21. Kinetic Theory of Ideal Gases

During the isobaric part of the overall path from state d to state b for an ideal gas, how are the heat $q$ and the work $w$ related?

The pressure of a gas at the triple point of water is $1.35$ atm. If its volume remains unchanged, what will its pressure be at the temperature at which CO₂ solidifies?

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?

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).

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?

The total lung capacity of a typical adult is 5.0 L. Approximately 20% of the air is oxygen. At sea level and at a body temperature of 37°C, how many oxygen molecules do the lungs contain at the end of a strong inhalation?

In Problems $67,68,69,$ and $70$ you are given the equation(s) used to solve a problem. For each of these, you are to write a realistic problem for which this is the correct equation(s).

$(T_2+273)\text{ K}=\frac{200\text{ kPa}}{500\text{ kPa}}\times 1\times (400+273)\text{ K}$

Estimate how many air molecules rebound from a wall in a typical room per second, assuming an ideal gas of N molecules contained in a cubic room with sides of length ℓ at temperature T and pressure P.

(a) Show that the frequency f with which gas molecules strike a wall is ƒ = ($\overline{{\upsilon }_x}$ /2)(P/kT) ℓ² where $\overline{{\upsilon }_x}$ is the average x component of the molecule’s velocity.

(b) Show that the equation can then be written as ƒ≈ Pℓ² /$\sqrt{4mkT}$ where m is the mass of a gas molecule.

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. What is the pressure in the tank, in pascals and in atmospheres?

A cylinder contains nitrogen gas. A piston compresses the gas to half its initial volume. Afterward, has the mass density of the gas changed? If so, by what factor? If not, why not?

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.

0.0050 mol of gas undergoes the process 1→2→3 shown in FIGURE EX18.37. What are temperature T₁?

A gas at 100°C fills volume V₀. If the pressure is held constant, what is the volume if the Kelvin temperature is doubled?

Using the ideal gas law, find an expression for the mean free path ℓ_M that involves pressure and temperature instead of (N/V). Use this expression to find the mean free path for nitrogen molecules at a pressure of 7.5 atm and 300 K.

What is the partial pressure of water on a day when the temperature is 25°C and the relative humidity is 75%?