The order of reaction is an experimentally determined quanity. It may be zero, poistive, negative, or fractional. The kinetic equation of `nth` order reaction is
`k xx t = (1)/((n-1))[(1)/((a-x)^(n-1)) - (1)/(a^(n-1))]` …(i)
Half life of `nth` order reaction depends on the initial concentration according to the following relation:
`t_(1//2) prop (1)/(a^(n-1))` ...(ii)
The unit of the rate constant varies with the order but general relation for the unit of `nth` order reaction is
Units of `k = [(1)/(Conc)]^(n-1) xx "Time"^(-1)` ...(iii)
The differential rate law for `nth` order reaction may be given as:
`(dX)/(dt) = k[A]^(n)` ...(iv)
where `A` denotes the reactant.
The half life for a zero order reaction equals

Options

A. `(1)/(2) (k)/(a^(2))`
B. `(a^(2))/(2k)`
C. `(2k)/(a)`
D. `(a)/(2k)`

Explanation

To find the half-life of a zero-order reaction, we can follow these steps:

### Step 1: Understand the definition of half-life
The half-life (\(t_{1/2}\)) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial concentration.

### Step 2: Write the rate equation for a zero-order reaction
For a zero-order reaction, the rate of reaction is constant and can be expressed as:
\[ \text{Rate} = k = -\frac{d[A]}{dt} \]
Where \(k\) is the rate constant and \([A]\) is the concentration of the reactant.

### Step 3: Express the concentration change over time
For a zero-order reaction, the concentration of the reactant decreases linearly over time. The integrated rate law for a zero-order reaction is:
\[ [A] = [A]_0 - kt \]
Where \([A]_0\) is the initial concentration and \([A]\) is the concentration at time \(t\).

### Step 4: Set up the equation for half-life
To find the half-life, we set \([A]\) to \(\frac{[A]_0}{2}\):
\[ \frac{[A]_0}{2} = [A]_0 - kt_{1/2} \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ kt_{1/2} = [A]_0 - \frac{[A]_0}{2} \]
\[ kt_{1/2} = \frac{[A]_0}{2} \]

### Step 6: Solve for half-life
Now, we solve for \(t_{1/2}\):
\[ t_{1/2} = \frac{[A]_0}{2k} \]

### Conclusion
Thus, the half-life of a zero-order reaction is given by:
\[ t_{1/2} = \frac{[A]_0}{2k} \]