To derive an expression for the integrated rate law for a zero order reaction, we start with the general rate equation:

Rate = -d[A]/dt = k

For a zero order reaction, the rate is constant and does not depend on the concentration of the reactant. Thus, we can express this as:

-d[A] = k dt

Integrating both sides, we get:

∫d[A] = -k ∫dt

This results in:

[A] - [A]₀ = -kt

Rearranging gives us the integrated rate law for a zero order reaction:

[A] = [A]₀ - kt

Where [A]₀ is the initial concentration of the reactant, k is the rate constant, and t is time.

To derive the integrated rate law for a zero-order reaction, we start with the fundamental rate equation:

Understanding the Rate Equation

The rate of a zero-order reaction is expressed as:

Rate = -d[A]/dt = k

Here, k is the rate constant, and importantly, the rate remains constant regardless of the concentration of the reactant.

Setting Up the Equation

We can rewrite the equation as:

-d[A] = k dt

Integrating Both Sides

Next, we integrate both sides:

∫d[A] = -k ∫dt

This integration leads to:

[A] - [A]₀ = -kt

Final Expression

By rearranging the equation, we arrive at the integrated rate law for a zero-order reaction:

[A] = [A]₀ - kt

Key Variables

• [A]₀: Initial concentration of the reactant
• k: Rate constant
• t: Time

This expression shows how the concentration of the reactant decreases linearly over time in a zero-order reaction.