The rate of reaction for `N_(2)+3H_(2)to2NH_(3)` may be represented as

Options

A. `r=-(d[N_(2)])/(dt)=-1/3(d[H_(2)])/(dt)=+1/2(d[NH_(3)])/(dt)`
B. `r=-(d[N_(2)])/(dt)=1/3(d[H_(2)])/(dt)=+1/2(d[NH_(3)])/(dt)`
C. `r=-(d[N_(2)])/(dt)=3(d[H_(2)])/(dt)=+1/2(d[NH_(3)])/(dt)`
D. `r=-(d[N_(2)])/(dt)=-1/3(d[H_(2)])/(dt)=+2(d[NH_(3)])/(dt)`

Answer

To determine the rate of the reaction for the equation \( N_2 + 3H_2 \rightarrow 2NH_3 \), we can express the rate law based on the stoichiometry of the reactants and products involved. Here’s a step-by-step breakdown of how to derive the rate expression:

### Step 1: Write the general form of the rate law
The rate of a reaction can be expressed in terms of the change in concentration of the reactants and products over time. For a general reaction, the rate can be represented as:

\[ \text{Rate} = -\frac{d[N_2]}{dt} = -\frac{1}{3}\frac{d[H_2]}{dt} = \frac{1}{2}\frac{d[NH_3]}{dt} \]

### Step 2: Assign the stoichiometric coefficients
In the reaction \( N_2 + 3H_2 \rightarrow 2NH_3 \), the stoichiometric coefficients are:
- For \( N_2 \): 1
- For \( H_2 \): 3
- For \( NH_3 \): 2

### Step 3: Write the rate expression
Using the stoichiometric coefficients, we can write the rate expression as follows:

\[ \text{Rate} = -\frac{1}{1}\frac{d[N_2]}{dt} = -\frac{1}{3}\frac{d[H_2]}{dt} = \frac{1}{2}\frac{d[NH_3]}{dt} \]

This means that the rate of disappearance of \( N_2 \) is equal to the rate of disappearance of \( H_2 \) divided by 3, and the rate of formation of \( NH_3 \) divided by 2.

### Step 4: Combine the expressions
Thus, the complete rate expression can be summarized as:

\[ -\frac{d[N_2]}{dt} = \frac{1}{3} \left(-\frac{d[H_2]}{dt}\right) = \frac{1}{2} \left(\frac{d[NH_3]}{dt}\right) \]

### Step 5: Final rate law expression
The final rate law expression for the reaction is:

\[ \text{Rate} = k [N_2]^1 [H_2]^3 \]

Where \( k \) is the rate constant for the reaction.

### Conclusion
The rate of the reaction can be represented as:

\[ -\frac{1}{1}\frac{d[N_2]}{dt} = -\frac{1}{3}\frac{d[H_2]}{dt} = \frac{1}{2}\frac{d[NH_3]}{dt} \]