a. The relative content of radioactive atoms in a gas is small. Their number per unit volume increases linearly with height: $n = \alpha h$. The mass of an atom is $m$, its mean free path is $\lambda$, and the temperature is $T$. Estimate the density of these atoms on the ground.
b. Estimate the diffusion coefficient of water vapor in air at 20°C. The radius of water molecules is 0.21 nm. The radius of nitrogen and oxygen molecules is 0.18 nm.
### Solution
<b>Part a)</b>
Based on the given parameters (the presence of mass $m$ and temperature $T$), the problem requires estimating not just the concentration, but the "flux density" of atoms to the ground.
Let us consider a horizontal surface of unit area located at height $h$. It is crossed by two opposing fluxes of atoms.
The atoms crossing the surface from top to bottom experienced, on average, their last collision at a distance of the mean free path $\lambda$ above it, that is, at height $h + \lambda$. The density of this flux is directed downwards and is equal to:
$$W_1 \approx\frac{1}{2} n_{h+\lambda}\bar{v}_z$$
where $n_{h+\lambda} = \alpha(h + \lambda)$ is the concentration of atoms at height $(h + \lambda)$, and $\bar{v}_z \approx\sqrt{\frac{kT}{m}}$ is the characteristic thermal velocity of the directed motion of molecules along the $z$-axis.
Similarly, the flux of atoms going from bottom to top is formed by particles flying from height $h - \lambda$:
Converting to square millimeters, we obtain $D \approx 12 \text{ mm}^2/\text{s}$.
+
<b>Theory Reference:</b>
+
For a detailed derivation of the mean free path utilized in this solution, you can refer to <i>Physics</i> by Halliday, Resnick, and Krane (HRK), Volume 1, Chapter 22, Section 22-3 ("The mean free path"), page 502. It thoroughly describes the effective cross-section concept ($\sigma = \pi d^2$) and the concentration relation from the ideal gas law ($1/n = kT/P$) that form the basis of these molecular-kinetic equations.
+
#### Answer
a. $W = W_1 - W_2 \approx\alpha\lambda\sqrt{kT/m}$
b. $D \approx 12 \text{ mm}^2/\text{s}$
### Statement
### Statement
$5.3.8.$
$5.3.8.$
a. The relative content of radioactive atoms in a gas is small. Their number per unit volume increases linearly with height: $n = \alpha h$. The mass of an atom is $m$, its mean free path is $\lambda$, and the temperature is $T$. Estimate the density of these atoms on the ground.
a. The relative content of radioactive atoms in a gas is small. Their number per unit volume increases linearly with height: $n = \alpha h$. The mass of an atom is $m$, its mean free path is $\lambda$, and the temperature is $T$. Estimate the density of these atoms on the ground.
b. Estimate the diffusion coefficient of water vapor in air at 20°C. The radius of water molecules is 0.21 nm. The radius of nitrogen and oxygen molecules is 0.18 nm.
b. Estimate the diffusion coefficient of water vapor in air at 20°C. The radius of water molecules is 0.21 nm. The radius of nitrogen and oxygen molecules is 0.18 nm.
### Solution
### Solution
<b>Part a)</b>
<b>Part a)</b>
Based on the given parameters (the presence of mass $m$ and temperature $T$), the problem requires estimating not just the concentration, but the "flux density" of atoms to the ground.
Based on the given parameters (the presence of mass $m$ and temperature $T$), the problem requires estimating not just the concentration, but the "flux density" of atoms to the ground.
Let us consider a horizontal surface of unit area located at height $h$. It is crossed by two opposing fluxes of atoms.
Let us consider a horizontal surface of unit area located at height $h$. It is crossed by two opposing fluxes of atoms.
The atoms crossing the surface from top to bottom experienced, on average, their last collision at a distance of the mean free path $\lambda$ above it, that is, at height $h + \lambda$. The density of this flux is directed downwards and is equal to:
The atoms crossing the surface from top to bottom experienced, on average, their last collision at a distance of the mean free path $\lambda$ above it, that is, at height $h + \lambda$. The density of this flux is directed downwards and is equal to:
$$W_1 \approx\frac{1}{2} n_{h+\lambda}\bar{v}_z$$
$$W_1 \approx\frac{1}{2} n_{h+\lambda}\bar{v}_z$$
where $n_{h+\lambda} = \alpha(h + \lambda)$ is the concentration of atoms at height $(h + \lambda)$, and $\bar{v}_z \approx\sqrt{\frac{kT}{m}}$ is the characteristic thermal velocity of the directed motion of molecules along the $z$-axis.
where $n_{h+\lambda} = \alpha(h + \lambda)$ is the concentration of atoms at height $(h + \lambda)$, and $\bar{v}_z \approx\sqrt{\frac{kT}{m}}$ is the characteristic thermal velocity of the directed motion of molecules along the $z$-axis.
Similarly, the flux of atoms going from bottom to top is formed by particles flying from height $h - \lambda$:
Similarly, the flux of atoms going from bottom to top is formed by particles flying from height $h - \lambda$:
Converting to square millimeters, we obtain $D \approx 12 \text{ mm}^2/\text{s}$.
Converting to square millimeters, we obtain $D \approx 12 \text{ mm}^2/\text{s}$.
<b>Theory Reference:</b>
For a detailed derivation of the mean free path utilized in this solution, you can refer to <i>Physics</i> by Halliday, Resnick, and Krane (HRK), Volume 1, Chapter 22, Section 22-3 ("The mean free path"), page 502. It thoroughly describes the effective cross-section concept ($\sigma = \pi d^2$) and the concentration relation from the ideal gas law ($1/n = kT/P$) that form the basis of these molecular-kinetic equations.
#### Answer
#### Answer
a. $W = W_1 - W_2 \approx\alpha\lambda\sqrt{kT/m}$
a. $W = W_1 - W_2 \approx\alpha\lambda\sqrt{kT/m}$
