I'm wondering if it is possible to find an analytical solution for a diffusion reaction equation where:
A + B ---> R
R + B ---> S
So in terms of molecular diffusion and reaction (no advection in this problem):
$$\frac{\partial c_A}{\partial t} = D\frac{\partial ^2 c_A}{\partial x^2} - k_1c_Ac_B$$
$$\frac{\partial c_B}{\partial t} = D\frac{\partial ^2 c_B}{\partial x^2} - k_1c_Ac_B - k_2c_Rc_B$$
$$\frac{\partial c_R}{\partial t} = D\frac{\partial ^2 c_R}{\partial x^2} + k_1c_Ac_B - k_2c_Rc_B$$
Let:
$$z = c_A + c_R$$ $$w = c_B - c_A$$
Subtracting 3 from 4 yields:
- $$\frac{\partial (c_B - c_A)}{\partial t} = D\frac{\partial ^2 (c_B - c_A)}{\partial x^2} - k_2c_Bc_R$$
The first reaction is virtually instantaneous so A and B cannot coexist in the same domain. So, where A exists (and B is absent), $w < 0$ and where B exists (and A is absent), $w > 0$. So. in this case $w = c_B$ and $z = c_R$. Equation 6 can, I think, be written as:
- $$\frac{\partial w}{\partial t} = D\frac{\partial ^2 w}{\partial x^2} - k_2wz$$
In a similar manner if equations 3 and 5 are added:
- $$\frac{\partial z}{\partial t} = D\frac{\partial ^2 z}{\partial x^2} - k_2wz$$
The physical setup for this reaction is as follows:
Initially there is a slab of A next to a slab of B. Both slabs are of equal thickness, $\delta$. The initial conditions are:
at $t = 0$ and $-\delta \leq x \leq 0$,
$$c_A = c_{A0}$$
at $t = 0$ and $0 \leq x \leq +\delta$,
$$c_B = c_{B0}$$
The boundary conditions are: at $x = -\delta$
$$\frac{\partial c_A}{\partial x} = 0$$
and at $x = +\delta$
$$\frac{\partial c_B}{\partial x} = 0$$
With assistance from this site, I recently was able to come up with a step-by-step solution for a similar problem that involved a single instantaneous, irreversible reaction. That problem used separation of variables to achieve an analytical solution. A video by the Khan academy suggests that separation of variables might work for this problem but it is initially inhomogenous. Since I am weak at solving differential equations, I'm hoping to get some insight as to whether or not this problem has an analytical solution and if so, how might I go about solving it.
