Chemical Kinetics and Gases: Core Concepts and Applications

Chemical Kinetics

Introduction to Chemical Kinetics

Chemical kinetics is the study of the rates at which chemical processes occur and the factors that influence these rates. Understanding kinetics provides insight into reaction mechanisms, which describe the stepwise molecular pathway from reactants to products.

• Reaction rate: The speed at which reactants are converted to products.

• Reaction mechanism: The sequence of elementary steps by which a chemical change occurs.

Factors Affecting Reaction Rates

Several factors influence how quickly a reaction proceeds:

• Physical state of reactants: Homogeneous reactions (all reactants in the same phase) are typically faster than heterogeneous reactions. Increasing the surface area of solids increases reaction rate.

• Reactant concentrations: Higher concentrations generally increase reaction rates due to more frequent collisions.

• Temperature: Raising temperature increases molecular kinetic energy, leading to more frequent and energetic collisions.

• Catalysts: Catalysts speed up reactions by providing alternative pathways with lower activation energy, without being consumed in the reaction.

Measuring Reaction Rates

Reaction rate is defined as the change in concentration of a reactant or product per unit time:

• Average rate: Measured over a time interval.

• Instantaneous rate: The rate at a specific moment (slope of the tangent to the concentration vs. time curve).

• Initial rate: The instantaneous rate at time zero.

Mathematically: $\text{Rate} = -\frac{\Delta [\text{Reactant}]}{\Delta t}$

Relative Rates and Stoichiometry

The rate at which reactants are consumed and products are formed depends on the reaction's stoichiometry. For example, in the reaction $2\,\mathrm{O}_3 \rightarrow 3\,\mathrm{O}_2$, the rate of disappearance of O3 is related to the rate of appearance of O2 by their stoichiometric coefficients.

Determining Rate Laws

The rate law expresses the relationship between the reaction rate and the concentrations of reactants:

• General form: $\text{Rate} = k [A]^x [B]^y$

• The exponents x and y are the reaction orders with respect to each reactant and are determined experimentally.

First-Order Reactions

For first-order reactions, the rate depends linearly on the concentration of one reactant:

• Rate law: $\text{Rate} = k [A]$

• Integrated form: $\ln [A] = -kt + \ln [A]_0$

• A plot of $\ln [A]$ vs. $t$ yields a straight line with slope $-k$.

Half-Life of First-Order Reactions

The half-life ($t_{1/2}$) is the time required for the concentration of a reactant to decrease by half. For first-order reactions:

$t_{1/2} = \frac{0.693}{k}$

Second-Order Reactions

For second-order reactions, the rate depends on the square of the concentration of one reactant:

• Rate law: $\text{Rate} = k [A]^2$

• Integrated form: $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$

• A plot of $1/[A]$ vs. $t$ is linear for second-order reactions.

Half-Life of Second-Order Reactions

For second-order reactions, the half-life depends on the initial concentration:

$t_{1/2} = \frac{1}{k [A]_0}$

Zero-Order Reactions

In zero-order reactions, the rate is independent of the concentration of the reactant:

• Rate law: $\text{Rate} = k$

• Concentration decreases linearly with time.

Temperature and Reaction Rate

Increasing temperature generally increases reaction rates. The rate constant $k$ is temperature-dependent and typically doubles with every 10°C rise in temperature.

The Collision Model

The collision model explains reaction rates based on the kinetic molecular theory:

• Molecules must collide to react.

• More frequent and energetic collisions increase reaction rate.

• Proper orientation during collision is necessary for reaction.

Activation Energy and Transition State

The minimum energy required for a reaction to occur is the activation energy ($E_a$). The highest energy arrangement of atoms during a reaction is the transition state (or activated complex).

Energy Distribution and Temperature

At higher temperatures, a greater fraction of molecules have enough energy to overcome the activation energy barrier, leading to faster reactions.

Arrhenius Equation

The Arrhenius equation relates the rate constant $k$ to temperature and activation energy:

$k = A e^{-E_a/RT}$

Linearized form: $\ln k = -\frac{E_a}{R} \frac{1}{T} + \ln A$

Reaction Mechanisms and Molecularity

A reaction mechanism is a sequence of elementary steps. The molecularity of a step refers to the number of molecules involved:

Molecularity	Elementary Reaction	Rate Law
Unimolecular	A → products	Rate = k[A]
Bimolecular	A + A → products	Rate = k[A]2
Bimolecular	A + B → products	Rate = k[A][B]
Termolecular	A + A + A → products	Rate = k[A]3
Termolecular	A + A + B → products	Rate = k[A]2[B]
Termolecular	A + B + C → products	Rate = k[A][B][C]

Catalysts and Enzymes

Catalysts increase reaction rates by lowering activation energy and providing alternative reaction pathways. Enzymes are biological catalysts with specific active sites for substrates.

Gases

Characteristics of Gases

Gases are composed mainly of nonmetallic elements with simple formulas and low molar masses. They expand to fill their containers, are highly compressible, and have low densities. Gases mix homogeneously in any proportion.

Properties Defining the State of a Gas

• Temperature (T)

• Pressure (P)

• Volume (V)

• Amount (n, in moles)

Pressure

Pressure is the force exerted per unit area:

$P = \frac{F}{A}$

Units of Pressure

• Pascal (Pa): $1\,\mathrm{Pa} = 1\,\mathrm{N/m}^2$

• Bar: $1\,\mathrm{bar} = 10^5\,\mathrm{Pa} = 100\,\mathrm{kPa}$

• mm Hg or torr: $1\,\mathrm{atm} = 760\,\mathrm{torr} = 760\,\mathrm{mmHg} = 101.325\,\mathrm{kPa}$

Gas Laws

Boyle’s Law

At constant temperature, the volume of a fixed amount of gas is inversely proportional to its pressure:

$P_1V_1 = P_2V_2$

Charles’s Law

At constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature:

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$

Avogadro’s Law

At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles:

$\frac{V_1}{n_1} = \frac{V_2}{n_2}$

Ideal Gas Law

Combining the above laws yields the ideal gas equation:

$PV = nRT$

• P = pressure (atm)

• V = volume (L)

• n = moles of gas

• R = ideal gas constant (0.0821 L·atm·mol−1·K−1)

• T = temperature (K)

Density and Molar Mass of Gases

The density (d) of a gas can be calculated using:

$d = \frac{MP}{RT}$

The molar mass (M) can be found from:

$M = \frac{mRT}{PV}$

Dalton’s Law of Partial Pressures

The total pressure of a mixture of non-reacting gases is the sum of the partial pressures of each gas:

$P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$

Mole Fraction and Partial Pressure

The mole fraction ($\chi$) of a component is the ratio of its moles to the total moles in the mixture. The partial pressure is:

$P_1 = X_1 P_{\text{total}}$

Kinetic-Molecular Theory of Gases

This theory explains the observed properties of gases:

• Gases consist of particles in constant, random motion.

• The volume of gas particles is negligible compared to the container volume.

• There are no significant attractive or repulsive forces between particles.

• Collisions are elastic; average kinetic energy is proportional to temperature.

Speeds of Gas Molecules

At a given temperature, gas molecules have a range of speeds. The most probable speed ($u_{mp}$), average speed ($u_{av}$), and root-mean-square speed ($u_{rms}$) are key descriptors.

Effusion and Diffusion

Effusion is the escape of gas molecules through a tiny hole; diffusion is the mixing of gases. Graham’s Law relates the rates of effusion/diffusion to molar mass:

$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

Real Gases and Deviations from Ideal Behavior

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and finite molecular volume.

Van der Waals Equation

The van der Waals equation corrects the ideal gas law for intermolecular attractions (a) and molecular volume (b):

$\left(P + \frac{n^2 a}{V^2}\right)(V - nb) = nRT$

Substance	a (L2·atm/mol2)	b (L/mol)
He	0.0341	0.0237
Ne	0.211	0.0171
Ar	1.34	0.0322
Kr	2.32	0.0398
Xe	4.19	0.0510
N2	1.39	0.0391
O2	1.36	0.0318
CO2	3.59	0.0427
CH4	2.25	0.0428
NH3	4.17	0.0371
H2O	5.46	0.0305