# 5.12: Chemical Kinetics (Summary)

- Page ID
- 81888

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)These is a summary of key concepts of the chapter in the Textmap created for "Chemistry: The Central Science" by Brown et al.

## 14.1: Factors that Affect Reaction Rates

**chemical kinetics** – area of chemistry dealing with speeds/rates of reactions

- rates of reactions affected by four factors
- concentrations of reactants
- temperature at which reaction occurs
- presence of a catalyst
- surface area of solid or liquid reactants and/or catalysts

## 14.2: Reaction Rates

**reaction rate**– speed of a chemical reaction

\[\displaystyle \textit{average rate} = \frac{\textit{change #moles B}}{\textit{change in time}}= \frac{\Delta\textit{moles B}}{\Delta t}\textit{ if }A \to B \nonumber \]

\[\Delta\textit{moles B} = \textit{moles B at final time}- \textit{moles B at initial time} \nonumber \]

\[\displaystyle \textit{average rate} = -\frac{\Delta\textit{moles A}}{\Delta t}\textit{ if }A \to B \nonumber \]

### 14.2.1 Rates in Terms of Concentrations

- rate calculated in units of M/s
- brackets around a substance indicate the concentration
**instantaneous rate**– rate at a particular time- instantaneous rate obtained from the straight line tangent that touches the curve at a specific point
- slopes give instantaneous rates
- instantaneous rate also referred to as the rate

### 14.2.2 Reaction Rates and Stoichiometry

- for the irreversible reaction \(aA+bB\to cC+dD\)

\[\displaystyle\textit{rate} = -\frac{1}{a}\frac{\Delta [A]}{\Delta t} = -\frac{1}{b}\frac{\Delta [B]}{\Delta t} = \frac{1}{c}\frac{\Delta [C]}{\Delta t} = \frac{1}{d}\frac{\Delta [D]}{\Delta t} \nonumber \]

## 14.3: Concentration and Rate

- equation used only if C and D only substances formed
- Rate = k[A][B]
**Rate law**– expression that shows that rate depends on concentrations of reactants- k = rate constant

### 14.3.1 Reaction Order

- Rate = k[reactant 1]
^{m}[reactant 2]^{n} - m, n are called reaction orders
- m+n, overall reaction order
- reaction orders do not have to correspond with coefficients in balanced equation
- values of reaction order determined experimentally
- reaction order can be fractional or negative

### 14.3.2 Units of Rates Constants

- units of rate constant depend on overall reaction order of rate law
- for reaction of second order overall
- units of rate = (units of rate constant)(units of concentration)
^{2} - units of rate constant = M
^{-1}s^{-1}

### 14.3.3 Using Initial Rates to Determine Rate Laws

**zero order**– no change in rate when concentration changed**first order**– change in concentration gives proportional changes in rate**second order**– change in concentration changes rate by the square of the concentration change, such as 2^{2}or 3^{2}, etc…- rate constant does not depend on concentration

## 14.4: The Change of Concentration with Time

- rate laws can be converted into equations that give concentrations of reactants or products

### 14.4.1 First-Order Reactions

\[\textit{rate} = -\frac{\Delta [A]}{\Delta t} = k[A] \nonumber \]

and in integral form:

\[\ln[A]_t - \ln[A]_0 =-kt \nonumber \]

or

\[\ln\frac{[A]_t}{[A]_0} = -kt \nonumber \]

\[\ln[A]_t = - kt + \ln[A]_0 \nonumber \]

- corresponds to a straight line with \(y = mx + b\)
- equations used to determine:
- concentration of reactant remaining at any time
- time required for given fraction of sample to react
- time required for reactant concentration to reach a certain level

### 14.3.2 Half-Life

- half-life of first order reaction

\[\displaystyle t_{\frac{1}{2}} = -\frac{\ln\frac{1}{2}}{k} = \frac{0.693}{k} \nonumber \]

**half-life**– time required for concentration of reactant to drop to one-half of initial value- \(t_{1/2}\) of first order independent of initial concentrations
- half-life same at any given time of reaction
- in first order reaction – concentrations of reactant decreases by ½ in each series of regularly spaced time intervals

### 14.3.3 Second-Order Reactions

- rate depends on reactant concentration raised to second power or concentrations of two different reactants each raised to first power

\[\text{Rate} = k[A]^2 \nonumber \]

\[\displaystyle\frac{1}{[A]_t} = kt + \frac{1}{[A]_0} \nonumber \]

\[\displaystyle\textit{half life} = t_{\frac{1}{2}} = \frac{1}{k[A]_0} \nonumber \]

- half life dependent on initial concentration of reactant

## 14.5: Temperature and Rate

- rate constant must increase with increasing temperature, thus increasing the rate of reaction

### 14.5.1 The Collision Model

**collision model**– molecules must collide to react- greater frequency of collisions the greater the reaction rate
- for most reactions only a small fraction of collisions leads to a reaction

### 14.5.2 Activation Energy

- Svante August Arrhenius
- Molecules must have a minimum amount of energy to react
- Energy comes from kinetic energy of collisions
- Kinetic energy used to break bonds
- Activation energy, E
_{a}– minimum energy required to initiate a chemical reaction - Activated complex or transition state – atoms at the top of the energy barrier
- Rate depends on temperature and E
_{a} - Lower E
_{a}means faster reaction - Reactions occur when collisions between molecules occur with enough energy and proper orientation

### 14.5.3 The Arrhenius Equation

- reaction rate data:
- theArrhenius Equation:

\[\displaystyle k = A e^{\frac{-E_a}{RT}} \nonumber \]

- \(k\) = rate constant, \(E_a\) = activation energy, \(R\) = gas constant (8.314 J/(mol K)), \(T\) = absolute temperature, \(A\) = frequency factor
- \(A\) relates to frequency of collisions, favorable orientations

\[\displaystyle \ln k = -\frac{E_a}{RT} + \ln A \nonumber \]

- the \(\ln k\) vs. \(1/t\) graph (also known as an Arrhenius plot) has a slope \(–E_a/R\) and the y-intercept \(\ln A\)
- for two temperatures:

\[\displaystyle \ln \frac{k_1}{k_2} = \frac{E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \nonumber \]

- used to calculate rate constant, \(k_1\) and \(T_1\)

## 14.6: Reaction Mechanisms

**reaction mechanism**– process by which a reaction occurs

### 14.6.1 Elementary Steps

**elementary steps**– each step in a reaction**molecularity**– if only one molecule involved in step**unimolecular**– if only one molecule involved in step**bimolecular**– elementary step involving collision of two reactant molecules**termolecular**– elementary step involving simultaneous collision of three molecules- elementary steps in multi-step mechanism must always add to give chemical equation of overall process
**intermediate**– product formed in one step and consumed in a later step

### 14.6.2 Rate Laws of Elementary Steps

- if reaction is known to be an elementary step then the rate law is known
- rate of unimolecular step is first order (Rate = k[A])
- rate of bimolecular steps is second order (Rate = k[A][B])
- first order in [A] and [B]
- if double [A] than number of collisions of A and B will double

### 14.6.3 Rate Laws of Multi-step Mechanisms

**rate-determining step**– slowest elementary step- determines rate law of overall reaction

### 14.6.4 Mechanisms with an Initial Slow Step vs. Mechanisms with an Initial Fast Step

- intermediates are usually unstable, in low concentration, and difficult to isolate
- when a fast step precedes a slow one, solve for concentration of intermediate by assuming that equilibrium is established in fast step

## 14.7: Catalysis

**catalyst**– substance that changes speed of chemical reaction without undergoing a permanent chemical change

### 14.7.1 Homogeneous Catalysis

**homogeneous catalyst**– catalyst that is present in same phase as reacting molecule- catalysts alter E
_{a}or A - generally catalysts lowers overall E
_{a}for chemical reaction - catalysts provides a different mechanism for reaction

### 14.7.2 Heterogeneous Catalysis

- exists in different phase from reactants
- initial step in heterogeneous catalyst is adsorption
**adsorption**– binding of molecules to surface- adsorption occurs because ions/atoms at surface of solid extremely reactive

### 14.7.3 Enzymes

- biological catalysts
- large protein molecules with molecular weights 10,000 – 1 million amu
**catalase**– enzyme in blood and liver that decomposes hydrogen peroxide into water and oxygen**substrates**– substances that undergo reaction at the active site**lock-and-key model**– substrate molecules bind specifically to the active site**enzyme-substrate complex**– combination of enzyme and substrate- binding between enzyme and substrate involves intermolecular forces (dipole-dipole, hydrogen bonding, and London dispersion forces)
- product from reaction leaves enzyme allowing for another substrate to enter enzyme
**enzyme inhibitors**– molecules that bind strongly to enzymes**turnover number**– number of catalyzed reactions occurring at a particular active site- large turnover numbers = low activation energies