For a chemical reaction A ⟶product, the mechanism of the reaction postulated was as follows. A k

For a chemical reaction A ⟶product, the mechanism of the...

For a chemical reaction $\mathrm{A} \longrightarrow$ product, the mechanism of the reaction postulated was as follows. $$ \mathrm{A} \stackrel{\mathrm{k}_{1}}{\mathrm{~g}_{2}} 3 \mathrm{~B} \frac{\mathrm{k}_{\mathrm{s}}}{\text { R.D. }}{\mathrm{\longrightarrow}} \mathrm{C}_{\mathrm{g}} $$ If the reaction occurred with individual rate constants $\mathrm{k}_{1}, \mathrm{k}_{2}$ and $\mathrm{k}_{3}$, determine activation energy for the overall reaction if the activation energies associated with these rate constants are $\mathrm{E}_{a_{1}}=180 \mathrm{~kJ} \mathrm{~mol}^{-1}, \mathrm{E}_{a_{2}}=90 \mathrm{~kJ}$ $\mathrm{mol}^{-1}$ and $\mathrm{E}_{a_{3}}=40 \mathrm{~kJ} \mathrm{~mol}^{-1}$ (a) $70 \mathrm{~kJ}$ (b) $-10 \mathrm{~kJ}$ (c) $310 \mathrm{~kJ}$ (d) $130 \mathrm{~kJ}$

For a chemical reaction $\mathrm{A} \longrightarrow$ product, the mechanism of the reaction postulated was as follows.
$$
\mathrm{A} \stackrel{\mathrm{k}_{1}}{\mathrm{~g}_{2}} 3 \mathrm{~B} \frac{\mathrm{k}_{\mathrm{s}}}{\text { R.D. }}{\mathrm{\longrightarrow}} \mathrm{C}_{\mathrm{g}}
$$
If the reaction occurred with individual rate constants $\mathrm{k}_{1}, \mathrm{k}_{2}$ and $\mathrm{k}_{3}$, determine activation energy for the overall reaction if the activation energies associated with these rate constants are $\mathrm{E}_{a_{1}}=180 \mathrm{~kJ} \mathrm{~mol}^{-1}, \mathrm{E}_{a_{2}}=90 \mathrm{~kJ}$
$\mathrm{mol}^{-1}$ and $\mathrm{E}_{a_{3}}=40 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(a) $70 \mathrm{~kJ}$
(b) $-10 \mathrm{~kJ}$
(c) $310 \mathrm{~kJ}$
(d) $130 \mathrm{~kJ}$

Key Concepts

Reaction Mechanism

A reaction mechanism describes the sequence of elementary steps by which a chemical reaction occurs. Understanding the mechanism provides insight into how reactants are transformed into products through one or more intermediate stages, each characterized by its own rate constant and activation energy.

Arrhenius Equation

The Arrhenius equation relates the rate constant for an elementary reaction to its activation energy and temperature. It is expressed as k = A exp(–Ea/RT), where A is the pre?exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature. This relationship underpins the quantitative analysis of how individual steps contribute to overall reaction kinetics.

Overall Activation Energy

The overall activation energy for a reaction that proceeds through multiple elementary steps can, under certain conditions, be obtained by considering the individual activation energies. When the overall rate constant depends on the product of the individual rate constants (as in sequences where each step is indispensable and multiplicative), the effective activation energy is the sum of the activation energies from each step.

Rate?Determining Step

In a multi?step reaction mechanism, the rate?determining step is the slowest step and thus exerts the greatest influence on the overall reaction rate. Although many elementary steps may occur, the kinetic behavior of the overall reaction is often controlled by the step with the highest activation energy or the lowest rate constant, making its kinetics crucial for understanding the reaction’s energy profile.

Transcript

00:02 In this problem we have given that for a chemical reaction, a gives products.

00:15 The mechanism of the reaction postulated was as follows.

00:25 Gaseous a reacts reversibly with rate constant k1 and k2 to give three moles of b, which gives c having rate constant k3, and this step is rate determining step that is rds so we can write rate expression as follows rate equals to rate constant k3 into concentration of b because rate expression can be written as according to rate determining step while equilibrium constant k c is equal to k1 divide by k2 which is equal to concentration of b raised to the power 3 divide by concentration of a so we can write b is equal to k1 k2 raise to the power 1 by 3 into concentration of a raised to the power 1 by 3.

02:18 Now overall rate can be written as rate equals to k3 into k1 divide by k2 raised to the power 1 by 3 into concentration of a raised to the power 1 by 3.

03:07 So we can write experimental rate constant k is equal to k 3 into k1 divide by k2 raised to the power 1 by 3 this is experimental rate constant now we know that rate constant k is equal to a into e raise to the power minus ea by r t from argeny equation so k3 is equal to a 3 into a 3 into e raise to the power minus e a by r t by r t from ar heinous equation so k 3 is equal to a 3 into e raised to the power minus e a 3 divide by rt into k1 is equal to a1 into e raise to the power minus ea1 divide by ea1 divide by a2 into e raised to the power minus e82 divide by rt raised to the power 1 by 3.

04:53 Now we can rearrange this expression and get a3 into e1 divide by a2 raised to the power 1 by 3 into e raised to the power minus ea by rt into e raised to the power ea3 by rt divide into e raised to the power minus ea3 by rt divide into e -rae -a -1 divided by 3 r t divide by e -raise to the power minus e a 2 divide by 3r t so we can rearrange e -ra -ra -to -the -power activation energy divide by r t in this manner and get e -raise to the power minus e .a 3 minus ea 1 divide by 3 plus ea2 divide by 3 into 1 pi rt.

06:59 So equating both side, we get activation as the ea is equal to it is e raised to the power minus ea...

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