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# Chemical Kinetics ## Reaction Rates ### Definition The reaction rate represents the change in the concentration of reactants or products with respect to time. It is typically expressed in units of $mol \cdot L^{-1} \cdot s^{-1}$ or $M/s$. ### Rate Expression For a general reaction: $aA + bB...

# Chemical Kinetics ## Reaction Rates ### Definition The reaction rate represents the change in the concentration of reactants or products with respect to time. It is typically expressed in units of $mol \cdot L^{-1} \cdot s^{-1}$ or $M/s$. ### Rate Expression For a general reaction: $aA + bB \rightarrow cC + dD$ The rate can be expressed as: $Rate = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$ ### Factors Affecting Reaction Rates 1. **Concentration of Reactants**: Generally, increasing the concentration of reactants increases the reaction rate. 2. **Temperature**: Higher temperatures usually increase the reaction rate. 3. **Surface Area**: For reactions involving solids, increased surface area increases the reaction rate. 4. **Catalysts**: Catalysts speed up reactions without being consumed. 5. **Pressure**: For gaseous reactions, increasing pressure usually increases the reaction rate. ## Rate Laws ### Definition A rate law is an equation that relates the reaction rate to the concentrations of reactants. ### General Form For a reaction: $aA + bB \rightarrow cC + dD$ The rate law generally takes the form: $Rate = k[A]^m[B]^n$ Where: * $k$ is the rate constant * $[A]$ and $[B]$ are the concentrations of reactants * $m$ and $n$ are the reaction orders with respect to A and B, respectively. The overall reaction order is $m + n$. ### Determining Rate Laws Rate laws must be determined experimentally. Common methods include: 1. **Method of Initial Rates**: Measuring the initial rate of a reaction for different initial concentrations of reactants. 2. **Integrated Rate Laws**: Analyzing concentration-time data to determine the order of the reaction. ### Common Rate Laws 1. **Zero-Order**: $Rate = k$ 2. **First-Order**: $Rate = k[A]$ 3. **Second-Order**: $Rate = k[A]^2$ or $Rate = k[A][B]$ ## Integrated Rate Laws ### Definition Integrated rate laws relate the concentration of reactants to time. ### Equations 1. **Zero-Order**: $[A]_t = -kt + [A]_0$ 2. **First-Order**: $ln[A]_t = -kt + ln[A]_0$ 3. **Second-Order**: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$ Where: * $[A]_t$ is the concentration of A at time t * $[A]_0$ is the initial concentration of A ### Half-Life The half-life ($t_{1/2}$) is the time required for the concentration of a reactant to decrease to one-half of its initial value. * **Zero-Order**: $t_{1/2} = \frac{[A]_0}{2k}$ * **First-Order**: $t_{1/2} = \frac{0.693}{k}$ * **Second-Order**: $t_{1/2} = \frac{1}{k[A]_0}$ ## Reaction Mechanisms ### Definition A reaction mechanism is the step-by-step sequence of elementary reactions by which an overall chemical change occurs. ### Elementary Reactions Elementary reactions are single-step reactions that cannot be broken down into simpler steps. The molecularity of an elementary reaction is the number of reactant molecules involved in the reaction (e.g., unimolecular, bimolecular). ### Rate-Determining Step The rate-determining step is the slowest step in the reaction mechanism and determines the overall rate of the reaction. ### Catalysis Catalysts provide an alternative reaction pathway with a lower activation energy, thereby increasing the reaction rate. Catalysts are not consumed in the reaction. * **Homogeneous Catalysis**: The catalyst is in the same phase as the reactants. * **Heterogeneous Catalysis**: The catalyst is in a different phase from the reactants. ## Temperature and Reaction Rate ### Arrhenius Equation The Arrhenius equation relates the rate constant $k$ to the temperature $T$: $k = Ae^{-E_a/RT}$ Where: * $A$ is the pre-exponential factor (frequency factor) * $E_a$ is the activation energy * $R$ is the gas constant ($8.314 J \cdot mol^{-1} \cdot K^{-1}$) * $T$ is the temperature in Kelvin ### Activation Energy Activation energy ($E_a$) is the minimum energy required for a reaction to occur. A higher activation energy corresponds to a slower reaction rate. ### Graphical Determination of $E_a$ Taking the natural logarithm of the Arrhenius equation: $ln(k) = -\frac{E_a}{R}\frac{1}{T} + ln(A)$ Plotting $ln(k)$ versus $\frac{1}{T}$ yields a straight line with a slope of $-\frac{E_a}{R}$, from which $E_a$ can be determined.