Liquid helium has a temperature of only $4.20 \text{~K}$ and is in equilibrium with its vapor at atmospheric pressure. Calculate the rms speed of a helium atom in the vapor at this temperature, and comment on the result.

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?

"If a vessel contains equal amounts, by mass, of helium and argon, which of the following are true? $(b)$ The average speed of a helium atom is the same as that of an argon atom."

"If a vessel contains equal amounts, by mass, of helium and argon, which of the following are true? $(d)$ None of the above."

Two different gases are at the same temperature. What can be said about the average translational kinetic energies of the molecules? What can be said about the rms speeds of the gas molecules?

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?

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.

After nitrogen $\left(\text{N}_2\right)$ and oxygen $\left(\text{O}_2\right)$, the most abundant molecule in Earth's atmosphere is water, $\text{H}_2 \text{O}$. However, the fraction of $\text{H}_2 \text{O}$ molecules in a given volume of air varies dramatically, from practically zero percent under the driest conditions to a high of $4$ percent where it is very humid. $(b)$ What is the difference in mass, at room temperature and atmospheric pressure, between a cubic meter of air with no water vapor molecules and a cubic meter of air in which 4 percent of the molecules are water vapor molecules?

After nitrogen $\left(\text{N}_2\right)$ and oxygen $\left(\text{O}_2\right)$, the most abundant molecule in Earth's atmosphere is water, $\text{H}_2 \text{O}$. However, the fraction of $\text{H}_2 \text{O}$ molecules in a given volume of air varies dramatically, from practically zero percent under the driest conditions to a high of $4$ percent where it is very humid. $(a)$ At a given temperature and pressure, would air be denser when its water vapor content is large or small?