Rate Of A Chemical Reaction

Chemical kinetics is the branch of chemistry that studies the rates of chemical reactions, the factors influencing these rates, and their mechanisms.

While thermodynamics predicts the feasibility ($\Delta \textsf{G} < 0$) and extent (equilibrium constant) of a reaction, it does not provide information about the speed at which the reaction occurs.

Chemical reactions vary greatly in their speed:

• Very fast reactions: Occur almost instantaneously (e.g., ionic reactions like precipitation of $\textsf{AgCl}$ upon mixing aqueous $\textsf{AgNO}_3$ and $\textsf{NaCl}$).
• Very slow reactions: Take a long time to complete (e.g., rusting of iron).
• Moderate speed reactions: Proceed at a measurable rate (e.g., inversion of cane sugar, hydrolysis of starch).

The rate of a chemical reaction is defined as the change in concentration of a reactant or product per unit time. It can be expressed in two ways:

1. Rate of decrease in the concentration of any one of the reactants.
2. Rate of increase in the concentration of any one of the products.

Consider a simple hypothetical reaction where one mole of reactant R converts to one mole of product P:

$\textsf{R} \to \textsf{P}$

If $[\textsf{R}]_1$ and $[\textsf{P}]_1$ are concentrations at time $\textsf{t}_1$, and $[\textsf{R}]_2$ and $[\textsf{P}]_2$ are concentrations at time $\textsf{t}_2$, where $\Delta \textsf{t} = \textsf{t}_2 - \textsf{t}_1$:

Change in concentration of R = $\Delta [\textsf{R}] = [\textsf{R}]_2 - [\textsf{R}]_1$ (Note: $\Delta [\textsf{R}]$ is negative as reactant concentration decreases).

Change in concentration of P = $\Delta [\textsf{P}] = [\textsf{P}]_2 - [\textsf{P}]_1$ (Note: $\Delta [\textsf{P}]$ is positive as product concentration increases).

The average rate of reaction over the time interval $\Delta \textsf{t}$ (denoted as $\textsf{r}_{\text{av}}$) is:

$\textsf{r}_{\text{av}} = \textsf{Rate of disappearance of R} = -\frac{\Delta [\textsf{R}]}{\Delta \textsf{t}}$

$\textsf{r}_{\text{av}} = \textsf{Rate of appearance of P} = +\frac{\Delta [\textsf{P}]}{\Delta \textsf{t}}$

The negative sign for the reactant rate ensures that the rate is always a positive quantity. The average rate is calculated over a specific time interval.

The units of the rate of reaction are typically concentration per unit time. Common units are mol L$^{-1}$ s$^{-1}$ or mol L$^{-1}$ min$^{-1}$. For gaseous reactions, where concentration is often expressed as partial pressure, units can be atm s$^{-1}$ or bar s$^{-1}$.

Answer:

The instantaneous rate of reaction ($\textsf{r}_{\text{inst}}$) is the rate at a specific moment in time. It is determined by taking the average rate over an infinitesimally small time interval $\text{dt}$ (i.e., as $\Delta \text{t} \to 0$).

$\textsf{r}_{\text{inst}} = \lim_{\Delta \text{t} \to 0} -\frac{\Delta [\textsf{R}]}{\Delta \text{t}} = -\frac{\text{d}[\textsf{R}]}{\text{dt}}$

$\textsf{r}_{\text{inst}} = \lim_{\Delta \text{t} \to 0} +\frac{\Delta [\textsf{P}]}{\Delta \text{t}} = +\frac{\text{d}[\textsf{P}]}{\text{dt}}$

Graphically, the instantaneous rate at time 't' is the slope of the tangent drawn to the concentration vs time curve at that specific time point (Fig. 4.1).

For reactions where the stoichiometric coefficients are not all equal to one, the rate of disappearance or appearance of each species is divided by its respective stoichiometric coefficient to define the overall rate of reaction.

Consider the reaction: $\textsf{2HI(g)} \to \textsf{H}_2\text{(g)} + \textsf{I}_2\text{(g)}$

Rate of reaction $= -\frac{1}{2}\frac{\Delta [\textsf{HI}]}{\Delta \text{t}} = +\frac{\Delta [\textsf{H}_2]}{\Delta \text{t}} = +\frac{\Delta [\textsf{I}_2]}{\Delta \text{t}}$ (for average rate)

Rate of reaction $= -\frac{1}{2}\frac{\text{d}[\textsf{HI}]}{\text{dt}} = +\frac{\text{d}[\textsf{H}_2]}{\text{dt}} = +\frac{\text{d}[\textsf{I}_2]}{\text{dt}}$ (for instantaneous rate)

In general, for a reaction: $\textsf{aA} + \textsf{bB} \to \textsf{cC} + \textsf{dD}$

Rate $= -\frac{1}{\textsf{a}}\frac{\text{d}[\textsf{A}]}{\text{dt}} = -\frac{1}{\textsf{b}}\frac{\text{d}[\textsf{B}]}{\text{dt}} = +\frac{1}{\textsf{c}}\frac{\text{d}[\textsf{C}]}{\text{dt}} = +\frac{1}{\textsf{d}}\frac{\text{d}[\textsf{D}]}{\text{dt}}$

For gaseous reactions at constant temperature, concentration is proportional to partial pressure. Thus, rate can also be expressed in terms of partial pressures:

Rate $= -\frac{1}{\textsf{a}}\frac{\text{dp}_\text{A}}{\text{dt}} = -\frac{1}{\textsf{b}}\frac{\text{dp}_\text{B}}{\text{dt}} = +\frac{1}{\textsf{c}}\frac{\text{dp}_\text{C}}{\text{dt}} = +\frac{1}{\textsf{d}}\frac{\text{dp}_\text{D}}{\text{dt}}$

Answer:

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Answer:

Factors Influencing Rate Of A Reaction

The rate of a chemical reaction is influenced by several experimental conditions:

• Concentration of reactants: Higher concentration generally leads to faster reaction.
• Temperature: Rate usually increases with increasing temperature.
• Pressure: Significant for reactions involving gaseous reactants. Higher pressure means higher concentration.
• Catalyst: A substance that increases the rate without being consumed.
• Surface area of reactants: Important for heterogeneous reactions (e.g., solid reactant with liquid/gas). Larger surface area means faster rate.
• Presence of light: Some reactions are photochemical, initiated or accelerated by light.

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Dependence Of Rate On Concentration

Experimental studies show that the rate of a reaction at a given temperature depends on the concentration of reactants. As reactant concentrations decrease over time, the reaction rate typically slows down.

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Rate Expression And Rate Constant

The representation of the rate of a reaction in terms of the concentrations of reactants is called the rate law or rate expression or rate equation.

For a general reaction: $\textsf{aA} + \textsf{bB} \to \textsf{cC} + \textsf{dD}$

The rate law is expressed as:

$\textsf{Rate} \propto [\textsf{A}]^{\text{x}} [\textsf{B}]^{\text{y}}$

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

where $\textsf{k}$ is the proportionality constant called the rate constant (or rate coefficient). The exponents $\textsf{x}$ and $\textsf{y}$ are determined experimentally and may or may not be equal to the stoichiometric coefficients $\textsf{a}$ and $\textsf{b}$.

The rate law is the expression that relates the reaction rate to the molar concentrations of reactants, with each concentration term raised to some power determined experimentally.

The differential rate equation for the reaction is:

$-\frac{\text{d}[\textsf{R}]}{\text{dt}} = \textsf{k} [\textsf{A}]^{\text{x}} [\textsf{B}]^{\text{y}}$ (where R is a reactant whose rate is being expressed).

Example: Reaction $2\textsf{NO(g)} + \textsf{O}_2\text{(g)} \to 2\textsf{NO}_2\text{(g)}$. Experimental data (Table 4.2) shows:

• Doubling $[\textsf{NO}]$ while keeping $[\textsf{O}_2]$ constant quadruples the rate. This suggests Rate $\propto [\textsf{NO}]^2$.
• Doubling $[\textsf{O}_2]$ while keeping $[\textsf{NO}]$ constant doubles the rate. This suggests Rate $\propto [\textsf{O}_2]^1$.

The experimental rate law is: Rate $= \textsf{k} [\textsf{NO}]^2 [\textsf{O}_2]^1$.

In this specific case, the exponents match the stoichiometric coefficients. However, this is not always true.

Examples where exponents do not match stoichiometric coefficients:

• $\textsf{CHCl}_3 + \textsf{Cl}_2 \to \textsf{CCl}_4 + \textsf{HCl}$; Rate $= \textsf{k} [\textsf{CHCl}_3] [\textsf{Cl}_2]^{1/2}$
• $\textsf{CH}_3\text{COOC}_2\text{H}_5 + \textsf{H}_2\text{O} \to \textsf{CH}_3\text{COOH} + \textsf{C}_2\text{H}_5\text{OH}$; Rate $= \textsf{k} [\textsf{CH}_3\text{COOC}_2\text{H}_5]^1 [\textsf{H}_2\text{O}]^0$ (If water is in large excess, as in pseudo first order reactions).

Therefore, the rate law and the exponents must always be determined experimentally.

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Order Of A Reaction

In the rate law expression, Rate $= \textsf{k} [\textsf{A}]^{\text{x}} [\textsf{B}]^{\text{y}}$, the exponents $\textsf{x}$ and $\textsf{y}$ represent the order of the reaction with respect to reactant A and order with respect to reactant B, respectively. The sum of these exponents ($\textsf{x} + \textsf{y}$) is the overall order of the reaction.

The order of a reaction is the sum of the powers of the concentration terms of the reactants in the rate law expression.

Order is an experimental quantity and can be an integer (0, 1, 2, 3) or a fraction.

• Zero order reaction: Rate is independent of reactant concentration (exponent is 0). Rate $= \textsf{k} [\textsf{A}]^0 = \textsf{k}$.
• First order reaction: Overall order is 1 (sum of exponents is 1).
• Second order reaction: Overall order is 2 (sum of exponents is 2).

Answer:

Units of rate constant (k): From the rate law Rate $= \textsf{k} [\textsf{A}]^{\text{x}} [\textsf{B}]^{\text{y}}$, the units of $\textsf{k}$ depend on the overall order (n = $\textsf{x} + \textsf{y}$).

$\textsf{k} = \frac{\textsf{Rate}}{[\textsf{A}]^{\text{x}} [\textsf{B}]^{\text{y}}} = \frac{\text{Concentration/Time}}{(\text{Concentration})^{\text{n}}} = (\text{Concentration})^{1-\text{n}} \text{ Time}^{-1}$

Using typical units (mol L$^{-1}$ for concentration, s for time):

Reaction Order (n)	Units of Rate Constant (k)
0 (Zero order)	(mol L$^{-1}$)$^{1-0}$ s$^{-1}$ = mol L$^{-1}$ s$^{-1}$
1 (First order)	(mol L$^{-1}$)$^{1-1}$ s$^{-1}$ = (mol L$^{-1}$)$^0$ s$^{-1}$ = s$^{-1}$
2 (Second order)	(mol L$^{-1}$)$^{1-2}$ s$^{-1}$ = (mol L$^{-1}$)$^{-1}$ s$^{-1}$ = L mol$^{-1}$ s$^{-1}$
3 (Third order)	(mol L$^{-1}$)$^{1-3}$ s$^{-1}$ = (mol L$^{-1}$)$^{-2}$ s$^{-1}$ = L$^2$ mol$^{-2}$ s$^{-1}$
n (nth order)	(mol L$^{-1}$)$^{1-\text{n}}$ s$^{-1}$

Answer:

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Molecularity Of A Reaction

The molecularity of a reaction is the number of reacting species (atoms, ions, or molecules) that collide simultaneously in a single step of a chemical reaction to bring about the reaction.

• Elementary reactions: Reactions that occur in one single step. Molecularity is defined only for elementary reactions.
• Complex reactions: Reactions that occur in a sequence of multiple elementary steps. For complex reactions, molecularity of individual steps is defined, but the overall molecularity of the complex reaction is not meaningful.

Based on molecularity, elementary reactions are classified:

• Unimolecular: One reacting species involved in the collision (e.g., decomposition of ammonium nitrite $\textsf{NH}_4\text{NO}_2 \to \textsf{N}_2 + 2\textsf{H}_2\text{O}$).
• Bimolecular: Two reacting species collide simultaneously (e.g., dissociation of hydrogen iodide $2\textsf{HI} \to \textsf{H}_2 + \textsf{I}_2$, or combination of two species $\textsf{A} + \textsf{B} \to \textsf{P}$).
• Trimolecular (or Termolecular): Three reacting species collide simultaneously (e.g., $2\textsf{NO} + \textsf{O}_2 \to 2\textsf{NO}_2$). Trimolecular collisions are rare because the probability of three particles colliding at the exact same time with proper orientation is very low.

Reactions involving more than three molecules simultaneously are extremely rare. Complex reactions whose overall stoichiometric equation suggests a high number of reacting molecules actually proceed through a series of elementary steps.

For complex reactions, the overall rate is determined by the rate of the slowest step, called the rate-determining step.

Example: Decomposition of hydrogen peroxide catalysed by iodide ion ($ \textsf{I}^{-}$) in alkaline medium: $2\textsf{H}_2\text{O}_2 \xrightarrow{\text{I}^-} 2\textsf{H}_2\text{O} + \textsf{O}_2$. The experimentally determined rate law is Rate $= \textsf{k} [\textsf{H}_2\text{O}_2] [\textsf{I}^{-}]$. The order is 1 with respect to $\textsf{H}_2\text{O}_2$ and 1 with respect to $\textsf{I}^{-}$, for an overall second order (1+1=2).

The proposed mechanism involves two elementary bimolecular steps:

1. $\textsf{H}_2\text{O}_2 + \textsf{I}^{-} \to \textsf{H}_2\text{O} + \textsf{IO}^{-}$ (slow step, molecularity = 2)

2. $\textsf{H}_2\text{O}_2 + \textsf{IO}^{-} \to \textsf{H}_2\text{O} + \textsf{I}^{-} + \textsf{O}_2$ (fast step, molecularity = 2)

$\textsf{IO}^{-}$ is an intermediate. The first step is the slow, rate-determining step. The rate law is determined by the stoichiometry of the rate-determining elementary step, consistent with the experimental rate law.

Summary of distinctions between Order and Molecularity:

Feature	Order of Reaction	Molecularity of Reaction
Definition	Sum of powers of concentration terms in the experimentally determined rate law.	Number of reacting species colliding simultaneously in an elementary reaction step.
Determination	Determined experimentally; cannot be predicted from the balanced equation.	Determined theoretically by counting reacting species in an elementary step; applies only to elementary reactions.
Value	Can be 0, 1, 2, 3, or a fraction.	Must be a whole number (1, 2, or 3); cannot be zero or a fraction.
Applicability	Applicable to elementary and complex reactions.	Applicable only to elementary reactions.
For Complex Reactions	Order is determined by the slowest (rate-determining) step.	Molecularity has no meaning for the overall complex reaction; it applies to individual elementary steps. The molecularity of the slowest step is often equal to the overall order.

Answer:

Integrated Rate Equations

The differential rate equations express the instantaneous rate as a function of concentration. However, it is often more convenient to work with concentrations measured at different times. Integrated rate equations provide a relationship between reactant concentrations at different times and the rate constant (k). These equations are derived by integrating the differential rate equations.

The form of the integrated rate equation depends on the order of the reaction. We will look at integrated rate equations for zero and first order reactions.

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Zero Order Reactions

For a zero order reaction, the rate is proportional to the zero power of the reactant concentration. Consider the reaction: R $\to$ P.

Differential rate law: Rate $= -\frac{\text{d}[\textsf{R}]}{\text{dt}} = \textsf{k} [\textsf{R}]^0 = \textsf{k}$

Rearranging: $\text{d}[\textsf{R}] = -\textsf{k} \text{ dt}$.

Integrating both sides: $\int \text{d}[\textsf{R}] = \int -\textsf{k} \text{ dt}$

$[\textsf{R}] = -\textsf{k} \text{t} + \text{I}$ (where I is the integration constant).

To find I, use the initial condition: at time $\text{t} = 0$, the concentration of R is the initial concentration, $[\textsf{R}] = [\textsf{R}]_0$.

$[\textsf{R}]_0 = -\textsf{k} (0) + \text{I} \implies \text{I} = [\textsf{R}]_0$

Substitute I back into the integrated equation:

$[\textsf{R}] = -\textsf{k} \text{t} + [\textsf{R}]_0$

This is the integrated rate equation for a zero order reaction. It has the form of a straight line ($\textsf{y} = \textsf{mx} + \textsf{c}$). If we plot $[\textsf{R}]$ (y-axis) against $\text{t}$ (x-axis), we get a straight line with slope $= -\textsf{k}$ and y-intercept $= [\textsf{R}]_0$.

From the integrated rate equation, the rate constant $\textsf{k}$ can be determined:

$\textsf{k} = \frac{[\textsf{R}]_0 - [\textsf{R}]}{\text{t}}$

Examples of zero order reactions include some enzyme-catalysed reactions and reactions occurring on saturated metal surfaces (where the rate is limited by the available surface area rather than concentration), such as the decomposition of ammonia on hot platinum at high pressure.

$2\textsf{NH}_3\text{(g)} \xrightarrow{\text{Pt catalyst}} \textsf{N}_2\text{(g)} + 3\textsf{H}_2\text{(g)}$ ; Rate $= \textsf{k} [\textsf{NH}_3]^0 = \textsf{k}$.

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First Order Reactions

For a first order reaction, the rate is proportional to the first power of the reactant concentration. Consider the reaction: R $\to$ P.

Differential rate law: Rate $= -\frac{\text{d}[\textsf{R}]}{\text{dt}} = \textsf{k} [\textsf{R}]^1 = \textsf{k} [\textsf{R}]$

Rearranging: $\frac{\text{d}[\textsf{R}]}{[\textsf{R}]} = -\textsf{k} \text{ dt}$.

Integrating both sides: $\int \frac{\text{d}[\textsf{R}]}{[\textsf{R}]} = \int -\textsf{k} \text{ dt}$

$\ln [\textsf{R}] = -\textsf{k} \text{t} + \text{I}$ (where I is the integration constant).

To find I, use the initial condition: at time $\text{t} = 0$, $[\textsf{R}] = [\textsf{R}]_0$.

$\ln [\textsf{R}]_0 = -\textsf{k} (0) + \text{I} \implies \text{I} = \ln [\textsf{R}]_0$

Substitute I back into the integrated equation:

$\ln [\textsf{R}] = -\textsf{k} \text{t} + \ln [\textsf{R}]_0$

This is one form of the integrated rate equation for a first order reaction. It also has the form of a straight line ($\textsf{y} = \textsf{mx} + \textsf{c}$). If we plot $\ln [\textsf{R}]$ (y-axis) against $\text{t}$ (x-axis), we get a straight line with slope $= -\textsf{k}$ and y-intercept $= \ln [\textsf{R}]_0$.

Rearranging the equation $\ln [\textsf{R}] = -\textsf{k} \text{t} + \ln [\textsf{R}]_0$ gives:

$\ln [\textsf{R}] - \ln [\textsf{R}]_0 = -\textsf{k} \text{t}$

$\ln \frac{[\textsf{R}]}{[\textsf{R}]_0} = -\textsf{k} \text{t}$

$\ln \frac{[\textsf{R}]_0}{[\textsf{R}]} = \textsf{k} \text{t}$

From this, the rate constant $\textsf{k}$ is:

$\textsf{k} = \frac{1}{\text{t}} \ln \frac{[\textsf{R}]_0}{[\textsf{R}]}$

Converting from natural logarithm ($\ln$) to base 10 logarithm ($\log$): $\ln x = 2.303 \log x$.

$\textsf{k} = \frac{2.303}{\text{t}} \log \frac{[\textsf{R}]_0}{[\textsf{R}]}$

This is another common form of the integrated rate equation for a first order reaction. If we plot $\log \frac{[\textsf{R}]_0}{[\textsf{R}]}$ (y-axis) against $\text{t}$ (x-axis), we get a straight line passing through the origin with slope $= \textsf{k}/2.303$.

The equation $\ln [\textsf{R}] = -\textsf{k} \text{t} + \ln [\textsf{R}]_0$ can also be written in exponential form:

$[\textsf{R}] = [\textsf{R}]_0 \text{e}^{-\textsf{kt}}$

Examples of first order reactions: Hydrogenation of ethene (Rate $= \textsf{k} [\textsf{C}_2\text{H}_4]$), radioactive decay of unstable nuclei (always first order), decomposition of $\textsf{N}_2\text{O}_5$ and $\textsf{N}_2\text{O}$.

Answer:

Integrated rate equation for a first order gas phase reaction $\textsf{A(g)} \to \textsf{B(g)} + \textsf{C(g)}$ in terms of partial pressures:

Let $\textsf{p}_\text{i}$ be the initial pressure of A at $\textsf{t} = 0$. Let $\textsf{p}_\text{t}$ be the total pressure at time $\text{t}$.

Reaction: $\textsf{A(g)} \to \textsf{B(g)} + \textsf{C(g)}$

Initial pressure (t=0): $\textsf{p}_\text{i}$ 0 0

Pressure at time t: $\textsf{p}_\text{A}$ $\textsf{p}_\text{B}$ $\textsf{p}_\text{C}$

Total pressure at time t: $\textsf{p}_\text{t} = \textsf{p}_\text{A} + \textsf{p}_\text{B} + \textsf{p}_\text{C}$.

If x is the decrease in pressure of A at time t, then by stoichiometry, B and C pressures increase by x.

Initial pressure (t=0): $\textsf{p}_\text{i}$ 0 0

Pressure at time t: $(\textsf{p}_\text{i}-\textsf{x})$ $\textsf{x}$ $\textsf{x}$

Total pressure $\textsf{p}_\text{t} = (\textsf{p}_\text{i}-\textsf{x}) + \textsf{x} + \textsf{x} = \textsf{p}_\text{i} + \textsf{x}$.

So, $\textsf{x} = \textsf{p}_\text{t} - \textsf{p}_\text{i}$.

The pressure of reactant A at time t is $\textsf{p}_\text{A} = \textsf{p}_\text{i} - \textsf{x} = \textsf{p}_\text{i} - (\textsf{p}_\text{t} - \textsf{p}_\text{i}) = 2\textsf{p}_\text{i} - \textsf{p}_\text{t}$.

For a first order reaction, the rate constant $\textsf{k} = \frac{2.303}{\text{t}} \log \frac{[\textsf{R}]_0}{[\textsf{R}]}$. Using partial pressures (which are proportional to concentrations for gases at constant temperature):

$\textsf{k} = \frac{2.303}{\text{t}} \log \frac{\textsf{p}_\text{initial, A}}{\textsf{p}_\text{t, A}} = \frac{2.303}{\text{t}} \log \frac{\textsf{p}_\text{i}}{(2\textsf{p}_\text{i} - \textsf{p}_\text{t})}$

Answer:

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Half-Life Of A Reaction

The half-life ($t_{1/2}$) of a reaction is the time required for the concentration of a reactant to be reduced to one half of its initial concentration.

For a zero order reaction: Integrated rate law is $[\textsf{R}] = -\textsf{kt} + [\textsf{R}]_0$.

At $t = t_{1/2}$, $[\textsf{R}] = [\textsf{R}]_0/2$.

$[\textsf{R}]_0/2 = -\textsf{k} t_{1/2} + [\textsf{R}]_0$

$\textsf{k} t_{1/2} = [\textsf{R}]_0 - [\textsf{R}]_0/2 = [\textsf{R}]_0/2$

$t_{1/2} = \frac{[\textsf{R}]_0}{2\textsf{k}}$

The half-life of a zero order reaction is directly proportional to the initial concentration of the reactant.

For a first order reaction: Integrated rate law is $\textsf{k} = \frac{2.303}{\text{t}} \log \frac{[\textsf{R}]_0}{[\textsf{R}]}$.

At $t = t_{1/2}$, $[\textsf{R}] = [\textsf{R}]_0/2$.

$\textsf{k} = \frac{2.303}{t_{1/2}} \log \frac{[\textsf{R}]_0}{[\textsf{R}]_0/2} = \frac{2.303}{t_{1/2}} \log 2$.

$\log 2 \approx 0.3010$.

$\textsf{k} = \frac{2.303}{t_{1/2}} \times 0.3010 = \frac{0.693}{t_{1/2}}$

$t_{1/2} = \frac{0.693}{\textsf{k}}$

The half-life of a first order reaction is constant and independent of the initial concentration of the reactant. This is a key characteristic used to identify a first order reaction.

Answer:

Answer:

Summary of integrated rate laws for zero and first order reactions:

Order	Reaction	Differential Rate Law	Integrated Rate Law (Form 1)	Integrated Rate Law (Form 2, straight line plot)	Straight Line Plot Axes	Slope	Half-Life ($t_{1/2}$)	Units of k
0	R $\to$ P	$-\frac{\text{d}[\textsf{R}]}{\text{dt}} = \textsf{k}$	$[\textsf{R}] = -\textsf{kt} + [\textsf{R}]_0$	$[\textsf{R}]_0 - [\textsf{R}] = \textsf{kt}$	$[\textsf{R}]$ vs t	-k	$\frac{[\textsf{R}]_0}{2\textsf{k}}$	mol L$^{-1}$ s$^{-1}$
1	R $\to$ P	$-\frac{\text{d}[\textsf{R}]}{\text{dt}} = \textsf{k}[\textsf{R}]$	$\ln [\textsf{R}] = -\textsf{kt} + \ln [\textsf{R}]_0$	$\ln \frac{[\textsf{R}]_0}{[\textsf{R}]} = \textsf{kt}$ or $\log \frac{[\textsf{R}]_0}{[\textsf{R}]} = \frac{\textsf{k}}{2.303}\text{t}$	$\ln [\textsf{R}]$ vs t OR $\log \frac{[\textsf{R}]_0}{[\textsf{R}]}$ vs t	-k OR $\frac{\textsf{k}}{2.303}$	$\frac{0.693}{\textsf{k}}$	s$^{-1}$

Pseudo First Order Reactions: Reactions that appear to be of a higher order but follow first order kinetics under specific conditions, typically when one reactant is present in very large excess (its concentration change during the reaction is negligible).

Example: Hydrolysis of ethyl acetate ($\textsf{CH}_3\text{COOC}_2\text{H}_5$) in water: $\textsf{CH}_3\text{COOC}_2\text{H}_5 + \textsf{H}_2\text{O} \xrightarrow{\text{Acid}} \textsf{CH}_3\text{COOH} + \textsf{C}_2\text{H}_5\text{OH}$. This is a second order reaction, Rate $= \textsf{k}' [\textsf{CH}_3\text{COOC}_2\text{H}_5][\textsf{H}_2\text{O}]$. If water is in large excess, $[\textsf{H}_2\text{O}]$ is essentially constant. The rate becomes Rate $= \textsf{k} [\textsf{CH}_3\text{COOC}_2\text{H}_5]$, where $\textsf{k} = \textsf{k}'[\textsf{H}_2\text{O}]$ (pseudo first order rate constant). This reaction then behaves as first order with respect to ethyl acetate.

Example: Inversion of cane sugar: $\textsf{C}_{12}\text{H}_{22}\text{O}_{11} + \textsf{H}_2\text{O} \xrightarrow{\text{Acid}} \textsf{C}_6\text{H}_{12}\text{O}_6 \text{(glucose)} + \textsf{C}_6\text{H}_{12}\text{O}_6 \text{(fructose)}$. Water is in large excess, so Rate $= \textsf{k} [\textsf{C}_{12}\text{H}_{22}\text{O}_{11}]$.

Answer:

Temperature Dependence Of The Rate Of A Reaction

The rate of most chemical reactions increases with increasing temperature. A general observation is that the rate constant approximately doubles for every $10^\circ\textsf{C}$ increase in temperature.

The dependence of the rate constant ($\textsf{k}$) on temperature (T) is described by the Arrhenius equation:

$\textsf{k} = \textsf{A} \text{e}^{-\textsf{E}_\text{a}/\textsf{RT}}$

Where:

• $\textsf{A}$ is the Arrhenius factor or frequency factor (also called pre-exponential factor). It is a constant specific to the reaction and related to the frequency of collisions.
• $\textsf{E}_\text{a}$ is the activation energy, measured in J mol$^{-1}$. It is the minimum energy required for reactant molecules to overcome the energy barrier and form products.
• $\textsf{R}$ is the gas constant (8.314 J K$^{-1}$ mol$^{-1}$).
• $\textsf{T}$ is the absolute temperature in Kelvin.

Arrhenius explained that reactant molecules must overcome an energy barrier to react. When reactant molecules collide, they form a high-energy, unstable intermediate structure called the activated complex or transition state (Fig. 4.6). The energy difference between the activated complex and the reactants is the activation energy ($\textsf{E}_\text{a}$). The activated complex exists briefly before breaking down into products.

The distribution of kinetic energy among gas molecules is described by the Maxwell-Boltzmann distribution curve (Fig. 4.8). The curve shows the fraction of molecules with a given kinetic energy at a specific temperature. Increasing the temperature shifts the curve to higher kinetic energies and broadens it, significantly increasing the fraction of molecules possessing energy greater than or equal to the activation energy ($\textsf{E}_\text{a}$) (Fig. 4.9).

The term $\text{e}^{-\textsf{E}_\text{a}/\textsf{RT}}$ in the Arrhenius equation represents the fraction of molecules that have kinetic energy greater than or equal to the activation energy, which are capable of reacting upon collision.

Taking the natural logarithm of the Arrhenius equation:

$\ln \textsf{k} = \ln (\textsf{A} \text{e}^{-\textsf{E}_\text{a}/\textsf{RT}}) = \ln \textsf{A} + \ln (\text{e}^{-\textsf{E}_\text{a}/\textsf{RT}})$

$\ln \textsf{k} = \ln \textsf{A} - \frac{\textsf{E}_\text{a}}{\textsf{RT}}$

This equation is in the form of a straight line ($\textsf{y} = \textsf{c} + \textsf{mx}$). A plot of $\ln \textsf{k}$ (y-axis) against $1/\textsf{T}$ (x-axis) yields a straight line with slope $= -\frac{\textsf{E}_\text{a}}{\textsf{R}}$ and y-intercept $= \ln \textsf{A}$.

The activation energy ($\textsf{E}_\text{a}$) and the Arrhenius factor ($\textsf{A}$) can be determined from the slope and intercept of this plot.

If the rate constants $\textsf{k}_1$ and $\textsf{k}_2$ are known at two different temperatures $\textsf{T}_1$ and $\textsf{T}_2$, the Arrhenius equation can be used to calculate $\textsf{E}_\text{a}$ and $\textsf{A}$.

$\ln \textsf{k}_1 = \ln \textsf{A} - \frac{\textsf{E}_\text{a}}{\textsf{RT}_1}$ (at $\textsf{T}_1$)

$\ln \textsf{k}_2 = \ln \textsf{A} - \frac{\textsf{E}_\text{a}}{\textsf{RT}_2}$ (at $\textsf{T}_2$)

Subtracting the first equation from the second:

$\ln \textsf{k}_2 - \ln \textsf{k}_1 = \left( \ln \textsf{A} - \frac{\textsf{E}_\text{a}}{\textsf{RT}_2} \right) - \left( \ln \textsf{A} - \frac{\textsf{E}_\text{a}}{\textsf{RT}_1} \right)$

$\ln \frac{\textsf{k}_2}{\textsf{k}_1} = \frac{\textsf{E}_\text{a}}{\textsf{RT}_1} - \frac{\textsf{E}_\text{a}}{\textsf{RT}_2} = \frac{\textsf{E}_\text{a}}{\textsf{R}} \left( \frac{1}{\textsf{T}_1} - \frac{1}{\textsf{T}_2} \right) = \frac{\textsf{E}_\text{a}}{\textsf{R}} \left( \frac{\textsf{T}_2 - \textsf{T}_1}{\textsf{T}_1 \textsf{T}_2} \right)$

Converting to base 10 logarithm:

$\log \frac{\textsf{k}_2}{\textsf{k}_1} = \frac{\textsf{E}_\text{a}}{2.303 \textsf{R}} \left( \frac{\textsf{T}_2 - \textsf{T}_1}{\textsf{T}_1 \textsf{T}_2} \right)$

This equation is used to calculate $\textsf{E}_\text{a}$ from rate constants at two temperatures or predict $\textsf{k}$ at a different temperature if $\textsf{E}_\text{a}$ and $\textsf{k}$ at one temperature are known.

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Effect Of Catalyst

A catalyst is a substance that increases the rate of a chemical reaction without undergoing any permanent chemical change itself. Substances that reduce the rate are called inhibitors.

Catalysts increase reaction rates by providing an alternate reaction pathway or mechanism that has a lower activation energy ($\textsf{E}_\text{a}$) compared to the uncatalysed reaction pathway (Fig. 4.11).

According to the Arrhenius equation ($\textsf{k} = \textsf{A} \text{e}^{-\textsf{E}_\text{a}/\textsf{RT}}$), lowering $\textsf{E}_\text{a}$ significantly increases the rate constant ($\textsf{k}$) and thus the reaction rate.

Key points about catalysts:

• They lower the activation energy.
• A small amount of catalyst can affect the rate of a large amount of reactants.
• Catalysts do not change the Gibbs energy ($\Delta \textsf{G}$) of a reaction.
• They catalyze spontaneous reactions but not non-spontaneous ones.
• Catalysts do not change the equilibrium constant ($\textsf{K}_\text{c}$) of a reaction. They speed up both the forward and backward reactions equally, helping the system reach equilibrium faster, but the position of equilibrium remains unchanged.

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Collision Theory Of Chemical Reactions

Collision theory, developed by Max Trautz and William Lewis, provides a theoretical explanation for reaction rates based on kinetic theory of gases. It views reactant molecules as hard spheres that must collide to react.

Basic postulates:

1. Reactant molecules are assumed to be hard spheres.
2. Reactions occur when molecules collide.
3. The number of collisions per second per unit volume of the reaction mixture is called the collision frequency (Z).

For a bimolecular elementary reaction A + B $\to$ Products, the rate is related to the collision frequency between A and B (Z$_{AB}$) and the fraction of molecules with sufficient energy to react (e$^{-Ea/RT}$):

Rate $= \textsf{Z}_{\text{AB}} \text{e}^{-\textsf{E}_\text{a}/\textsf{RT}}$

Comparing this to the Arrhenius equation ($\textsf{k} = \textsf{A} \text{e}^{-\textsf{E}_\text{a}/\textsf{RT}}$), the Arrhenius factor A is related to the collision frequency Z.

However, this simple model doesn't perfectly predict rates for complex molecules. Not all collisions lead to product formation. For a collision to be effective and lead to a reaction, two conditions must be met:

1. Energy barrier: The colliding molecules must possess kinetic energy equal to or greater than the activation energy ($\textsf{E}_\text{a}$) or threshold energy* ($\textsf{E}_0$). Collisions with insufficient energy are ineffective.
2. Orientation barrier: The colliding molecules must have the proper orientation relative to each other at the moment of collision to allow bonds to break and form. Collisions with improper orientation are ineffective and molecules just bounce back.

*Threshold energy = Activation Energy + average energy of reacting species.

To account for the orientation requirement, a factor P, called the probability factor or steric factor, is included in the collision theory equation:

$\textsf{k} = \textsf{P} \text{Z}_{\text{AB}} \text{e}^{-\textsf{E}_\text{a}/\textsf{RT}}$

This modified equation incorporates both energy and orientation requirements for effective collisions. The value of P is usually less than 1.

Collision theory provides insights into reaction rates but has limitations (e.g., treating molecules as hard spheres, neglecting structural details). More advanced theories address these limitations.

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