Chemical Kinetics: Rate Laws | General Chemistry 2

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:

• Reaction rate of a product:

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

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
 

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 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:

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] = $\frac{{\left[\mathrm{A}\right]}_0}{2}$

Rate law:

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

Integrated rate law:

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

Half-life:

At t = t_1/2,  [A] = $\frac{{\left[\mathrm{A}\right]}_0}{2}$ 

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:

Integrated rate law:

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

Half-life:

At t = t_1/2, [A] = $\frac{{\left[\mathrm{A}\right]}_0}{2}$

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

          ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_0}\right)$ = - kt   ⇒    ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_0}\right)$ = - $\frac{\ln 2}{{\mathrm{t}}_{1/2}}$ 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: