Convergent First-order Kinetics

Scheme:

`Astackrel(k_1) rarrB stackrel(k_2)larrC`

Differential Equation:

`A'(t)=-k_1A` `B'(t)=k_1A+k_2C` `C'(t)=-k_2C`

Mass balance equation:

`A+B+C=A_0+C_0`

Initial conditions:

`A(0)=A_0`, `B(0)=0` and `C(0)=C_0`

Solution:

`A_t=A_0exp(-k_1t)` `B_t=A_0-A_0exp(-k_1t)+C_0-C_0exp(-k_2t)` `C_t=C_0exp(-k_2t)`

Graph:

Plotting A₀=2.1Mol, B₀=0, C₀=3.3Mol, k1=1e-2, k2=1.7e-2

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Derivation:

The solution can be obtained directly by integration of the separable equations. First `A'(t),`

`int_(A_0)^(A_t)(dA)/A=-k_1t`

`lnA_t=lnA_0-k_1t`

`A(t)=A_0exp(-k_1t)`

Next, `C'(t)`,

`int_(C_0)^(C_t)(dC)/C=-k_2t`

`lnC_t=lnC_0-k_2t`

`C(t)=C_0exp(-k_2t)`

We can now find `B(t)` through the mass-balance expression,

`B=A_0-A+C_0-C`

Thus,

`B(t)=A_0-A_0exp(-k_1t)+C_0-C_0exp(-k_2t)`