Chemical Kinetics

Chemical kinetics helps us to understand how chemical reaction occurs.

Rate of Reaction:

Rate of reaction is define in terms of concentrations of reaction product:

**Rate = (Change in Concentration)/(Change in Time)**

Rate = change in concentration per unit time of reactant or

(i) Rate of decrease in concentration of any one of reactant

(ii) The rate of increase in concentration of any one of product.

**Example: R ⟶ P **

One mole of R produces one mole of P.

If [R_{1}] and [P_{1}] are concentration at time t_{1} and [R_{2}] and [P_{2}] at time t_{2} :

Rate is always positive quantity since multiply by -1 for reactant concentration.

Rate of reaction at every moment is different the rate at any instant is :

Units of Rate of Reaction:

In gaseous reaction concentration expressed in terms of press so the unit is atm S^{-1}

Rate of reaction for general reactions: aA + bB → cC + dD

Rate Law:

The representation of rate of reaction in terms of concentration of reactants is known as rate law. It is also called rate expression or rate equation.

Rate of reaction depend upon concentration

Consider a reaction: aA + aB → cC + dD

a, b, c, d => stoichiometric coefficient

**Rate ∝ [A] ^{x} [B]^{y} → Rate Law **

Exponent x and y may be or may not be equal to stoichiometric coefficient

Differential Rate Equation or Empirical Rate Law:

k = proportionality constant or rate constant or velocity constant or specific reaction rate constant.

Rate constant is the rate of reaction when all the reactant are at unit (one) concentration.

Example:

**2NO(g) + O _{2}(g) → 2NO_{2} (g) Rate = k [NO]_{2} [O_{2}]**

**CHCl _{3} + Cl_{2} → CCl_{4} + HCl Rate = k [CHCl_{3}] [Cl_{2}]^{1/2}**

Order of reaction:

**Rate = k [A] ^{x} [B]^{y}**

Sum of these exponents (x+y) is called overall order of reaction

x = Order of reaction with respect to A

y = Order of reaction with respect to B

Sum of power of the concentration of reactant in rate law expression is called order of the "Chemical Reaction".

Order can be 0, 1, 2, 3 and fraction

A zero order reaction means that the rate of reaction is independent of concentration of reactant.

Calculate overall order of reaction which has rate expression:

**Rate =k [A] ^{1/2} [B]^{3/2} = 2 order**

**Rate =k [A] ^{3/2} [B]^{-1} = 1/2 order **

Unit of rate constant for various order of reaction:

Consider a general reaction:

**aA + aB → cC + dD **

**Rate = k [A] ^{x} [B]^{y} **

**Rate = k [conc.] ^{x+y} **

**x + y = overall order of reaction = q **

**Rate = k [conc.] ^{q} **

Examples:

Integrate Rate Equation:

In order to avoid difficulty in determine rate law and order we can integrated different rate expression to give relation between directly measured experimental data i.e. concentration at different time and rate constant.

It is different for different order .

1) Zero Order Reaction:

Consider Reaction : R → P

** Rate = (-d[R])/dt= k[R] ^{0} ----------(1)**

As any quantity raised to power zero is unity.

**(-d[R])/dt=k **

** ∴ -d[R] = kdt **

Taking integration on both sides with boundary conditions:

Compare this equation with equation of straight line :

y = mx + c where m = slope and c = intercept

Zero order reaction is uncommon but they occur under special condition.

Example:

1. Some enzyme catalyzed reaction which occur on metal surfaces are zero order

2. Decomposition of gaseous ammonia on hot platinum surfaces is zero order.

** Rate = k [NH _{3}]^{2} = k **

Half-Life of Reaction:

The half-life of reaction is the time in which the concentration of reactant is reduced to one half of its initial concentration.

It is representation by t_{1/2}.

Half-life for zero order reaction:

2) First Order Reaction:

** A → P **

Assuming order of reaction is first :

** Rate = (-d[A])/dt=k _{1}[A]-----(1) **

Separating variable:

**(-d[A])/([A])=k _{1} dt **

**(d[A])/([A])=-k _{1} dt **

Taking integration on both side

RULE:

Compare equation with y = mx

Example:

** C _{2}H_{4}(g) H_{2}(g) → C_{2}H_{6}(g) **

** Rate = k [C _{2}H_{4}] **

All natural and artificial radioactive decay of unstable nuclei place by first order kinetics.

** Rate = k [Ra] **

Let us consider typical 1^{st} order gas phase reaction :

** A (g) → B (g) + C(g) **

Let Pi = initial press of A

Pt = total press at time t

** Pt = PA + PB + Pc **

X is decrease in press after time t

X atm pressure decrease and B and C formed

** A (g) → B (g) + C(g)**

According to Dalton law:

** Pt = Pi – x + x + x **

** Pt = Pi + x **

** x = Pt – Pi **

** PA = Pi – x **

** i.e. PA = Pi – Pt + Pi **

** PA = 2 Pi – Pt **

** k = (2.303/t) log (Pi/(2 Pi-Pt) **

Half-Life of Reaction:

The half-life of reaction is the time in which the concentration of reactant is reduced to one half of its initial concentration.

It is representation by t_{1/2}.

Average Life Time Const ( Ʈ):

The time required for concentration of reactant to fall to 1/e of its initial value.

3. Second Order

Case 1 : Both reactant are same.

** A + B → P **

Here both reactants are some: A = B

Case 2 : When A ≠ B

Half - Life Period

Second order half-life time depending upon initial concentration.

4) Third order reaction

A reaction is said to be of third order if the rate is determined by the variation of three concentration terms. In other words, the minimum number of molecules necessary for the reaction to take place is three.

There may be three different cases in third order reaction.

(i) All the three species have equal concentrations. A + A + A → P

(ii) Two species have equal concentrations and one different. A + A + B → P

(iii) All three species have unequal concentrations. A + B + C → P

Case I: All species having equal concentration.

5) nth Order reaction

Substituting n= 0, 2, 3, 4, 5 , ... or any fraction or positive and negative as we can obtain integrated rate equation for any ordered other than 1st order

Similarly for half -life time :

n = 0, 2 ,3 , ... also or any fraction positive and negative except one.

Half-Life of Reactions

The half-life (t_{1/2}) of a reaction is the time it takes for the reactant concentration to decrease to one-half its initial value.

For Zero-Order reaction : t_{1/2} = [R]_{0}/2k

For First-Order reaction : t_{1/2} = 0.693/k

For Second-Order reactions : t_{1/2} = 1/(k[R]_{0})

Temperature Dependence Rate of Reaction:

Most of chemical reaction is accelerated by increase in temperature

Example:

1. Decomposition of N_{2}O_{5} the time taken for 1/2 of its original amount of material to decompose is 12 minute at 50 ^{0}c 5h at 25 ^{0}c and 10 days 0 ^{0}c

2. Mix of KMnO_{4} and oxalic acid decolorized faster at higher temperature.

For a chemical reaction with rise in temperature by 10 ^{0}c the rate constant is nearly double.

The Temperature dependence of the rate of chemical reaction can be accurately explain by Arrhenius equation. It was first proposed by J.H. Vant Hoff but Swedish chemist Arrhenius provided its physical justification.

** k = A e ^{- Ea/RT} --------------1 **

A = Arrhenius factor, frequency factor or preexponential factor . It is constant to a particular reaction

R = gas constant

Ea = activation of energy measured in (J mol^{-1})

e ^{- Ea/RT} = fraction of molecule that have K.E. greater than Ea.

A plot of ln(k) versus 1/T yields a straight line with slope = -E_{a}/R; (R = 8.314 J/mol.K).

The activation energy (E_{a}) can also be calculated using two values of the rate constant k according to the expression:

Parallel Reaction :

In the most of reaction reactant decay result in the production of only a signal species. However in many instances a single reactant become variety of product such reaction are referred to as parallel reaction.

Limiting Condition :

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