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 rate constant for zero order reaction is
where `c_(0)` and `c_(t)` are concentration of reactants at respective times.

Options

A. `k = (c_(0))/(2t)`
B. `k = (c_(0)-c_(t))/(t)`
C. `k = ln.(c_(0)-c_(t))/(2t)`
D. `k = (c_(0))/(c_(t))`

Explanation

To determine the rate constant for a zero-order reaction, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Zero-Order Reactions**:
In a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactants. The rate law can be expressed as:
\[ \frac{dX}{dt} = -k \]
where \( k \) is the rate constant.

2. **Setting Up the Differential Equation**:
Rearranging the rate law gives:
\[ -dX = k \, dt \]

3. **Integrating Both Sides**:
We integrate both sides to find the relationship between concentration and time. The limits for concentration will be from \( C_0 \) (initial concentration) to \( C_t \) (concentration at time \( t \)), and the limits for time will be from \( 0 \) to \( t \):
\[ \int_{C_0}^{C_t} -dX = \int_{0}^{t} k \, dt \]

4. **Performing the Integration**:
The left side integrates to:
\[ -(C_t - C_0) = -\Delta C \]
The right side integrates to:
\[ kt \]
Therefore, we have:
\[ C_0 - C_t = kt \]

5. **Rearranging for Rate Constant \( k \)**:
Rearranging the equation gives us the expression for the rate constant:
\[ k = \frac{C_0 - C_t}{t} \]

6. **Conclusion**:
The rate constant \( k \) for a zero-order reaction is given by:
\[ k = \frac{C_0 - C_t}{t} \]
This means that the rate constant is directly proportional to the change in concentration over time.