Gases

Appendices

Summary

• For a constant temperature the pressure p of a gas in a closed but variable volume V obeys the Boyle–Mariotte law: $p\cdot V=\mathrm{const}$

• The air pressure in an isothermal atmosphere decreases exponential with the altitude h above ground, due to the gravitational force.

• For the particle density $n(h)$ holds:

  $$\displaystyle N(h)=n(0)\cdot\exp[-mgh/kT]{\;}.$$

  Without mixing effects in the atmosphere the concentration of particles with larger mass m therefore decreases faster with h than for those with lighter mass.

• The real earth atmosphere in not isothermal. Due to upwards and downwards air currents the different layers of the atmosphere are mixed which leads to an equilibrium of the concentrations of different masses.

• The kinetic gas theory explains the macroscopic features of gases such as pressure and temperature by the average momentum and the kinetic energy of the gas molecules. With the Boltzmann constant k the mean kinetic energy of molecules with mass m is related to the temperature T by $({1}/{2})m\overline{*}{v^{2}}=(3/2)kT$.

• The velocity distribution $n(v)$ of gas molecules at thermal equilibrium is the Maxwell–Boltzmann distribution $n(v)\,\mathrm{d}v=v^{2}\cdot\exp[-({1}/{2})mv^{2}/kT]\,\mathrm{d}v$ for the magnitude $v=|\boldsymbol{v}|$ of the velocity. The distribution $n(v_{i})$, $(i=x,y,z)$ of the velocity components v _i is a Gaussian function, symmetric to $v_{i}=0$.

• These distributions can be experimentally determined in molecular beams using mechanical velocity selectors. The molecular beams are formed by expanding a gas from a reservoir through a small hole into the vacuum, where the mean free path is longer than the dimensions of the vacuum chamber. The beam can be collimated by small apertures which transmit only molecules with small transverse velocities.

• Always when gradients of concentrations in a gas exist, diffusion processes occur which try to equalize the concentrations. The mean diffusion particle flux $j_{\mathrm{D}}=-D\cdot\mathrm{grad}\,\,n$ is proportional to the gradient of the particle density n. The diffusion constant D depends on the kind of particles. Diffusion causes a mass transport from regions of high particle density n to those of low density.

• If velocity gradient in a gas flow appear, viscosity causes momentum transfer from particles with higher flow velocity to those with lower velocity.

• If temperature gradients appear in a gas, energy is transported by diffusing molecules from regions of higher temperature to those of lower temperature. For a one-dimensional temperature gradient $\,\mathrm{d}T/\,\mathrm{d}x$ the transferred heat power is $\,\mathrm{d}W/\,\mathrm{d}t=\lambda\cdot\,\mathrm{d}T/\,\mathrm{d}x$. The heat conductivity λ depends on the particle density n, the mean velocity $\overline{*}{v}$, and the mean free path Λ.

• The density distribution $n(h)$ in the atmosphere is determined by the common action of gravitational attraction of the air molecules by the earth and the diffusion current from regions with higher density to those with lower density. In the real earth atmosphere furthermore vertical and horizontal air currents occur caused by local heat sources due to absorption of sun radiation and infrared radiation from the earth surface. This convection leads to a mixing of different layers in the lower atmosphere.

Problems

7.1

What would be the density distribution in the atmosphere, if the dependence of the gravitational force on the altitude is taken into account?

7.2

At which altitude exists, according to (7.6a)–(7.6c), a pressure of 1 mbar, if the constant value $T=300\,\mathrm{K}$ is assumed for the temperature $T(h)$?

7.3

Calculate from (7.6a)–(7.6c) the pressure at $h=100\,\mathrm{k}\mathrm{m}$ and the density n for $T=250\,\mathrm{K}$.

7.4

A balloon with $V=3000\,\mathrm{m}^{3}$ floats at $h=1000\,\mathrm{m}$ and a temperature of $20\,{}^{\circ}\text{C}$. What is the maximum weight of balloon with ballast mass and passengers (without the weight of the filling gas) if one uses as filling gas

• a) helium

• b) hydrogen gas H${}_{2}$

at a pressure equal to the external air pressure.($\varrho_{\mathrm{air}}=1.293\,\mathrm{k}\mathrm{g}/\mathrm{m}^{3}$, $\varrho_{\mathrm{He}}=0.1785\,\mathrm{k}\mathrm{g}/\mathrm{m}^{3}$, $\varrho_{\mathrm{H}_{2}}=0.09\,\mathrm{k}\mathrm{g}/\mathrm{m}^{3}$ at $T=20\,{}^{\circ}\text{C}$ and $p=10^{5}\,\mathrm{Pa}$).

7.5

A shop for diving equipment offers for measuring the diving depth a glass tube with movable piston that compresses a gas volume $V=A\cdot x$. Down to which depth is the uncertainty of the device $\Delta z\leq 1\,\mathrm{m}$ if the piston edge can be read with an accuracy of $1\,\mathrm{m}\mathrm{m}$ and $x(p_{0})=0.2\,\mathrm{m}$.

7.6

Which fraction of all gas molecules has a free path that is larger than

• a) the mean free path Λ

• b) $2\Lambda$?

7.7

Calculate the probability that N${}_{2}$-molecules in a gas at $T=300\,\mathrm{K}$ have velocities within the interval $900\,\mathrm{m}/\mathrm{s}\leq v\leq 1000\,\mathrm{m}/\mathrm{s}$. What is the total number $N(v)$ of molecules with velocities within this interval in a volume $V=1\,\mathrm{m}^{3}$ at $T=300\,\mathrm{K}$ and $p=10^{5}\,\mathrm{Pa}$?

7.8

What is the thickness $\Delta z$ of an isothermal atmospheric layer at $T=280\,\mathrm{K}$ between the altitudes z ₁ and z ₂ with $p(z_{1})=1000\,\mathrm{h}\mathrm{Pa}$ and $p(z_{2})=900\,\mathrm{h}\mathrm{Pa}$?

7.9

What is the square root of the mean square relative velocities between two gas molecules

• a) for a Maxwell distribution

• b) if the magnitudes of all velocities are equal but the directions uniformly distributed?

7.10

The mean free path Λ in a gas at $p=10^{5}\,\mathrm{Pa}$ and $T=20\,{}^{\circ}\text{C}$ is for argon atoms $\Lambda_{\mathrm{Ar}}=1\cdot 10^{-7}\,\mathrm{m}$ and for N${}_{2}$ molecules $\Lambda_{\mathrm{N}_{2}}=2.7\cdot 10^{-7}\,\mathrm{m}$.

• a) What are the collision cross sections $\sigma_{\mathrm{Ar}}$ and $\sigma_{\mathrm{N}_{2}}$?

• b) How large are the mean times between two successive collisions?

7.11

In a container is $0.1\,\mathrm{k}\mathrm{g}$ helium at $p=10^{5}\,\mathrm{Pa}$ and $T=300\,\mathrm{K}$. Calculate

• a) the number of He-atoms,

• b) the mean free path Λ,

• c) the sum $\sum S_{i}$ of all path lengths S _i which is passed by all molecules in $1\,\mathrm{s}$. Give this sum in the units m and light years.

7.12

The rotating disc of a velocity selector with a slit allows N${}_{2}$ molecules with a Maxwellian distribution at $T=500\,\mathrm{K}$ to pass for a time interval $\Delta t=10^{-3}\,\mathrm{s}$. A detector at $1\,\mathrm{m}$ distance from the disc measures the time distribution of the molecules. What is the half width of this distribution?

7.13

What is the minimum velocity of a helium atom at $100\,\mathrm{k}\mathrm{m}$ above ground for leaving the earth into space? At which temperature would half of the N${}_{2}$-molecules above $100\,\mathrm{k}\mathrm{m}$ altitude escape into space?

7.14

The exhaust gases of a factory escaping out of a $50\,\mathrm{m}$ high smokestack have the density $\varrho=0.85\,\mathrm{k}\mathrm{g}/\mathrm{m}^{3}$. How large is the pressure difference at the base of the smokestack to that of the surrounding air with $\varrho_{\mathrm{air}}=1.29\,\mathrm{k}\mathrm{g}/\mathrm{m}^{3}$?

7.15

Up to which volume a children’s balloon (m = 10g) has to be blown and filled with helium at a pressure of 1.5 bar, in order to let it float in air?

7.16

In the centre of the sun the density of protons and electrons is estimated as $n=5\cdot 10^{29}/{\mathrm{m}}^{3}$ at a temperature of $1.5\cdot 10^{7}\,\mathrm{K}$.

• a) What is the mean kinetic energy of electrons and protons? Compare this with the ionization energy of the hydrogen atom ($E_{\mathrm{ion}}=13.6\,\mathrm{eV}$).

• b) What are the mean velocities?

• c) How large is the pressure?

7.17

Determine the total mass of the earth atmosphere from the pressure $p=1\,\mathrm{atm}=1013\,\mathrm{h}\mathrm{Pa}$ the atmosphere exerts onto the earth surface.

7.18

A research balloon has without filling a mass $m=300\,\mathrm{k}\mathrm{g}$. How large must be the volume of helium inside the balloon to let it rise up if the helium pressure at any height is always 0.1 bar higher than that of the surrounding air? ($T(h=0)=300\,\mathrm{K}$, $T(h=20\,\mathrm{k}\mathrm{m})=217\,\mathrm{K}$)

7.19

What would be the height of the earth atmosphere

• a) if the atmosphere is compressed with a pressure at the upper edge (assumed to be sharp) of 10 atm at a temperature of $300\,\mathrm{K}$?

• b) at $T=0\,\mathrm{K}$ where all gases are solidified?