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Derive integrated rate equation for rate constant for a first order reaction.

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SOLUTION :`R to P`
Rate`=-(d[R])/(dt)=k[R] or (d[R])/([R])=-kdt`
Integrating this equation
In [R]`=-kt+I ""…(i)`
where I is constant of integration.
When `t=0, R=[R]_(0)`, where `[R]_(0)` is initial concentration.
The equation (i) can be written as
`"In [R]_(0)= -k xx 0+I`
or `"In [R]_(0)=I`
Substituting the value of I in equation (i)
`"In [R]"= -kt +"In [R]"_(0) or -kt = "In [R]" -"In [R]"_(0)`
or `k=(1)/(t)"In"([R]_(0))/([R]) or k-(2.303)/(t)"log"([R]_(0))/([R])`


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