The reactions in which, the overall rate of the reaction is proportional to the first power of concentration of one of the reactants only are called as first order reactions. Consider the reaction

**Rate equation for first order
reactions**

The reactions in which, the overall rate of the
reaction is proportional to the first power of concentration of one of the
reactants only are called as first order reactions. Consider the reaction

A --* ^{k}*

Rate of reaction = *-d*[A]
/ *dt
= *= *k* [A]^{1.0}

where *k*_{1}
is the rate constant of the first order reaction.

At the beginning of the reaction, time ' *t*' = 0, let the concentration of A be ' *a*' mole.lit ^{-1} . After the
reaction has proceeded for some time '*t*',
let the concentration of A that has reacted be *x* mole.lit^{-1} . The concentration of unreacted A
remaining at time ' *t*' will be ( *a* - *x*)
mole.lit^{-1} . The rate of the reaction will be dx/dt. For a first
order reaction, rate = *dx*/*dt*
= *k*_{1} (*a* - *x*) ……. (2)

upon integrating, equation 2 becomes,

which is, *-ln(a
- x) = k* _{1}t + c

*c *= integration constant

at time, *t*
= 0, *x* = 0.

in equation 3,

- ln ( *a*
- 0) = *k*_{1} ´ 0 +

C or C = -ln *a*.

Substituting C value in equation 3

Unit of *k*_{1}
is sec^{-1}

This equation is known as the first order rate
constant equation.

This equation can be used for determining the
rate constant of a first order reaction based on the experimental data of (*a*) and (*a* - *x*) at different
periods of time '*t*'. Sometimes, the
following expression is also used.

*k*_{1} = ( 2
303 / t_{2}-t_{1} ) log (a-x_{1 }/ a-x_{2})

where *x*_{1}
and *x*_{2} are the amounts
reacted in *t*_{1}* *and*
t*_{2}* *periods of time.

**Characteristics of first
order reaction**

When the
concentration of the reactant is increased by '*n*' times, the rate of reaction is also increased by n times. That
is, if the concentration of the reactant is doubled, the rate is doubled.

The unit of rate constant of a first order
reaction is sec^{-1} or time^{-1}
.

*k*_{1}* *= rate /
( *a* - *x* ) = mol.lit ^{-}^{1} sec^{-}^{1 } / mol.lit ^{-}^{1 }= sec^{-1}

The time required to complete a definite
fraction of reaction is independent of the initial concentration, of the
reactant if *t*1/u is the time of one '
*u*^{'th} fraction of reaction
to take place then from equation 4,

*x *= a /* u and*

^{t}_{1/u} ^{=} (2 303 / *k*_{1}) log ( a/(a-a/u) )

^{t}_{1/u} ^{=} (2 303 / *k*_{1}) log ( u/(u-1) )

since *k*_{1}
= rate constant, *t*_{1/u} is
independent of initial concentration '*a*'.

**Examples of first order reactions**

1. All radioactive transformations follow first
order kinetics. For example,

_{92}U^{238 }-- -- >^{ }_{ 90}Th^{234} + _{2}He^{4}

Decomposition of sulphuryl chloride in the gas
phase proceed by first order kinetics.

^{SO}2^{Cl}2(g) ^{ -- -- >
SO}2(g) ^{+ Cl}2(g)

Inversion
of sucrose in acidic aqueous medium follows first order reaction.

C_{12}H_{22}O_{11}
+ H_{2}O --^{H+} --- >^{
}C_{6}H_{12}O_{6} (Glucose)
+ C_{6}H_{12}O_{6}
(Fructose)Decomposition of nitrogen pentoxide in CCl_{4} medium also
exhibits first order kinetics.

N_{2}O_{5} --- > 2 NO_{2}
+ ½ O_{2}

There are many other reactions that proceed by
first order kinetics. We shall study some of the reactions that are
experimentally followed by first order kinetic expressions including the
parameters that change with concentration of the reactants or products which
change with time of progress of the reaction.

Let us consider some of the first order
reactions in detail :

**1. Decomposition of nitrogen
pentoxide in CCl _{4}**

N_{2}O_{2} ---- ^{k1}
--- > 2NO_{2} + 1/2O_{2}

At time *t*
= 0, the volume of oxygen liberated is zero. Let V_{t} and V_{¥} be the
measured volumes of oxygen liberated after the reactant has reacted in '*t*' time and at completion (*t* = ¥). Initial concentration of N_{2}O_{5}
is proportional to total volume of oxygen liberated (i.e.,) (V_{¥}).

(V_{¥}-V_{t}) is proportional to undecomposed
N_{2}O_{5} at time ' *t*'.

K_{1} = (2.303 / t) log ( V_{¥} /(V_{¥}-V_{t}) )sec^{-1}

**2. Decomposition of H _{2}O_{2}
in aqueous solution**

H_{2}O_{2} --- ^{Pt}
--- > H_{2}O + ½ O_{2}

The decomposition of H_{2}O_{2}
in aqueous medium in the presence of Pt catalyst follows a first order
reaction. The progress of the reaction is followed by titrating equal volumes
of the reaction mixture at regular time intervals against standard KMnO_{4}
solution.

Since volume of KMnO_{4} used in the
titration is a measure of concentration of undecomposed H_{2}O_{2},
volume of KMnO_{4} consumed at *t *=
0 is 'V* *_{o}' which
proportional to '*a*', the initial
concentration of H* *_{2}O_{2}.
V_{t}* *is proportional to
unreacted H_{2}O_{2} which is similarly (*a - x* ). Similarly (V_{o}-V_{t}) is proportional to
' *x*', the concentration of H_{2}O_{2}
reacted in time interval '*t*'. V_{t}
is the volume of KMnO_{4} consumed after time ' *t*' of the reaction.

The first order rate constant ' *k*_{1}' of the reaction is,

K_{1} = (2.303 / t) log ( V_{0} / V_{t} )sec^{-1}

**Half life period 't _{½}'**

Half life period, 't _{½}', of a
reaction is defined as the time required to reduce the concentration of a
reactant to one half of its initial value. t_{½} values are calculated
by using the integrated rate equation of any order of a reaction.

For first order reaction,

K_{1} = (2.303 / t) log ( a / (a-x) )sec^{-1}

if amount reacted *x* = _{2} then *t= t _{1/2}*

*t _{1/2} = *(2.303 / k

*t _{1/2} = *(2.303 / k

*t _{1/2} = *0.693/ k

Thus half life period of a first order reaction
is independent of the initial concentration of the reactant and also, inversely
proportional to the rate constant of the reaction.

^{ }

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11th 12th std standard Class Organic Inorganic Physical Chemistry Higher secondary school College Notes : Rate equation for first order reactions |

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