For the reaction, `2A + B rarr 3C + D`, which of the following does not express the reaction rate

Options

A. `-(d[C])/(3dt)`
B. `-(d[B])/(dt)`
C. `(d[D])/(dt)`
D. `-(d[A])/(2dt)`

Answer

To determine which expression does not represent the reaction rate for the reaction \(2A + B \rightarrow 3C + D\), we need to understand how the reaction rate is defined in terms of the stoichiometric coefficients of the reactants and products.

### Step-by-Step Solution:

1. **Identify the Reaction:**
The given reaction is:
\[ 2A + B \rightarrow 3C + D \]

2. **Define Reaction Rate:**
The rate of a reaction can be expressed in terms of the change in concentration of reactants and products over time. For a general reaction:
\[ aA + bB \rightarrow cC + dD \]
The rate can be expressed as:
\[ \text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt} \]

3. **Apply to Given Reaction:**
For our reaction:
- The stoichiometric coefficients are:
- \(A: 2\)
- \(B: 1\)
- \(C: 3\)
- \(D: 1\)

Therefore, the reaction rate can be expressed as:
\[ \text{Rate} = -\frac{1}{2}\frac{d[A]}{dt} = -\frac{1}{1}\frac{d[B]}{dt} = \frac{1}{3}\frac{d[C]}{dt} = \frac{1}{1}\frac{d[D]}{dt} \]

4. **Evaluate Each Option:**
Now, let's evaluate the options provided to see which one does not express the reaction rate correctly:
- **Option 1:** \(-\frac{1}{3}\frac{d[C]}{dt}\)
- This is incorrect because the rate of appearance of \(C\) should be \(\frac{1}{3}\frac{d[C]}{dt}\) (not negative).
- **Option 2:** \(-\frac{d[B]}{dt}\)
- This is correct as it represents the rate of disappearance of \(B\).
- **Option 3:** \(\frac{d[D]}{dt}\)
- This is correct as it represents the rate of appearance of \(D\).
- **Option 4:** \(-\frac{1}{2}\frac{d[A]}{dt}\)
- This is correct as it represents the rate of disappearance of \(A\).

5. **Conclusion:**
The expression that does not represent the reaction rate is:
\[ \text{Option 1: } -\frac{1}{3}\frac{d[C]}{dt} \]