General Chemistry: Chemical Kinetics and Nuclear Chemistry Study Notes

Chemical Kinetics

Reaction Orders and Rate Laws

Chemical kinetics studies the speed of chemical reactions and the factors that affect them. The rate law expresses the relationship between the rate of a reaction and the concentration of its reactants.

• Order with respect to a reactant: The exponent of the concentration term for that reactant in the rate law. Determined experimentally.

• Overall order: The sum of the exponents in the rate law.

• Example: For a reaction where the rate triples when [BrO3-] triples, the order with respect to BrO3- is 1.

Key Terms:

• Rate constant (k): Proportionality constant in the rate law, specific to a reaction at a given temperature.

• Experimental determination: Orders are found by varying concentrations and measuring rate changes.

Example Calculation:

• If doubling [Br-] doubles the rate, the order with respect to Br- is 1.

• If doubling [H+] quadruples the rate, the order with respect to H+ is 2.

Implicitly held constant: When determining rates as a function of different starting concentrations, temperature is typically held constant.

Consecutive Reactions and Intermediates

Some reactions proceed through a series of steps, forming intermediates. The kinetics of these reactions can be more complex.

• Consecutive reaction: A → B → C, with different rate constants for each step.

• Intermediate: A species formed in one step and consumed in another.

• Concentration vs. time: The concentration of an intermediate (e.g., [B]) often rises and then falls, not simply linearly or exponentially.

Example: For A → B → C, the rate of change of [B] is:

• $\frac{d[B]}{dt} = k_1[A] - k_2[B]$

Interpretation of rate constants:

• If $k_2 > k_1$, B converts to C faster than A converts to B.

Arrhenius Equation and Temperature Dependence

Arrhenius Equation

The Arrhenius equation describes how the rate constant (k) depends on temperature (T):

• $k = Ae^{-E_a/(RT)}$

• Where:

  • A = frequency factor (pre-exponential factor)

  • Ea = activation energy (J/mol or kJ/mol)

  • R = gas constant (8.314 J/mol·K)

  • T = temperature (K)

Graphical Analysis: Plotting $\ln k$ vs. $1/T$ yields a straight line with slope $-E_a/R$ and intercept $\ln A$.

Interpreting Arrhenius Plots

• Slope: $\text{slope} = -E_a/R$

• Intercept: $\text{intercept} = \ln A$

• Example: If the best-fit line is $y = -2766.4x + 25.105$, then:

  • $E_a = -\text{slope} \times R = 2766.4 \times 8.314 = 2.3 \times 10^4$ J/mol

  • $A = e^{25.105} = 8.0 \times 10^{10}$ (units depend on reaction order)

Frequency Factor (A): A small A suggests the reaction has large steric requirements or a small probability of effective collisions.

Half-Life Calculations

The half-life ($t_{1/2}$) is the time required for half of a reactant to be consumed.

• For a first-order reaction: $t_{1/2} = \frac{0.693}{k}$

• For a second-order reaction: $t_{1/2} = \frac{1}{k[A]_0}$

• Example: For the decomposition of cyclobutane with given Arrhenius parameters, calculate k at a specific T, then use the first-order half-life formula.

Nuclear Chemistry: Radiocarbon Dating

Radiocarbon Dating and Maximum Measurable Age

Radiocarbon dating uses the decay of $^{14}$C to estimate the age of organic materials. The maximum measurable age depends on the half-life and the precision of measurement.

• Half-life of $^{14}$C: 5700 years

• Maximum age: Determined by the smallest detectable amount, often a few half-lives.

• Example: With 0.5% error, the maximum age is about 43,560 years.

Thermodynamics and Kinetics

Thermodynamic Parameters from Kinetics

Thermodynamic quantities such as activation enthalpy ($\Delta H^\ddagger$) and activation entropy ($\Delta S^\ddagger$) can sometimes be determined from kinetic data.

• If A (frequency factor) is known, $\Delta S^\ddagger$ can be computed.

• If $E_a$ is known, $\Delta H^\ddagger$ can be computed.

Comparing Reaction Rates

Given two reactions with different activation energies and entropies, the rate at a given temperature can be compared using the Arrhenius equation.

• Lower $E_a$ and higher $A$ (or $\Delta S^\ddagger$) generally mean a faster reaction.

• Example: If reaction 1 has a lower $E_a$ and higher $\Delta S^\ddagger$ than reaction 2, reaction 1 is faster at the same temperature.

Summary Table: Key Kinetic and Thermodynamic Quantities

Quantity	Symbol	How Determined	Units
Rate constant	k	Experimentally, from rate law	varies (e.g., s-1, M-1s-1)
Activation energy	Ea	From Arrhenius plot slope	J/mol or kJ/mol
Frequency factor	A	From Arrhenius plot intercept	varies
Half-life	t1/2	From k and reaction order	time (s, min, yr)
Activation enthalpy	ΔH‡	From Ea	kJ/mol
Activation entropy	ΔS‡	From A	J/mol·K