4.3. Common Rate Laws: Zero, First, and Second Order

Today, we're starting with zero-order reactions. In a zero-order reaction, the rate is constant and independent of the concentration of the reactants. This often occurs when a catalyst's surface is saturated.

How is the rate law for a zero-order reaction defined?

Great question! The rate law is simply Rate = k, where k is the rate constant. Can you tell me what the integrated form looks like?

Is it [A]_t = [A]_0 - kt?

Exactly! And what about the half-life for a zero-order reaction? Does it depend on the initial concentration?

Yes, it depends on [A]_0, right? It’s t₁₋₂ = [A]_0/(2k).

Yes, perfect! If we plot [A] versus time, what do we expect to see?

A straight line with a slope of -k.

Great job! Zero-order reactions are important when the concentration doesn’t affect the rate, typically involving enzyme saturation or catalyst reactions.

Now, let's delve into first-order reactions. In a first-order reaction, the rate is directly proportional to the concentration of one reactant: Rate = k[A].

What is the differential form for a first-order reaction?

It is d[A]/dt = -k[A]. Can anyone tell me the integrated form?

It’s ln([A]_t) = ln([A]_0) - kt!

Correct! And the half-life? What’s unique about it?

The half-life is t₁₋₂ = 0.693/k and it’s independent of the initial concentration [A]_0.

Exactly! This means the half-life remains constant regardless of how much reactant you start with. What kind of plot can we use to confirm first-order kinetics?

A plot of ln([A]) versus time, which should yield a straight line.

Well done! First-order reactions are common in processes like radioactive decay.

Now, let's discuss second-order reactions. These can come in two forms: either two molecules of the same reactant or one of two different reactants.

What’s the rate law for two identical reactants?

For two identical reactants, it’s Rate = k[A]^2. And what about for two different reactants?

That would be Rate = k[A][B]!

Excellent! Can anyone share the differential form for a second-order reaction with one reactant?

It’s -d[A]/dt = k[A]^2.

Right! And how do we integrate this for two identical reactants?

The integrated form would be 1/[A]_t = 1/[A]_0 + kt.

Very good! What's the half-life formula for this reaction?

It’s t₁₋₂ = 1/(k[A]_0); notice it depends on the initial concentration.

Exactly! Hence, second-order kinetics reactants will produce a plot of 1/[A] versus time that shows a straight line confirming the reaction order.

To wrap up today's lesson, can anyone identify the major differences between zero, first, and second-order reactions?

Sure! Zero-order has a constant rate regardless of concentration, first-order's rate changes with concentration, and second-order's rate depends on the square of the concentration.

Correct! What about the half-life in each case?

For zero-order, it depends on the initial concentration; first-order is constant, and for second-order, it also depends on concentration.

Excellent summary! Understanding these differences is critical for predicting reaction behavior in chemical kinetics.