From the van der Waals equation of state, show that the critical temperature and pressure are given by T_cr = 8a / 27bR , P_cr = a / 27b². [Hint: Use the fact that the P versus V curve has an inflection point at the critical point so that the first and second derivatives are zero.]

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.

Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At constant temperature, pressure is inversely proportional to the square of the volume.

Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At constant pressure, the volume varies directly with the 2/3 power of the temperature.

Assume that in an alternate universe, the laws of physics are very different from ours and that “ideal” gases behave as follows: At 273.15 K and 1.00 atm pressure, 1.00 mole of an ideal gas is found to occupy 22.4 L. Obtain the form of the ideal gas law in this alternate universe, including the value of the gas constant R.

Use the ideal gas law to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to β = 1/ T, where T is the kelvin temperature. Compare to Table 17–1 for gases at T = 293 K.

Show that the bulk modulus (Section 12–5) for an ideal gas held at constant temperature is B = P, where P is the pressure.

A sauna has 7.8 m³ of air volume, and the temperature is 85°C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 18–2.)

Why snorkels are not 4 feet long. Snorkelers breathe through short tubular “snorkels” while swimming under water very near the surface (Fig. 17–24). One end of the snorkel is in the snorkeler’s mouth and the other end protrudes just above the water’s surface. Unfortunately, snorkels cannot support breathing to any great depth: it is said that a typical snorkeler below a water depth of only about 30 cm cannot draw a breath through a snorkel. Based on this observation, what is the approximate change in a typical person’s lung pressure (in atm) when drawing a breath? (Note that your diaphragm muscles, which expand your lungs, must work also against the extra water pressure.)

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