What is half-life `(t_(1//2))` of a reaction ? Derive the equations for the 'half-life' value of zero and first order reactions.

Explanation

The time required for the initial concentration of the reactants to become half of it's value during the progress of the reaction is called half life `(t_(1//2))` of reaction
Eg: The radio active of C-14 is exponential with a half life of 5730 years.
Half life of zero order reaction :
Rate expression `k=([R]_(0)-[R])/(t)`
`At t = t_(1//2), [R] =([R_(0)])/(2)`
`k=([R]_(0)-[R_(0)]//2)/(t_(1//2))implies t_(1//2)=([R_(0)])/(2k)`
Half-life of first order reaction :
Rate expression `k=(2.303)/(t)log ""([R]_(0))/([R])`
`Att =t_(1//2), R=([R]_(0))/(2)`
`therefore k=(2.303)/(t_(1//2))log ""([R]_(0))/([R]_(0//2))=(2303)/(t_(1//2))log 2`
`k=(0.693)/(t_(1//2)),t_(1//2)=(0.693)/(k)`