21. Kinetic Theory of Ideal Gases

A cylinder of nitrogen and a cylinder of neon are at the same temperature and pressure. The mean free path of a nitrogen molecule is 150 nm. What is the mean free path of a neon atom?

Laboratory environments can achieve pressures of 3.5×10^-13 atm and temperatures of 300K. Calculate the mean free path (in km) of air molecules, which you can assume are diatomic.

Calculate the mean free path of air molecules at $3.50\times {10}^{-13}$ atm and $300$ K. (This pressure is readily attainable in the laboratory; see Exercise $18.23$.) As in Example $18.8$, model the air molecules as spheres of radius $2.0\times {10}^{-10}$ m.

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 (CO₂), the pressure at the surface is a staggering 93 atm, and the average temperature is 470℃. 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 450℃. 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 CO₂ molecule is 1.5 x 10^-10 m.

Photons of light scatter off molecules, and the distance you can see through a gas is proportional to the mean free path of photons through the gas. Photons are not gas molecules, so the mean free path of a photon is not given by Equation 20.3, but its dependence on the number density of the gas and on the molecular radius is the same. Suppose you are in a smoggy city and can barely see buildings 500 m away. How far would you be able to see if all the molecules around you suddenly doubled in volume?

A mad engineer builds a cube, 2.5 m on a side, in which 6.2-cm-diameter rubber balls are constantly sent flying in random directions by vibrating walls. He will award a prize to anyone who can figure out how many balls are in the cube without entering it or taking out any of the balls. You decide to shoot 6.2-cm-diameter plastic balls into the cube, through a small hole, to see how far they get before colliding with a rubber ball. After many shots, you find they travel an average distance of 1.8 m. How many rubber balls do you think are in the cube?

Integrated circuits are manufactured in vacuum chambers in which the air pressure is 1.0 x 10^-10 of Hg. What are (a) the number density and (b) the mean free path of a molecule? Assume T = 20℃.

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

The pressure inside a tank of neon is 150 atm. The temperature is 25℃. On average, how many atomic diameters does a neon atom move between collisions?

At about what pressure would the mean free path of air molecules be 0.30 m? Assume T = 20° C.

A very small amount of hydrogen gas is released into the air. If the air is at 1.0 atm and 15°C, estimate the mean free path for an H₂ molecule. What assumptions did you make?

The density of atoms, mostly hydrogen, in interstellar space is about one per cubic centimeter. Estimate the mean free path of the hydrogen atoms, assuming an atomic diameter of 1.0 x 10^-10 m.

Below a certain threshold pressure, the air molecules (0.3-nm diameter) within a research vacuum chamber are in the “collision-free regime,” meaning that a particular air molecule is as likely to cross the container and collide with the opposite wall as it is to collide with another air molecule. Estimate the threshold pressure for a vacuum chamber of side 1.0 m at 20°C.

A monatomic gas is adiabatically compressed to 1/8 of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not? The mean free path.