A $20.0$-L tank contains $4.86\times {10}^{-4}$ kg of helium at $18.0$°C. The molar mass of helium is $4.00$ g/mol. How many moles of helium are in the tank?

A $20.0$-L tank contains $4.86\times10^{-4}$ kg of helium at $18.0$°C. The molar mass of helium is $4.00$ g/mol. What is the pressure in the tank, in pascals and in atmospheres?

Helium gas with a volume of $3.20$ L, under a pressure of $0.180$ atm and at $41.0$°C, is warmed until both pressure and volume are doubled. What is the final temperature?

Helium gas with a volume of $3.20$ L, under a pressure of $0.180$ atm and at $41.0$°C, is warmed until both pressure and volume are doubled. How many grams of helium are there? The molar mass of helium is $4.00$ g/mol.

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains $0.110$ m³ of air at a pressure of $0.355$ atm. The piston is slowly pulled out until the volume of the gas is increased to $0.390$ m³. If the temperature remains constant, what is the final value of the pressure?

Calculate the density of the atmosphere at the surface of Mars (where the pressure is $650$ Pa and the temperature is typically $253$ K, with a CO₂ atmosphere), Venus (with an average temperature of $730$ K and pressure of $92$ atm, with a CO₂ atmosphere), and Saturn's moon Titan (where the pressure is $1.5$ atm and the temperature is $-178$°C, with a N₂ atmosphere).

You have several identical balloons. You experimentally determine that a balloon will break if its volume exceeds $0.900$ L. The pressure of the gas inside the balloon equals air pressure ($1.00$ atm). If the air inside the balloon is at a constant $22.0$°C and behaves as an ideal gas, what mass of air can you blow into one of the balloons before it bursts?

A large cylindrical tank contains $0.750$ m³ of nitrogen gas at $27$°C and $7.50\times {10}^3$ Pa (absolute pressure). The tank has a tight-fitting piston that allows the volume to be changed. What will be the pressure if the volume is decreased to $0.410$ m³ and the temperature is increased to $157$°C?

If a certain amount of ideal gas occupies a volume V at STP on earth, what would be its volume (in terms of V) on Venus, where the temperature is $1003$°C and the pressure is $92$ atm?

Martian Climate. The atmosphere of Mars is mostly CO2 (molar mass 44.0 g/mol) under a pressure of 650 Pa, which we shall assume remains constant. In many places the temperature varies from 0.0°C in summer to -100°C in winter. Over the course of a Martian year, what are the ranges of (b) the density (in mol/m^3) of the atmosphere?

At an altitude of $11,000$ m (a typical cruising altitude for a jet airliner), the air temperature is $-56.5$°C and the air density is $0.364$ kg/m³ . What is the pressure of the atmosphere at that altitude? (Note: The temperature at this altitude is not the same as at the surface of the earth, so the calculation of Example $18.4$ in Section $18.1$ doesn't apply.)

How many moles are in a $1.00$-kg bottle of water? How many molecules? The molar mass of water is $18.0$ g/mol.

A large organic molecule has a mass of $1.41\times {10}^{-21}$ kg. What is the molar mass of this compound?

Modern vacuum pumps make it easy to attain pressures of the order of ${10}^{-13}$ atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. At a pressure of $9.00\times {10}^{-14}$ atm and an ordinary temperature of $300.0$ K, how many molecules are present in a volume of $1.00$ cm³?

Modern vacuum pumps make it easy to attain pressures of the order of $10^{-13}$ atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. How many molecules would be present at the same temperature but at $1.00$ atm instead?

In a gas at standard conditions, what is the length of the side of a cube that contains a number of molecules equal to the population of the earth (about $7\times {10}^9$ people)?

Consider an ideal gas at $27$°C and $1.00$ atm. To get some idea how close these molecules are to each other, on the average, imagine them to be uniformly spaced, with each molecule at the center of a small cube. What is the length of an edge of each cube if adjacent cubes touch but do not overlap?

What is the total translational kinetic energy of the air in an empty room that has dimensions $8.00$ m$\times 12.00$ m$\times 4.00$ m if the air is treated as an ideal gas at $1.00$ atm?

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare the average kinetic energies of the three types of atoms.

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare the root-mean-square speeds. (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

We have two equal-size boxes, A and B. Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box A is at $50$°C while the gas in box B is at $10$°C. This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? Explain your reasoning.

(a) The pressure in A is higher than in B.

(b) There are more molecules in A than in B.

(c) A and B do not contain the same type of gas.

(d) The molecules in A have more average kinetic energy per molecule than those in B.

(e) The molecules in A are moving faster than those in B.

The atmosphere of Mars is mostly CO₂ (molar mass $44.0$ g/mol) under a pressure of $650$ Pa, which we shall assume remains constant. In many places the temperature varies from $0.0$°C in summer to $-100$°C in winter. Over the course of a Martian year, what are the ranges of the rms speeds of the CO₂ molecules.

Oxygen (O₂) has a molar mass of $32.0$ g/mol. What is the average translational kinetic energy of an oxygen molecule at a temperature of $300$ K?

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

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?

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.

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.)

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.

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?