AP Chem integrated rate laws and half-life review

Integrated rate law definition?

An integrated rate law expresses the concentration of reactants or products as a function of time, derived from the differential rate law.

First-order rate law equation?

$\displaystyle [A] = [A]_0 e^{-kt}$ where $\displaystyle [A]$ is concentration, $\displaystyle k$ is the rate constant.

Second-order rate law general form?

$\displaystyle \frac{1}{[A]} = \frac{1}{[A]_0} + kt$ indicates dependence on the square of concentration.

True or False: Zero-order reactions depend on concentration.

False. Zero-order reactions have a rate independent of the concentration of the reactants.

What does the rate constant (k) indicate?

The rate constant indicates how fast a reaction occurs and is specific to temperature and reaction type.

For a first-order reaction, how does doubling the concentration affect rate?

Rate remains the same; first-order reactions depend only on the concentration of one reactant.

What is the unit of the rate constant (k) for a second-order reaction?

The unit for $\displaystyle k$ is typically M$\displaystyle ^{-1}$s$\displaystyle ^{-1}$ (molarity inverse times seconds).

Fill in the blank: The integrated rate law for zero-order reactions is _____.

$\displaystyle [A] = [A]_0 - kt$.

Compare first-order and second-order rate laws.

First-order depends on one concentration; second-order depends on the square of the concentration.

What is half-life for first-order reactions?

Half-life ($\displaystyle t_{1/2}$) is constant and given by $\displaystyle t_{1/2} = \frac{0.693}{k}$.

True or False: Integrated rate laws only apply at equilibrium.

False. Integrated rate laws apply throughout the reaction process, not just at equilibrium.

What is the general form of the integrated rate law for second-order reactions?

$\displaystyle \frac{1}{[A]} = \frac{1}{[A]_0} + kt$ shows dependency on concentration squared.

Cause → Effect: Increase in temperature → ?

Increases reaction rate, often increasing the value of k.

How to determine reaction order experimentally?

Use the method of initial rates or integrated rate laws to observe concentration vs. time.

Worked example: If k = 0.05 s$\displaystyle ^{-1}$, find the half-life of a first-order reaction.

$\displaystyle t_{1/2} = \frac{0.693}{0.05} = 13.86$ seconds.

What is the slope of a concentration vs. time graph for zero-order reactions?

The slope equals -k, indicating a linear relationship.

First-order vs. zero-order: which has a constant rate?

Zero-order has a constant rate regardless of concentration.

What is the integrated rate law for first-order reactions?

The integrated rate law for first-order reactions is: t = \frac{1}{k} imes ext{ln}\frac{[A_0]}{[A]} - Where [A_0] is initial concentration - [A] is concentration at time t - k is the rate constant.

What is half-life?

Half-life is the time required for half of a sample of a radioactive substance to decay.

Calculate the half-life of a substance if it takes 10 years to decay from 100g to 50g.

Half-life = 10 years.

True or False: Half-life depends on the initial amount of substance.

False. Half-life is a constant value for a given substance regardless of the initial amount.

If a substance has a half-life of 5 years, how much remains after 15 years?

After 15 years: 12.5% remains. (100g → 50g → 25g → 12.5g)

Fill in the blank: After one half-life, ____ of the original sample remains.

50%.

What is the formula for half-life in first-order reactions?

The formula is: $\displaystyle t_{1/2} = \\frac{0.693}{k}$, where $\displaystyle k$ is the rate constant.

Compare first-order and zero-order half-lives.

First-order: half-life is constant; Zero-order: half-life decreases as concentration decreases.

Calculate the remaining mass after 3 half-lives if starting with 80g.

After 3 half-lives: 10g remains. (80g → 40g → 20g → 10g)

What happens to the half-life of a radioactive isotope over time?

The half-life remains constant; it does not change as the isotope decays.

True or False: Half-lives can vary for different isotopes.

True. Different isotopes have unique half-lives.

What is the half-life of Carbon-14?

The half-life of Carbon-14 is approximately 5,730 years.

If a sample has a half-life of 8 years, how long until only 25g remains from 100g?

24 years (3 half-lives).

Cause → Effect: What happens to the amount of a substance after each half-life?

The amount of the substance halves.

Using the half-life of 6 years, how much of a 200g sample remains after 18 years?

25g remains. (200g → 100g → 50g → 25g)

What does it mean if a substance has a half-life of 1 day?

It means that every day, half of the substance decays.

Fill in the blank: The concept of half-life is crucial in ____ dating.

radiocarbon.

Calculate the number of half-lives for a sample that decays from 80g to 10g.

3 half-lives.

What is the relationship between half-life and stability of isotopes?

Longer half-lives indicate more stable isotopes.