GATE Life Sciences Reaction Kinetics Syllabus

What is the key distinguishing feature of the half-life of a first-order reaction?

The half-life is constant and independent of the initial concentration ($t_{1/2} = \frac{0.693}{k}$). It takes the same time to halve, regardless of starting amount.

Why is radioactive decay a first-order process, and what is its rate constant called?

Radioactive decay rate is proportional to the number of undecayed nuclei: $-\frac{dN}{dt} = \lambda N$, a first-order law. The rate constant $\lambda$ is the decay constant, with $t_{1/2} = \frac{0.693}{\lambda}$.

Give three examples of first-order reactions.

Radioactive decay; decomposition of $\ce{N2O5}$ ($\ce{2N2O5 -> 4NO2 + O2}$); decomposition of $\ce{H2O2}$; hydrogenation/decomposition reactions and many isomerizations. Also the basis of carbon-14 dating.

What fraction of a first-order reactant remains after $n$ half-lives?

$$\frac{[A]}{[A]_{0}} = \left(\frac{1}{2}\right)^{n}$$ After $n$ half-lives, $(1/2)^{n}$ of the original reactant remains.

What is a pseudo-first-order reaction?

A reaction that is intrinsically higher order but behaves as first order because one reactant is present in large excess (or is a catalyst/solvent), so its concentration stays effectively constant. Example: acid-catalysed hydrolysis of an ester or cane-sugar inversion, where $[\ce{H2O}]$ is essentially constant: $\text{rate} = k'[A]$ with $k' = k[\ce{H2O}]$.

Define a second-order reaction.

A reaction whose overall order is $2$. Either rate $= k[A]^{2}$ (one reactant, second order in it) or rate $= k[A][B]$ (first order in each of two reactants).

What is the differential rate law for a second-order reaction in a single reactant?

$$-\frac{d[A]}{dt} = k[A]^{2}$$

What is the integrated rate law for a second-order reaction of the type $2A \to$ products (rate $=k[A]^2$)?

$$\frac{1}{[A]} = \frac{1}{[A]_{0}} + kt$$

For a second-order reaction ($\text{rate}=k[A]^2$), which plot is linear and what is its slope and intercept?

A plot of $\frac{1}{[A]}$ versus $t$ is linear, with slope $= +k$ and y-intercept $= \frac{1}{[A]_{0}}$.

What are the units of the rate constant $k$ for a second-order reaction?

$\text{L mol}^{-1}\,\text{s}^{-1}$ (i.e. concentration$^{-1}\cdot$ time$^{-1}$), commonly $\text{M}^{-1}\text{s}^{-1}$.

State the half-life expression for a second-order reaction (rate $=k[A]^2$).

$$t_{1/2} = \frac{1}{k[A]_{0}}$$ The half-life is inversely proportional to the initial concentration.

How does the half-life of a second-order reaction depend on initial concentration?

It is inversely proportional to initial concentration: $t_{1/2} = \frac{1}{k[A]_{0}}$. The more dilute the start, the longer the half-life; successive half-lives keep getting longer.

Write the integrated rate law for a second-order reaction $A + B \to$ products with equal initial concentrations $[A]_0=[B]_0$.

With equal initial concentrations the kinetics reduce to the single-reactant form: $$\frac{1}{[A]} = \frac{1}{[A]_{0}} + kt$$

Write the integrated rate law for $A + B \to$ products (rate $=k[A][B]$) with unequal initial concentrations.

$$\frac{1}{[A]_{0}-[B]_{0}}\ln\frac{[B]_{0}[A]}{[A]_{0}[B]} = kt$$

Give two examples of second-order reactions.

Decomposition of $\ce{NO2}$ ($\ce{2NO2 -> 2NO + O2}$); gas-phase $\ce{2HI -> H2 + I2}$; alkaline ester hydrolysis (saponification, $\ce{CH3COOC2H5 + OH^- -> CH3COO^- + C2H5OH}$); the $\ce{SN2}$ reaction $\ce{CH3Br + OH^- -> CH3OH + Br^-}$.

Compare the half-life dependence on initial concentration for zero, first, and second order reactions.

Zero order: $t_{1/2} = \frac{[A]_{0}}{2k}$ (proportional to $[A]_0$). First order: $t_{1/2} = \frac{0.693}{k}$ (independent of $[A]_0$). Second order: $t_{1/2} = \frac{1}{k[A]_{0}}$ (inversely proportional to $[A]_0$).

Summarize which concentration-vs-time plot is linear for each order.

Zero order: $[A]$ vs $t$ (slope $-k$). First order: $\ln[A]$ vs $t$ (slope $-k$). Second order: $\frac{1}{[A]}$ vs $t$ (slope $+k$). This graphical method is used to determine reaction order experimentally.

Summarize the units of $k$ for zero, first, and second order reactions.

Zero order: $\text{mol L}^{-1}\text{s}^{-1}$. First order: $\text{s}^{-1}$. Second order: $\text{L mol}^{-1}\text{s}^{-1}$. General: order-$n$ has units $(\text{mol L}^{-1})^{1-n}\,\text{s}^{-1}$.

State the general formula for the half-life of an $n$-th order reaction ($n \ne 1$).

$$t_{1/2} \propto \frac{1}{[A]_{0}^{\,n-1}}$$ Specifically $t_{1/2} = \frac{2^{n-1}-1}{(n-1)k[A]_{0}^{n-1}}$ for $n\neq 1$.

What is the difference between molecularity and order of a reaction?

Molecularity is the number of reactant species colliding in a single elementary step; it is a theoretical integer ($\geq 1$) defined only for elementary reactions. Order is experimentally determined from the rate law, can be zero, fractional, or negative, and applies to overall reactions.

Can order and molecularity differ? Give an example.

Yes, for multi-step (non-elementary) reactions. Example: acid-catalysed ester hydrolysis is bimolecular in mechanism but pseudo-first order experimentally because water is in large excess. The rate-determining step controls the observed order.

What does the rate constant $k$ depend on?

$k$ depends on temperature and the presence of a catalyst (and the activation energy), but NOT on reactant concentrations. Its temperature dependence is given by the Arrhenius equation.

State the Arrhenius equation and identify its terms.

$$k = A\,e^{-E_{a}/RT}$$ where $A$ is the pre-exponential (frequency) factor, $E_{a}$ is the activation energy, $R$ is the gas constant, and $T$ is absolute temperature.

Write the logarithmic (linear) form of the Arrhenius equation and describe its plot.

$$\ln k = \ln A - \frac{E_{a}}{R}\cdot\frac{1}{T}$$ A plot of $\ln k$ versus $\frac{1}{T}$ is linear with slope $-\frac{E_{a}}{R}$ and intercept $\ln A$.

How is the activation energy obtained from rate constants at two temperatures?

$$\ln\frac{k_{2}}{k_{1}} = \frac{E_{a}}{R}\left(\frac{1}{T_{1}} - \frac{1}{T_{2}}\right)$$

What is the rate-determining step and how does it relate to overall order?

The rate-determining step is the slowest elementary step in a reaction mechanism. It controls the overall rate, and the experimentally observed rate law (and hence order) corresponds to this slowest step (including species up to and including it).

For the first-order decay $[A]=[A]_0 e^{-kt}$, what does the reciprocal of $k$ represent?

$\tau = \frac{1}{k}$ is the mean lifetime (relaxation time): the time for the concentration to fall to $\frac{1}{e}$ ($\approx 36.8\%$) of its initial value. Note $t_{1/2} = \tau\ln 2 = 0.693\,\tau$.

In carbon-14 dating, why is first-order kinetics essential, and what equation is used?

Because radioactive decay is first order, $t_{1/2}$ is constant regardless of amount, allowing dating from remaining activity. Age is found from $t = \frac{2.303}{\lambda}\log\frac{N_{0}}{N}$, with $\lambda = \frac{0.693}{t_{1/2}}$ and $t_{1/2}(\ce{^{14}C}) \approx 5730$ years.

A reaction is zero order. If the initial rate is $k$ and $[A]_0 = 0.50\,\text{M}$ with $k = 0.01\,\text{M s}^{-1}$, find $t_{1/2}$.

$$t_{1/2} = \frac{[A]_{0}}{2k} = \frac{0.50}{2(0.01)} = 25\ \text{s}$$

How can you experimentally distinguish first-order from second-order kinetics using successive half-lives?

Measure successive half-lives. For first order they remain constant ($t_{1/2}$ unchanged as reaction proceeds). For second order each successive half-life doubles (since $t_{1/2}\propto \frac{1}{[A]_{0}}$ and concentration keeps falling). For zero order each successive half-life is halved.