21. Kinetic Theory of Ideal Gases

What is the temperature of a sample of CO₂ molecules whose rms speed is 300 m/s? The molecular mass for Carbon and Oxygen is 12.01 g/mol and 16 g/mol, respectively.

Oxygen (O₂) has a molar mass of $32.0$ g/mol. How many oxygen molecules traveling at this speed are necessary to produce an average pressure of $1$ atm?

Oxygen (O₂) has a molar mass of $32.0$ g/mol. Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel $0.10$ m on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.)

For diatomic carbon dioxide gas (CO₂, molar mass $44.0$ g/mol) at $T=300$ K, calculate the root-mean-square speed $v_{rms}$.

Smoke particles in the air typically have masses of the order of ${10}^{-16}$ kg. The Brownian motion (rapid, irregular movement) of these particles, resulting from collisions with air molecules, can be observed with a microscope. Find the root-mean-square speed of Brownian motion for a particle with a mass of $3.00\times {10}^{-16}$ kg in air at $300$ K.

At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at $20.0$°C? (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element. The molar mass of H₂ is twice the molar mass of hydrogen atoms, and similarly for N₂.)

Oxygen (O₂) has a molar mass of $32.0$ g/mol. What is the momentum of an oxygen molecule traveling at this speed?

The atmosphere of Mars is mostly CO₂ (molar mass $44.0$ g/mol) under a pressure of $650$ Pa, which we shall assume remains constant. In many places the temperature varies from $0.0$°C in summer to $-100$°C in winter. Over the course of a Martian year, what are the ranges of the rms speeds of the CO₂ molecules.

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare the root-mean-square speeds. (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

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 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?

Eleven molecules have speeds 15, 16, 17, …, 25 m/s. Calculate (a) v_avg and (b) v_rms.

A gas cylinder has a piston at one end that is moving outward at speed v_piston during an isobaric expansion of the gas. Find an expression for the rate at which v_rms is changing in terms of v_piston, the instantaneous value of v_rms, and the instantaneous value L of the length of the cylinder.

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

A cylinder contains gas at a pressure of 2.0 atm and a number density of 4.2 x 10²⁵ m^-3. The rms speed of the atoms is 660 m/s. Identify the gas.

The rms speed of molecules in a gas is 600 m/s. What will be the rms speed if the gas pressure and volume are both halved?