For the reaction `B+2D to 3T, -(d[B])/(dt)=k[B][D]^(2)`. The expression for `-(d[D])/(dt)` will be

Options

A. 1. `2k[B][D]^(2)`
B. 2. `(1//2)k[B][D]^(2)`
C. 3. `4k[B][D]^(2)`
D. 4. `(1//4)k[B][D]^(2)`

Answer

To derive the expression for \(-\frac{d[D]}{dt}\) from the given reaction \(B + 2D \rightarrow 3T\) and the rate equation \(-\frac{d[B]}{dt} = k[B][D]^2\), we can follow these steps:

### Step 1: Understand the Stoichiometry of the Reaction
The balanced chemical equation is:
\[ B + 2D \rightarrow 3T \]
From this equation, we can see that for every 1 mole of \(B\) that reacts, 2 moles of \(D\) are consumed.

### Step 2: Relate the Rate of Change of Reactants
The rate of reaction for each reactant can be expressed in terms of their stoichiometric coefficients:
- For \(B\):
\[ -\frac{d[B]}{dt} = k[B][D]^2 \]
- For \(D\):
Since 2 moles of \(D\) are consumed for every mole of \(B\), we can relate the rate of change of \(D\) to that of \(B\):
\[ -\frac{d[D]}{dt} = \frac{1}{2} \left(-\frac{d[B]}{dt}\right) \]

### Step 3: Substitute the Expression for \(-\frac{d[B]}{dt}\)
Now, we can substitute the expression for \(-\frac{d[B]}{dt}\) into the equation for \(-\frac{d[D]}{dt}\):
\[ -\frac{d[D]}{dt} = \frac{1}{2} \left(k[B][D]^2\right) \]

### Step 4: Final Expression
Thus, the final expression for the rate of change of \(D\) is:
\[ -\frac{d[D]}{dt} = \frac{1}{2} k[B][D]^2 \]