Chapter 20: The Micro/Macro Connection

Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms ,$\mathrm{H}$not${\mathrm{H}}_2$. The number density is about$1\mathrm{atom}/{\mathrm{cm}}^3$and the temperature is about$3\mathrm{K}$.

a. Estimate the pressure in interstellar space. Give your answer in $\mathrm{Pa}$and in localid="1648635470965" $\mathrm{atm}$.

b. What is the rms speed of the atoms?

c. What is the edge lengthlocalid="1648635477647" $\mathrm{L}$of anlocalid="1648637118909" $\mathrm{L}\times \mathrm{L}\times \mathrm{L}$cube of gas withlocalid="1648635499087" $1.0\mathrm{J}$of thermal energy?

You are watching a science fiction movie in which the hero shrinks down to the size of an atom and fights villains while jumping from air molecule to air molecule. In one scene, the hero's molecule is about to crash head-on into the molecule on which a villain is riding. The villain's molecule is initially$50$molecular radii away and, in the movie, it takes$3.5\mathrm{s}$for the molecules to collide. Estimate the air temperature required for this to be possible. Assume the molecules are nitrogen molecules, each traveling at the rms speed. Is this a plausible temperature for air?

a. Find an expression for the${\mathrm{v}}_{\mathrm{rms}}$of gas molecules in terms of$\mathrm{p}$,$\mathrm{V}$and the total mass of the gas $\mathrm{M}$.

b. A gas cylinder has a piston at one end that is moving outward at speed $v_{\text{piston}}$during an isobaric expansion of the gas. Find an expression for the rate at which is changing in terms of ${\mathrm{v}}_{\text{piston}}$, the instantaneous value of ${\mathrm{v}}_{\text{rms}}$, and the instantaneous value $\mathrm{L}$of the length of the cylinder.

c. A cylindrical sample chamber has a piston moving outward at $0.50\mathrm{m}/\mathrm{s}$during an isobaric expansion. The rms speed of the gas molecules is localid="1648640672000" $450\mathrm{m}/\mathrm{s}$at the instant the chamber length is localid="1648640676590" $1.5\mathrm{m}$. At what rate is localid="1648640708264" ${\mathrm{v}}_{\mathrm{rms}}$changing?

Equation $20.3$is the mean free path of a particle through a gas of identical particles of equal radius. An electron can be thought of as a point particle with zero radius.

a. Find an expression for the mean free path of an electron through a gas.

b. Electrons travel $3\mathrm{km}$through the Stanford Linear Accelerator. In order for scattering losses to be negligible, the pressure inside the accelerator tube must be reduced to the point where the mean free path is at least $50\mathrm{km}$. What is the maximum possible pressure inside the accelerator tube, assuming $T={20}^{°}C$$?$Give your answer in both $\mathrm{Pa}$and $\mathrm{atm}$.

Uranium has two naturally occurring isotopes. ${}^{238}\mathrm{U}$ has a natural abundance of $99.3\%$ and${}^{235}\mathrm{U}$ has an abundance of $0.7\%$. It is the rarer ${}^{235}\mathrm{U}$that is needed for nuclear reactors. The isotopes are separated by forming uranium hexafluoride, role="math" ${\mathrm{UF}}_6$, which is a gas, then allowing it to diffuse through a series of porous membranes. ${}^{235}\mathrm{UF}_6$ has a slightly larger rms speed than ${}^{238}\mathrm{UF}_6$ and diffuses slightly faster. Many repetitions of this procedure gradually separate the two isotopes. What is the ratio of the rms speed of ${}^{235}\mathrm{UF}_6$ to that of${}^{238}\mathrm{UF}_6$$?$

On earth, STP is based on the average atmospheric pressure at the surface and on a phase change of water that occurs at an easily produced temperature, being only slightly cooler than the average air temperature. The atmosphere of Venus is almost entirely carbon dioxide $\left({\mathrm{CO}}_2\right)$, the pressure at the surface is a staggering $93\mathrm{atm}$, and the average temperature is localid="1648638013375" $470°\mathrm{C}$. Venusian scientists, if they existed, would certainly use the surface pressure as part of their definition of STP. To complete the definition, they would seek a phase change that occurs near the average temperature. Conveniently, the melting point of the element tellurium is localid="1648638019185" $450°\mathrm{C}$. What are (a) the rms speed and (b) the mean free path of carbon dioxide molecules at Venusian STP based on this phase change in tellurium? The radius of a ${\mathrm{CO}}_2$molecule islocalid="1648638027654" $1.5\times {10}^{-10}\mathrm{m}$.

$5.0\times {10}^{23}$nitrogen molecules collide with a $10c{m}^2$ wall each second. Assume that the molecules all travel with a speed of $400m/s$ and strike the wall head-on. What is the pressure on the wall?

A $10cm\times 10cm\times 10cm$ box contains $0.010mol$ of nitrogen at $20°C$. What is the rate of collisions (collisions/s) on one wall of the box?

The mean free path of a molecule in a gas is $300\mathrm{nm}$. What will the mean free path be if the gas temperature is doubled at (a) Constant volume and (b) Constant pressure?

If the pressure of a gas is really due to the random collisions of molecules with the walls of the container, why do pressure gauges— even very sensitive ones—give perfectly steady readings? Shouldn’t the gauge be continually jiggling and fluctuating? Explain.