Derive an expression to calculate time required time required for completion of zero order reaction.

Explanation

To derive the expression for the time required for the completion of a zero-order reaction, we will follow these steps:

### Step 1: Define the Reaction and Initial Conditions
Let's consider a zero-order reaction where reactant A converts to product B:
\[ A \rightarrow B \]
At time \( t = 0 \), the concentration of A is \( [A]_0 = R_0 \) and the concentration of B is \( [B]_0 = 0 \).

### Step 2: Express Concentration at Time \( t \)
At a later time \( t \), let the amount of A that has reacted be \( x \). Therefore, the concentration of A at time \( t \) can be expressed as:
\[ [A] = R_0 - x \]
And the concentration of B at time \( t \) is:
\[ [B] = x \]

### Step 3: Write the Rate Law for Zero-Order Reaction
For a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactant. The rate can be expressed as:
\[ \text{Rate} = k \]
where \( k \) is the rate constant.

### Step 4: Relate Rate to Change in Concentration
The rate of change of concentration of A can also be expressed as:
\[ -\frac{d[A]}{dt} = k \]
Substituting \( [A] = R_0 - x \), we have:
\[ -\frac{d(R_0 - x)}{dt} = k \]
This simplifies to:
\[ \frac{dx}{dt} = k \]

### Step 5: Integrate to Find Time
Now, we can integrate both sides. The left side will be integrated with respect to \( x \) from 0 to \( x \), and the right side will be integrated with respect to \( t \) from 0 to \( t \):
\[ \int_0^x dx = \int_0^t k \, dt \]
This gives:
\[ x = kt \]

### Step 6: Substitute Back to Find Time for Completion
At completion of the reaction, the concentration of A will be zero:
\[ R_0 - x = 0 \]
Thus, \( x = R_0 \). Substituting this into our equation:
\[ R_0 = kt \]
Rearranging for \( t \):
\[ t = \frac{R_0}{k} \]

### Final Expression
The time required for the completion of a zero-order reaction is:
\[ t = \frac{R_0}{k} \]