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.)

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?

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?

An automobile tire is filled to a gauge pressure of $200 \text{kPa}$ when its temperature is $20^{\circ} \text{C}$. (Gauge pressure is the difference between the actual pressure and atmospheric pressure.) After the car has been driven at high speeds, the tire temperature increases to $50^{\circ} \text{C}$. $(b)$ Calculate the gauge pressure if the tire expands so the volume of the enclosed air increases by 10 percent.

An automobile tire is filled to a gauge pressure of $200 \text{kPa}$ when its temperature is $20^{\circ} \text{C}$. (Gauge pressure is the difference between the actual pressure and atmospheric pressure.) After the car has been driven at high speeds, the tire temperature increases to $50^{\circ} \text{C}$. $(a)$ Assuming that the volume of the tire does not change and that air behaves as an ideal gas, find the gauge pressure of the air in the tire.

A closed container with a volume of $6.00 \text{~L}$ holds $10.0 \text{~g}$ of liquid helium at $25.0 \text{~K}$ and enough air to fill the rest of its volume at a pressure of $1.00 \text{~atm}$. The helium then evaporates and the container warms to room temperature $(293 \text{~K})$. What is the final pressure inside the container?

Imagine that $10.0 \text{~g}$ of liquid helium, initially at $4.20 \text{~K}$, evaporate into an empty balloon that is kept at $1.00$-atm pressure. What is the volume of the balloon at $(b)$ $293 \text{~K}$?