

### Step 6: Simplify the Units
Now, simplifying the units:
\[ k = \frac{\text{liter}^2}{\text{moles}^2 \cdot \text{minute}} \]

### Final Answer
Thus, the units of the rate constant \( k \) are:
\[ \text{L}^2 \cdot \text{mol}^{-2} \cdot \text{min}^{-1} \]