For the reaction
$$a\ce{A} +b\ce{B} \ce{->} c\ce{C} +d\ce{D}$$The reaction rate is
$$\begin{align}rate=-\frac{1}{a}\frac{\Delta [\ce{A}]}{\Delta t}=-\frac{1}{b}\frac{\Delta [\ce{B}]}{\Delta t}=\frac{1}{c}\frac{\Delta [\ce{C}]}{\Delta t}=\frac{1}{d}\frac{\Delta [\ce{D}]}{\Delta t}\end{align}$$Requirements
- Write the rate expressions for given reactions in terms of the disappearance of the reactants and the appearance of the products. Or write a balanced equation for a reaction whose rate is given.
- Knowing the consuming rate of a reactant (or the generating rate of a product), tell the rate of the consentration changes of other reactants and products.
The rate law expresses the relationship of the rate of a reaction to the rate constant ($k$) and the concentrations of the reactants raised to some powers.
$$a\ce{A} +b\ce{B} \ce{->} c\ce{C} +d\ce{D}$$$$\begin{align}rate=k[\ce{A}]^x[\ce{B}]^y\end{align}$$$x$ and $y$ are NOT related with $a$ and $b$. The values of $x$ and $y$ are determined experimentally.
$k$ depends on the nature of the reactants and the temperature, but does not depend on the concentrations of the reactants.
The unit of $k$ is to keep the unit balanced on both sides.
| Trial | [$\ce{NO}$] (mol/L) | [$\ce{Cl2}$] (mol/L) | $-\frac{\Delta[\ce{NO}]}{\Delta t}$ (mol L$^{-1}$ s$^{-1}$) |
|---|---|---|---|
| 1 | 0.10 | 0.10 | 0.00300 |
| 2 | 0.10 | 0.15 | 0.00450 |
| 3 | 0.15 | 0.10 | 0.00675 |
Requirements
- Determine the values of $x$, $y$ and $k$ from experimental data.
A first-order reaction is a reaction whose rate depends on the reactant concentration raised to the first power.
$$\ce{A}\ce{->} \ce{B} $$
$$\begin{align}
& rate = k[\ce{A}] \\
& rate =-\frac{\Delta[\ce{A}]}{\Delta t} \\
& k[\ce{A}] =-\frac{\Delta[\ce{A}]}{\Delta t} \\
& k=\frac{rate}{[\ce{A}]}=\frac{\text{M/s}}{\text{M}}=\text{1/s}\ \text{or}\ \text{s}^{-1} \\
\end{align}$$
The concentration of $\ce{A}$ and time $t$ can be correlated by the following equations.
$$\begin{align} & \ln\frac{[\ce{A}]_t}{[\ce{A}]_0}=-kt \\ & \ln [\ce{A}]_t=-kt+ \ln [\ce{A}]_0 \end{align}$$where $[\ce{A}]_t$ is the concentration of $\ce{A}$ at time $t$; $[\ce{A}]_0$ is the concentration of $\ce{A}$ at time $0$.
Half-life
The half-life of a reaction, $t_{1/2}$, is the time requied for the concentration of a reactant to decrease to half of its initial concentration.
Half-life of a first-order reaction is
$$\begin{align}t_{1/2}=\frac{\ln 2}{k}\end{align}$$
Requirements
- Calculate half-life of a first-order reaction from known rate constant.
- Calculate $k$ from the plot of $\ln [\ce{A}]$ vs $t$.
- Knowing $k$ and $[\ce{A}]_0$, calculate the concentration of $\ce{A}$ after time $t$; or knowing $k$, $[\ce{A}]_0$ and $[\ce{A}]_t$, calculate the time $t$.
A second-order reaction is a reaction whose rate depends on the concentration of one reactant raised to the second power or on the concentrations of two different reactants, each raised to the first power.
$$\ce{A}\ce{->} \ce{B} \\$$$$\begin{align} & rate = k[\ce{A}]^2 \\ & rate =-\frac{\Delta[\ce{A}]}{\Delta t} \\ & k[\ce{A}]^2 =-\frac{\Delta[\ce{A}]}{\Delta t} \\ & k=\frac{rate}{[\ce{A}]^2}=\frac{\text{M/s}}{\text{M}^2}=\text{1/M}\cdot\text{s}\ \text{or}\ \text{M}^{-1}\text{s}^{-1} \\ \end{align}$$The concentration of $\ce{A}$ and time $t$ can be correlated by the following equation.
$$\begin{align} \frac{1}{[\ce{A}]_t}=\frac{1}{[\ce{A}]_0}+kt \end{align}$$where $[\ce{A}]_t$ is the concentration of $\ce{A}$ at time $t$; $[\ce{A}]_0$ is the concentration of $\ce{A}$ at time $0$.
The half-life of a second-order reaction is
$$\begin{align} t_{1/2}=\frac{1}{k[\ce{A}]_0} \end{align}$$Requirements
- Calculate $k$ from the plot of $\frac{1}{[\ce{A}]}$ vs $t$.
- Calculate half-life of a second-order reaction from known rate constant and initial concentration.
- Knowing $k$ and $[\ce{A}]_0$, calculate the concentration of $\ce{A}$ after time $t$; or knowing $k$, $[\ce{A}]_0$ and $[\ce{A}]_t$, calculate the time $t$.
The concentration of $\ce{A}$ and time $t$ can be correlated by the following equation.
$$\begin{align} [\ce{A}]_t=[\ce{A}]_0-kt \end{align}$$where $[\ce{A}]_t$ is the concentration of $\ce{A}$ at time $t$; $[\ce{A}]_0$ is the concentration of $\ce{A}$ at time $0$.
The half-life of a second-order reaction is
$$\begin{align} t_{1/2}=\frac{[\ce{A}]_0}{2k} \end{align}$$| Zero-Order | First-Order | Second-Order | |
|---|---|---|---|
| rate law | $\text{rate}=k$ | $\text{rate}=k[\ce{A}]$ | $\text{rate}=k[\ce{A}]^2$ |
| units of rate constant | $\text{M s}^{-1}$ | $\text{s}^{-1}$ | $\text{M}^{-1}\text{ s}^{-1}$ |
| integrated rate law | $[\ce{A}]=[\ce{A}]_0-kt$ | $\ln[\ce{A}]=\ln[\ce{A}]_0+kt$ | $\frac{1}{[\ce{A}]}=\frac{1}{[\ce{A}]_0}+kt$ |
| half-life | $t_{1/2}=\frac{[\ce{A}]_0}{2k}$ | $t_{1/2}=\frac{\ln 2}{k}$ | $t_{1/2}=\frac{1}{[\ce{A}]_0 k}$ |
Requirements
- Understand what is a zero-order reaction.
Collision Theory
Activiation Energy is the minimum amount of energy required to initiate a chemical reaction.
Transition State refers to the species temporarily formed by the reactant molecules as a result of the collision before they form the product.
Arrhenius Equation
$$\begin{align}
& k=Ae^{-E_a/RT} \\
& \ln k = \ln A - \frac{E_a}{RT} \\
\end{align}$$
Alternate form of Arrhenius Equation $$\ln\frac{k_1}{k_2}=\frac{E_a}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$$
Requirements
- Understand collision theory; know what is activation energy and what is transition state.
- Calculate rate constans at different temperatures using Arrhenius Equation.
# Knowing k at one temperature, calculate k at another temperature. Ea is given.
from numpy import*
Ea = 110 # in kJ/mol
T1=27 # in Celsius
T2=42 # in Celsius
k1=3.27e-6
k2=k1/exp(Ea*1000/8.314*(1/(T2+273)-1/(T1+273)))
k2
2.6706322527649514e-05
# Knowing A, Ea and T, calculate k
A = 7500000
Ea = 33.1 # in kJ/mol
T=25 # in Celsius
k=A*exp(-Ea/0.008314/(T+273))
k
11.829065218689623
Elementary Reactions are a series of simple reactions that represent the progress of the overall reaction at the molecular levle.
Reaction Mechanism is the sequence of elementary reactions that leads to product formation.
An Example from the Textbook:
The overall reaction
$$\ce{2NO(g) + O2(g) -> 2NO2(g)}$$
is actually a sum of two elementary reactions
$$\ce{NO +NO -> N2O2}$$
$$\ce{N2O2 + O2 -> 2NO2}$$
The sum of the elementary steps must give the overall balanced equation for the reaction.
Intermediats (are not the same as transition states) are formed in early elementary steps and consumed in later elementary steps. Thus intermediats do not appear in the overall ballanced equation.
molecularity of a reaction is the number of molecules reacting in an elementary step, such as unimolecular reaction, bimolecular reaction and termolecular reaction.
Rate Laws of Elementary Reactions
$$a\ce{A} +b\ce{B} \ce{->} c\ce{C} +d\ce{D}$$
If this is an elementary reaction, then
$$\begin{align}rate=k[\ce{A}]^a[\ce{B}]^b\end{align}$$
Since an elementary reaction usually involves only one or two molecules, there are only a few possible types of elementary reactions, which are:
\begin{align*} \ce{A -> B}\qquad & \text{rate}=k[\ce{A}]\\ \ce{2A -> B}\qquad & \text{rate}=k[\ce{A}]^2\\ \ce{A + B -> C}\qquad & \text{rate}=k[\ce{A}][\ce{B}] \end{align*}Termolecular reaction is possible, but is rare. Kinetics of termolecular reactions is not discussed here.
Rate-determining step is the slowest step in the sequence of steps leading to product formation.
The rate-determining step should predict the same rate law as is determined experimentally.
Potential energy profile for a multi-step reaction
Requirements
- Given elementary steps, identify the intermediates and write the overall balanced reaction equation.
- Knowing the rate-determining step of a reaction, tell the rate law of the overall reaction. Or knowing the rate law of a reaction and the elementary steps, tell which is the rate-determining step.
- Given the potential energy profile of a multi-step reaction, tell whether the overal reaction is exothermic or endothermic, the number of elementary steps, the intermediate(s), the transition states, the activation energies of each step, the rate-determining step.
Catalyst is a substance that increases the rate of a chemical reaction without itself undergoing any permanent chemical change. A catalyst lowers the activation energy for both the forward and reverse reactions.
Three types of catalysis: heterogeneous catalysis, homogeneous catalysis, and enzyme catalysis.
Requirements
- Know what is a catalyst and its role in a reaction.
- Know the three types of catalysis.
| Trial | [$\ce{NO}$] (mol/L) | [$\ce{H2}$] (mol/L) | $-\frac{\Delta[\ce{NO}]}{\Delta t}$ (mol L$^{-1}$ s$^{-1}$) |
|---|---|---|---|
| 1 | $5.0\times10^{-3}$ | $2.0\times10^{-3}$ | $1.25\times10^{-5}$ |
| 2 | $10.0\times10^{-3}$ | $2.0\times10^{-3}$ | $5.0\times10^{-5}$ |
| 3 | $10.0\times10^{-3}$ | $4.0\times10^{-3}$ | $10.0\times10^{-5}$ |
from IPython.core.display import HTML
def css_styling():
styles = open("custom.css", "r").read()
return HTML(styles)
css_styling()