Edits to “Problem”, “Solution”, “Answer”
en/5.3.11.md
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| @@ -1,11 +1,40 @@ | |||
| − | ### | ||
| + | ### Problem | ||
| − | $5.3.11.$ [Insert the problem statement] | ||
| + | $5.3.11.$ <b>a.</b> The air temperature of the Earth's atmosphere linearly increases with height $h$, $T = T_0 + \alpha h$. In this case, the relative change in temperature $\alpha h/T_0$ remains much less than unity. The mean free path of air molecules is $\lambda$, the mass of each molecule is $m$, and the number of molecules per unit volume of air is $n$. Estimate the heat flux density to the Earth. Will the density of this flux change if the number of molecules per unit volume of air increases?\ | ||
| + | <b>b.</b> How many times is the thermal conductivity of hydrogen greater than the thermal conductivity of air? The radius of hydrogen molecules is $0.14$ nm, the radius of nitrogen and oxygen molecules is $0.18$ nm. The temperature of the gases is the same. | ||
| ### Solution | |||
| − | Studio Cyborg Squad presents | ||
| + | <b>a. Heat flux density</b><br> | ||
| + | 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. | ||
| − | #### Answer | ||
| + | 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: | ||
| + | $$ q \sim n \langle v \rangle \Delta E \sim n \sqrt{\frac{k T_0}{m}} \cdot k \alpha \lambda = n \lambda k \alpha \sqrt{\frac{k T_0}{m}} $$ | ||
| − | |||
| + | <b>Dependence on concentration:</b> 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. | ||
| + | <i>Conclusion:</i> as the number of molecules per unit volume increases, the heat flux density <b>will not change</b>. | ||
| + | |||
| + | <b>b. Comparison of thermal conductivities</b><br> | ||
| + | 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. | ||
| + | |||
| + | #### Answer | ||
| + | <b>a.</b> $q \sim n \lambda k \alpha \sqrt{\frac{k T_0}{m}}$. Will not change. | ||
| + | <b>b.</b> By a factor of approximately $6.3$. | ||
| @@ -1,11 +1,40 @@ | |||
| ### |
### Problem | ||
| $5.3.11.$ [Insert the problem statement] | $5.3.11.$ <b>a.</b> The air temperature of the Earth's atmosphere linearly increases with height $h$, $T = T_0 + \alpha h$. In this case, the relative change in temperature $\alpha h/T_0$ remains much less than unity. The mean free path of air molecules is $\lambda$, the mass of each molecule is $m$, and the number of molecules per unit volume of air is $n$. Estimate the heat flux density to the Earth. Will the density of this flux change if the number of molecules per unit volume of air increases?\ | ||
| <b>b.</b> How many times is the thermal conductivity of hydrogen greater than the thermal conductivity of air? The radius of hydrogen molecules is $0.14$ nm, the radius of nitrogen and oxygen molecules is $0.18$ nm. The temperature of the gases is the same. | |||
| ### Solution | ### Solution | ||
| Studio Cyborg Squad presents | <b>a. Heat flux density</b><br> | ||
| 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. | |||
| #### Answer | 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: | |||
| $$ q \sim n \langle v \rangle \Delta E \sim n \sqrt{\frac{k T_0}{m}} \cdot k \alpha \lambda = n \lambda k \alpha \sqrt{\frac{k T_0}{m}} $$ | |||
| <b>Dependence on concentration:</b> 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. | |||
| <i>Conclusion:</i> as the number of molecules per unit volume increases, the heat flux density <b>will not change</b>. | |||
| <b>b. Comparison of thermal conductivities</b><br> | |||
| 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. | |||
| #### Answer | |||
| <b>a.</b> $q \sim n \lambda k \alpha \sqrt{\frac{k T_0}{m}}$. Will not change. | |||
| <b>b.</b> By a factor of approximately $6.3$. | |||