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.

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.

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.)

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?

How much heat does it take to increase the temperature of $1.80$ mol of an ideal gas by $50.0$ K near room temperature if the gas is held at constant volume and is diatomic?

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

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.)