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. 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?

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 monatomic?

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.

Compute the specific heat at constant volume of nitrogen (N₂) gas, and compare it with the specific heat of liquid water. The molar mass of N₂ is $28.0$ g/mol.

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.)