The difference rate law for the reaction
`H_(2) +I_(2) to 2HI` is

Options

A. `(-d[H_(2)])/(dt) =(-d[I_(2)])/(dt)=(-d[HI])/(dt)`
B. `(d[H_(2)])/(dt) =(d[I_(2)])/(dt)=(d[HI])/(dt)`
C. `(1)/(2)(d[H_(2)])/(dt) =(1)/(2)(d[I_(2)])/(dt)=(-d[HI])/(dt)`
D. `-2(d[H_(2)])/(dt) =-2(d[I_(2)])/(dt)=(d[HI])/(dt)`

Explanation

To derive the differential rate law for the reaction \( H_2 + I_2 \rightarrow 2HI \), we will analyze the rates of disappearance of the reactants and the rate of appearance of the product.

### Step 1: Write the rate expressions for the reactants and products.

For the reaction, we can express the rates of disappearance of the reactants and the rate of appearance of the product as follows:

- The rate of disappearance of \( H_2 \):
\[ \text{Rate} = -\frac{d[H_2]}{dt} \]

- The rate of disappearance of \( I_2 \):
\[ \text{Rate} = -\frac{d[I_2]}{dt} \]

- The rate of appearance of \( HI \):
\[ \text{Rate} = \frac{d[HI]}{dt} \]

### Step 2: Relate the rates using stoichiometry.

According to the stoichiometry of the reaction, for every mole of \( H_2 \) that reacts, one mole of \( I_2 \) reacts, and two moles of \( HI \) are produced. Therefore, we can relate these rates as follows:

\[ -\frac{d[H_2]}{dt} = -\frac{d[I_2]}{dt} = \frac{1}{2} \frac{d[HI]}{dt} \]

### Step 3: Write the differential rate law.

From the relationships established in Step 2, we can express the differential rate law. Since the rates of disappearance of the reactants are equal and related to the rate of appearance of the product, we can write:

\[ -\frac{d[H_2]}{dt} = -\frac{d[I_2]}{dt} = \frac{1}{2} \frac{d[HI]}{dt} \]

This indicates that the rate of the reaction can be expressed in terms of the concentration changes of the reactants and products.

### Step 4: Finalize the differential rate law expression.

To express the rate law in a standard form, we can denote the rate of the reaction as \( r \):

\[ r = k[H_2]^a[I_2]^b \]

Where \( k \) is the rate constant, and \( a \) and \( b \) are the orders of the reaction with respect to \( H_2 \) and \( I_2 \), respectively. For this reaction, the stoichiometry suggests that:

\[ r = k[H_2]^1[I_2]^1 \]

Thus, the differential rate law for the reaction can be summarized as:

\[ -\frac{d[H_2]}{dt} = -\frac{d[I_2]}{dt} = \frac{1}{2} \frac{d[HI]}{dt} = k[H_2][I_2] \]