First-order Kinetics (the Unimolecular Reaction)

Scheme:

`Astackrel(k) rarrB`

Differential Equation:

`A'(t)=-kA`

Mass balance equation:

`A_0=A+B`

Initial conditions:

`A(0)=A_0` and `B(0)=B_0`

Separable Equation:

`(dA)/A=-kt`

Solution:

`A_t=A_0exp(-kt)`

`B_t=A_0-A_0exp(-kt)`

Graph:

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Derivation:

The solution can be obtained directly by integration of the separable equation,

`int_(A_0)^(A_t)(dA)/A=-kt`

`[lnA]_(A_0)^(A_t)=-kt`

`lnA_t=lnA_0-kt`