Reaction Rates & Rate Equations - Comprehensive Study Guide

Concentration versus Time and Reaction Kinetics

Plot of Concentration versus Time for the Reaction Between NO(g) and O2(g)

Collision Theory Foundation:
- In order for a chemical reaction to occur, reactant molecules must physically collide with each other.
- The frequency and energy of collisions dictate the rate at which reactants are transformed into products.
Concentration Changes over Time:
- In the gas-phase reaction $2NO(g) + O_2(g) \rightarrow 2NO_2(g)$ , concentrations of reactants ( $[NO]$ and $[O_2]$ ) decrease over time, while the concentration of product ( $[NO_2]$ ) increases.
- The rate of change for nitrogen monoxide ( $NO$ ) decreases from one time interval to the next, as shown by the flattening curve over time.
Relationship Between Reactant Concentration and Reaction Rate:
- As the reaction proceeds, the rate of reaction slows down directly in tandem with the decreasing concentration of reactants ( $NO$ or $O_2$ ).
- Lower concentration means fewer reactant particles per unit volume.
- Fewer particles per unit volume result in fewer collisions per unit time between reactant molecules, directly causing the reaction rate to decrease.

Writing Rate Equations and Determining Reaction Orders

General Rate Law Form:
- For a general reaction involving reactants $A$ and $B$ , the rate equation is expressed as: $\text{Rate} = k[A]^m[B]^n$
- $k$ is the reaction rate constant.
- $m$ is the reaction order with respect to reactant $A$ .
- $n$ is the reaction order with respect to reactant $B$ .
- The overall reaction order is the sum of individual reaction orders: $m + n$
Writing Rate Equations from Explicit Orders (Practice 4):
- Group Member 01 (GM01):
- Reaction order: Zero order in $A$ ( $m = 0$ ), First order in $B$ ( $n = 1$ ).
- Rate Equation: $\text{Rate} = k[A]^0[B]^1 = k[B]$
- Overall Reaction Order: $0 + 1 = 1$
- Group Member 02 (GM02):
- Reaction order: First order in $A$ ( $m = 1$ ), Second order in $B$ ( $n = 2$ ).
- Rate Equation: $\text{Rate} = k[A]^1[B]^2$
- Overall Reaction Order: $1 + 2 = 3$
- Group Member 03 (GM03):
- Reaction order: Second order in $A$ ( $m = 2$ ), First order in $B$ ( $n = 1$ ).
- Rate Equation: $\text{Rate} = k[A]^2[B]^1$
- Overall Reaction Order: $2 + 1 = 3$
- Group Member 04 (GM04):
- Reaction order: Second order in $A$ ( $m = 2$ ), Zero order in $B$ ( $n = 0$ ).
- Rate Equation: $\text{Rate} = k[A]^2[B]^0 = k[A]^2$
- Overall Reaction Order: $2 + 0 = 2$
Determining Orders from Concentration-Rate Changes (Practice 5):
- Group Member 01 (GM01):
- Behavior: Rate doubles when $[A]$ is doubled ( $2^x = 2 \Rightarrow x = 1$ ); Rate remains constant when $[B]$ is doubled ( $2^y = 1 \Rightarrow y = 0$ ).
- Rate Equation: $\text{Rate} = k[A]^1[B]^0$
- Overall Order: $1 + 0 = 1$
- Group Member 02 (GM02):
- Behavior: Rate quadruples when $[A]$ is doubled ( $2^x = 4 \Rightarrow x = 2$ ); Rate doubles when $[B]$ is doubled ( $2^y = 2 \Rightarrow y = 1$ ).
- Rate Equation: $\text{Rate} = k[A]^2[B]^1$
- Overall Order: $2 + 1 = 3$
- Group Member 03 (GM03):
- Behavior: Rate does not change when $[A]$ is doubled ( $2^x = 1 \Rightarrow x = 0$ ); Rate quadruples when $[B]$ is doubled ( $2^y = 4 \Rightarrow y = 2$ ).
- Rate Equation: $\text{Rate} = k[A]^0[B]^2$
- Overall Order: $0 + 2 = 2$
- Group Member 04 (GM04):
- Behavior: Rate doubles when $[A]$ is doubled ( $2^x = 2 \Rightarrow x = 1$ ); Rate doubles when $[B]$ is doubled ( $2^y = 2 \Rightarrow y = 1$ ).
- Rate Equation: $\text{Rate} = k[A]^1[B]^1$
- Overall Order: $1 + 1 = 2$

Method of Initial Rates: Reaction of Nitrogen Dioxide and Carbon Monoxide

Reaction Equation: $NO_2(g) + CO(g) \rightarrow NO(g) + CO_2(g)$
Experimental Data (Table 1):
- Experiment 1: $[NO_2] = 0.10\,M$ , $[CO] = 0.10\,M$ , Initial Rate = $0.0021\,M\,s^{-1}$
- Experiment 2: $[NO_2] = 0.20\,M$ , $[CO] = 0.10\,M$ , Initial Rate = $0.0082\,M\,s^{-1}$
- Experiment 3: $[NO_2] = 0.20\,M$ , $[CO] = 0.20\,M$ , Initial Rate = $0.0083\,M\,s^{-1}$
Derivation & Results (Practice 6):
- Order with respect to $NO_2$ : Comparing Exp 1 and Exp 2, $[NO_2]$ doubles while $[CO]$ is constant; rate increases by a factor of $\frac{0.0082}{0.0021} \approx 4 = 2^2$ . Thus, order with respect to $NO_2$ is $2$ .
- Order with respect to $CO$ : Comparing Exp 2 and Exp 3, $[CO]$ doubles while $[NO_2]$ is constant; rate remains unchanged ( $0.0082\,M\,s^{-1}$ to $0.0083\,M\,s^{-1}$ ). Thus, order with respect to $CO$ is 0$.\n * **Part a - Rate Law**: \text{Rate} = k[NO_2]^2[CO]^0\n * **Part b - Rate Constant (k)**:\n k = \frac{\text{Rate}}{[NO_2]^2} = \frac{0.0021\,M\,s^{-1}}{(0.10\,M)^2} = 0.21\,M^{-1}\,s^{-1}\n * **Part c - Overall Order**: 2 + 0 = 2\n\n# Method of Initial Rates: Reaction of Nitrogen Monoxide and Molecular Hydrogen\n\n* **Reaction Equation**:\n 2NO(g) + 2H_2(g) \rightarrow N_2(g) + 2H_2O(g)\n\n* **Group Member 01 Data & Results (GM01)**:\n * **Table 2 Data**:\n * Experiment 1: [NO] = 0.420\,M $,$ [H_2] = 0.122\,M $, Initial Rate =$ 0.136\,M\,s^{-1}\n * Experiment 2: [NO] = 0.210\,M $,$ [H_2] = 0.122\,M $, Initial Rate =$ 0.0339\,M\,s^{-1}\n * Experiment 3: [NO] = 0.210\,M $,$ [H_2] = 0.244\,M $, Initial Rate =$ 0.0678\,M\,s^{-1}\n * **Part a - Rate Law**: \text{Rate} = k[NO]^2[H_2]^1\n * **Part b - Rate Constant (k)**:\n k = \frac{0.136\,M\,s^{-1}}{(0.420\,M)^2(0.122\,M)} = 6.30\,M^{-2}\,s^{-1}\n * **Part c - Overall Order**: 2 + 1 = 3\n\n* **Group Member 02 Data & Results (GM02)**:\n * **Table 2 Data**:\n * Experiment 1: [NO] = 0.420\,M $,$ [H_2] = 0.122\,M $, Initial Rate =$ 0.0678\,M\,s^{-1}\n * Experiment 2: [NO] = 0.210\,M $,$ [H_2] = 0.122\,M $, Initial Rate =$ 0.0339\,M\,s^{-1}\n * Experiment 3: [NO] = 0.210\,M $,$ [H_2] = 0.244\,M $, Initial Rate =$ 0.1356\,M\,s^{-1}\n * **Part a - Rate Law**: \text{Rate} = k[NO]^1[H_2]^2\n * **Part b - Rate Constant (k)**:\n k = \frac{0.0678\,M\,s^{-1}}{(0.420\,M)^1(0.122\,M)^2} = 10.8\,M^{-2}\,s^{-1}\n * **Part c - Overall Order**: 1 + 2 = 3\n\n* **Group Member 03 Data & Results (GM03)**:\n * **Table 2 Data**:\n * Experiment 1: [NO] = 0.420\,M $,$ [H_2] = 0.122\,M $, Initial Rate =$ 0.1356\,M\,s^{-1}\n * Experiment 2: [NO] = 0.210\,M $,$ [H_2] = 0.122\,M $, Initial Rate =$ 0.0339\,M\,s^{-1}\n * Experiment 3: [NO] = 0.210\,M $,$ [H_2] = 0.244\,M $, Initial Rate =$ 0.0339\,M\,s^{-1}\n * **Part a - Rate Law**: \text{Rate} = k[NO]^2[H_2]^0\n * **Part b - Rate Constant (k)**:\n k = \frac{0.1356\,M\,s^{-1}}{(0.420\,M)^2} = 0.768\,M^{-1}\,s^{-1}\n * **Part c - Overall Order**: 2 + 0 = 2\n\n* **Group Member 04 Data & Results (GM04)**:\n * **Table 2 Data**:\n * Experiment 1: [NO] = 0.420\,M $,$ [H_2] = 0.122\,M $, Initial Rate =$ 0.0678\,M\,s^{-1}\n * Experiment 2: [NO] = 0.210\,M $,$ [H_2] = 0.122\,M $, Initial Rate =$ 0.0339\,M\,s^{-1}\n * Experiment 3: [NO] = 0.210\,M $,$ [H_2] = 0.244\,M $, Initial Rate =$ 0.0678\,M\,s^{-1}\n * **Part a - Rate Law**: \text{Rate} = k[NO]^1[H_2]^1\n * **Part b - Rate Constant (k)**:\n k = \frac{0.0678\,M\,s^{-1}}{(0.420\,M)(0.122\,M)} = 1.32\,M^{-1}\,s^{-1}\n * **Part c - Overall Order**: 1 + 1 = 2\n\n# Method of Initial Rates: Reaction of Unknown Compounds A and B\n\n* **Reaction Equation**:\n A(g) + 2B(g) \rightarrow C(g)\n\n* **Experimental Data (Table 1)**:\n * Experiment 1: [A] = 0.113\,M $,$ [B] = 0.115\,M $, Initial Rate =$ 0.0213\,M\,s^{-1}\n * Experiment 2: [A] = 0.339\,M $,$ [B] = 0.115\,M $, Initial Rate =$ 0.0639\,M\,s^{-1}\n * Experiment 3: [A] = 0.339\,M $,$ [B] = 0.288\,M $, Initial Rate =$ 0.3994\,M\,s^{-1}\n\n* **Derivation & Results (Practice 8)**:\n * **Order with respect to A $**: Comparing Exp 1 and Exp 2,$ [A] $triples ($ \frac{0.339}{0.113} = 3 $) while$ [B] $remains constant. Initial rate increases by a factor of 3 ($ \frac{0.0639}{0.0213} = 3 $). Therefore, the reaction is first order in$ A $($ m = 1).\n * **Order with respect to B $**: Comparing Exp 2 and Exp 3,$ [B] $increases by a factor of approximately$ 2.504 $($ \frac{0.288}{0.115} $) while$ [A] $is constant. Initial rate increases by a factor of approximately$ 6.25 $($ \frac{0.3994}{0.0639} $). Since$ 2.504^2 \approx 6.27 $, the reaction is second order in$ B $($ n = 2).\n * **Part a - Rate Law**: \text{Rate} = k[A]^1[B]^2\n * **Part b - Rate Constant (k)**:\n k = \frac{0.0213\,M\,s^{-1}}{(0.113\,M)^1(0.115\,M)^2} = 14.3\,M^{-2}\,s^{-1}\n * **Part c - Overall Order**: 1 + 2 = 3$$