Solution

Let's determine the thermal conductivity coefficients of gases in a mixture, denote them $\kappa'_{1}$ and $\kappa'_{2}$. Since the gases are monatomic and the temperatures are the same, we can assume that:

$$\kappa\sim \frac{1}{\sigma\cdot \sqrt{\mu}}$$

This means that after all the transformations we can obtain:

$$\frac{\kappa_{1}}{\kappa_{2}}=\sqrt{\frac{\mu_{2}}{\mu_{1}}}\cdot \frac{r_{2}^2}{r_{1}^2}$$

Where $r_{1}$ and $r_{2}$ are the radii of the gas atoms.

$$r_{2}= r_{1}\cdot (\frac{\kappa_{1}}{\kappa_{2}}\cdot \sqrt{\frac{\mu_{1}}{\mu_{2}}})^{\frac{1}{2}}$$

The concentrations of gases in a mixture are related as $$\frac{n_{1}}{n_{2}}=\alpha$$

The number of collisions of molecules of gas 1 and gas 2 in the mixture:

$$\nu_{1}=4\cdot \pi\cdot r_{1}^{2}\cdot n_{1}\cdot <\upsilon_{sq_{1}}> + \frac{n_{1}}{\alpha}\cdot \pi\cdot (r_{1} + r_{2})^{2}\cdot <\upsilon_{sq_{1}}>$$

$$\nu_{2}=4\cdot \pi\cdot r_{2}^{2}\cdot n_{2}\cdot <\upsilon_{sq_{2}}> + n_{2}\cdot \alpha\cdot \pi\cdot (r_{1} + r_{2})^{2}\cdot <\upsilon_{sq_{2}}>$$

Let's write down the ratio of thermal conductivities "before" to "after", after transformations we get:

$$\frac{\kappa_{1}}{\kappa'_{1}}=\frac{4\cdot \pi\cdot r_{1}^{2}\cdot n_{1} + \frac{n_{1}}{\alpha}\cdot \pi\cdot (r_{1} + r_{2})^{2}}{4\cdot \pi\cdot r_{1}^{2}\cdot n_{1}}$$

We substitute the expression for $r_{2}$, complete the transformations and express the new thermal conductivity of gas 1:

$$\kappa'_{1} = \frac{\kappa_{1}}{1 + \frac{1}{4\cdot \alpha}\cdot [1 + (\frac{\kappa_{1}}{\kappa_{2}}\cdot \sqrt{\frac{\mu_{1}}{\mu_{2}}})^\frac{1}{2}]^{2}}$$

Similarly for gas 2:

$$\kappa'_{2} = \frac{\kappa_{2}}{1 + \frac{\alpha}{4}\cdot [1 + (\frac{\kappa_{2}}{\kappa_{1}}\cdot \sqrt{\frac{\mu_{2}}{\mu_{1}}})^\frac{1}{2}]^{2}}$$

Let a temperature gradient $\frac{\partial T}{\partial x}$ be created within the mixture. The heat flux through the mixture is:

$$q_{12} = \kappa_{12}\cdot \frac{\partial T}{\partial x}\cdot S = q_{1} + q_{2} = \kappa'_{1}\cdot \frac{\partial T}{\partial x}\cdot S + \kappa'_{2}\cdot \frac{\partial T}{\partial x}\cdot S$$

From here:
$$\kappa_{12} = \kappa'_{1} + \kappa'_{2}$$

$$\kappa_{12} = \frac{\kappa_{1}}{1 + \frac{1}{4\cdot \alpha}\cdot [1 + (\frac{\kappa_{1}}{\kappa_{2}}\cdot \sqrt{\frac{\mu_{1}}{\mu_{2}}})^\frac{1}{2}]^{2}} + \frac{\kappa_{2}}{1 + \frac{\alpha}{4}\cdot [1 + (\frac{\kappa_{2}}{\kappa_{1}}\cdot \sqrt{\frac{\mu_{2}}{\mu_{1}}})^\frac{1}{2}]^{2}}$$