Chemical Kinetics & Radioactivity – Chemistry Study Notes

Rate of Reaction and Rate Law

When chemical reactions occur, reactants are consumed while products are formed. The rate of a reaction is defined as the change in concentration of any reactant or product per unit time.

Mathematically, for a general reaction $\text{A} \rightarrow \text{Products}$, the average rate is expressed as $-\frac{\Delta[\text{A}]}{\Delta t}$, and the instantaneous rate is expressed as $-\frac{d[\text{A}]}{dt}$. The negative sign is crucial because reactant concentration decreases over time, and rates must always be positive values.

The speed at which a reaction proceeds is not constant; it typically starts fast and slows down as reactants are depleted. To quantify this dependence, we turn to the rate law (also known as the rate equation or rate expression). The rate law expresses the reaction rate as a mathematical function of the molar concentrations of the reactants, raised to some powers.

For a reaction $a\text{A} + b\text{B} \rightarrow c\text{C} + d\text{D}$, the rate law is written as:

Rate = $k[\text{A}]^x [\text{B}}^y$

In this expression, $k$ represents the rate constant, while $x$ and $y$ are the partial orders of the reaction with respect to reactants A and B, respectively. It is vital for competitive exam aspirants to remember that $x$ and $y$ are experimentally determined values and do not necessarily equal the stoichiometric coefficients $a$ and $b$ derived from a balanced chemical equation. The sum of these powers, $n = x + y$, gives the overall order of a reaction.

Integrated Rate Equations: Zero and First-Order Reactions

Differential rate laws describe how rate changes with concentration at a given instant, but chemists often need to know how concentrations change over extended periods. This is achieved using integrated rate equations.

For a zero-order reaction, the rate is completely independent of the concentration of the reactants. The rate law is written as $\text{Rate} = -\frac{d[\text{A}]}{dt} = k[\text{A}]^0 = k$. Integrating this differential equation from time $t = 0$ (initial concentration $[\text{A}]_0$) to time $t$ (concentration $[\text{A}]_t$) yields the integrated zero-order equation:

• Zero-order integrated equation: $[\text{A}]_t = -kt + [\text{A}]_0$
• Half-life of zero-order ($t_{1/2}$): $\frac{[\text{A}]_0}{2k}$
• Unit of rate constant ($k$): $\text{mol L}^{-1}\text{s}^{-1}$

In contrast, a first-order reaction has a rate that is directly proportional to the concentration of a single reactant raised to the first power. The differential equation is $-\frac{d[\text{A}]}{dt} = k[\text{A}]$. Integrating this expression across the appropriate limits provides the logarithmic and exponential forms widely utilized in physical chemistry problem-solving:

• First-order integrated equation: $k = \frac{2.303}{t} \log\left(\frac{[\text{A}]_0}{[\text{A}]_t}\right)$ or $[\text{A}]_t = [\text{A}]_0 e^{-kt}$
• Half-life of first-order ($t_{1/2}$): $\frac{0.693}{k}$ (Notice that the half-life is completely independent of the initial concentration)
• Unit of rate constant ($k$): $\text{s}^{-1}$

Temperature Dependence and the Arrhenius Equation

Reaction rates generally increase significantly with rising temperature. As a rough rule of thumb, a temperature increase of $10^\circ\text{C}$ approximately doubles or triples the rate constant of a chemical reaction.

This observation led Swedish chemist Svante Arrhenius to quantitatively connect the rate constant $k$ with absolute temperature $T$ through the famous Arrhenius equation:

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

In this expression, $A$ represents the Arrhenius factor (or pre-exponential factor/frequency factor), $R$ is the universal gas constant, and $E_a$ is the activation energy — defined as the minimum extra amount of energy required by reactant molecules to undergo a chemical reaction. To make this equation easier to plot and manipulate linearly, we take the natural logarithm of both sides:

$\ln k = \ln A – \frac{E_a}{RT}$ or $\log k = \log A – \frac{E_a}{2.303 RT}$

If you plot $\ln k$ against $\frac{1}{T}$, you obtain a straight line with a slope equal to $-\frac{E_a}{R}$ and a y-intercept equal to $\ln A$. Furthermore, when evaluating the rate constants at two different temperatures ($T_1$ and $T_2$), the integrated form of the Arrhenius equation becomes:

$\log\left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.303 R} \left(\frac{T_2 – T_1}{T_1 T_2}\right)$

Radioactive Decay Kinetics

Nuclear chemistry and radioactivity share identical mathematical formulations with chemical kinetics, specifically following first-order kinetics. Because radioactive decay is a random process at the single-nucleus level, the rate of disintegration of a radioactive isotope is directly proportional to the number of undecayed nuclei present at that time. The rate law for radioactive decay is expressed as $-\frac{dN}{dt} = \lambda N$, where $N$ is the number of active nuclei and $\lambda$ is the decay constant.

Integration of this nuclear decay expression yields equations that mirror chemical first-order equations, substituting concentrations with nuclear counts or masses:

• Decay Law Equation: $\lambda = \frac{2.303}{t} \log\left(\frac{N_0}{N_t}\right)$
• Radioactive Half-Life: $t_{1/2} = \frac{0.693}{\lambda}$
• Average Life ($\tau$): $\tau = \frac{1}{\lambda} = 1.44 \times t_{1/2}$

For competitive exams, problems often involve calculating the percentage of a radioactive substance remaining after a given time period or determining the age of archaeological artifacts using carbon dating ($^{14}\text{C}$ half-life $\approx 5730\text{ years}$). Always ensure that the units of time for $t$, $\lambda$, and half-lives are consistent throughout your calculations.

Key Points to Remember

• Reaction order can be zero, fractional, or integer values, whereas molecularity is always an integer (1, 2, or 3) and applies only to elementary steps.
• The unit of the rate constant $k$ depends entirely on the overall order ($n$) of the reaction: units of $k = (\text{concentration})^{1-n} \cdot \text{time}^{-1}$.
• For a first-order reaction, the time required for completion of 99.9% of the reaction is approximately 10 times its half-life ($t_{99.9\%} \approx 10 \times t_{1/2}$).
• A catalyst lowers the activation energy ($E_a$) of a forward and reverse reaction equally, thereby increasing the rate constant without altering the standard Gibbs free energy change ($\Delta G^\circ$) or equilibrium constant ($K_{eq}$).
• Radioactive disintegration is unaffected by external physical conditions such as temperature, pressure, or concentration, as it is a nuclear phenomenon.
• Collision theory posits that for a reaction to occur, molecules must collide with proper orientation and possess energy equal to or greater than the activation energy.
• The temperature coefficient of a reaction is defined as the ratio of rate constants at temperatures differing by $10^\circ\text{C}$ (typically $k_{T+10}/k_T \approx 2 \text{ to } 3$).
• Pseudounimolecular reactions are second-order reactions that behave as first-order reactions because one reactant is present in large excess (e.g., acid hydrolysis of an ester).

Important Facts / Formulas

Previous Year Question Hints

• Hint 1 (Activation Energy Calculation): If a question gives rate constants at two distinct temperatures ($k_1$ at $T_1$ and $k_2$ at $T_2$), apply the two-point Arrhenius logarithmic formula directly to isolate and solve for $E_a$. Be careful with temperature conversions to Kelvin and verify whether $R$ should be used in $\text{J K}^{-1}\text{mol}^{-1}$ ($8.314$) or $\text{cal K}^{-1}\text{mol}^{-1}$ ($2.0$).
• Hint 2 (Fractional Life Problems): For first-order reactions, questions often ask for the time required to complete 75% or 87.5% of a reaction. Remember that 75% completion equals $2 \times t_{1/2}$, and 87.5% completion equals $3 \times t_{1/2}$.
• Hint 3 (Radioactive Dating): When tackling isotope decay problems, set up the ratio of remaining mass to initial mass ($N_t / N_0$) as fractions like $(1/2)^n$, where $n$ represents the number of half-lives elapsed. This trick often avoids tedious logarithmic calculations during exams.

Quick Revision Summary

• Rate of reaction measures the change in concentration of reactants or products over time, quantified by the rate law expression.
• Reaction order is an experimental quantity representing the sum of powers of concentration terms in the rate law.
• Zero-order reactions feature a half-life directly proportional to initial concentration, whereas first-order half-life is entirely independent of initial concentration.
• The Arrhenius equation links temperature to the rate constant via activation energy ($E_a$) and frequency factor ($A$).
• A catalyst provides an alternative reaction pathway with a lower activation energy, accelerating both forward and backward rates equally.
• Radioactive decay follows pure first-order kinetics, where the decay constant $\lambda$ replaces the rate constant $k$.
• Half-life formulas are indispensable shortcuts: $t_{1/2} = 0.693/k$ for first-order systems.
• Graphical plots of kinetic data allow easy extraction of rate constants (from slopes) and activation energies.