Pseudo First-Order & Special Topics

CHEIntegrated Rate Laws & Calculations

A pseudo first-order reaction is genuinely higher order but behaves as first order because one reactant is in large excess. This subtopic also gathers conceptual 'why' questions on temperature, catalysts, and the energetic fraction of molecules.

Pseudo first-order rate law

\text{rate} = k[\text{ester}][H_2O] = k'[\text{ester}], \quad k' = k[H_2O]

[H_2O]

effectively constant in large excess

Fraction of molecules with energy at least Ea

f = e^{-E_a/RT}

E_a

activation energy,

R=8.314\,J\,mol^{-1}K^{-1}

T

in kelvin

A pseudo first-order reaction has true order $>1$ but observed order $1$ , because a reactant present in large excess has effectively constant concentration.
Acid hydrolysis of ethyl acetate: $CH_3COOC_2H_5 + H_2O \rightarrow CH_3COOH + C_2H_5OH$ ; water is in vast excess, so $[H_2O]$ is constant and the rate depends only on $[\text{ester}]$ .
Inversion of cane sugar (sucrose) is another standard pseudo first-order example, since water again swamps the reaction and stays effectively constant.
Effectively $\text{rate} = k'[\text{ester}]$ where $k' = k[H_2O]$ ; the true second-order $k$ is hidden inside the observed pseudo-constant $k'$ .
Rate rises with temperature mainly because a much larger fraction of molecules acquires energy $\ge E_a$ (Maxwell-Boltzmann tail), not chiefly because collisions become more frequent.
The fraction of molecules with energy $\ge E_a$ is $e^{-E_a/RT}$ — compute it directly when $E_a$ , $R$ , and $T$ are given.
A catalyst lowers $E_a$ by offering an alternative pathway but does not change $\Delta H$ , since the energies of reactants and products are untouched.
Inversion of sucrose is named because the optical rotation changes from dextro (sucrose) to a net laevo mixture of glucose and fructose — a way to monitor the reaction.
A pseudo first-order reaction is truly bimolecular; calling it 'first order' refers only to the observed kinetics under excess-reactant conditions.
Acid hydrolysis of esters is also catalysed by $H^+$ , whose concentration stays constant, reinforcing the simple first-order dependence on ester concentration.
Because $[H_2O]$ is folded into $k'$ , the experimentally measured constant has first-order units ( $s^{-1}$ ) even though the intrinsic constant $k$ is second order.
When evaluating $f=e^{-E_a/RT}$ , compute the exponent $-E_a/RT$ first (in consistent units), then take the exponential using the supplied value of $e$ .

Where the marks go

Calling a pseudo first-order reaction 'genuinely first order' — its true molecularity/order is higher; the excess reactant merely masks it.
Forgetting that the observed $k'$ already contains $[H_2O]$ , so $k'$ has units $s^{-1}$ while the real $k$ is second order.
Explaining the temperature effect only as 'molecules collide more often' — the dominant cause is the rise in the fraction with energy $\ge E_a$ .
Mixing $E_a$ (kJ) and $R$ (J) when computing $e^{-E_a/RT}$ — convert units first or the exponent is off by $1000$ .
Stating a catalyst changes $\Delta H$ — it changes only $E_a$ ; reactant and product energies are unchanged.

How the board asks it

Define / statepseudo first-order definition and excess-reactant masking
What is meant by a pseudo first-order reaction? Explain with the help of the acid hydrolysis of ethyl acetate, $CH_3COOC_2H_5 + H_2O \rightarrow CH_3COOH + C_2H_5OH$ , why it is observed to be first order.
Give reasonstemperature effect via the maxwell-boltzmann tail
Account for the fact that the rate of a reaction nearly doubles for a $10^\circ C$ rise in temperature, even though the increase in the number of collisions is comparatively small.
Give reasonscatalyst lowers $E_a$ but leaves $\Delta H$ unchanged
Give reasons: a catalyst increases the rate of a reaction but does not alter the enthalpy change $\Delta H$ of the reaction.
Numericalfraction of molecules with energy at least $E_a$
Calculate the fraction of molecules having energy equal to or greater than the activation energy $E_a = 100\ kJ\ mol^{-1}$ at $T = 300\ K$ , given $R = 8.314\ J\ K^{-1}\ mol^{-1}$ and using $f = e^{-E_a/RT}$ .
Give reasonsunits of $k'$ versus the true second-order $k$
Explain why the experimentally measured rate constant for the acid hydrolysis of an ester has units of $s^{-1}$ , although the reaction is genuinely bimolecular.
Give reasonsinversion of sucrose monitored by optical rotation
Why is the acid-catalysed hydrolysis of sucrose called the 'inversion of cane sugar', and how can this reaction be followed experimentally?

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.