I have the following mathematical expression:
$ D_{k,mixture} = \left( 1-X_k \right) \left(\sum\limits^N_{\substack{j=1 \\ j \neq k}} \frac{X_j}{D_{k,j}} \right)^{-1} $
Which is one way of expressing a diffusion coefficient from species $k$ towards a mixture. Inside the expression, we have molar fractions $X$ and also binary diffusion coefficients $D_{k,j}$ between species $k$ and $j$. These binary diffusion coefficients $\textbf{do not}$ depend on molar fractions $X$. Of paramount importance is the following additional information:
$ \sum_{m=1}^N X_m = 1 $
Which is the physical trivial constraint that molar fractions from all species in a mixture when summed must be unity. I want to prove that the following limit:
$ \mathcal{L} = \lim\limits_{X_k \to 1} \left( 1-X_k \right) \left(\sum\limits^N_{\substack{j=1 \\ j \neq k}} \frac{X_j}{D_{k,j}} \right)^{-1} $
either converges to a specific value or at least that it is bounded between the smallest $D_{k,j}, k \neq j$ and the largest $D_{k,j}, k \neq j$ which is my current suspicion from numerical experiments. My understanding is that this limit can be evaluated using L'Hopital's rule (please do correct me if I'm wrong). My first approach was to do:
$ \mathcal{L} = \lim\limits_{X_k \to 1} \left( 1-X_k \right) \left(\sum\limits^N_{\substack{j=1 \\ j \neq k}} \frac{X_j}{D_{k,j}} \right)^{-1} = \lim\limits_{X_k \to 1} \frac{\frac{\partial}{\partial X_k} (1-X_k)}{\frac{\partial}{\partial X_k} \left(\sum\limits^N_{\substack{j=1 \\ j \neq k}} \frac{X_j}{D_{k,j}} \right)} = \lim\limits_{X_k \to 1} \frac{-1}{\frac{\partial}{\partial X_k} \left(\sum\limits^N_{\substack{j=1 \\ j \neq k}} \frac{1-X_k-\sum\limits^N_{\substack{i=1 \\ i \neq j, i \neq k}} X_i}{D_{k,j}} \right)} $
in which the last substitution, inside the denominator, comes from the unity constraint, or the fact that $X_j+X_k+\sum\limits^N_{\substack{i=1 \\ i \neq j, i \neq k}} X_i=1$. Taking the differentiation of this specific expression would lead to:
$ \mathcal{L} = {\left(\sum\limits^N_{\substack{j=1 \\ j \neq k}} \frac{1}{D_{k,j}} \right)}^{-1} $
However, my numerical computations do not lead to this specific result. I suspect the fallacy comes from the substitution I did (I selectively only substitute the explicit dependency on $X_k$ only for $X_j$, and not for all other $X_i$). I'm in need of help from fellow math guys and girls. I do apologize in advance if anything is not clear and for any obvious mistakes I've made - I come from a physics/engineering background and therefore am quite rusty regarding limit computations. Thanks in advance!