21. Kinetic Theory of Ideal Gases

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.)

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.

A space vehicle returning from the Moon enters the Earth’s atmosphere at a speed of about 42,000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

(I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

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 ${}^{238}U{F}_6$?

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.)

Estimate the rms speed of an amino acid, whose molecular mass is 89 u, in a living cell at 37°C.

What is the root mean square velocity of a nitrogen molecule ($N_2$) at temperature $T$ in terms of the Boltzmann constant $k$ and the mass of the molecule $m$?

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.

By what factor does the rms speed of a molecule change if the temperature is increased from 10℃ to 1000℃?

(II) A group of 25 particles have the following speeds: two have speed 10 m/s, seven have 15 m/s, four have 20 m/s, three have 25 m/s, six have 30 m/s, one has 35 m/s, and two have 40 m/s. Determine the rms speed.

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.

Dust particles are ≈ 10 μm in diameter. They are pulverized rock, with ρ ≈ 2500 kg/m³. If you treat dust as an ideal gas, what is the rms speed of a dust particle at 20℃?

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?

(I) Twelve molecules have the following speeds, given in arbitrary units: 6.0, 2.0, 4.0, 6.0, 0.0, 4.0, 1.0, 8.0, 5.0, 3.0, 7.0, and 8.0. Calculate the mean speed.