# Chemical Kinetics: Rate Laws | General Chemistry 2

## Reaction Rates

**Reaction rate (in mol.L ^{-1}.s^{-1}):**

The change in concentration of reactants or products per unit time. Reaction rate refers to the rate at which a chemical reaction occurs. The concentration of reactants, temperature, surface area, and the presence of catalyst all influence the rate of reaction

- Reaction rate of a reactant:

- $\frac{1}{\mathrm{\alpha}}\frac{\u2206\left[\mathrm{reactant}\right]}{\u2206\mathrm{t}}$

α = stoichiometric coefficient

[reactant] = reactant concentration (in mol.L^{-1})

Δt = time period (in s)

- Reaction rate of a product:

$\frac{1}{\mathrm{\alpha}}\frac{\u2206\left[\mathrm{product}\right]}{\u2206\mathrm{t}}$

α = stoichiometric coefficient

[product] = product concentration (in mol.L^{-1})

Δt = time period (in s)

aA +bB →cC

reaction rate = - $\frac{1}{\mathrm{a}}\frac{\u2206\left[\mathrm{A}\right]}{\u2206\mathrm{t}}$ = - $\frac{1}{\mathrm{b}}\frac{\u2206\left[\mathrm{B}\right]}{\u2206\mathrm{t}}$ = $\frac{1}{\mathrm{c}}\frac{\u2206\left[\mathrm{C}\right]}{\u2206\mathrm{t}}$

## Rate Laws

**Rate law:**

An equation relating the reaction rate to the concentration of the reactants. The reaction rate can be expressed as a product of the concentration of reactants and a constant of proportionality k for a given reaction, called the rate constant, which depends only on temperature

*a* A + *b *B → products

reaction rate = k [A]^{α }[B]^{β}

k = rate constant

*α* = reaction order with respect to A

*β* = reaction order with respect to B

*α* + *β* = overall reaction order

**Reaction order:**

The power at which the concentration of a given reactant is raised in the rate law equation. The overall reaction order is the sum of the reaction orders of each reactant. The reaction order should be determined by comparing changes in the initial rate with changes in concentrations of the starting reactants

## Integrated Rate Laws

**Integrated rate law**

The rate laws discussed so far relate rate and reactant concentrations. If the concentrations of the reactants can be written as a function of a single reactant:

- $\frac{1}{a}$ $\frac{\left[\mathrm{A}\right]}{\u2206\mathrm{t}}$ = k [A]^{α}

*a* = stoichiometric coefficient of A

[A] = concentration of A (in mol.L^{-1})

k = rate constant

*α* = reaction order with respect to A

By integrating this equation, we can determine a form of rate law that relates reactant concentrations and time. This law is called an integrated rate law and can be used to determine:

- The concentrations of reactants after a specified period of time
- The time required to reach a specified reactant concentration

**Half-life t _{1/2}:**

The time required for half of the reactant to react. The reactant concentration drops to half of its initial value

At t = t_{1/2}, [A] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$

## First-Order Reactions

**Rate law:**

The reaction rate of a first-order reaction is proportional to the concentration of a single reactant:

reaction rate = k [A]

k = rate constant (in s^{-1})

[A] = concentration of A (in mol.L^{-1})

**Integrated rate law:**

Time dependence of [A] for a first-order reaction: - $\frac{\u2206\left[\mathrm{A}\right]}{\u2206\mathrm{t}}$ = k [A]

After integrating:

ln[A] = ln[A]_{0} – kt

⇒ ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}\right)$ = -kt

⇒ $\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}$ = e^{-kt}

Test plot of first-order reactions: ln[A] versus t

**Half-life:**

At t = t_{1/2}, [A] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$

ln $\left(\frac{{\left[\mathrm{A}\right]}_{0}}{2{\left[\mathrm{A}\right]}_{0}}\right)$ = - k t_{1/2}

⇒ t_{1/2} = $\frac{\mathrm{ln}2}{\mathrm{k}}$

t_{1/2} is independent of [A]_{0 }⇒ the half-life of a first-order reaction is independent of the initial reactant concentration

## Second-Order Reactions

**Rate law:**

The reaction rate of a second-order reaction is proportional to the product of 2 reactant concentration [A][B], or on the concentration of a single reactant squared [A]^{2}:

reaction rate = k [A]^{2}

k = rate constant (in mol^{-1}.L.s^{-1})

[A] = concentration of [A] (in mol.L^{-1})

**Integrated rate law:**

Time dependence of [A] for a second-order reaction: - $\frac{\u2206\left[\mathrm{A}\right]}{\u2206\mathrm{t}}$ = k [A]^{2}

After integrating:

$\frac{1}{\left[\mathrm{A}\right]}$ = $\frac{1}{{\left[\mathrm{A}\right]}_{0}}$ + kt

Test plot of second-order reactions: $\frac{1}{\left[\mathrm{A}\right]}$ versus t

**Half-life:**

At t = t_{1/2}, [A] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$

$\frac{2}{{\left[\mathrm{A}\right]}_{0}}$ = $\frac{1}{{\left[\mathrm{A}\right]}_{0}}$ + k t_{1/2}

⇒ t_{1/2} = $\frac{1}{\mathrm{k}{\left[\mathrm{A}\right]}_{0}}$

t_{1/2 }is dependent of [A]_{0} for a second-order reaction

## Radioactive Decay

**Radioactivity:**

The spontaneous emission of small particles (α-particles, β-particles) or radiation (γ-rays) from unstable nuclei. A radioactive decay series is a sequence of nuclear reactions that ultimately results in the formation of a stable isotope. Radioactive decay is a first-order process:

t_{1/2} = $\frac{\mathrm{ln}2}{\mathrm{k}}$ ⇒ k = $\frac{\mathrm{ln}2}{{\mathrm{t}}_{1/2}}$

ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}\right)$ = - kt ⇒ ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}\right)$ = - $\frac{\mathrm{ln}2}{{\mathrm{t}}_{1/2}}$ x t

**Relationship between half-life and number of particles in first-order nuclear decay:**

The number of radioactive nuclei N is proportional to the concentration of the radioactive species:

$\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}$ = $\frac{\mathrm{N}}{{\mathrm{N}}_{0}}$

ln $\left(\frac{\mathrm{N}}{{\mathrm{N}}_{0}}\right)$ = ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}\right)$ = - $\frac{\mathrm{ln}2}{{\mathrm{t}}_{1/2}}$ x t