21. Kinetic Theory of Ideal Gases

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.

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

The escape speed from the Earth is 1.12 x 10⁴ m/s (Section 8–7). So a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. Can you explain why our atmosphere contains oxygen but not helium?

(I) By what factor will the rms speed of gas molecules increase if the temperature is increased from 0°C to 160°C?

For an ideal gas at temperature T show that$\frac{d{v}_{rms}}{dT}=\frac{1}{2}\left(\frac{v_{rms}}{T}\right)$, and using the approximation $\Delta v_{rms}\approx \left(\frac{d{v}_{rms}}{dT}\right)\Delta T$, show that $\frac{\Delta v_{rms}}{v_{rms}}\approx \frac{1}{2}\left(\frac{\Delta T}{T}\right)$.

The escape speed from the Earth is 1.12 x 10⁴ m/s (Section 8–7). So a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. At what temperature is the rms speed of oxygen molecules?

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.

Calculate the total water vapor pressure in the air on the following day: a hot summer day, with the temperature 30°C and the relative humidity at 75%.

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.)

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

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

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?

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?