Problem 5.3.11

Solution

a. Heat flux density

Heat flux arises due to molecules transferring kinetic energy from warmer layers to cooler ones.
Let us estimate the average thermal velocity of molecules near the Earth's surface:
$$\langle v \rangle \sim \sqrt{\frac{k T_0}{m}}$$
where $k$ is the Boltzmann constant. Since $\alpha h \ll T_0$, we can consider this velocity to be characteristic of the entire considered layer.

The number of molecules passing through a unit area per unit time is proportional to $n \langle v \rangle$. These molecules bring energy from a layer located at a distance equal to the mean free path $\lambda$. The temperature difference between the exchanging layers is:
$$\Delta T \approx \frac{dT}{dh} \lambda = \alpha \lambda$$
The energy difference carried by one molecule is:
$$\Delta E \sim k \Delta T = k \alpha \lambda$$
Then the heat flux density $q$ is estimated as the product of the particle flux and the transferred energy difference:
$$q \sim n \langle v \rangle \Delta E \sim n \sqrt{\frac{k T_0}{m}} \cdot k \alpha \lambda = n \lambda k \alpha \sqrt{\frac{k T_0}{m}}$$

Dependence on concentration: It is known that the mean free path $\lambda$ is inversely proportional to the concentration of molecules ($\lambda \sim 1/n$). Therefore, the product $n \lambda$ remains constant.
Conclusion: as the number of molecules per unit volume increases, the heat flux density will not change.

b. Comparison of thermal conductivities

The thermal conductivity coefficient of a gas $\kappa$ is determined by the formula of the kinetic theory:
$$\kappa \sim n \langle v \rangle \lambda c_1$$
where $c_1$ is the heat capacity of one molecule. Since hydrogen ($H_2$) and air ($N_2, O_2$) are diatomic gases, the heat capacity of one molecule is the same for both ($c_1 = \frac{5}{2}k$).
Given that $\lambda \sim \frac{1}{n r^2}$ (where $r$ is the molecular radius), the concentration $n$ cancels out:
$$\kappa \sim \langle v \rangle \frac{1}{r^2}$$
The thermal velocity $\langle v \rangle$ is inversely proportional to the square root of the molar mass $\mu$ ($\langle v \rangle \sim 1/\sqrt{\mu}$). The final dependence is:
$$\kappa \sim \frac{1}{r^2 \sqrt{\mu}}$$
Let us find the ratio of the thermal conductivities of hydrogen and air. The molar mass of hydrogen is $\mu_{H_2} \approx 2$ g/mol, and the molar mass of air is $\mu_{air} \approx 29$ g/mol:
$$\frac{\kappa_{H_2}}{\kappa_{air}} = \left( \frac{r_{air}}{r_{H_2}} \right)^2 \sqrt{\frac{\mu_{air}}{\mu_{H_2}}}$$
Substituting the numerical values:
$$\frac{\kappa_{H_2}}{\kappa_{air}} = \left( \frac{0.18}{0.14} \right)^2 \sqrt{\frac{29}{2}} = \left( \frac{9}{7} \right)^2 \sqrt{14.5} \approx 1.653 \cdot 3.808 \approx 6.3$$
The thermal conductivity of hydrogen is approximately $6.3$ times greater.