For the reaction `x+2y to3z` the rate at a given instant of time can be represented as:

Options

A. `+(d[x])/(dt)=+(d[y])/(dt)=-(d[z])/(dt)`
B. `-(d[x])/(dt)=-1/2(d[y])/(dt)=+1/3(d[z])/(dt)`
C. `-1/2 (d[x])/(dt)=+1/2(2d[y])/(dt)=-1/3(3d[z])/(dt)`
D. `-2(d[x])/(dt)=+1/2(2d[y])/(dt)=-(3d[z])/(dt)`

Answer

To solve the problem regarding the reaction \( x + 2y \rightarrow 3z \) and to express the rate at a given instant of time, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Reaction Components**:
The reaction involves the reactants \( x \) and \( y \), and the product \( z \). The stoichiometry of the reaction is important for writing the rate expressions.

2. **Write the Rate of Reaction**:
The rate of reaction can be expressed in terms of the change in concentration of the reactants and products over time. The general form is:
\[ \text{Rate} = -\frac{1}{a} \frac{d[x]}{dt} = -\frac{1}{b} \frac{d[y]}{dt} = \frac{1}{c} \frac{d[z]}{dt} \]
where \( a, b, c \) are the stoichiometric coefficients of \( x, y, z \) respectively.

3. **Apply Stoichiometric Coefficients**:
For the given reaction:
- The coefficient of \( x \) is 1.
- The coefficient of \( y \) is 2.
- The coefficient of \( z \) is 3.

Therefore, we can express the rate as:
\[ \text{Rate} = -\frac{1}{1} \frac{d[x]}{dt} = -\frac{1}{2} \frac{d[y]}{dt} = \frac{1}{3} \frac{d[z]}{dt} \]

4. **Express the Rate of Change**:
From the above expressions, we can write:
\[ \text{Rate} = -\frac{d[x]}{dt} = -\frac{1}{2} \frac{d[y]}{dt} = \frac{1}{3} \frac{d[z]}{dt} \]

5. **Final Rate Expression**:
The instantaneous rate of the reaction can be summarized as:
\[ \text{Rate} = -\frac{d[x]}{dt} = -\frac{1}{2} \frac{d[y]}{dt} = \frac{1}{3} \frac{d[z]}{dt} \]

### Conclusion:
The correct representation of the rate at a given instant for the reaction \( x + 2y \rightarrow 3z \) is:
\[ \text{Rate} = -\frac{d[x]}{dt} = -\frac{1}{2} \frac{d[y]}{dt} = \frac{1}{3} \frac{d[z]}{dt} \]