## First order reaction- source : askIITians

## First order reaction-

#### Ex.1) Inversion of cane sugar-

C12 H22 O11 + H2O ————> C6H12O6 + C6H12O6

Rate = K [C12 H22 O11]

#### Ex.2) Decomposition of nitrogen penta oxide-

N2O5 (g)  ——– > 2 NO2 (g) + 1/2 O2 (g)

Rate = K [N2O5]

#### Ex.3) Decomposition of  ammonium nitrite in aqueous solution –

NH4NO2   ——– >  N2 (g) + 2 H2O

Rate = K [NH4NO2]

Consider the reaction,

A      ————->   B          +      C

initial conc.                         a                                 0                   0

conc. after’t’ time             a- x                              x                   x

Rate = dx / dt = K[A] n

n = 1

dx / dt = K[A]

‘K ‘ is rate constant or velocity constant

dx / dt = K ( a- x)

dx /(a- x) = K.dt

Taking integration of both sides,

∫ dx /(a- x) = K ∫ dt

– ln( a- x) = Kt + C        ———-  eq. 1

C = integration constant

If  t = 0  then x = 0

– ln( a- 0) = K x 0 + C

-ln a = C    ——— eq.2

Putting the value of ‘C’ from eq.2  to eq. 1

– ln( a- x) = Kt  – ln a

ln a – ln( a- x) = Kt

K = 1/ t [ ln a / (a- x)]

#### K =  2.303/ t  [ log a / (a- x)]

If  log [a / (a-x)] is plotted against ‘t’, then a straight line passing through the origin is obtained and slope is ‘ 2.303/ K’.

## Characteristics of first order reaction –

1) Unit of ‘K’-

K =  2.303/ t  [ log a / (a- x)]

unit of ‘K’ = 1 / time ( mole  litre-1 / mole  litre-1 )

unit of ‘K’ =  time-1.

Time may be in hour , minute or second etc.

#### unit of ‘K’ =  hour-1  or minute -1 or second -1

2) The unit of velocity constant is independent of the units of concentration because,

unit of ‘K’ =  time-1

3) The time taken to complete a  half reaction is independent of the initial concentration of the reactant.

K =  2.303/ t  [ log a / (a- x)]

If t = t1/2 then x = a/2

K =  2.303/ t1/2  [ log a / (a- a/2)]

K =  2.303/ t1/2  [ log a / ( a/2)]

K =  2.303/ t1/2  [ log 2]

K =  2.303/ t1/2  [ 0.3010]

K = 2.303 x 0.3010 / t 1/2