Definitions [10]
The rate of a chemical reaction may be defined as the change in concentration of any of the reactants or any of the products per unit time.
Rate of Reaction = `"Change in concentration of a reactant or a prodect"/"Time taken for the change"`
Define the following term:
Pseudo first-order reaction
The reactions that have higher order true rate law but are found to behave as first order are called pseudo first order reactions.
\[\ce{CH3COOCH3 + H2O - CH3COOH + CH3OH}\]
Define “zero order reaction”.
Zero order reaction is the reaction whose rate is independent of the reactant concentration and remains constant throughout the course of the reaction.
Define the half-life of a first-order reaction.
The time in which concentration of reactant becomes half of its initial concentration is called half Life. It is denoted by `t_(1/2)`.
A reaction is zero order if the rate is independent of the concentration of the reactant.
Define first-order reaction.
A chemical reaction in which the rate of reaction depends solely linearly on the concentration of one ingredient is referred to as a first-order reaction.
A first-order reaction is a reaction whose rate depends upon the first power of the concentration of reactants, i.e., the rate is directly proportional to the concentration of reactants.
The half-life t1/2 is the time required for the concentration of a reactant to fall to half its initial value.
\[t_{1/2}\propto\frac{1}{[A_0]^{n-1}}\]
Define half life of a reaction.
Half life of a reaction is defined as the time required for the reactant concentration to reach one half of its initial value.
Define activation energy.
Activation energy is the lowest energy necessary to commence a chemical reaction by disrupting the bonds of reactant molecules and creating the activated complex or transition state. It signifies the energy threshold that must be surmounted for a reaction to transpire. Activation energy is typically represented as Ea.
Activation energy may be defined as the excess energy that the reactant molecules (having energy less than the threshold energy) must acquire in order to cross the energy barrier and to change into the products.
The Arrhenius equation is a mathematical expression to give a quantitative relationship between the rate constant and temperature.
Formulae [3]
\[\mathrm{Rate}=\frac{\text{Decrease in concentration of Reactant}}{\text{Time interval}}\]
\[=-\frac{\Delta[R]}{\Delta T}\]
\[\mathrm{Rate}=\frac{\text{Increase in concentration of Product}}{\text{Time interval}}\]
\[=+\frac{\Delta\left[P\right]}{\Delta T}\]
For a general reaction aA + bB → cC + dD:
\[\frac{dx}{dt}=-\frac{1}{a}\frac{d[A]}{dt}=-\frac{1}{b}\frac{d[B]}{dt}=+\frac{1}{c}\frac{d[C]}{dt}=+\frac{1}{d}\frac{d[D]}{dt}\]
Theorems and Laws [1]
Collision Theory explains why and how temperature increases the rate of reaction.
Microscopic Factors:
Factor 1: Collisional Frequency (Z):
- The number of collisions taking place per second per unit volume of the reaction mixture.
- Effective collision: Only those collisions that actually produce the products.
\[\mathrm{Rate}=\frac{dx}{dt}=Z\times\text{(fraction of effective collisions)}\]
Factor 2: Activation Energy:
- The minimum amount of extra energy required by a reacting molecule to get converted into an activated molecule (transition state).
- Ea = Threshold energy − Average energy of reactant molecules
Conditions for Effective Collision:
- Colliding molecules must possess energy ≥ threshold energy.
- Colliding molecules must have proper orientation at the time of collision.
Drawback of Collision Theory: It considers atoms/molecules to be hard spheres and ignores their structural features.
Key Points
The rate of a reaction depends on:
| Factor | Effect on Rate |
|---|---|
| Concentration of reactants | Higher conc. → more collisions → higher rate |
| Temperature | Higher T → more energetic collisions → higher rate |
| Physical state and surface area | Greater surface area → higher rate |
| Catalyst | Lowers activation energy → higher rate |
| Pressure (gaseous reactions) | Higher pressure → higher rate |
| Light / electromagnetic radiation | Provides energy for photochemical reactions |
| Concept | Zero Order Reaction | First Order Reaction |
|---|---|---|
| Rate law | Rate = k | Rate = k[A] |
| Differential form | \[-\frac{\mathrm{d[A]}}{[\mathrm{dt]}}=\mathrm{k}[\mathrm{A}]^{0}=\mathrm{k}\] | \[-\frac{\mathrm{d[A]}}{[\mathrm{dt]}}=\mathrm{k[A]}\] |
| Integrated form | \[\mathrm{k}=\frac{\left[\mathrm{A}\right]_{0}-\left[\mathrm{A}\right]_{t}}{\mathrm{t}}\] | \[\mathrm{k=\frac{2.303}{t}\log_{10}\frac{\left[A\right]_{0}}{\left[A\right]_{t}}}\] |
| Unit of k | mol L⁻¹ s⁻¹ | s⁻¹ |
| Half-life (t₁/₂) | \[\mathrm{t}_{1/2}=\frac{[\mathrm{A}]_0}{2\mathrm{k}}\] | t₁/₂ = 0.693 / k |
| Dependence | Independent of concentration | Depends on concentration |
A reaction is first order if the rate depends on the first power of concentration of one reactant.
For A → Products:
| Time | Concentration |
|---|---|
| t = 0 | a |
| t = t | a − x |
\[k=\frac{2.303}{t}\log\frac{a}{a-x}\quad\mathrm{or}\quad k=\frac{2.303}{t}\log\frac{[A]_0}{[A]}\]
Also: \[[A]=[A]_0\cdot e^{-kt}\]
Half-life:
\[t_{1/2}=\frac{0.693}{k}\]
- Half-life is independent of initial concentration — a defining feature of first order reactions.
- \[t_{75\%}=2\times t_{1/2}\]
Temperature Coefficient:
The temperature coefficient μμ is the ratio of rate constants at two temperatures differing by 10°C:
\[\mu=\frac{k_{T+10}}{k_T}=2\mathrm{~to~3}\]
The two reference temperatures are typically 35°C (308 K) and 25°C (298 K).
If R1 = reaction rate at T1 and R2 = reaction rate at T2:
\[\frac{R_1}{R_2}=\frac{\mu T}{10}\]
Arrhenius Equation:
\[k=A\cdot e^{-E_a/RT}\]
where:
- k = rate constant
- A = pre-exponential factor (frequency factor)
- EaEa = activation energy (J mol⁻¹)
- R = gas constant (8.314 J mol⁻¹ K⁻¹)
- T = temperature in Kelvin
The factor \[e^{-E_a/RT}\] is called the Boltzmann factor.
Logarithmic form:
\[\log k=\log A-\frac{E_a}{2.303RT}\]
A plot of log k vs 1/T is a straight line with:
\[\mathrm{Slope}=-\frac{E_a}{2.303R}\]
Intercept = log A
This is of the form y = mx + c.
Two-Temperature Form:
\[\log\frac{k_2}{k_1}=\frac{E_a}{2.303R}\left(\frac{1}{T_1}-\frac{1}{T_2}\right)=\frac{E_a}{2.303R}\left(\frac{T_2-T_1}{T_1T_2}\right)\]
