Diffusion in one dimension

Question

I was wondering where the negative sign comes from in the following simplification for diffusion in 1 dimension. The initial equation is given as follows:

$$\frac{\partial C_i}{\partial t} A \Delta x= (N_{ix}|_{x} - N_{ix}|_{x+\Delta x}A + R_iA \Delta x )$$

Which simplifies to:

$$ \frac{\partial C_i}{\partial t} = -\frac{\partial N_{ix}}{\partial x} + R_i$$

Answer 1

Because the definition of derivative is:

$\lim_{\Delta x\rightarrow0}\frac{f_{x+\Delta x}-f_x}{\Delta x}=\frac{\partial f}{\partial x}$

Applied to this case:

$\lim_{\Delta x\rightarrow0}\frac{N_{ix}|_{x}-N_{ix}|_{x+\Delta x}}{\Delta x}=-lim_{\Delta x\rightarrow0}{\Delta x}\frac{N_{ix}|_{x+\Delta x}-N_{ix}|_{x}}{\Delta x}=-\frac{\partial N}{\partial x}$

Answer 2

because you go from high to low. For example, if you want to go from 10 to 1, you "move forward -9 steps"