Conisder the chemical reaction
`N_(2)(g) + 3H_(2)(g) rarr 2NH_(3)(g)`
The rate of this reaction can be expressed in terms of time derivatives of the concentration of `N_(2)(g), H_(2)(g)`, or `NH_(3)(g)`. Identify the correct relationship among the rate expresisons.

Options

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

Explanation

To solve the problem, we need to express the rate of the reaction in terms of the concentrations of the reactants and products involved in the given chemical equation:

**Chemical Reaction:**
\[ N_2(g) + 3H_2(g) \rightarrow 2NH_3(g) \]

### Step-by-Step Solution:

1. **Identify the Rate of Reaction:**
The rate of a chemical reaction can be expressed in terms of the change in concentration of the reactants and products over time. The general form is:
\[ \text{Rate} = -\frac{1}{\nu} \frac{d[\text{Reactant}]}{dt} = \frac{1}{\nu} \frac{d[\text{Product}]}{dt} \]
where \(\nu\) is the stoichiometric coefficient of the substance.

2. **Write the Rate Expression for Each Substance:**
- For \(N_2\):
\[ \text{Rate} = -\frac{d[N_2]}{dt} \]
- For \(H_2\):
\[ \text{Rate} = -\frac{1}{3} \frac{d[H_2]}{dt} \]
- For \(NH_3\):
\[ \text{Rate} = \frac{1}{2} \frac{d[NH_3]}{dt} \]

3. **Establish Relationships Among the Rate Expressions:**
From the above expressions, we can equate the rates:
\[ -\frac{d[N_2]}{dt} = -3 \left(\frac{1}{3} \frac{d[H_2]}{dt}\right) = 2 \left(\frac{1}{2} \frac{d[NH_3]}{dt}\right) \]

4. **Final Relationships:**
Therefore, we can summarize the relationships as:
\[ \text{Rate} = -\frac{d[N_2]}{dt} = -3 \frac{d[H_2]}{dt} = 2 \frac{d[NH_3]}{dt} \]

### Conclusion:
The correct relationships among the rate expressions for the reaction \(N_2 + 3H_2 \rightarrow 2NH_3\) are:
\[ -\frac{d[N_2]}{dt} = 3 \left(-\frac{d[H_2]}{dt}\right) = 2 \frac{d[NH_3]}{dt} \]