Show that the mean free path for a molecule in an ideal gas at temperature $T$ and pressure $P$ is given by $\lambda=k T /\left(\sqrt{2} P \pi d^2\right)$.

Estimate the rms speed and the average kinetic energy of a hydrogen atom in a gas at a temperature of $1.0 \times 10^7 \text{~K}$. (At this temperature, which is approximately the temperature in the interior of a star, hydrogen atoms are ionized and become protons.)

A plot of pressure P versus absolute temperature T for a process that takes a sample of an ideal gas from point A to point B. What happens to the volume of the gas during this process?

Find the total translational kinetic energy of the molecules of $1.0 \text{~L}$ of oxygen gas at a temperature of $0.0^{\circ} \text{C}$ and a pressure of $1.0 \text{~atm}$.

(a) Find $v_{\text {tims }}$ for an argon atom if $1 \mathrm{~mol}$ of the gas is confined to a 1-L container at a pressure of $10 \mathrm{~atm}$. (For argon, $M=40 \times 10^{-3} \mathrm{~kg} / \mathrm{mol}$.) (b) Compare this with $v_{\text {rins }}$ for a helium atom under the same conditions. (For helium, $M=$ $4 \times 10^{-3} \mathrm{~kg} / \mathrm{mol}$.)

Compare your answer to the rms speed for helium atoms under the same conditions.

One mole of argon gas is confined to a $1.0\text{-liter}$ container at a pressure of $10 \text{~atm}$. What is the rms speed of the argon atoms?

A helium balloon is used to lift a load of $110 \text{~N}$. The weight of the envelope of the balloon is $50.0 \text{~N}$ and the volume of the helium when the balloon is fully inflated is $32.0 \text{~m}^3$. The temperature of the air is $0^{\circ} \text{C}$ and the atmospheric pressure is $1.00 \text{~atm}$. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is $30.0$ N. Neglect any effects due to the changes of temperature as the altitude changes. $(d)$ If the answer to Part $(c)$ is "Yes," what is the maximum altitude attained by the balloon?

A helium balloon is used to lift a load of $110 \text{~N}$. The weight of the envelope of the balloon is $50.0 \text{~N}$ and the volume of the helium when the balloon is fully inflated is $32.0 \text{~m}^3$. The temperature of the air is $0^{\circ} \text{C}$ and the atmospheric pressure is $1.00 \text{~atm}$. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is $30.0$ N. Neglect any effects due to the changes of temperature as the altitude changes. $(c)$ Does the balloon ever reach the altitude at which it is fully inflated?

A helium balloon is used to lift a load of $110 \text{~N}$. The weight of the envelope of the balloon is $50.0 \text{~N}$ and the volume of the helium when the balloon is fully inflated is $32.0 \text{~m}^3$. The temperature of the air is $0^{\circ} \text{C}$ and the atmospheric pressure is $1.00 \text{~atm}$. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is $30.0$ N. Neglect any effects due to the changes of temperature as the altitude changes. $(b)$ At what altitude will the balloon be fully inflated?

A helium balloon is used to lift a load of $110 \text{~N}$. The weight of the envelope of the balloon is $50.0 \text{~N}$ and the volume of the helium when the balloon is fully inflated is $32.0 \text{~m}^3$. The temperature of the air is $0^{\circ} \text{C}$ and the atmospheric pressure is $1.00 \text{~atm}$. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is $30.0$ N. Neglect any effects due to the changes of temperature as the altitude changes. $(a)$ How many moles of helium gas is contained in the balloon?

Calculate the mass density of air at a temperature of $24^{\circ} \mathrm{C}$ and a pressure of $1 \mathrm{~atm}\left(1.01 \times 10^5 \mathrm{~N} / \mathrm{m}^2\right)$ using the ideal-gas law. Air is roughly 74 percent $\mathrm{N}_2$ and 26 percent $\mathrm{O}_2$.

A hot-air balloon is open at the bottom. The balloon, which has a volume of $446 \text{~m}^3$, is filled with air that has an average temperature of $100^{\circ} \text{C}$. The air outside the balloon has a temperature of $20.0^{\circ} \text{C}$ and a pressure of $1.00 \text{~atm}$. How large a payload (including the envelope of the balloon itself) can the balloon lift? Use $29.0 \text{~g/mol}$ for the molar mass of air. (Neglect the volume of both the payload and the envelope of the balloon.)

A scuba diver is $40 \text{~m}$ below the surface of a lake, where the temperature is $5.0^{\circ} \text{C}$. He releases an air bubble that has a volume of $15 \text{~cm}^3$. The bubble rises to the surface, where the temperature is $25^{\circ} \text{C}$. Assume that the air in the bubble is always in thermal equilibrium with the surrounding water, and assume that there is no exchange of molecules between the bubble and the surrounding water. What is the volume of the bubble right before it breaks the surface? Hint: Remember that the pressure also changes.