Find pH and Kw for water with given values of conductivity At $\pu{25 ^\circ C}$, $\lambda_\infty (\ce{H+}) = \pu{3.5 \times 10^{-2} S m2 mol-1}$ and $\lambda_\infty (\ce{OH-}) = \pu{2 \times 10^{-2} S m2 mol-1}$. Determine $\mathrm{pH}$ and $K_\mathrm{w}$.


Given: Specific conductance $= \pu{5.5 \times 10^{-6} Sm-1}$ for $\ce{H2O}$

My Attempt: 

I know I need to find the concentration of $\ce{H+}$ or $\ce{OH-}$ for $\mathrm{pH}$. $K_\mathrm{w}$ can be found subsequently. However, I have no clue as to how do I find that with the given information. 

I am also aware of the relation between conductivity, Specific conductance, and molarity. It is easy to find the molarity from the given information, but that would be of no use as we need the concentration of the individual ions rather than the molarity of the solution.

Any help would be appreciated.
We know,

$$\lambda_c = \frac{(\text{Specific Conductivity})}{1000 \times c}$$

$$\alpha\lambda_\infty = \lambda_c$$ 

(where $c$ is concentration in moles per litres of the electrolyte and $\alpha$ is degree of dissociation of the electrolyte)

$\lambda_{\infty}$ of water (the electrolyte) is the sum of $\lambda_{\infty}$ of $\ce{H+}$ and $\ce{OH-}$.

From equations above, one can easily find  the the value of $c\alpha$ is $10^{-7}$, which is the concentration of $\ce{H+}$.