## First order reaction-

source : askIITians

## First order reaction-

#### ” A first order reaction is that in which the rate of reaction is determined by the change of one concentration term of reactant only.”

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

C_{12} H_{22} O_{11} + H_{2}O ————> C_{6}H_{12}O_{6} + C_{6}H_{12}O_{6}

Rate = K [C_{12} H_{22} O_{11}]

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

N_{2}O_{5} (g) ——– > 2 NO_{2} (g) + 1/2 O_{2} (g)

Rate = K [N_{2}O_{5}]

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

NH_{4}NO_{2} ——– > N_{2} (g) + 2 H_{2}O

Rate = K [NH_{4}NO_{2}]

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}

^{-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 = t _{1/2} then x = a/2*

*K = 2.303/ t _{1/2} [ log a / (a- a/2)]*

*K = 2.303/ t _{1/2} [ log a / ( a/2)]*

*K = 2.303/ t _{1/2} [ log 2]*

*K = 2.303/ t _{1/2} [ 0.3010]*

*K = 2.303 x 0.3010 / t _{1/2}*

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

_{1/2}= 0.693 / K