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 how to represent the rate at a given instant of time, we will follow these steps:

### Step 1: Understand the Reaction
The reaction involves the reactants \( x \) and \( y \) forming the product \( z \). The coefficients in the balanced equation indicate the stoichiometric relationships between the reactants and products.

### Step 2: Define 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. For the given reaction, we can define the rates as follows:
- The rate of disappearance of \( x \) is given by \( -\frac{dx}{dt} \).
- The rate of disappearance of \( y \) is given by \( -\frac{1}{2} \frac{dy}{dt} \) (since 2 moles of \( y \) are consumed for every mole of \( x \)).
- The rate of appearance of \( z \) is given by \( \frac{1}{3} \frac{dz}{dt} \) (since 3 moles of \( z \) are produced for every mole of \( x \)).

### Step 3: Write the Rate Expressions
From the definitions above, we can write the following relationships:
\[ -\frac{dx}{dt} = -\frac{1}{2} \frac{dy}{dt} = \frac{1}{3} \frac{dz}{dt} \]

### Step 4: Combine the Rate Expressions
We can express the rates in a unified equation:
\[ -\frac{dx}{dt} = \frac{1}{2} \frac{dy}{dt} = \frac{1}{3} \frac{dz}{dt} \]

### Step 5: Final Representation
Thus, the final representation of the rates at a given instant of time can be summarized as:
\[ -\frac{dx}{dt} = \frac{1}{2} \left(-\frac{dy}{dt}\right) = \frac{1}{3} \frac{dz}{dt} \]

### Step 6: Identify the Correct Option
From the options provided, the correct representation of the rate of reaction is:
\[ -\frac{dx}{dt} = -\frac{1}{2} \frac{dy}{dt} + \frac{1}{3} \frac{dz}{dt} \]
This matches with option B from the video transcript.